CN113282096B - Control method for nonlinear error of relative position of geostationary orbit game spacecraft - Google Patents

Control method for nonlinear error of relative position of geostationary orbit game spacecraft Download PDF

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CN113282096B
CN113282096B CN202110625147.XA CN202110625147A CN113282096B CN 113282096 B CN113282096 B CN 113282096B CN 202110625147 A CN202110625147 A CN 202110625147A CN 113282096 B CN113282096 B CN 113282096B
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张雅声
张海涛
李智
王伟林
于金龙
梁爽
请求不公布姓名
郭威
万新民
陈松
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Peoples Liberation Army Strategic Support Force Aerospace Engineering University
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Abstract

The invention discloses a control method for nonlinear errors of relative positions of a geostationary orbit game spacecraft, which comprises the following steps: establishing an LVLH coordinate system by taking the mass center of a reference spacecraft as an origin; respectively calculating initial state vectors of the tracking spacecraft and the escape spacecraft relative to the reference spacecraft in an LVLH coordinate system; adding correction vectors into initial state vectors of the tracking spacecraft and the escape spacecraft relative to the reference spacecraft respectively, substituting the correction vectors into a correction CW equation, and obtaining state vectors of the tracking spacecraft and the escape spacecraft; the state vector of the tracking spacecraft is subtracted from the state vector of the escape spacecraft to obtain the relative position of the geostationary orbit game spacecraft after nonlinear errors are controlled. According to the method, a second-order nonlinear term of a potential function is used as virtual control thrust in the relative motion process of the geostationary orbit game spacecraft, a state equation is solved, nonlinear deviation accumulated along with time is converted into a correction term for initial state parameters, and accumulation of the nonlinear deviation along with time is avoided.

Description

Control method for nonlinear error of relative position of geostationary orbit game spacecraft
Technical Field
The invention relates to a control method for nonlinear errors of relative positions of a geostationary orbit game spacecraft, and belongs to the field of spacecraft orbit dynamics.
Background
Methods of studying relative movement of games can be divided into two categories: relative motion based on kinematics and relative motion based on dynamics. The general practice of kinematics-based relative motion is: firstly, establishing a conversion relation between a difference value of absolute orbit numbers of a master spacecraft and a slave spacecraft and a relative orbit number, and generally obtaining an accurate conversion matrix; secondly, establishing a coordinate system which is more favorable for describing formation problems, and researching a conversion matrix of the relative track number and the state quantity under the coordinate system; thirdly, the relation between the pulse control force in three directions and the absolute track root number difference value can be obtained according to the Gaussian perturbation equation, so that the conversion relation among the control force, the absolute track root number difference value, the relative track root number and the state quantity is established, and fourth, the time and the magnitude of the control force application are optimized, so that the method can be used for analyzing the formation construction and the reconstruction optimal control problem.
Representative of the comparison is the equation of elliptical relative motion derived by Yan Jianfeng, han Chao et al based on the relative orbit elements. Yan Jianfeng, han Chao et al, have proposed a set of methods from the kinematic approach that can be used for near-circular, elliptical formation design, reconstruction, optimal control: the relative orbital elements are strictly defined in literature ([ 1] Korean tide, yan Jianfeng. Relative motion of elliptical orbit satellites based on relative orbital elements study [ J ]. Aviation journal, 2011,32 (12): 2244-2258.Han Chao,Yin Jianfeng.Study of satellite relative motion in elliptical orbit using relative orbit elements[j ]. Acta Aeronautica et Astronautica sinica,2011,32 (12): 2244-2258.) and a transformation matrix between them and absolute orbital elements is derived for analysis of the convoy relative motion relationship; the conversion matrix and inverse conversion matrix of relative orbit elements to mass center orbit system state quantity are deduced in the literature (1]Yin J,Rao Y,Han C.Inverse Transformation of Elliptical Relative State Transition Matrix[J, international Journal of Astronomy & Astrochysics, 2014,04 (3): 419-428), and a conversion equation of pulse velocity increment and state quantity in the orbit system is established; in the literature (1]Yin J,Han C.Elliptical formation control based on relative orbit elements[J, chinese aviation journal (english edition), 2013,26 (006): 1554-1567.), control in the track plane and control in the track normal direction are decoupled based on the relative track number, state control is realized by four track maneuvers, and the timing and control amount of control are optimized by numerical method. Yan Jianfeng and Han Chao have reference significance in the unilateral optimization control of games, and have the following defects: the state equation based on the relative orbit element kinematics formation relative motion equation can not establish the state equation of state quantity and time differentiation, and is inconvenient for the research of the interactive game optimization strategy of both spacecraft.
The advantages of studying relative motion based on kinematics are: the model has high precision, is convenient for introducing the influence of perturbation on the relative motion, and can effectively control the accumulation of model errors along with time. The defects are that: the system cannot be described through a differential equation, and in the differential game problem, the interactive game optimization countermeasures of both parties cannot be conveniently researched.
The relative motion is studied based on a dynamic method, and the Clohessy-Wiltshire equation, abbreviated as cw equation, is a very widely applied method. General idea of studying relative motion based on dynamics: firstly, selecting a proper state quantity, and determining a conversion relation between a difference value of absolute track numbers and the state quantity; secondly, establishing a differential equation of the state quantity with respect to time, namely a state equation; thirdly, analyzing differential equations for formation construction, reconstruction and optimal control.
The CW equation is:
Figure BDA0003101870740000021
wherein X (t) represents a state vector at time t, X (t) 0 ) Representing an initial state vector, U is the applied control force, B is a constant matrix,
Figure BDA0003101870740000031
s is the sign of the integral argument, U(s) is the control force at time s, phi (t, s) is the state transition matrix from s to t, phi (t, t) 0 ) Is t 0 A state transition matrix to t;
Figure BDA0003101870740000032
wherein: τ=ω (t-t) 0 ) Representing the reference spacecraft from t 0 By the angle by which t is rotated,
Figure BDA0003101870740000033
a c for reference of the orbit semi-long axis of the spacecraft, μ= 3.986005 ×10 14 m 3 /s 2 Is the gravitational constant.
The defects are that: the CW equation error is derived from linearization error and perturbation error, and the CW equation also has small eccentricity assumption error, and the error is larger, in particular, the error accumulated with time is not easy to control. The advantages are that: after the state equation is established, the theory of modern control and optimal control is conveniently used for researching the relative motion, and the problem of interactive differential game between two parties is particularly conveniently solved.
For the relative motion model of GEO spacecraft, the STM model is representative. STM models are developed over a period of many years from initial build-up. D' Amico ([ 1)]Montenbruck,O.,Kirschner,M.,D’Amico,S.,and Bettadpur,S.,“E/I-Vector Separation for Safe Switching ofthe GRACE Formation,”Aerospace Science and Technology,Vol.10,No.7,2006,pp.628–635.doi:10.1016/j.ast.2006.04.001；[2]D’Amico,S.,and Montenbruck,O.,“Proximity Operations of Formation-Flying Spacecraft Using an Eccentricity/Inclination Vector Separation,”Journal of Guidance,Control,and Dynamics,Vol.29,No.3,2006,pp.554–563.doi:10.2514/1.15114；[3]D' Amico, S., "Autonomous Formation Flying in Low Earth Orbit," Ph.D.) studied a relative orbit root formation dynamics model STM model based on relative E/I vectors, in which J was considered 2 Term perturbation and research on formation safety path problems; gaias, G (Gaias, G., ardaens, J. -S., and Montenbruck, O., "Model of J2 Perturbed Satellite Relative Motion with Time-Varying Differential Drag," Celestial Mechanics and Dynamical Astronomy, vol.123, no.4,2015, pp.411-433.doi:10.1007/s 10569-015-9643-2) introduced atmospheric resistance perturbation, analyzed the long term effects of atmospheric resistance perturbation on the semi-long axis of the track, and improved the STM Model; spiridonova, S (Spiridonova, S., "Formation Dynamics in Geostationary Ring," Celestial Mechanics and Dynamical Astronomy, vol.125, no.4,2016, pp.485-500.Doi: 10.1007/S10569-016-9693-0) adds solar pressure and trisomy to the STM model, eliminates all long term and long period terms of relative track root drift, perfects the STM model, and simulation shows that the model also has higher accuracy after 10 days, making the STM model suitable for GEO formation problems involving on-orbit service tasks such as close-range spatial operations. Although the model has higher precision, the STM model after the spironova, S is complete is complex, contains nonlinear parts, cannot establish a linear state control equation, and is inconvenient for the GEO-spacecraft game problem.
Disclosure of Invention
The invention provides a control method for nonlinear errors of relative positions of geostationary orbit game spacecrafts, which aims at the technical problems that the calculated relative positions are inaccurate and cannot normally finish a spaceflight task due to nonlinear errors existing in a Clohessy-Wilthhire equation, and improves the precision of the CW equation by correcting the CW equation.
In order to achieve the above purpose, the invention provides a method for controlling nonlinear errors of relative positions of a geostationary orbit game spacecraft, which comprises the following steps:
step one, selecting a reference spacecraft, wherein the orbit semi-long axis of the reference spacecraft is a GEO nominal orbit semi-long axis, the eccentricity is 0, the orbit inclination angle is 0, the right ascent and intersection point, the near-site depression angle and the flat near-site angle are the same as those of a tracking spacecraft or an escape spacecraft, and establishing an LVLH coordinate system by taking the mass center of the reference spacecraft as an origin;
step two, respectively calculating initial state vectors of the tracking spacecraft and the escape spacecraft relative to the reference spacecraft in the LVLH coordinate system;
step three, adding a correction vector into an initial state vector of the tracking spacecraft relative to the reference spacecraft, and substituting the correction vector into a correction CW equation to obtain a state vector of the tracking spacecraft; adding a correction vector into an initial state vector of the escape spacecraft relative to the reference spacecraft, and substituting the correction vector into a correction CW equation to obtain a state vector of the escape spacecraft; subtracting the state vector of the tracking spacecraft from the state vector of the escape spacecraft to obtain the relative position of the geostationary orbit game spacecraft after controlling the nonlinear error;
the modified CW equation is:
Figure BDA0003101870740000051
wherein: x (t) represents a state vector at time t, X correction (t 0 ) Representing the initial state vector after addition of the correction term, U being the applied control force, B being a constant matrix,
Figure BDA0003101870740000052
s is the sign of the integral argument, U(s) is the control force applied at time s, phi (t, s) is the state transition matrix from time s to time t, phi (t, t 0 ) Is t 0 State transition matrix from time to time t.
In the second step, in the LVLH coordinate system, the method for calculating the initial state vector of the tracking spacecraft relative to the reference spacecraft includes:
Figure BDA0003101870740000061
/>
wherein, (x) P (t 0 ),y P (t 0 ),z P (t 0 ) Tracking the position coordinates of the spacecraft in the LVLH coordinate system at the initial moment;
Figure BDA0003101870740000062
is x P ,y P ,z P The value of the time derivative function at the initial moment; t represents a transpose;
in an LVLH coordinate system, a method of calculating an initial state vector of an escape spacecraft relative to a reference spacecraft includes:
Figure BDA0003101870740000063
wherein, (x) E (t 0 ),y E (t 0 ),z E (t 0 ) Tracking the position coordinates of the spacecraft in the LVLH coordinate system for the initial moment,
Figure BDA0003101870740000064
is x E ,y E ,z E The value of the time derivative function at the initial moment; t represents the transpose.
In the third step, the correction vector added in the initial state vector of the tracking spacecraft relative to the reference spacecraft is as follows:
Figure BDA0003101870740000065
wherein ρ is P To track the distance of a spacecraft relative to the initial moment of a reference spacecraft, alpha P0 For tracking the position phase of the initial moment of the spacecraft, n is the average angular velocity of the reference spacecraft, μ= 3.986005 ×10 14 m 3 /s 2 Is the constant of the gravitational force, a c Is the orbit semi-long axis of the reference spacecraft;
the correction vectors added in the initial state vector of the escape spacecraft relative to the reference spacecraft are as follows:
Figure BDA0003101870740000066
wherein ρ is E For escaping the distance of the spacecraft relative to the initial moment of the reference spacecraft, alpha E0 For escaping the position phase of the spacecraft at the initial moment, a c For the orbit semi-long axis of the reference spacecraft, n is the average angular velocity of the reference spacecraft, μ= 3.986005 ×10 14 m 3 /s 2 Is the gravitational constant.
In the third step, a correction vector is added into an initial state vector of the tracking spacecraft relative to the reference spacecraft, and the initial state vector of the tracking spacecraft relative to the reference spacecraft is corrected as follows:
Figure BDA0003101870740000067
wherein ρ is P To track the distance of a spacecraft relative to the initial moment of a reference spacecraft, alpha P0 For tracking the position phase of the initial moment of the spacecraft, n is the average angular velocity of the reference spacecraft, μ= 3.986005 ×10 14 m 3 /s 2 Is the constant of the gravitational force, a c Is the orbit semi-long axis of the reference spacecraft;
correcting the initial state quantity of the escape spacecraft relative to the reference spacecraft as follows:
Figure BDA0003101870740000071
wherein ρ is E For escaping the distance of the spacecraft relative to the initial moment of the reference spacecraft, alpha E0 For escaping the position phase of the spacecraft at the initial moment, a c For the orbit semi-long axis of the reference spacecraft, n is the average angular velocity of the reference spacecraft, μ= 3.986005 ×10 14 m 3 /s 2 Is the gravitational constant.
In the third step, a correction vector is added into an initial state vector of the tracking spacecraft relative to the reference spacecraft, and is substituted into a correction CW equation
Figure BDA0003101870740000072
In the method, the state vector of the spacecraft can be tracked at the time t>
Figure BDA0003101870740000073
Wherein, (x) P (t),y P (t),z P (t)) tracking the position coordinates of the spacecraft in the LVLH coordinate system for the moment t;
Figure BDA0003101870740000077
is x P ,y P ,z P For the value of the time derivative function at the moment T, T represents transposition;
adding a correction vector into an initial state vector of the escape spacecraft relative to the reference spacecraft, and substituting a correction CW equation
Figure BDA0003101870740000074
State vector of escape spacecraft at time t
Figure BDA0003101870740000075
Wherein, (x) E (t),y E (t),z E (t)) is the position coordinate of the escape spacecraft at the moment t in the LVLH coordinate system;
Figure BDA0003101870740000076
is x E ,y E ,z E For the value of the time derivative function at the moment T, T represents transposition;
(x E (t)-x P (t),y E (t)-y P (t),z E (t)-z P (t)) is the relative position of the GEO-game spacecraft after controlling the aspheric perturbation error.
The LVLH coordinate system is a local horizontal and local vertical coordinate system, and refers to a positive direction taking the mass center of the reference spacecraft as an origin and the sagittal diameter from the earth center to the reference spacecraft as an x-axis, wherein the z-axis points to the positive normal direction of the orbit of the reference spacecraft, and the direction of the y-axis is determined according to a right-hand rule.
The tracking spacecraft can be used for carrying out a capturing task on a space target;
escaping spacecraft and avoiding capturing of tracking spacecraft;
the reference spacecraft is a virtual spacecraft which does not do orbital maneuver.
The embodiment of the invention has the following beneficial effects:
compared with a CW equation, the nonlinear error propagation of the CW equation is avoided, and the nonlinear error propagation is reflected in:
a. aiming at the nonlinear error problem existing in the CW equation, the method provided by the invention considers the influence of the abandoned second-order nonlinear term in the linearization process of the potential function, calculates error propagation caused by the second-order nonlinear term, wherein the error comprises three types of a constant term, a time accumulation term and a short period term, further calculates a correction term for eliminating the time accumulation term, and avoids the propagation of the second-order nonlinear error by correcting the initial state quantity,
b. aiming at the problem of eccentricity error in a CW equation, when the reference spacecraft is selected, the eccentricity of the reference spacecraft is made to be 0, and the propagation of the eccentricity error in the method is avoided.
Description of the drawings:
FIG. 1 is a graph showing the change of the x state quantity with time in the method and CW equation according to the present invention;
FIG. 2 is a graph showing the y state quantity over time in the method and CW equation according to the present invention;
FIG. 3 is a graph showing the difference between the y state quantity in the CW equation and the method according to the present invention over time;
FIG. 4 is a graph showing the change of the z state quantity with time in the method and CW equation according to the present invention;
FIG. 5 is a graph showing the difference between the x state quantity and the true value in the CW equation according to the method of the present invention;
FIG. 6 is a graph showing the difference between the y state quantity and the true value in the CW equation over time according to the method of the present invention;
FIG. 7 is a graph showing the difference between the z state quantity and the true value in the CW equation over time according to the method of the present invention;
FIG. 8 is a graph showing the change of the x state quantity with time in the method and CW equation according to the present invention;
FIG. 9 is a graph showing the y state quantity over time in the method and CW equation according to the present invention;
FIG. 10 is a graph showing the difference between the y state quantity in the CW equation and the method according to the present invention over time;
FIG. 11 is a graph showing the change of the z state quantity with time in the method and CW equation according to the present invention;
FIG. 12 is a graph showing the difference between the x state quantity and the true value in the CW equation over time according to the method of the present invention;
FIG. 13 is a graph showing the difference between the y state quantity and the true value in the CW equation over time according to the method of the present invention;
fig. 14 is a graph showing the time-dependent difference between the z state quantity and the true value in the CW equation and the method according to the present invention.
Detailed Description
In this embodiment, the reference spacecraft is also called master spacecraft and i is slave spacecraft.
A control method for nonlinear errors of relative positions of a geostationary orbit game spacecraft comprises the following steps:
step one, selecting a reference spacecraft, wherein the orbit semi-long axis of the reference spacecraft is a GEO nominal orbit semi-long axis, the eccentricity is 0, the orbit inclination angle is 0, the right ascent and intersection point, the near-site depression angle and the flat near-site angle are the same as those of a tracking spacecraft or an escape spacecraft, and establishing an LVLH coordinate system by taking the mass center of the reference spacecraft as an origin;
step two, respectively calculating initial state vectors of the tracking spacecraft and the escape spacecraft relative to the reference spacecraft in the LVLH coordinate system;
step three, adding a correction vector into an initial state vector of the tracking spacecraft relative to the reference spacecraft, and substituting the correction vector into a correction CW equation to obtain a state vector of the tracking spacecraft; adding a correction vector into an initial state vector of the escape spacecraft relative to the reference spacecraft, and substituting the correction vector into a correction CW equation to obtain a state vector of the escape spacecraft; subtracting the state vector of the tracking spacecraft from the state vector of the escape spacecraft to obtain the relative position of the geostationary orbit game spacecraft after controlling the nonlinear error;
the modified CW equation is:
Figure BDA0003101870740000101
wherein: x (t) represents a state vector at time t, X correction (t 0 ) Representing the initial state vector after addition of the correction term, U being the applied control force, B being a constant matrix,
Figure BDA0003101870740000102
s is the sign of the integral argument, U(s) is the control force applied at time s, phi (t, s) is the state transition matrix from time s to time t, phi (t, t 0 ) Is t 0 State transition matrix from time to time t.
In the second step, in the LVLH coordinate system, the method for calculating the initial state vector of the tracking spacecraft relative to the reference spacecraft includes:
Figure BDA0003101870740000103
wherein, (x) P (t 0 ),y P (t 0 ),z P (t 0 ) Tracking the position coordinates of the spacecraft in the LVLH coordinate system at the initial moment;
Figure BDA0003101870740000104
is x P ,y P ,z P The value of the time derivative function at the initial moment; t represents a transpose;
in an LVLH coordinate system, a method of calculating an initial state vector of an escape spacecraft relative to a reference spacecraft includes:
Figure BDA0003101870740000111
wherein, (x) E (t 0 ),y E (t 0 ),z E (t 0 ) Tracking the position coordinates of the spacecraft in the LVLH coordinate system for the initial moment,
Figure BDA0003101870740000112
is x E ,y E ,z E The value of the time derivative function at the initial moment; t represents the transpose. />
For example: r is recorded LVLH To track the position vector of spacecraft P in LVLH coordinate system, r ECI To track the position vector of a spacecraft P in an Earth-centered-inertial frame ECI (Earth-inertial frame), A ECI→LVLH A is a coordinate transformation matrix from an ECI coordinate system to an LVLH coordinate system LVLH→ECI The coordinate transformation matrix from the LVLH coordinate system to the ECI coordinate system is as follows:
r ECI ＝A LVLH→ECI ·r LVLH (1)
r LVLH ＝A ECI→LVLH ·r ECI (2)
Figure BDA0003101870740000113
Figure BDA0003101870740000114
wherein:
Figure BDA0003101870740000115
Figure BDA0003101870740000116
Figure BDA0003101870740000117
for column vector a= [ a ] 1 a 2 a 3 ] T And b= [ b ] 1 b 2 b 3 ] T Exists in the presence of
Figure BDA0003101870740000118
Wherein the method comprises the steps of：
Figure BDA0003101870740000119
A ECI→LVLH And A LVLH→ECI The rate of change over time is:
Figure BDA0003101870740000121
Figure BDA0003101870740000122
wherein n is the angular velocity of rotation of the LVLH coordinate system relative to the ECI coordinate system, and:
Figure BDA0003101870740000123
Figure BDA0003101870740000124
Figure BDA0003101870740000125
Figure BDA0003101870740000126
in [ δr ] ECI T δv ECI T ] T To track the difference of the position velocity vectors of spacecraft P and reference spacecraft c in the ECI coordinate system, [ δr ] LVLH T δv LVLH T ] T Namely, tracking the state quantity of the spacecraft P relative to the reference spacecraft c, wherein the state quantity at the initial moment is the required initial state quantity
Figure BDA0003101870740000127
The same method can obtain the state quantity of the escape spacecraft E relative to the reference spacecraft c, and the state quantity at the initial moment is the required initial state quantity
Figure BDA0003101870740000128
/>
In the third step, the correction vector added in the initial state vector of the tracking spacecraft relative to the reference spacecraft is as follows:
Figure BDA0003101870740000129
wherein ρ is P To track the distance of a spacecraft relative to the initial moment of a reference spacecraft, alpha P0 For tracking the position phase of the initial moment of the spacecraft, n is the average angular velocity of the reference spacecraft, μ= 3.986005 ×10 14 m 3 /s 2 Is the constant of the gravitational force, a c Is the orbit semi-long axis of the reference spacecraft;
the correction vectors added in the initial state vector of the escape spacecraft relative to the reference spacecraft are as follows:
Figure BDA0003101870740000131
wherein ρ is E For escaping the distance of the spacecraft relative to the initial moment of the reference spacecraft, alpha E0 For escaping the position phase of the spacecraft at the initial moment, a c For the orbit semi-long axis of the reference spacecraft, n is the average angular velocity of the reference spacecraft, μ= 3.986005 ×10 14 m 3 /s 2 Is the gravitational constant.
In the third step, a correction vector is added into an initial state vector of the tracking spacecraft relative to the reference spacecraft, and the initial state vector of the tracking spacecraft relative to the reference spacecraft is corrected as follows:
Figure BDA0003101870740000132
wherein ρ is P To track the distance of a spacecraft relative to the initial moment of a reference spacecraft, alpha P0 For tracking the position phase of the initial moment of the spacecraft, n is the average angular velocity of the reference spacecraft, μ= 3.986005 ×10 14 m 3 /s 2 Is the constant of the gravitational force, a c Is the orbit semi-long axis of the reference spacecraft;
correcting the initial state quantity of the escape spacecraft relative to the reference spacecraft as follows:
Figure BDA0003101870740000133
wherein ρ is E For escaping the distance of the spacecraft relative to the initial moment of the reference spacecraft, alpha E0 For escaping the position phase of the spacecraft at the initial moment, a c For the orbit semi-long axis of the reference spacecraft, n is the average angular velocity of the reference spacecraft, μ= 3.986005 ×10 14 m 3 /s 2 Is the gravitational constant.
In the third step, a correction vector is added into an initial state vector of the tracking spacecraft relative to the reference spacecraft, and is substituted into a correction CW equation
Figure BDA0003101870740000134
In the method, the state vector of the spacecraft can be tracked at the time t>
Figure BDA0003101870740000135
Wherein, (x) P (t),y P (t),z P (t)) tracking the position coordinates of the spacecraft in the LVLH coordinate system for the moment t;
Figure BDA0003101870740000136
is x P ,y P ,z P For the value of the time derivative function at the moment T, T represents transposition;
adding a correction vector into an initial state vector of the escape spacecraft relative to the reference spacecraft, and substituting a correction CW equation
Figure BDA0003101870740000141
Can escape at time tState vector for spacecraft
Figure BDA0003101870740000142
Wherein, (x) E (t),y E (t),z E (t)) is the position coordinate of the escape spacecraft at the moment t in the LVLH coordinate system;
Figure BDA0003101870740000143
is x E ,y E ,z E For the value of the time derivative function at the moment T, T represents transposition; />
(x E (t)-x P (t),y E (t)-y P (t),z E (t)-z P (t)) is the relative position of the GEO-game spacecraft after controlling the aspheric perturbation error.
The tracking spacecraft can be used for carrying out a capturing task on a space target; escaping spacecraft and avoiding capturing of tracking spacecraft; the reference spacecraft is a virtual spacecraft which does not do orbital maneuver.
The LVLH coordinate system is a local horizontal and local vertical coordinate system, and refers to a positive direction taking the mass center of the reference spacecraft as an origin and the sagittal diameter from the earth center to the reference spacecraft as an x-axis, wherein the z-axis points to the positive normal direction of the orbit of the reference spacecraft, and the direction of the y-axis is determined according to a right-hand rule.
A local vertical and local horizontal coordinate system (Local Vertical Local Horizontal, LVLH) with the centre of mass of the spacecraft as the origin of coordinates in the direction e from the centre of earth to the centre of mass of the spacecraft x In the direction of the x-axis, in the positive normal direction e of the orbit plane of the spacecraft z Direction e of the y-axis, which is the direction of the z-axis y Determined by the following formula:
e y ＝e z ×e x (5)
selecting a virtual non-orbiting spacecraft which is not far away from the tracking spacecraft P and the escape spacecraft E as a reference spacecraft, marking the reference spacecraft as c, establishing an LVLH coordinate system taking the mass center of the reference spacecraft as an origin, and marking as:
Figure BDA0003101870740000144
the spacecraft moves relatively closely, especially for two spacecrafts in game, the relative distance is relatively close, when:
Figure BDA0003101870740000145
wherein: ρ represents the relative distance of the two spacecraft and a represents the orbit semi-major axis of the spacecraft.
LVLH coordinate system with center of mass of reference spacecraft as origin
Figure BDA0003101870740000146
In the method, the relative state quantity of the slave spacecraft is recorded as
Figure BDA0003101870740000151
Thrust acceleration u= (U) x ,u y ,u z ) T Then:
Figure BDA0003101870740000152
wherein:
Figure BDA0003101870740000153
Figure BDA0003101870740000154
t 0 at the moment, the relative state quantity of the slave spacecraft is:
Figure BDA0003101870740000155
then, at time t, the slave spacecraft state is: />
Figure BDA0003101870740000156
Wherein:
Figure BDA0003101870740000157
φ rr (t,t 0 )、φ rv (t,t 0 )、φ vr (t,t 0 ) And phi vv (t,t 0 ) Are all 3 x 3 matrices and:
Figure BDA0003101870740000158
wherein: τ=ω (t-t) 0 )，s＝sinτ，c＝cosτ。
In order to obtain a linearized state equation, the principal and subordinate spacecraft are subjected to Taylor expansion to discard the second-order and high-order small quantities in the earth potential function under the condition that the discarded nonlinear terms can lead the state quantity of the relative motion to be in
Figure BDA00031018707400001612
The y-axis direction of the coordinate system produces errors that accumulate over time. In the method provided by the invention, the influence of a second-order term in the earth potential function on the relative motion is studied, and the error accumulated over time caused by the second-order term is eliminated by adding a correction term in the initial state quantity.
The method provided by the invention has the following processing steps of:
(1) Preserving the second order term in the taylor expansion of the gravitational potential function, and taking the second order term as
Figure BDA00031018707400001610
Virtual control force u=u in (a) nl ；
(2) Solving homogeneous state equation
Figure BDA00031018707400001611
Solution X of (2) cw X is taken as cw Substitution U nl Then solve +.>
Figure BDA0003101870740000161
Solution of->
Figure BDA0003101870740000162
The error propagation along with time is obtained, and X is made nl The cumulative term over time in (t) is 0, and the correction amount +_of the initial state quantity is obtained>
Figure BDA0003101870740000163
(3) Order the
Figure BDA0003101870740000164
Replacing the initial value X of the CW equation cw (0) Further, a state quantity X (t) which changes with time is obtained, and the X (t) has no error accumulated with time in the y-axis direction.
The method provided by the invention eliminates the demonstration of linearization errors.
And (3) proving:
the processing steps for eliminating linearization errors according to the method provided by the invention are as follows: x is X cw Is that
Figure BDA00031018707400001613
Solution of (1) linearization error->
Figure BDA0003101870740000165
When the correction is not performed,
Figure BDA0003101870740000166
at this time, errors in the y-axis direction accumulate over time, and errors in the state quantity are:
Figure BDA0003101870740000167
/>
selecting a proper one
Figure BDA0003101870740000168
Make->
Figure BDA0003101870740000169
No error accumulated in the y-axis direction with time, let +.>
Figure BDA0003101870740000171
Correcting the initial value of the second step, wherein the linearization error is +.>
Figure BDA0003101870740000172
The y-axis direction is free of errors accumulated with time, and the verification is obtained.
The syndrome is known.
Two spacecraft for geostationary orbit gaming: the method is characterized in that the tracking spacecraft P and the escape spacecraft E are frequently maneuvered in the game process, so that a relative motion dynamics model of the tracking spacecraft P relative to the escape spacecraft E or the escape spacecraft E relative to the tracking spacecraft P is inconvenient to establish, and a virtual spacecraft which is not far away from the tracking spacecraft P and the escape spacecraft E and does not maneuver in an orbit is selected as a reference spacecraft to establish. The reference spacecraft c is in a nominal geostationary orbit, i.e. the orbit semi-major axis of the reference spacecraft c is a c The tilt angle and the eccentricity are both 0, and:
Figure BDA0003101870740000173
the reference spacecraft is also called a master spacecraft, wherein i is a slave spacecraft. ρ ic Representing the distance from spacecraft i to reference spacecraft c. The subscripts i and c in this section denote the slave spacecraft i and the reference spacecraft c, respectively.
Instantaneous position and velocity vectors of spacecraft in geocentric inertial system are r and r, respectively
Figure BDA0003101870740000178
According to newton's second law:
Figure BDA0003101870740000174
wherein R is the earth attraction potential function of the spacecraft at R. Establishing an LVLH coordinate system by taking the centroid of a reference spacecraft c as an origin
Figure BDA0003101870740000175
Slave spacecraft i and reference spacecraft c and relative position vector are noted as ρ=r i -r c ρ is +.>
Figure BDA0003101870740000176
The coordinates of (a) are:
Figure BDA0003101870740000177
for slave spacecraft i and reference spacecraft c, there are:
Figure BDA0003101870740000181
Figure BDA0003101870740000182
Figure BDA0003101870740000183
in the formula, deltaf represents the difference value of perturbation acceleration suffered by the master-slave spacecraft, and only J is considered in the method provided by the invention 2 And J 22 Term perturbation, Δf=0, u i Indicating the control acceleration from spacecraft i, let u be i =0. Conversion of the geocentric inertial system to
Figure BDA0003101870740000189
In (a):
Figure BDA0003101870740000184
wherein: omega c ＝[0 0 n] T ，
Figure BDA0003101870740000185
r c ＝[r c 0 0] T Wherein: />
Figure BDA0003101870740000186
And (3) finishing to obtain:
Figure BDA0003101870740000187
in the CW equation, the equation is reduced to:
Figure BDA0003101870740000188
the state quantity in the CW equation is noted as:
Figure BDA0003101870740000191
solving to obtain:
Figure BDA0003101870740000192
in the method, in the process of the invention,
Figure BDA0003101870740000193
is the square of the relative distance between the master spacecraft and the slave spacecraft, alpha 0 For the initial phase, defined as: />
Figure BDA0003101870740000194
The y-axis of the system is along the-z directionAngle of rotation. Initial value of step two:
Figure BDA0003101870740000195
/>
reserving a second-order term on the right side in the middle, discarding a third-order term and higher-order terms above the third-order term, and finishing to obtain the three-order and higher-order:
Figure BDA0003101870740000196
from equation (25), considering the influence of the second order term in the taylor expansion of the gravitational potential function corresponds to the application of the virtual control force in step two:
Figure BDA0003101870740000197
substituting formula (23) into formula (26) to obtain:
Figure BDA0003101870740000201
formula (25) can be written as:
Figure BDA0003101870740000202
equation (28) is the linearized error propagation state equation of step two. The linearization error of the second step is as follows:
Figure BDA0003101870740000203
the initial value is recorded as:
Figure BDA0003101870740000204
using the conclusion of formula (10), the solution of formula (28) is:
Figure BDA0003101870740000205
X nl the expression of (t) is shown in appendix A. From this, it can be seen that the linearization error in step two can be divided into three categories: constant term, cumulative over time term, and short period term. The cumulative term over time only appears in the y-axis direction, i.e.:
Figure BDA0003101870740000206
in GEO-spacecraft gaming, the duration is long and the accumulated items over time can have a large impact on the outcome of the game. Therefore, in order to avoid accumulation of errors in the y-axis direction, let y nl-long =0, yielding:
Figure BDA0003101870740000207
thus in the initial state quantity X of the relative movement cw (0) Adding correction amount
Figure BDA0003101870740000208
Immediate use->
Figure BDA0003101870740000209
Replacement X cw (0) The linearization error in step two can be eliminated, wherein:
Figure BDA0003101870740000211
i.e. X in step two i (0) The correction is as follows:
Figure BDA0003101870740000212
a preferred embodiment
In order to analyze the errors of the method provided by the invention under different initial conditions, the number of orbits of the slave spacecraft i is set as shown in table 1.
TABLE 1 orbit count from spacecraft i
Figure BDA0003101870740000213
The number of orbits of spacecraft c is shown in table 2.
TABLE 2 reference orbit count for spacecraft c
Figure BDA0003101870740000214
Spacecraft i is at
Figure BDA0003101870740000216
The initial state amounts in (a) are shown in table 3.
TABLE 3 from spacecraft i
Figure BDA0003101870740000217
Initial state of (a)
Figure BDA0003101870740000215
Figure BDA0003101870740000221
The relative motion relation between the No. 1 spacecraft and the No. 2 spacecraft in the table 1 and the reference spacecraft c is simulated by applying the method and the CW equation provided by the invention respectively. Comparing the change condition of the x, y and z state quantity of the method and the CW equation with time, calculating the change condition of the difference value of the y state quantity of the method and the CW equation with time, and calculating the change condition of the difference value of the state quantity and the true value calculated by the method and the CW equation with time.
a. Case 1: ρ=200 km, α 0 ＝180°
The relative motion relation between the No. 1 spacecraft and the reference spacecraft c is analyzed by using the method and the CW equation provided by the invention. The method and the change of the x state quantity in the CW equation with time are shown in figure 1, wherein the solid line in the figure represents the x state quantity in the method provided by the invention, and the dotted line represents the x state quantity in the CW equation.
The change of the y state quantity in the method and the CW equation with time is shown in FIG. 2, wherein the solid line in the graph represents the y state quantity in the method and the broken line represents the x state quantity in the CW equation.
The time-dependent variation of the difference between the method provided by the invention and the y state quantity in the CW equation is shown in FIG. 3
The change of the z state quantity in the method and the CW equation with time is shown in FIG. 4, wherein the solid line in the diagram represents the z state quantity in the method and the broken line represents the x state quantity in the CW equation.
The time-dependent change of the difference between the x state quantity and the true value in the method and the CW equation is shown in FIG. 5, wherein the solid line in the graph represents the time-dependent change of the difference between the x state quantity and the true value in the method and the broken line represents the time-dependent change of the difference between the x state quantity and the true value in the CW equation.
The time-dependent change of the difference between the y state quantity and the true value in the method and the CW equation is shown in FIG. 6, wherein the solid line in the graph represents the time-dependent change of the difference between the y state quantity and the true value in the method and the broken line represents the time-dependent change of the difference between the y state quantity and the true value in the CW equation.
The time-dependent change of the difference between the z state quantity and the true value in the method and the CW equation is shown in FIG. 7, wherein the solid line in the graph represents the time-dependent change of the difference between the z state quantity and the true value in the method and the broken line represents the time-dependent change of the difference between the z state quantity and the true value in the CW equation.
b. Case 2: ρ=200 km, α 0 ＝270°
The relative motion relation between the No. 2 spacecraft and the reference spacecraft c is analyzed by using the method and the CW equation provided by the invention. The time-dependent changes of the x state quantity in the method and the CW method provided by the invention are shown in FIG. 8, wherein the solid line in the graph represents the x state quantity in the method provided by the invention, and the dotted line represents the x state quantity in the CW equation.
The method and the change of the y state quantity in the CW equation with time are shown in FIG. 9, wherein the solid line in the figure represents the x state quantity in the method provided by the invention, and the dotted line represents the x state quantity in the CW equation.
The time-dependent variation of the difference between the method provided by the invention and the y state quantity in the CW equation is shown in FIG. 10
The change of the z state quantity with time in the method and the CW equation provided by the invention is shown in FIG. 11, wherein the solid line in the figure represents the z state quantity in the method provided by the invention, and the dotted line represents the z state quantity in the CW equation.
The time-dependent change of the difference between the x state quantity and the true value in the method and the CW equation is shown in FIG. 12, wherein the solid line in the graph represents the time-dependent change of the difference between the x state quantity and the true value in the method and the broken line represents the time-dependent change of the difference between the x state quantity and the true value in the CW equation.
The time-dependent change of the difference between the y state quantity and the true value in the method and the CW equation is shown in FIG. 13, wherein the solid line in the graph represents the time-dependent change of the difference between the y state quantity and the true value in the method and the broken line represents the time-dependent change of the difference between the y state quantity and the true value in the CW equation.
The time-dependent change of the difference between the z state quantity and the true value in the method and the CW equation is shown in FIG. 14, wherein the solid line in the graph represents the time-dependent change of the difference between the z state quantity and the true value in the method and the broken line represents the time-dependent change of the difference between the z state quantity and the true value in the CW equation.
The game initial formation configuration parameters formed by spacecraft i and reference spacecraft c and the time accumulated deviation in the y-axis direction after 10 cycles (861640 s) are obtained through the four cases, and are shown in table 4, d ic (0) Is the initial distance of the master spacecraft and the slave spacecraft, y nl-long Is the time-cumulative bias in the y-axis direction in the CW equation,
Figure BDA0003101870740000241
is the time accumulated deviation in the y-axis direction in the method provided by the invention.
Table 4 game configuration parameters from spacecraft i and reference spacecraft c
Figure BDA0003101870740000242
The CW equation does not control the accumulated bias in the y-axis direction over time, in the 2 cases described above: the primary distance is 200km,100km of the primary and secondary spacecraft, after 10 orbit periods, respectively generate deviation of 112km,22km, obviously if the error accumulated along with time is not controlled, the task will fail.
The method provided by the invention is used for controlling the accumulated deviation of the y-axis direction along with time, and in the 2 cases: the initial distances are respectively 200km and 100km of master-slave spacecraft, the deviation is only 0.12km and 0.07km after 10 orbit periods, and compared with a CW equation, the accumulated deviation of the y-axis direction along time is not more than 0.7% of the deviation of the CW equation, so that the game task research requirement of the GEO spacecraft can be met.
The embodiment of the invention discloses a method for improving the accuracy of a game dynamics model of an geostationary orbit spacecraft. For a model for researching relative motion in the chase game, the relative state vector is used as a model of a state quantity, so that the problem can be conveniently resolved, but the accuracy is lower due to nonlinear errors. Aiming at the defect that nonlinear errors exist in the relative positions among game spacecrafts obtained by using a CW equation in the relative motion process of the earth stationary orbit game spacecrafts, the invention takes a second-order nonlinear item of a potential function as virtual control thrust, solves a state equation, converts nonlinear deviation accumulated along with time into a correction item for initial state parameters, and avoids the accumulation of the nonlinear deviation along with time. In simulation, the method provided by the invention reduces the deviation after 10 track periods from hundred kilometers to hundred meters. On the premise of not changing the linear characteristic of the equation, the accuracy of the method provided by the invention is improved, and the game strategy of the spacecraft is conveniently researched by using a variation method.
The embodiment of the invention has the following beneficial effects:
compared with a CW equation, the method provided by the invention avoids error propagation and is characterized in that:
a. aiming at the nonlinear error problem existing in the CW equation, the method provided by the invention considers the influence of the abandoned second-order nonlinear term in the linearization process of the potential function, calculates error propagation caused by the second-order nonlinear term, wherein the error comprises three types of a constant term, a time accumulation term and a short period term, further calculates a correction term for eliminating the time accumulation term, and avoids the propagation of the second-order nonlinear error by correcting the initial state quantity,
b. aiming at the problem of eccentricity error in a CW equation, when c is selected, the eccentricity of c is made to be 0, and the propagation of the eccentricity error in the method is avoided.
The method provided by the invention does not change the linear characteristic of the equation while improving the accuracy of the equation, and is convenient for researching the game strategy of the spacecraft by combining a variational method.

Claims (6)

1. A control method for nonlinear errors of relative positions of a geostationary orbit game spacecraft is characterized by comprising the following steps:
step one, selecting a reference spacecraft, wherein the orbit semi-long axis of the reference spacecraft is a GEO nominal orbit semi-long axis, the eccentricity is 0, the orbit inclination angle is 0, the right ascent and intersection point, the near-site depression angle and the flat near-site angle are the same as those of a tracking spacecraft or an escape spacecraft, and establishing an LVLH coordinate system by taking the mass center of the reference spacecraft as an origin;
step two, respectively calculating initial state vectors of the tracking spacecraft and the escape spacecraft relative to the reference spacecraft in the LVLH coordinate system;
step three, adding a correction vector into an initial state vector of the tracking spacecraft relative to the reference spacecraft, and substituting the correction vector into a correction CW equation to obtain a state vector of the tracking spacecraft; adding a correction vector into an initial state vector of the escape spacecraft relative to the reference spacecraft, and substituting the correction vector into a correction CW equation to obtain a state vector of the escape spacecraft; subtracting the state vector of the tracking spacecraft from the state vector of the escape spacecraft to obtain the relative position of the geostationary orbit game spacecraft after controlling the nonlinear error;
the modified CW equation is:
Figure FDA0004212680790000011
wherein: x (t) represents a state vector at time t, X correction (t 0 ) Representing the initial state vector after addition of the correction term, U being the applied control force, B being a constant matrix,
Figure FDA0004212680790000012
s is the sign of the integral argument, U(s) is the control force applied at time s, phi (t, s) is the state transition matrix from time s to time t, phi (t, t 0 ) A state transition matrix from time to time t;
in the third step, the correction vector added in the initial state vector of the tracking spacecraft relative to the reference spacecraft is as follows:
Figure FDA0004212680790000013
wherein ρ is P To track the distance of a spacecraft relative to the initial moment of a reference spacecraft, alpha P0 For tracking the position phase of the initial moment of the spacecraft, n is the average angular velocity of the reference spacecraft, μ= 3.986005 ×10 14 m 3 /s 2 Is the constant of the gravitational force, a c Is the orbit semi-long axis of the reference spacecraft;
the correction vectors added in the initial state vector of the escape spacecraft relative to the reference spacecraft are as follows:
Figure FDA0004212680790000021
wherein ρ is E For escaping the distance of the spacecraft relative to the initial moment of the reference spacecraft, alpha E0 For escaping the position phase of the spacecraft at the initial moment, a c For the orbit semi-long axis of the reference spacecraft, n is the average angular velocity of the reference spacecraft, μ=3.986005×10 14 m 3 /s 2 Is the gravitational constant.
2. The method of claim 1, wherein in the second step, the method for calculating an initial state vector of the tracking spacecraft with respect to the reference spacecraft in the LVLH coordinate system comprises:
Figure FDA0004212680790000022
wherein, (x) P (t 0 ),y P (t 0 ),z P (t 0 ) Tracking the position coordinates of the spacecraft in the LVLH coordinate system at the initial moment;
Figure FDA0004212680790000023
is x P ,y P ,z P The value of the time derivative function at the initial moment; t represents a transpose; />
In an LVLH coordinate system, a method of calculating an initial state vector of an escape spacecraft relative to a reference spacecraft includes:
Figure FDA0004212680790000024
wherein, (x) E (t 0 ),y E (t 0 ),z E (t 0 ) Tracking the position coordinates of the spacecraft in the LVLH coordinate system for the initial moment,
Figure FDA0004212680790000025
is x E ,y E ,z E The value of the time derivative function at the initial moment; t represents the transpose.
3. Method according to one of the claims 1-2, characterized in that in step three, a correction vector is added to the initial state vector of the tracking spacecraft with respect to the reference spacecraft, and the initial state vector of the tracking spacecraft with respect to the reference spacecraft is corrected as:
Figure FDA0004212680790000026
wherein ρ P To track the distance of the spacecraft relative to the initial moment of the reference spacecraft, alpha P0 For tracking the position phase of the initial moment of the spacecraft, n is the average angular velocity of the reference spacecraft, μ= 3.986005 ×10 14 m 3 /s 2 Is the constant of the gravitational force, a c Is the orbit semi-long axis of the reference spacecraft;
correcting the initial state quantity of the escape spacecraft relative to the reference spacecraft as follows:
Figure FDA0004212680790000031
wherein ρ is E For escaping the distance of the spacecraft relative to the initial moment of the reference spacecraft, alpha E0 For escaping the position phase of the spacecraft at the initial moment, a c For the orbit semi-long axis of the reference spacecraft, n is the average angular velocity of the reference spacecraft, μ= 3.986005 ×10 14 m 3 /s 2 Is the gravitational constant.
4. Method according to one of the claims 1-2, characterized in that in step three, a correction vector is added to the initial state vector of the tracking spacecraft with respect to the reference spacecraft and substituted into the correction CW equation
Figure FDA0004212680790000032
In the method, the state vector of the spacecraft can be tracked at the time t
Figure FDA0004212680790000033
Wherein, (x) P (t),y P (t),z P (t)) tracking the position coordinates of the spacecraft in the LVLH coordinate system for the moment t;
Figure FDA0004212680790000034
is x P ,y P ,z P For the value of the time derivative function at the moment T, T represents transposition;
adding a correction vector into an initial state vector of the escape spacecraft relative to the reference spacecraft, and substituting a correction CW equation
Figure FDA0004212680790000035
State vector of escape spacecraft at time t
Figure FDA0004212680790000036
Wherein, (x) E (t),y E (t),z E (t)) is the position coordinate of the escape spacecraft at the moment t in the LVLH coordinate system;
Figure FDA0004212680790000037
is x E ,y E ,z E For the value of the time derivative function at the moment T, T represents transposition;
(x E (t)-x P (t),y E (t)-y P (t),z E (t)-z P (t)) is the relative position of the GEO-game spacecraft after controlling the aspheric perturbation error.
5. The method of claim 1, wherein the LVLH coordinate system is a local horizontal and local vertical coordinate system, and is defined by taking a centroid of the reference spacecraft as an origin, taking a sagittal diameter from the earth center to the reference spacecraft as a forward direction of an x-axis, wherein a z-axis points to a forward direction of an orbit of the reference spacecraft, and wherein a direction of a y-axis is determined according to a right-hand rule.
6. Method according to one of claims 1-2, wherein the tracking spacecraft is operable to perform a capture task on a space object;
escaping spacecraft and avoiding capturing of tracking spacecraft;
the reference spacecraft is a virtual spacecraft which does not do orbital maneuver.
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