CN113282096A - Control method for relative position nonlinear error of geostationary orbit game spacecraft - Google Patents

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

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CN113282096A
CN113282096A CN202110625147.XA CN202110625147A CN113282096A CN 113282096 A CN113282096 A CN 113282096A CN 202110625147 A CN202110625147 A CN 202110625147A CN 113282096 A CN113282096 A CN 113282096A
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CN113282096B (en
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张雅声
张海涛
李智
王伟林
于金龙
梁爽
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 method for controlling nonlinear errors of relative positions of geostationary orbit game spacecrafts, which comprises the following steps: establishing an LVLH coordinate system by taking the mass center of the reference spacecraft as an origin; respectively calculating initial state vectors of the tracking spacecraft and the escaping spacecraft relative to the reference spacecraft in an LVLH coordinate system; respectively adding correction vectors into the initial state vectors of the tracking spacecraft and the escaping spacecraft relative to the reference spacecraft, substituting the correction vectors into a correction CW equation, and obtaining the state vectors of the tracking spacecraft and the escaping spacecraft; and subtracting the state vector of the tracking spacecraft from the state vector of the escaping spacecraft to obtain the relative position of the geostationary orbit game spacecraft after the nonlinear error is controlled. Aiming at the relative motion process of the earth stationary orbit game spacecraft, the second-order nonlinear term of the gravitational potential function is used as virtual control thrust, the state equation is solved, the nonlinear deviation accumulated along with time is converted into a correction term of initial state parameters, and the accumulation of the nonlinear deviation along with time is avoided.

Description

Control method for relative position nonlinear error of geostationary orbit game spacecraft
Technical Field
The invention relates to a control method for a relative position nonlinear error of a geostationary orbit game spacecraft, belonging to the field of spacecraft orbit dynamics.
Background
Methods of studying game relative motion can be divided into two categories: kinematic-based relative motion, kinetic-based relative motion. The kinematics-based relative motion is generally done by: firstly, establishing a conversion relation between the difference value of absolute orbit numbers of a master spacecraft and a slave spacecraft and the 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 number of relative tracks and state quantities under the coordinate system; thirdly, the relation between the pulse control force in three directions and the absolute track root difference value can be obtained according to the Gaussian perturbation equation, so that the conversion relation among the control force, the absolute track root difference value, the relative track root and the state quantity is established, and fourthly, the time and the size of the application of the control force are optimized, so that the method can be used for analyzing the construction of formation and reconstructing the optimal control problem.
Typical of the motion equation is an elliptic relative motion equation derived by invar abundance, korea tide and the like based on relative orbit elements. Invar and creation abundance, korea tide, etc. propose a set of methods for designing, reconstructing and optimally controlling near-circular and elliptical formation from the kinematics mode: the method comprises the following steps of strictly defining relative orbit elements in the literature ([1] Hanchao, Yijiafeng, relative motion research of elliptic orbit satellites based on the relative orbit elements [ J ]. aeronautical report, 2011,32(12):2244-2258.Han Chao, Yin Jianfeng.study of satellite relative motion in an analytical use of relative orbit elements [ J ]. Acta Aeronautica et astroautica site, 2011,32(12):2244 @ 2258.), and deriving a conversion matrix between the relative orbit elements and the absolute orbit elements for analyzing the formation relative motion relation; deriving a conversion Matrix and a conversion inverse Matrix of State quantities of Relative track elements to a centroid track system in a document ([1] Yin J, Rao Y, Han C.inverse Transformation of orthogonal State Transformation Matrix [ J ]. International Journal of Astronomy & Astronomy, 2014,04(3):419-428.), and establishing a conversion equation of a pulse velocity increment and the State quantities in the track system; in the literature ([1] Yin J, Han c. electrolytic formation controlled on relative orbit elements [ J ]. chinese aeronautical bulletin (english edition), 2013,26(006): 1554-. The work of invar creation abundance and korea tide has reference significance in the optimization control of the single prescription of the game, and the defects are as follows: the kinematic formation relative motion equation based on the relative orbit elements cannot establish a state equation of state quantity differential to time, and is inconvenient for the research of interactive game optimization strategies of both sides of the spacecraft.
The advantage of studying relative motion based on kinematics is: the model has high precision, is convenient for introducing the influence of perturbation on relative motion, and can effectively control the accumulation of model errors along with time. The disadvantages are that: the system cannot be described through a differential equation, and interactive game optimization strategies of two parties are inconvenient to research in the differential game problem.
The method is widely applied to research on relative motion based on a kinetic method, namely Clohessy-Wiltshire equation, which is abbreviated as cw equation. General idea for the kinetic-based study of relative motion: 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 to the time, namely a state equation; and thirdly, analyzing the differential equation for constructing, reconstructing and optimizing control of formation.
The CW equation is:
Figure BDA0003101870740000021
wherein X (t) represents a state vector at time t, X (t)0) Representing the 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, φ (t, s) is the state transition matrix from s to t, φ (t, t)0) Is t0A state transition matrix to t;
Figure BDA0003101870740000032
wherein: τ ═ ω (t-t)0) Denotes a reference spacecraft from t0By the angle of the turn-over of t,
Figure BDA0003101870740000033
acfor reference to the orbit semi-major axis of the spacecraft, mu is 3.986005 × 1014m3/s2Is the earth's gravitational constant.
The defects are as follows: the errors of the CW equation are derived from linearization errors and perturbation errors, the CW equation also has small eccentricity assumed errors, the errors are large, and particularly, the errors accumulated along with time are not easy to control. Has the advantages that: after the state equation is established, the research on the relative motion is conveniently carried out by applying the theory of modern control and optimal control, and the interactive differential game problem of the two parties is particularly conveniently solved.
For a relative motion model of a GEO spacecraft, an STM model is representative. The STM model was developed from the first set up over a period of more than a decade. 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.) researches a relative Orbit root Formation kinetic model STM model based on relative E/I vectors, and J is considered in the STM model2Item perturbation, and the problem of formation safety path is researched; gaias, G (Gaias, G., ardens, J. -S., and Montenbrock, O., "Model of J2 approved Satellite Relative Motion with Time-Varying Differential drags," cellular mechanisms and dynamic asymmetry, Vol.123, No.4,2015, pp.411-433. doi:10.1007/s 10569-015-; spiridonova, S (Spiridonova, S., "Formation Dynamics in Geostationary Ring," Celestial Mechanics and dynamic Astronomy, Vol.125, No.4,2016, pp.485-500. doi: 10.1007/S10569-016. 9693-0) adds sunlight pressure and three-body perturbation in the STM model, eliminates all long-term and long-term terms of relative track root drift, perfects the STM model, and the simulation shows that the model has higher precision after 10 days, so that the STM model is suitable for GEO Formation problems of in-track service tasks including near space operation and the like. Although the accuracy is high, the STM model after Spiridonova and S are complete is complex, comprises a nonlinear part, cannot establish a linear state control equation, and is not convenient to use for GEO spacecraft game problems.
Disclosure of Invention
The invention provides a control method of a non-linear error of a relative position of an geostationary orbit game spacecraft, which aims at the technical problem that a space mission cannot be normally completed due to the fact that a Clohessy-Wiltshire equation, namely the non-linear error existing in a CW equation, is inaccurate in calculated relative position.
In order to achieve the aim, the invention provides a method for controlling the relative position nonlinear error of a geostationary orbit gaming spacecraft, which comprises the following steps:
selecting a reference spacecraft, wherein the orbit semi-major axis of the reference spacecraft is a GEO nominal orbit semi-major axis, the eccentricity is 0, the orbit inclination angle is 0, the ascension of a rising intersection point, the depression angle of a near place and the horizontal near place 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;
respectively calculating initial state vectors of the tracked spacecraft and the escaped spacecraft relative to the reference spacecraft in an LVLH coordinate system;
adding a correction vector into the initial state vector of the tracking spacecraft relative to the reference spacecraft, and substituting the correction vector into a correction CW equation to obtain the state vector of the tracking spacecraft; adding a correction vector into the initial state vector of the escaping spacecraft relative to the reference spacecraft, and substituting the correction vector into a correction CW equation to obtain the state vector of the escaping 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 nonlinear error control;
the modified CW equation is:
Figure BDA0003101870740000051
wherein: x (t) represents the state vector at time t, Xcorrection(t0) Representing the initial state vector after the 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, φ (t, s) is the state transition matrix from time s to time t, φ (t, t)0) Is t0The state transition matrix from time to time t.
In the second step, the method for calculating the initial state vector of the tracking spacecraft relative to the reference spacecraft in the LVLH coordinate system includes:
Figure BDA0003101870740000061
wherein (x)P(t0),yP(t0),zP(t0) For initial time tracking the position coordinates of the spacecraft in the LVLH coordinate system;
Figure BDA0003101870740000062
is xP,yP,zPThe value of the time derivative function at the initial moment; t represents transposition;
in the LVLH coordinate system, the method for calculating the initial state vector of the escape spacecraft relative to the reference spacecraft comprises the following steps:
Figure BDA0003101870740000063
wherein (x)E(t0),yE(t0),zE(t0) To track the position coordinates of the spacecraft in the LVLH coordinate system for an initial moment,
Figure BDA0003101870740000064
is xE,yE,zEThe value of the time derivative function at the initial moment; t denotes transposition.
In the third step, the correction vector added to the initial state vector of the tracking spacecraft relative to the reference spacecraft is as follows:
Figure BDA0003101870740000065
where ρ isPFor tracking the distance of the spacecraft with respect to the initial moment of the reference spacecraft, αP0To track the position phase of the spacecraft at the initial moment, n is the average angular velocity of the reference spacecraft, μ 3.986005 × 1014m3/s2Is the constant of the earth's gravity, acIs the orbit semi-major axis of the reference spacecraft;
the correction vector added in the initial state vector of the escape spacecraft relative to the reference spacecraft is as follows:
Figure BDA0003101870740000066
where ρ isEIs the distance, alpha, of the escaping spacecraft relative to the initial time of the reference spacecraftE0For the position phase at the initial moment of the escaping spacecraft, acIs the orbit semi-major axis of the reference spacecraft, n is the average angular velocity of the reference spacecraft, mu is 3.986005 multiplied by 1014m3/s2Is the earth's gravitational constant.
In the third step, a correction vector is added into the 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 into:
Figure BDA0003101870740000067
where ρ isPFor tracking the distance of the spacecraft with respect to the initial moment of the reference spacecraft, αP0To track the position phase of the spacecraft at the initial moment, n is the average angular velocity of the reference spacecraft, μ 3.986005 × 1014m3/s2Is the constant of the earth's gravity, acIs the orbit semi-major axis of the reference spacecraft;
correcting the initial state quantity of the escape spacecraft relative to the reference spacecraft into:
Figure BDA0003101870740000071
where ρ isEIs the distance, alpha, of the escaping spacecraft relative to the initial time of the reference spacecraftE0For the position phase at the initial moment of the escaping spacecraft, acIs the orbit semi-major axis of the reference spacecraft, n is the average angular velocity of the reference spacecraft, mu is 3.986005 multiplied by 1014m3/s2Is the earth's gravitational constant.
In the third step, a correction vector is added into the initial state vector of the tracking spacecraft relative to the reference spacecraft and substituted into a correction CW equation
Figure BDA0003101870740000072
In the method, the state vector of the tracked spacecraft at the time t can be obtained
Figure BDA0003101870740000073
Wherein (x)P(t),yP(t),zP(t)) tracking the position coordinates of the spacecraft in the LVLH coordinate system at the time t;
Figure BDA0003101870740000077
is xP,yP,zPFor the value of the time derivative function at the time T, T represents transposition;
adding a correction vector into the initial state vector of the escape spacecraft relative to the reference spacecraft, and substituting the correction vector into a correction CW equation
Figure BDA0003101870740000074
The state vector of the escaping spacecraft at the moment t can be obtained
Figure BDA0003101870740000075
Wherein (x)E(t),yE(t),zE(t)) is the position coordinate of the escaping spacecraft in the LVLH coordinate system at the moment t;
Figure BDA0003101870740000076
is xE,yE,zEFor the value of the time derivative function at the time T, T represents transposition;
(xE(t)-xP(t),yE(t)-yP(t),zE(t)-zP(t)) is the relative position of the GEO gaming spacecraft after controlling the non-spherical perturbation error.
The LVLH coordinate system is a local horizontal local vertical coordinate system, and is defined by taking the centroid of the reference spacecraft as an origin and the sagittal diameter from the geocentric to the reference spacecraft as the positive direction of an x axis, wherein a z axis points to the positive normal direction of the orbit of the reference spacecraft, and the direction of a y axis is determined according to a right-hand rule.
Wherein the tracking spacecraft can be used for performing a capture task on a space target;
the escaping spacecraft avoids catching of the 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 the CW equation, the propagation of the nonlinear error of the CW equation is avoided, and the method is embodied in that:
a. aiming at the problem of nonlinear errors in a CW equation, the method provided by the invention considers the influence of a second-order nonlinear term discarded in the process of linearization of a gravitational potential function, calculates the error propagation caused by the second-order nonlinear term, and further calculates a correction term for eliminating the time accumulation term, and avoids the propagation of the second-order nonlinear error by correcting an initial state quantity,
b. aiming at the eccentricity error problem existing in the CW equation, the eccentricity of the reference spacecraft is set to be 0 when the reference spacecraft is selected by the method provided by the invention, so that the eccentricity error is prevented from being spread in the method provided by the invention.
Description of the drawings:
FIG. 1 is a schematic diagram of the method provided by the present invention and the variation of x state quantity in CW equation with time;
FIG. 2 is a schematic diagram of the method provided by the present invention and the variation of the y-state quantity in the CW equation with time;
FIG. 3 is a schematic diagram of the difference between the method provided by the present invention and the y state quantity in the CW equation over time;
FIG. 4 is a schematic diagram of the method provided by the present invention and the variation of the z-state quantity in the CW equation over time;
FIG. 5 is a schematic diagram showing the variation of the difference between the x state quantity and the true value in the CW equation with time according to the method of the present invention;
FIG. 6 is a schematic diagram showing the variation of the difference between the y-state quantity and the true value in the CW equation with time according to the method of the present invention;
FIG. 7 is a schematic diagram showing the variation of the difference between the z-state quantity and the true value in the CW equation with time according to the method of the present invention;
FIG. 8 is a schematic diagram of the method provided by the present invention and the variation of x state quantity in CW equation with time;
FIG. 9 is a schematic diagram of the method provided by the present invention and the variation of the y-state quantity in the CW equation over time;
FIG. 10 is a schematic diagram of the difference between the method provided by the present invention and the y state quantity in the CW equation over time;
FIG. 11 is a schematic diagram of the method provided by the present invention and the variation of the z-state quantity in the CW equation over time;
FIG. 12 is a schematic diagram showing the variation of the difference between the x-state quantity and the true value in the CW equation with time according to the method of the present invention;
FIG. 13 is a schematic diagram showing the variation of the difference between the y-state quantity and the true value in the CW equation with time according to the method of the present invention;
fig. 14 is a schematic diagram showing the variation of the difference between the z-state quantity and the true value in the CW equation with time according to the method of the present invention.
Detailed Description
In this embodiment, the reference spacecraft is also called the master spacecraft and i is the slave spacecraft.
A method for controlling the nonlinear error of the relative position of a geostationary orbit game spacecraft comprises the following steps:
selecting a reference spacecraft, wherein the orbit semi-major axis of the reference spacecraft is a GEO nominal orbit semi-major axis, the eccentricity is 0, the orbit inclination angle is 0, the ascension of a rising intersection point, the depression angle of a near place and the horizontal near place 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;
respectively calculating initial state vectors of the tracked spacecraft and the escaped spacecraft relative to the reference spacecraft in an LVLH coordinate system;
adding a correction vector into the initial state vector of the tracking spacecraft relative to the reference spacecraft, and substituting the correction vector into a correction CW equation to obtain the state vector of the tracking spacecraft; adding a correction vector into the initial state vector of the escaping spacecraft relative to the reference spacecraft, and substituting the correction vector into a correction CW equation to obtain the state vector of the escaping 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 nonlinear error control;
the modified CW equation is:
Figure BDA0003101870740000101
wherein: x (t) represents the state vector at time t, Xcorrection(t0) Representing the initial state vector after the 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, φ (t, s) is the state transition matrix from time s to time t, φ (t, t)0) Is t0The state transition matrix from time to time t.
In the second step, the method for calculating the initial state vector of the tracking spacecraft relative to the reference spacecraft in the LVLH coordinate system includes:
Figure BDA0003101870740000103
wherein (x)P(t0),yP(t0),zP(t0) For initial time tracking the position coordinates of the spacecraft in the LVLH coordinate system;
Figure BDA0003101870740000104
is xP,yP,zPThe value of the time derivative function at the initial moment; t represents transposition;
in the LVLH coordinate system, the method for calculating the initial state vector of the escape spacecraft relative to the reference spacecraft comprises the following steps:
Figure BDA0003101870740000111
wherein (x)E(t0),yE(t0),zE(t0) To track the position coordinates of the spacecraft in the LVLH coordinate system for an initial moment,
Figure BDA0003101870740000112
is xE,yE,zEThe value of the time derivative function at the initial moment; t denotes transposition.
For example: note rLVLHTo track the position vector of the spacecraft P in the LVLH coordinate system, rECITo track the position vector of the spacecraft P in the Earth's inertial frame ECI (Earth centered-inertial frame), AECI→LVLHIs a coordinate transformation matrix from an ECI coordinate system to an LVLH coordinate system, ALVLH→ECIIs a coordinate transformation matrix from LVLH coordinate system to ECI coordinate system, so:
rECI＝ALVLH→ECI·rLVLH (1)
rLVLH＝AECI→LVLH·rECI (2)
Figure BDA0003101870740000113
Figure BDA0003101870740000114
wherein:
Figure BDA0003101870740000115
Figure BDA0003101870740000116
Figure BDA0003101870740000117
for a column vector a ═ a1 a2 a3]TAnd b ═ b1 b2 b3]TExistence of
Figure BDA0003101870740000118
Wherein:
Figure BDA0003101870740000119
AECI→LVLHand ALVLH→ECIThe rate of change over time is:
Figure BDA0003101870740000121
Figure BDA0003101870740000122
where 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 the formula [ delta rECI T δvECI T]TTo track the difference in the ECI coordinate system of the position velocity vectors of the spacecraft P and the reference spacecraft c, [ δ r [ ]LVLH T δvLVLH T]TNamely the state quantity of the tracking spacecraft P relative to the reference spacecraft c, 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 escaping 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 to the initial state vector of the tracking spacecraft relative to the reference spacecraft is as follows:
Figure BDA0003101870740000129
where ρ isPFor tracking the distance of the spacecraft with respect to the initial moment of the reference spacecraft, αP0To track the position phase of the spacecraft at the initial moment, n is the average angular velocity of the reference spacecraft, μ 3.986005 × 1014m3/s2Is the constant of the earth's gravity, acIs the orbit semi-major axis of the reference spacecraft;
the correction vector added in the initial state vector of the escape spacecraft relative to the reference spacecraft is as follows:
Figure BDA0003101870740000131
where ρ isEIs the distance, alpha, of the escaping spacecraft relative to the initial time of the reference spacecraftE0For the position phase at the initial moment of the escaping spacecraft, acIs the orbit semi-major axis of the reference spacecraft, n is the average angular velocity of the reference spacecraft, mu is 3.986005 multiplied by 1014m3/s2Is the earth's gravitational constant.
In the third step, a correction vector is added into the 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 into:
Figure BDA0003101870740000132
where ρ isPFor tracking the distance of the spacecraft with respect to the initial moment of the reference spacecraft, αP0To track the position phase of the spacecraft at the initial moment, n is the average angular velocity of the reference spacecraft, μ 3.986005 × 1014m3/s2Is the constant of the earth's gravity, acIs the orbit semi-major axis of the reference spacecraft;
correcting the initial state quantity of the escape spacecraft relative to the reference spacecraft into:
Figure BDA0003101870740000133
where ρ isEIs the distance, alpha, of the escaping spacecraft relative to the initial time of the reference spacecraftE0For the position phase at the initial moment of the escaping spacecraft, acIs the orbit semi-major axis of the reference spacecraft, n is the average angular velocity of the reference spacecraft, mu is 3.986005 multiplied by 1014m3/s2Is the earth's gravitational constant.
In the third step, a correction vector is added into the initial state vector of the tracking spacecraft relative to the reference spacecraft and substituted into a correction CW equation
Figure BDA0003101870740000134
In the method, the state vector of the tracked spacecraft at the time t can be obtained
Figure BDA0003101870740000135
Wherein (x)P(t),yP(t),zP(t)) tracking the position coordinates of the spacecraft in the LVLH coordinate system at the time t;
Figure BDA0003101870740000136
is xP,yP,zPFor the value of the time derivative function at the time T, T represents transposition;
adding a correction vector into the initial state vector of the escape spacecraft relative to the reference spacecraft, and substituting the correction vector into a correction CW equation
Figure BDA0003101870740000141
The state vector of the escaping spacecraft at the moment t can be obtained
Figure BDA0003101870740000142
Wherein (x)E(t),yE(t),zE(t)) is the position coordinate of the escaping spacecraft in the LVLH coordinate system at the moment t;
Figure BDA0003101870740000143
is xE,yE,zEFor the value of the time derivative function at the time T, T represents transposition;
(xE(t)-xP(t),yE(t)-yP(t),zE(t)-zP(t)) is the relative position of the GEO gaming spacecraft after controlling the non-spherical perturbation error.
Wherein the tracking spacecraft can be used for performing a capture task on a space target; the escaping spacecraft avoids catching of the tracking spacecraft; the reference spacecraft is a virtual spacecraft which does not do orbital maneuver.
The LVLH coordinate system is a local horizontal local vertical coordinate system, and is defined by taking the centroid of the reference spacecraft as an origin and the sagittal diameter from the geocentric to the reference spacecraft as the positive direction of an x axis, wherein a z axis points to the positive normal direction of the orbit of the reference spacecraft, and the direction of a y axis is determined according to a right-hand rule.
A Local Vertical Local Horizontal coordinate system (LVLH) having the centroid of the spacecraft as the origin of coordinates and a direction e from the geocentric to the centroid of the spacecraftxIs the direction of the x-axis and is in the positive normal direction e of the orbit plane of the spacecraftzIs the direction of the z-axis, the direction of the y-axis eyIs determined by the following formula:
ey＝ez×ex (5)
selecting a virtual spacecraft which is not subjected to orbital maneuver and has a short distance between a tracking spacecraft P and an escape spacecraft E as a reference spacecraft, recording the reference spacecraft as c, establishing an LVLH coordinate system with the mass center of the reference spacecraft as an origin, and recording the LVLH coordinate system as:
Figure BDA0003101870740000144
the relative motion of the spacecrafts in close range, particularly for two spacecrafts gaming, is close to the relative distance when:
Figure BDA0003101870740000145
in the formula: ρ represents the relative distance of the two spacecraft and a represents the orbit semi-major axis of the spacecraft.
LVLH coordinate system with mass center of reference spacecraft as origin
Figure BDA0003101870740000146
In the specification, relative state quantities of slave spacecrafts are recorded as
Figure BDA0003101870740000151
Thrust acceleration U ═ Ux,uy,uz)TAnd then:
Figure BDA0003101870740000152
in the formula:
Figure BDA0003101870740000153
Figure BDA0003101870740000154
t0at the moment, the relative state quantities of the slave spacecraft are:
Figure BDA0003101870740000155
then, at time t, the state of the slave spacecraft is:
Figure BDA0003101870740000156
in the formula:
Figure BDA0003101870740000157
φrr(t,t0)、φrv(t,t0)、φvr(t,t0) And phivv(t,t0) Are all 3 × 3 matrices, and:
Figure BDA0003101870740000158
wherein: τ ═ ω (t-t)0)，s＝sinτ，c＝cosτ。
In order to obtain a linearized equation of state, Taylor expansion is needed to be carried out on gravitational potential functions received by a master spacecraft and a slave spacecraft, second-order and high-order small quantities in the earth potential function are omitted under the condition of a formula, and the omitted nonlinear terms can enable state quantities of relative motion to be in the state quantities
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 relative motion is researched, and errors accumulated along with time caused by the second-order term are eliminated by adding a correction term in an initial state quantity.
The method provided by the invention has the following processing steps of eliminating the linearization error:
(1) the second order term in the Taylor expansion of the gravitational potential function is reserved and taken as
Figure BDA00031018707400001610
The virtual control force U is equal to Unl；
(2) Solving a homogeneous equation of state
Figure BDA00031018707400001611
Solution X ofcwIs mixing XcwSubstituted into UnlThen solve for
Figure BDA0003101870740000161
Solution of (2)
Figure BDA0003101870740000162
I.e. the propagation of the error over time, let Xnl(t) the cumulative term with time is 0, and the correction amount of the initial state quantity is obtained
Figure BDA0003101870740000163
(3) Order to
Figure BDA0003101870740000164
Replacing the initial value X of the CW equationcw(0) Further, the state quantity X (t) which changes with time is obtained, and the error which is not accumulated with time in the y-axis direction is not generated.
The method provided by the invention eliminates the proof of linearization error.
And (3) proving that:
according to the processing steps of the method for eliminating the linearization error, the method comprises the following steps: xcwIs that
Figure BDA00031018707400001613
Solution of (2), linearization error
Figure BDA0003101870740000165
When the correction is not carried out, the correction is carried out,
Figure BDA0003101870740000166
at this time, errors in the y-axis direction are accumulated over time, and errors in the state quantities are:
Figure BDA0003101870740000167
selecting the appropriate
Figure BDA0003101870740000168
So that
Figure BDA0003101870740000169
No error accumulated in the y-axis direction with time, so
Figure BDA0003101870740000171
Correcting the initial value in the second step, wherein the linearization error is
Figure BDA0003101870740000172
No error accumulated along with time exists in the y-axis direction, and the evidence is obtained.
After the syndrome is confirmed.
Two spacecrafts of geostationary orbit game: the method comprises the steps that a tracked spacecraft P and an escape spacecraft E are frequently maneuvered in the game process, so that a relative motion dynamic model of the tracked spacecraft P relative to the escape spacecraft E or the escape spacecraft E relative to the tracked spacecraft P is not convenient to establish, and a virtual spacecraft which is not maneuvered in an orbit and is not far away from the tracked spacecraft P and the escape spacecraft E is selected as a reference spacecraft to establish the method provided by the invention. The reference spacecraft c is in a nominal geostationary orbit, i.e. the orbit semi-major axis of the reference spacecraft c is acThe tilt angle and eccentricity are both 0, and:
Figure BDA0003101870740000173
the reference spacecraft is also called the master spacecraft, where i is the slave spacecraft. RhoicRepresenting the distance between the slave spacecraft i and the reference spacecraft c. Subscript i in this section andc denotes the slave spacecraft i and the reference spacecraft c, respectively.
The instantaneous position and velocity vectors of the spacecraft in the geocentric inertial system are r and
Figure BDA0003101870740000178
according to newton's second law:
Figure BDA0003101870740000174
wherein R is the earth gravitational potential function of the spacecraft at R. Establishing LVLH coordinate system by taking mass center of reference spacecraft c as origin
Figure BDA0003101870740000175
Let p ═ r be the vector of relative positions of slave spacecraft i and reference spacecraft ci-rcρ is in
Figure BDA0003101870740000176
The coordinates in (1) are:
Figure BDA0003101870740000177
for slave spacecraft i and reference spacecraft c, respectively:
Figure BDA0003101870740000181
Figure BDA0003101870740000182
Figure BDA0003101870740000183
in the formula, deltaf represents the difference of perturbation accelerations suffered by the master-slave spacecraft, and only J is considered in the method provided by the invention2And J22Term perturbation, so Δ f is 0, uiRepresents the control acceleration of the slave spacecraft i, and leads u to eliminate the system error of the method provided by the invention when researchingi0. Converting the formula in the earth's center inertial system to
Figure BDA0003101870740000189
The method comprises the following steps:
Figure BDA0003101870740000184
in the formula: omegac＝[0 0 n]T，
Figure BDA0003101870740000185
rc＝[r c 0 0]TWherein:
Figure BDA0003101870740000186
finishing to obtain:
Figure BDA0003101870740000187
in the CW equation, the equation is simplified as:
Figure BDA0003101870740000188
note that the state quantities in the CW equation are:
Figure BDA0003101870740000191
solving to obtain:
Figure BDA0003101870740000192
in the formula (I), the compound is shown in the specification,
Figure BDA0003101870740000193
is the square of the relative distance between the master and slave spacecraft, alpha0Initial phase, defined as:
Figure BDA0003101870740000194
the y-axis of the system is rotated through an angle in the-z direction. Initial values of the second step:
Figure BDA0003101870740000195
and (3) retaining the second-order terms on the right side in the formula, discarding the third-order and higher-order terms above the third order, and sorting to obtain:
Figure BDA0003101870740000196
from equation (25), considering the influence of the second order term in the gravitational potential function taylor expansion equation, it is equivalent to applying the virtual control force in step two:
Figure BDA0003101870740000197
by substituting formula (23) for formula (26), we obtain:
Figure BDA0003101870740000201
equation (25) can be written as:
Figure BDA0003101870740000202
equation (28) is the linearized error propagation state equation of step two. Let the linearization error of step two be:
Figure BDA0003101870740000203
the initial values are recorded as:
Figure BDA0003101870740000204
using the conclusion of equation (10), the solution of equation (28) is:
Figure BDA0003101870740000205
Xnlthe expression for (t) is shown in appendix A. It can be seen that the linearization error of step two can be divided into three categories: constant terms, accumulated over time terms, and short period terms. The cumulative term over time only occurs in the y-axis direction, i.e.:
Figure BDA0003101870740000206
in the GEO spacecraft game, the duration is long, and the accumulated items over time have a large influence on the result of the game. Therefore, to avoid the accumulation of errors in the y-axis direction, let ynl-longWhen the ratio is 0, the following:
Figure BDA0003101870740000207
so at the initial state quantity X of the relative movementcw(0) Correction amount of
Figure BDA0003101870740000208
Ready to use
Figure BDA0003101870740000209
Substitution of Xcw(0) Then the linearization error in step two can be eliminated, where:
Figure BDA0003101870740000211
namely X in the second stepi(0) The correction is as follows:
Figure BDA0003101870740000212
a preferred embodiment
In order to analyze the error of the method provided by the invention under different initial conditions, the number of orbits of a slave spacecraft i is set as shown in table 1.
TABLE 1 number of orbits from spacecraft i
Figure BDA0003101870740000213
The number of orbits of the spacecraft c is shown in table 2.
TABLE 2 number of orbits of reference spacecraft c
Figure BDA0003101870740000214
The spacecraft is at
Figure BDA0003101870740000216
The initial state quantities in (1) are shown in table 3.
TABLE 3 spacecraft from i
Figure BDA0003101870740000217
Initial state of
Figure BDA0003101870740000215
Figure BDA0003101870740000221
The relative motion relationship between the spacecraft No. 1 and No. 2 in the table 1 and the reference spacecraft c is simulated by respectively applying the method and the CW equation provided by the invention. Comparing the change of the x, y and z state quantities of the method and the CW equation with time, calculating the change of the difference value of the y state quantity of the method and the CW equation with time, and the change of the difference value of the state quantity calculated by the method and the CW equation with time.
a. Case 1: ρ 200km, α0＝180°
The method and the CW equation provided by the invention are respectively utilized to analyze the relative motion relation between the No. 1 spacecraft and the reference spacecraft c. The method provided by the invention and the change of the x state quantity in the CW equation along with time are shown in fig. 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 method provided by the invention and the change of the y state quantity in the CW equation along with the time are shown in fig. 2, wherein the solid line in the figure represents the y state quantity in the method provided by the invention, and the dotted line represents the x state quantity in the CW equation.
The variation of the difference between the method provided by the present invention and the y state quantity in the CW equation with time is shown in FIG. 3
The method provided by the invention and the change of the z state quantity in the CW equation along with the time are shown in FIG. 4, 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 x state quantity in the CW equation.
The method provided by the present invention and the change of the difference between the x state quantity and the true value in the CW equation with time are shown in fig. 5, in which the solid line represents the change of the difference between the x state quantity and the true value in the method provided by the present invention with time, and the dotted line represents the change of the difference between the x state quantity and the true value in the CW equation with time.
The variation with time of the difference between the y state quantity and the true value in the method and the CW equation provided by the present invention is shown in fig. 6, in which the solid line represents the variation with time of the difference between the y state quantity and the true value in the method provided by the present invention, and the dotted line represents the variation with time 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 provided by the present invention is shown in fig. 7, in which the solid line represents the time-dependent change of the difference between the z state quantity and the true value in the method provided by the present invention, and the dotted 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 200km, α0＝270°
The method and the CW equation provided by the invention are respectively utilized to analyze the relative motion relation between the No. 2 spacecraft and the reference spacecraft c. The variation of the x state quantity with time in the method and the CW method provided by the present invention is shown in fig. 8, in which the solid line represents the x state quantity in the method provided by the present invention, and the dotted line represents the x state quantity in the CW equation.
The method provided by the invention and the change of the y state quantity in the CW equation along with the 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 variation of the difference between the method provided by the present invention and the y state quantity in the CW equation with time is shown in FIG. 10
The method provided by the present invention and the change of the z state quantity in the CW equation with time are shown in fig. 11, in which the solid line represents the z state quantity in the method provided by the present invention, and the dotted line represents the z state quantity in the CW equation.
The variation with time of the difference between the x state quantity and the true value in the method and the CW equation provided by the present invention is shown in fig. 12, in which the solid line represents the variation with time of the difference between the x state quantity and the true value in the method provided by the present invention, and the dotted line represents the variation with time of the difference between the x state quantity and the true value in the CW equation.
The variation with time of the difference between the y state quantity and the true value in the method and the CW equation provided by the present invention is shown in fig. 13, in which the solid line represents the variation with time of the difference between the y state quantity and the true value in the method provided by the present invention, and the dotted line represents the variation with time 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 provided by the present invention is shown in fig. 14, in which the solid line represents the time-dependent change of the difference between the z state quantity and the true value in the method provided by the present invention, and the dotted line represents the time-dependent change of the difference between the z state quantity and the true value in the CW equation.
Through the four cases, the aerospace is obtainedThe game initial formation configuration parameters of the device i and the reference spacecraft c and the time accumulated deviation of the y-axis direction after 10 periods (861640s) are shown in the table 4, and dic(0) Is the initial distance, y, of the master and slave spacecraftnl-longIs the time accumulated deviation in the y-axis direction in the CW equation,
Figure BDA0003101870740000241
is the accumulated time 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 deviation over time in the y-axis direction, in the 2 cases: after 10 orbit cycles, the deviation of 112km and 22km occurs for the master spacecraft and the slave spacecraft with the initial distances of 200km and 100km respectively, and obviously, if the error accumulated along with the time is not controlled, the task fails.
The method provided by the invention controls the accumulated deviation of the y-axis direction along with the time, and in the 2 cases: the initial distances of the main spacecraft and the slave spacecraft are respectively 200km and 100km, after 10 orbit periods, the deviation is respectively only 0.12km and 0.07km, compared with a CW equation, the deviation accumulated along the y-axis direction with time does not exceed 0.7 percent of the deviation of the CW equation, and the requirement of game mission research of the GEO spacecraft can be met.
The embodiment of the invention discloses a method for improving the game dynamic model precision of a geostationary orbit spacecraft. For a model for researching relative motion in the pursuit game, a relative state vector is used as a state quantity model, so that problem analysis is facilitated, but the precision is low due to the existence of nonlinear errors. Aiming at the defect that nonlinear errors exist in relative positions of game spacecrafts obtained by a CW (continuous wave) equation in the relative motion process of geostationary orbit game spacecrafts, a second-order nonlinear term of a gravitational potential function is used as a virtual control thrust, a state equation is solved, the nonlinear deviation accumulated along with time is converted into a correction term of initial state parameters, and the accumulation of the nonlinear deviation along with time is avoided. In simulation, the method provided by the invention reduces the deviation after 10 orbit periods from hundred kilometers magnitude to hundred meters magnitude. 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 a variational 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 embodied in that:
a. aiming at the problem of nonlinear errors in a CW equation, the method provided by the invention considers the influence of a second-order nonlinear term discarded in the process of linearization of a gravitational potential function, calculates the error propagation caused by the second-order nonlinear term, and further calculates a correction term for eliminating the time accumulation term, and avoids the propagation of the second-order nonlinear error by correcting an initial state quantity,
b. aiming at the problem of eccentricity error in the CW equation, the method provided by the invention makes the eccentricity of c be 0 when selecting c, thereby avoiding the transmission of the eccentricity error in the method provided by the invention.
The method provided by the invention improves the accuracy of the equation, does not change the linear characteristic of the equation, and is convenient for researching the game strategy of the spacecraft by combining the variational method.

Claims (7)

1. A method for controlling the nonlinear error of the relative position of a geostationary orbit game spacecraft is characterized by comprising the following steps:
selecting a reference spacecraft, wherein the orbit semi-major axis of the reference spacecraft is a GEO nominal orbit semi-major axis, the eccentricity is 0, the orbit inclination angle is 0, the ascension of a rising intersection point, the depression angle of a near place and the horizontal near place 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;
respectively calculating initial state vectors of the tracked spacecraft and the escaped spacecraft relative to the reference spacecraft in an LVLH coordinate system;
adding a correction vector into the initial state vector of the tracking spacecraft relative to the reference spacecraft, and substituting the correction vector into a correction CW equation to obtain the state vector of the tracking spacecraft; adding a correction vector into the initial state vector of the escaping spacecraft relative to the reference spacecraft, and substituting the correction vector into a correction CW equation to obtain the state vector of the escaping 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 nonlinear error control;
the modified CW equation is:
Figure FDA0003101870730000011
wherein: x (t) represents the state vector at time t, Xcorrection(t0) Representing the initial state vector after the addition of the correction term, U being the applied control force, B being a constant matrix,
Figure FDA0003101870730000013
s is the sign of the integral argument, U(s) is the control force applied at time s, φ (t, s) is the state transition matrix from time s to time t, φ (t, t)0) Is the state transition matrix from time to time t.
2. The method as claimed in claim 1, wherein in step two, the method of calculating the initial state vector of the tracked spacecraft relative to the reference spacecraft in the LVLH coordinate system comprises:
Figure FDA0003101870730000014
wherein (x)P(t0),yP(t0),zP(t0) For initial time tracking the position coordinates of the spacecraft in the LVLH coordinate system;
Figure FDA0003101870730000021
is xP,yP,zPThe value of the time derivative function at the initial moment; t represents transposition;
in the LVLH coordinate system, the method for calculating the initial state vector of the escape spacecraft relative to the reference spacecraft comprises the following steps:
Figure FDA0003101870730000022
wherein (x)E(t0),yE(t0),zE(t0) To track the position coordinates of the spacecraft in the LVLH coordinate system for an initial moment,
Figure FDA0003101870730000023
is xE,yE,zEThe value of the time derivative function at the initial moment; t denotes transposition.
3. The method according to claim 1, wherein in step three, the correction vector added to the initial state vector of the tracking spacecraft relative to the reference spacecraft is:
Figure FDA0003101870730000024
where ρ isPFor tracking the distance of the spacecraft with respect to the initial moment of the reference spacecraft, αP0To track the position phase of the spacecraft at the initial moment, n is the average angular velocity of the reference spacecraft, μ 3.986005 × 1014m3/s2Is the constant of the earth's gravity, acIs the orbit semi-major axis of the reference spacecraft;
the correction vector added in the initial state vector of the escape spacecraft relative to the reference spacecraft is as follows:
Figure FDA0003101870730000025
where ρ isEIs the distance, alpha, of the escaping spacecraft relative to the initial time of the reference spacecraftE0For the position phase at the initial moment of the escaping spacecraft, acFor reference navigationThe orbit semi-major axis of the spacecraft, n is the average angular velocity of the reference spacecraft, mu is 3.986005 multiplied by 1014m3/s2Is the earth's gravitational constant.
4. A method according to any one of claims 1 to 3, wherein in step three, a correction vector is added to the 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 to:
Figure FDA0003101870730000026
wherein,. rhoPTo track the distance of the spacecraft with respect to the initial moment of the reference spacecraft, αP0To track the position phase of the spacecraft at the initial moment, n is the average angular velocity of the reference spacecraft, μ 3.986005 × 1014m3/s2Is the constant of the earth's gravity, acIs the orbit semi-major axis of the reference spacecraft;
correcting the initial state quantity of the escape spacecraft relative to the reference spacecraft into:
Figure FDA0003101870730000031
where ρ isEIs the distance, alpha, of the escaping spacecraft relative to the initial time of the reference spacecraftE0For the position phase at the initial moment of the escaping spacecraft, acIs the orbit semi-major axis of the reference spacecraft, n is the average angular velocity of the reference spacecraft, mu is 3.986005 multiplied by 1014m3/s2Is the earth's gravitational constant.
5. Method according to one of claims 1 to 4, characterized in that in step three, a correction vector is added to the initial state vector of the tracking spacecraft relative to the reference spacecraft and substituted into the corrected CW equation
Figure FDA0003101870730000032
In the method, the state vector of the tracked spacecraft at the time t can be obtained
Figure FDA0003101870730000033
Wherein (x)P(t),yP(t),zP(t)) tracking the position coordinates of the spacecraft in the LVLH coordinate system at the time t;
Figure FDA0003101870730000034
is xP,yP,zPFor the value of the time derivative function at the time T, T represents transposition;
adding a correction vector into the initial state vector of the escape spacecraft relative to the reference spacecraft, and substituting the correction vector into a correction CW equation
Figure FDA0003101870730000035
The state vector of the escaping spacecraft at the moment t can be obtained
Figure FDA0003101870730000037
Wherein (x)E(t),yE(t),zE(t)) is the position coordinate of the escaping spacecraft in the LVLH coordinate system at the moment t;
Figure FDA0003101870730000036
is xE,yE,zEFor the value of the time derivative function at the time T, T represents transposition;
(xE(t)-xP(t),yE(t)-yP(t),zE(t)-zP(t)) is the relative position of the GEO gaming spacecraft after controlling the non-spherical perturbation error.
6. The method of claim 1, wherein the LVLH coordinate system, being a local horizontal local vertical coordinate system, is oriented with the centroid of the reference spacecraft as an origin and the sagittal diameter from the geocentric to the reference spacecraft as the positive x-axis, wherein the z-axis is oriented normal to the orbit of the reference spacecraft, and wherein the y-axis is oriented according to right-hand rules.
7. Method according to one of claims 1 to 6, characterized in that the tracking spacecraft is usable for performing capture tasks on a spatial target;
the escaping spacecraft avoids catching of the tracking spacecraft;
the reference spacecraft is a virtual spacecraft which does not do orbital maneuver.
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