CN113282097B - Control method for relative position non-spherical perturbation error of GEO game spacecraft - Google Patents
Control method for relative position non-spherical perturbation error of GEO game spacecraft Download PDFInfo
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Abstract
The invention discloses a control method of a relative position non-spherical perturbation error of a GEO game spacecraft, 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; substituting the initial state vectors of the tracked spacecraft and the escaping spacecraft relative to the reference spacecraft into a corrected CW equation to obtain the state vector of the tracked spacecraft and the state vector of 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 GEO game spacecraft after the non-spherical perturbation error is controlled. Aiming at the defect that the relative position between game spacecrafts obtained by a CW equation has non-spherical perturbation errors in the relative motion process of geostationary orbit game spacecrafts, the invention adds perturbation terms which have large influence on the relative motion to improve the precision of the CW equation, does not change the linear characteristic of the CW equation at the same time, and is convenient for researching the game strategy of the spacecrafts by a variational method.
Description
Technical Field
The invention relates to a method for controlling an aspheric perturbation error of a relative position 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, RaoY, Han C.inverse Transformation of Elliptical Relative State Transformation Matrix [ J ]. International Journal of Astromy & Astrophysics,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 that: 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 relative motion is studied based on a kinetic approach, Clohessy-Wiltshire equation, for short: the CW equation is a very widely used method. 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:
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,
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 time s to time t, φ (t, t) 0 ) Is t 0 A state transition matrix from the moment to the t moment;
wherein: τ ═ ω (t-t) 0 ) Denotes a reference spacecraft from t 0 The angle turned by the time t to the time t,
a c for reference to the orbit semi-major axis of the spacecraft, mu is 3.986005 × 10 14 m 3 /s 2 Is the earth's gravitational constant.
The defects are as follows: the errors of the CW equation are derived from linearization errors and perturbation errors, and the CW equation also has small eccentricity assumption errors (such as a CW equation), so that the errors are large, and particularly, the errors accumulated over 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 ofFormation-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 model 2 Item perturbation, and the problem of formation safety path is researched; gaias, G (Gaias, G., ardens, J. -S., and Montenbrock, O., "Model of J2 conditioned Satellite Relative Motion with Time-Varying Differential flag," cellular mechanisms and dynamic asymmetry, Vol.123, No.4,2015, pp.411-433. doi:10.1007/s10569-015- The long-term influence of the resistance perturbation on the semi-long axis of the track improves an STM model; 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 improved 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 an aspheric perturbation error of a relative position of a GEO game spacecraft, wherein a geostationary orbit is marked as GEO, and the control method is a control method of the aspheric perturbation error of the relative position of the GEO game spacecraft.
In order to achieve the aim, the invention provides a control method of an aspheric perturbation error of a relative position of a GEO game 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 angle of the near place 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 tracking spacecraft and the escaping spacecraft relative to the reference spacecraft in an LVLH coordinate system;
substituting the initial state vector of the tracking spacecraft relative to the reference spacecraft into a corrected CW equation to obtain a state vector of the tracking spacecraft; substituting the initial state vector of the escaping spacecraft relative to the reference spacecraft into a corrected 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 GEO game spacecraft after controlling the non-spherical perturbation error;
the modified CW equation is:
wherein X (t) represents a state vector at time t, X (t) 0 ) Representing the initial state vector, U the applied control force, B a constant matrix, 
ε is the sign of the integral argument, U (ε) is the control force at time ε, φ (t, ε) is the state transition matrix from time ε to time t, φ (t, t) 0 ) Is t 0 A state transition matrix from the moment to the t moment;
wherein: τ ═ ω (t-t) 0 ) Denotes a reference spacecraft from t 0 The angle turned by the time t to the time t,
a c is the orbit semi-major axis of the reference spacecraft;
in the formula, r E Is the radius of the earth, J 2 ＝-1082.627×10 -6 、J 22 ＝1.815528×10 -6 ，a c Is the orbit semi-major axis of the reference spacecraft, and lambda is the point-ground under the satellite of the reference spacecraftLongitude, mu is 3.986005 × 10 14 m 3 /s 2 Is the earth's gravitational constant.
Further, the LVLH coordinate system is a local horizontal local vertical coordinate system, and means a forward direction taking a centroid of the reference spacecraft as an origin and a radial direction from the centroid to the reference spacecraft as an x-axis, the z-axis points to the normal direction of the orbit of the reference spacecraft, and a direction of the y-axis is determined according to a right-hand rule.
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:
wherein (x) P (t 0 ),y P (t 0 ),z P (t 0 ) For initial time tracking the position coordinates of the spacecraft in the LVLH coordinate system;
is x P ,y P ,z P The 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:
Wherein (x) E (t 0 ),y E (t 0 ),z E (t 0 ) To track the position coordinates of the spacecraft in the LVLH coordinate system for an initial moment,
is x E ,y E ,z E The value of the time derivative function at the initial moment; t denotes transposition.
In the third step, the initial state of the tracking spacecraft relative to the reference spacecraft isVector substitution modified CW equation
In the method, the state vector of the tracked spacecraft at the time t can be obtained
Wherein (x) P (t),y P (t),z P (t)) tracking the position coordinates of the spacecraft in the LVLH coordinate system at the time t;
is x P ,y P ,z P For the value of the time derivative function at the time T, T represents transposition;
substituting the initial state vector of the escaping spacecraft relative to the reference spacecraft into a modified CW equation
The state vector of the escaping spacecraft at the moment t can be obtained
Wherein (x) E (t),y E (t),z E (t)) is the position coordinate of the escaping spacecraft in the LVLH coordinate system at the moment t;
is x E ,y E ,z E For the value of the time derivative function at the time 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 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 invention has the following beneficial effects:
compared with the relative position of the earth stationary orbit game spacecraft calculated by the CW equation, the method controls the propagation of perturbation errors and is embodied in that:
a. Aiming at the perturbation error problem existing in the traditional dynamic relative motion CW equation, the orbit resonance characteristic of the GEO spacecraft is considered, and J with the largest influence on relative motion is introduced 2 And J 22 The influence of non-spherical perturbation is avoided;
b. aiming at the eccentricity error problem existing in the traditional dynamic relative motion CW equation, the first step is to make the eccentricity of the reference spacecraft be 0 when the reference spacecraft is selected, so as to avoid the propagation of the eccentricity error.
Description of the drawings:
FIG. 1 is a schematic representation of the variation of μ'/μ with the longitude of the GEO spacecraft fix point.
Fig. 2 is a diagram showing the time-dependent variation of the difference between the x-state quantities in the present method and the CW equation.
Fig. 3 is a schematic diagram of the difference between the y state quantities in the method and the CW equation over time.
Fig. 4 is a graphical representation of the difference in z-state quantities over time in the present method and CW equation.
Fig. 5 is a diagram showing the difference between the x-state quantities in the method and the CW equation as a function of time.
Fig. 6 is a graph showing the difference between the y state quantities in the present method and the CW equation over time.
Fig. 7 is a graphical representation of the difference in z-state quantities over time in the present method and CW equation.
Fig. 8 is a graphical representation of the magnitude of the difference after 10T of the x-state quantity calculated by the present method and the CW equation.
Fig. 9 is a graphical representation of the magnitude of the difference after 10T of the y-state quantity calculated by the present method and the CW equation.
FIG. 10 is a graphical representation of the magnitude of the difference after 10T of the z-state quantity calculated by the present method and the CW equation.
Detailed Description
Aiming at the perturbation error problem in a CW equation, the precision of the CW equation is improved by correction, and when the GEO spacecraft game problem is researched, the method is used for calculating the relative position and speed among the spacecrafts, so that the propagation of the perturbation error of the geostationary orbit game spacecraft is controlled, and the precision of the relative position is improved.
In order to achieve the aim, the invention provides a control method of an aspheric perturbation error of a relative position of a GEO game 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 angle of the near place 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;
Substituting the initial state vector of the tracking spacecraft relative to the reference spacecraft into a correction CW equation to obtain a state vector of the tracking spacecraft; substituting the initial state vector of the escaping spacecraft relative to the reference spacecraft into a corrected 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 GEO game spacecraft after controlling the non-spherical perturbation error;
the modified CW equation is:
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,
ε is the sign of the integral argument, U (ε) is the control force at time ε, φ (t, ε) is the state transition matrix from time ε to time t, φ (t, t) 0 ) Is t 0 A state transition matrix from the moment to the t moment;
wherein: τ ═ ω (t-t) 0 ) Denotes a reference spacecraft from t 0 The angle turned by the time t to the time t,
a c is the orbit semi-major axis of the reference spacecraft;
in the formula, r E Is the radius of the earth, J 2 ＝-1082.627×10 -6 、J 22 ＝1.815528×10 -6 ，a c Is the orbit semi-major axis of the reference spacecraft, lambda is the geographic longitude of the subsatellite point of the reference spacecraft, mu is 3.986005 multiplied by 10 14 m 3 /s 2 Is the earth's gravitational constant.
Further, the LVLH coordinate system is a local horizontal local vertical coordinate system, and means a forward direction taking a centroid of the reference spacecraft as an origin and a radial direction from the centroid to the reference spacecraft as an x-axis, the z-axis points to the normal direction of the orbit of the reference spacecraft, and a direction of the y-axis is determined according to a right-hand rule.
Specifically, the earth's stationary orbit is denoted as GEO. 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 spacecraft x Is the direction of the x-axis and is in the positive normal direction e of the orbit plane of the spacecraft z Is the direction of the z-axis, the direction of the y-axis e y Is determined by the following formula:
e y ＝e z ×e x
and P is marked as a tracking spacecraft, and E is an escape spacecraft. Selecting a virtual space vehicle which is not subjected to orbital maneuver and has a less far distance i from { P, E } as a reference space vehicle, marking the reference space vehicle as c, establishing an LVLH coordinate system with the centroid of c as an origin, and marking the LVLH coordinate system as c：
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:
wherein (x) P (t 0 ),y P (t 0 ),z P (t 0 ) For initial time tracking the position coordinates of the spacecraft in the LVLH coordinate system;
is x P ,y P ,z P The 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:
wherein (x) E (t 0 ),y E (t 0 ),z E (t 0 ) To track the position coordinates of the spacecraft in the LVLH coordinate system for an initial moment,
is x E ,y E ,z E The value of the time derivative function at the initial moment; t denotes transposition.
Specifically, note r LVLH To track the position vector of the spacecraft P in the LVLH coordinate system, r ECI To track the position vector of the spacecraft P in the Earth's inertial frame ECI (Earth centered-inertial frame), A ECI→LVLH Is a coordinate transformation matrix from an ECI coordinate system to an LVLH coordinate system, A LVLH→ECI From LVLH to ECI coordinate systemsSo that:
r ECI ＝A LVLH→ECI ·r LVLH (1)
r LVLH ＝A ECI→LVLH ·r ECI (2)
wherein:
for a column vector a ═ a 1 a 2 a 3 ] T And b ═ b 1 b 2 b 3 ] T Existence of
Wherein:
A ECI→LVLH and A LVLH→ECI The rate of change over time is:
where n is the angular velocity of rotation of the LVLH coordinate system relative to the ECI coordinate system, and:
in the formula
To track the difference of the position velocity vectors of the spacecraft P and the reference spacecraft c in the ECI coordinate system,
namely 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
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
In the third step, tracking is performedCorrecting CW equation by substituting initial state vector of spacecraft relative to reference spacecraft
In the method, the state vector of the tracked spacecraft at the time t can be obtained
Wherein (x) P (t),y P (t),z P (t)) tracking the position coordinates of the spacecraft in the LVLH coordinate system at the time t; 
Is x P ,y P ,z P For the value of the time derivative function at the time T, T represents transposition;
substituting the initial state vector of the escaping spacecraft relative to the reference spacecraft into a modified CW equation
The state vector of the escaping spacecraft at the moment t can be obtained
Wherein (x) E (t),y E (t),z E (t)) is the position coordinate of the escaping spacecraft in the LVLH coordinate system at the moment t;
is x E ,y E ,z E For the value of the time derivative function at the time 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 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.
Specifically, the spacecraft can relatively move in a short distance, and particularly for two spacecrafts in a game, the relative distance is short, when:
in the formula: ρ represents the relative distance between two spacecrafts and a represents the orbit semi-major axis of one of the spacecrafts.
LVLH coordinate system with mass center of reference spacecraft as origin
In the specification, relative state quantities of slave spacecrafts are recorded as
Thrust acceleration U ═ U x ,u y ,u z ) T And then:
in the formula:
t 0 at the moment, the relative state quantities of the slave spacecraft are:
then, at time t, the state of the slave spacecraft is:
In the formula:
φ rr (t,t 0 )、φ rv (t,t 0 )、φ vr (t,t 0 ) And phi vv (t,t 0 ) Are all 3 × 3 matrices, and:
wherein: τ ═ ω (t-t) 0 )，s＝sinτ，c＝cosτ。
Taking the potential of the spacecraft at infinite distance from the earth as 0, the potential function of the earth gravitational field is as follows:
wherein, mu is 3.986005 multiplied by 10 14 m 3 /s 2 Is the gravitational constant, r is the distance of the spacecraft from the earth's center, r E Is the radius of the earth, J 2 ＝-1082.627×10 -6 、J 22 ＝1.815528×10 -6 、J 3 ＝2.532435×10 -6 、J 31 ＝2.2091169×10 -6 、J 33 ＝0.2213602×10 -6 、
after the arrangement, the potential function of the earth gravitational field is as follows:
the weights of the terms are:
normalized harmonic coefficient, divided by J 2 The magnitude of (A) is larger, and the remaining magnitudes are generally smaller. Aiming at the spacecraft near the earth stationary orbit, five items with the maximum weight are calculated as follows: j. the design is a square 2 、J 22 、J 3 、J 31 、J 33 The weight ratio is:
A 2 :A 22 :A 3 :A 31 :A 33 ＝11053:64:3:7:6
therefore, the two terms J with the largest weight in the earth non-spherical perturbation are introduced into the method 2 And J 22 An item. Consider J 2 And J 22 The term perturbed Earth's gravitational field potential function:
in the formula (I), the compound is shown in the specification,
the geographic latitude of the satellite point of the spacecraft is defined as the geographic latitude of the satellite point, because the GEO spacecraft participating in the game is relatively close in distance and the geographic longitude of the satellite point changes little in the game process, namely:
then:
in the formula, λ is the geographical longitude of the satellite point of the spacecraft, a s Is a nominal semi-major axis of the geostationary orbit spacecraft.
J 2 And J 22 The effect of the term on the CW equation is reflected in: the earth's gravity constant μ ' corrected in this method is no longer the earth's gravity constant μ in step two. Wherein:
Compared with CW equation, introduce J 2 And J 22 After the item perturbation, the absolute value of the geogravitational potential function of the GEO spacecraft becomes large, and the ratio of μ' to μ is shown in fig. 1 as the fixed point longitude changes. J. the design is a square 22 The term indicates that there are two minima points of gravitational potential for the earth's stationary orbit: 75.0716 ° E and 104.9284 ° W. J. the design is a square 22 The term causes the resonance characteristic of the geostationary orbit, and the geostationary orbit spacecraft with the fixed point of 14.9284 degrees W-165.0716 degrees E is J under the condition of no control 22 Is perturbed by drifting towards 75.0716 degrees E; the geostationary orbit spacecraft with the fixed point of 165.0716 degrees E-14.9284 degrees W is arranged at J 22 Is perturbed by a drift towards 104.9284 deg.w.
A preferred embodiment
In order to analyze the error of the method under different initial conditions, the number of orbits of the spacecraft No. 1 and the spacecraft No. 2 is set as shown in Table 1.
TABLE 1
The number of orbits of the reference spacecraft c is shown in table 2.
TABLE 2
Spacecraft No. 1 and spacecraft No. 2
The initial state quantities in (1) are shown in table 3.
TABLE 3
The method and the CW equation are respectively used for simulating the relative motion relation between the spacecraft No. 1 and the spacecraft No. 2 in the table 1 and the reference spacecraft, and the change conditions of the difference values of the x, y and z state quantities of the method and the CW equation along with time are compared. In the simulation case, the geographic longitude of the spacecraft's sub-satellite point was 165.0716 ° E.
a. Case 1: ρ 200km, α 0 ＝180°
In the present simulation case, fig. 2, fig. 3, and fig. 4 show the time variation of the difference between the x, y, and z state quantities in the present method and the CW equation, respectively. As can be seen from the figure: the difference value of the x, y and z state quantities in the method and the CW equation shows periodic change, the amplitude of the difference value is gradually enlarged, the phase difference of 90 degrees is formed between the change period of the x state quantity difference value and the case 2, and the change period of the z state quantity difference value is the same as the case.
b. Case 2: ρ 200km, α 0 ＝270°
In the present simulation case, fig. 5, fig. 6, and fig. 7 show the time variation of the difference between the x, y, and z state quantities in the present method and the CW equation, respectively. As can be seen from the figure: the method and the difference value of the x state quantity and the z state quantity in the CW equation show periodic change, the amplitude of the difference value is gradually enlarged, the phase difference of 90 degrees is formed between the change period of the difference value of the x state quantity and the case 3, and the change period of the difference value of the z state quantity is the same as the case; the difference value of the y state quantity comprises an accumulated term and a short period term along with time, the amplitude of the difference value is gradually enlarged, and the difference value is about 300m after 10 orbit periods.
c. Effect of different fixed point longitudes on magnitude of difference increase
From cases 1 and 2, α 0 Under two initial states of 180 degrees and 270 degrees, the difference value of the method and the x state quantity in the CW equation is periodically changed along with time, and the changed amplitude value is gradually increased; the difference in z-state quantities also varies periodically with time, with the magnitude of the variation increasing progressively but at a rate that is an order of magnitude less than the rate at which the difference in x-state quantities increases in magnitude.
α 0 In two initial states of 180 degrees, the difference value between the method and the y state quantity in the CW equation changes periodically along with time, the changed amplitude is gradually increased, and the rate of the amplitude increase is equivalent to the rate of the amplitude increase of the x state quantity difference value. Alpha (alpha) ("alpha") 0 In two initial states of 270 degrees, the difference value of the y state quantity comprises an accumulated term and a short period term along with time, and the amplitude of the difference value is gradually enlarged. Let ρ be 200km, α 0 The amplitude of the x, y, z state quantity 10T back difference calculated by the method and the CW equation for the game space at different fixed points and longitudes is analyzed as 270 degrees as shown in fig. 8, fig. 9 and fig. 10.
The amplitude value of the difference value after 10T of x, y and z state quantity calculated by the method and the CW equation from the game space at different fixed points and longitudes in the figure and J 22 The term perturbation potential function exhibits the same law of variation. Non-spherical perturbations cause a deviation of about 40.5m in the x-state quantity; non-spherical perturbations cause a deviation of about 311.5m in the y-state quantity; non-spherical perturbations cause a deviation of about 2.87m in the z-state quantity. The effect of non-spherical perturbation on the x, y state quantities is greater than the effect on the z state quantities.
Aiming at the defect that the relative position between game spacecrafts obtained by a CW equation has non-spherical perturbation errors in the relative motion process of geostationary orbit game spacecrafts, the invention adds perturbation terms which have large influence on the relative motion to improve the precision of the CW equation, does not change the linear characteristic of the CW equation at the same time, and is convenient for researching the game strategy of the spacecrafts by a variational method.
The embodiment of the invention has the following beneficial effects:
compared with a CW equation, the method avoids the propagation of errors of the CW equation and is embodied in that:
a. aiming at the perturbation error problem existing in the traditional dynamic relative motion CW equation, the orbit resonance characteristic of the GEO spacecraft is considered, and J with the largest influence on the relative motion is added in the method 2 And J 22 The influence of non-spherical perturbation is avoided;
b. aiming at the problem of eccentricity error in the traditional dynamic relative motion CW equation, when c is selected, the eccentricity of c is set to be 0, so that the propagation of the eccentricity error is avoided.
Claims (5)
1. A control method for an aspheric perturbation error of a relative position of a GEO 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 angle of the near place 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;
substituting the initial state vector of the tracking spacecraft relative to the reference spacecraft into a corrected CW equation to obtain a state vector of the tracking spacecraft; substituting the initial state vector of the escaping spacecraft relative to the reference spacecraft into a corrected 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 GEO game spacecraft after controlling the non-spherical perturbation error;
The modified CW equation is:
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,
ε is the sign of the integral argument, U (ε) is the control force at time ε, φ (t, ε) is the state transition matrix from time ε to time t, φ (t, t) 0 ) Is t 0 A state transition matrix from the moment to the t moment;
wherein: τ ═ ω (t-t) 0 ) Denotes a reference spacecraft from t 0 The angle turned by the time t to the time t,
a c is the orbit semi-major axis of the reference spacecraft;
in the formula, r E Is the radius of the earth, J 2 ＝-1082.627×10 -6 、J 22 ＝1.815528×10 -6 λ is the geographic longitude of the substellar point of the reference spacecraft, μ is 3.986005 × 10 14 m 3 /s 2 Is the earth's gravitational constant.
2. 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.
3. The method according to claim 1 or 2, wherein in step two, the method for calculating the initial state vector of the tracked spacecraft relative to the reference spacecraft in the LVLH coordinate system comprises:
Wherein (x) P (t 0 ),y P (t 0 ),z P (t 0 ) Track the position coordinates of the spacecraft in the LVLH coordinate system for the initial moment;
is x P ,y P ,z P The 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:
wherein (x) E (t 0 ),y E (t 0 ),z E (t 0 ) To track the position coordinates of the spacecraft in the LVLH coordinate system for an initial moment,
is x E ,y E ,z E The value of the time derivative function at the initial moment; t denotes transposition.
4. The method of claim 1, wherein in step three, the initial state vector of the tracking spacecraft relative to the reference spacecraft is substituted into the modified CW equation
In (1), can obtainState vector for tracking spacecraft at time t
Wherein (x) P (t),y P (t),z P (t)) tracking the position coordinates of the spacecraft in the LVLH coordinate system at the time t;
is x P ,y P ,z P For the value of the time derivative function at the time T, T represents transposition;
substituting the initial state vector of the escaping spacecraft relative to the reference spacecraft into a modified CW equation
The state vector of the escaping spacecraft at the moment t can be obtained
Wherein (x) E (t),y E (t),z E (t)) is the position coordinate of the escaping spacecraft in the LVLH coordinate system at the moment t;
is x E ,y E ,z E For the value of the time derivative function at the time 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 gaming spacecraft after controlling the non-spherical perturbation error.
5. The method of claim 1, wherein the tracking spacecraft is operable to perform a capture task on a spatial target;
the escaping spacecraft avoids catching the tracking spacecraft;
the reference spacecraft is a virtual spacecraft which does not do orbital maneuver.
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