CN111994304B - Method for keeping low-thrust long-term position of geostationary orbit satellite - Google Patents

Method for keeping low-thrust long-term position of geostationary orbit satellite Download PDF

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CN111994304B
CN111994304B CN202010898380.0A CN202010898380A CN111994304B CN 111994304 B CN111994304 B CN 111994304B CN 202010898380 A CN202010898380 A CN 202010898380A CN 111994304 B CN111994304 B CN 111994304B
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李林澄
张景瑞
陈国玖
张尧
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Beijing Institute of Technology BIT
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Abstract

The invention discloses a method for keeping a low-thrust long-term position of a geostationary orbit satellite, which comprises the steps of establishing an average orbit motion model in and out of a satellite orbit plane through a spherical coordinate; giving out the in-plane and out-of-plane long-period motion rules of the geostationary orbit satellite by a phase plane analysis method; on the basis, a fixed point position holding window of the geostationary orbit satellite is selected to obtain a periodic motion track of the satellite in an uncontrolled state in the fixed point window, namely a drift section track; and designing a low thrust control law, obtaining a motion track of the satellite in a controlled state, namely a thrust section track, and enabling the thrust section track and the drift section track to jointly form a closed-loop track, thereby finishing the low thrust long-term position maintenance of the geostationary orbit satellite.

Description

Method for keeping low-thrust long-term position of geostationary orbit satellite
Technical Field
The invention relates to a method for keeping a low-thrust long-term position of a geostationary orbit satellite, in particular to important technologies such as analysis of a long-term motion rule of the geostationary orbit satellite, keeping of a low-thrust position in/out of an orbit plane and the like, and belongs to the field of spacecraft orbit dynamics and control.
Background
The spacecraft running on the Geostationary Orbit (GEO) has the characteristics of large coverage area and high economic value relative to the ground stillness. The development of geostationary orbit spacecraft has mainly presented three "big" trends: firstly, flexible parts carried by the spacecraft are larger and larger (a 50-meter long solar panel, a hectometer-level antenna, two mechanical arms more than 2 meters and the like); secondly, the spacecraft body is getting bigger and bigger, and the quality of the spacecraft body reaches dozens or even hundreds of tons; thirdly, the economic volume is getting bigger and bigger, and the spacecraft such as the communication satellite on the GEO and the like have billions of RMB in cost, and the economic benefit generated every year is more considerable.
Due to the perturbation law of the stationary orbit, the longitude of the equator occupied by a single stationary orbit satellite is about +/-0.1 degree, so that only 1800 stationary orbit satellites can be operated in orbit at the same time theoretically. However, with the increase of space launch, the influence and potential threat of space debris such as failed spacecrafts on high orbit on large and complex stationary orbit satellites are not negligible. The geostationary orbit satellite ensures the normal function of the geostationary orbit satellite through position maintenance, but increasingly complex space environments and increasingly sophisticated in-orbit missions put higher requirements on the task of position maintenance of the geostationary orbit.
The position maintenance of GEO satellites is classified into two types, pulse position maintenance and low thrust position maintenance. The electric propulsion system providing the small thrust has the characteristic of high specific impulse, has smaller fuel consumption compared with a chemical propulsion system providing the pulse thrust, and plays an important role in promoting the effective load of the satellite and prolonging the on-orbit time of the satellite.
The method for keeping the position with the small thrust designed by part of the existing documents aims to eliminate the orbit drift caused by orbit perturbation through an efficient optimization algorithm and ensure the position keeping accuracy of the geostationary orbit satellite in a fixed-point window. The literature (Frederik J.de Bruijn, Stephan Theil, Daniel Choukroun, Eberhard Gill, geographic Satellite Station-Keeping Using Convex Optimization, Journal of Guidance Control and Dynamics,39(3),2016, pp.605-616) turns the low thrust position maintenance problem into a Convex Optimization problem, proposing a closed-loop Control law, but high-precision position maintenance can only be performed over a Control period of several days. A method for initially guessing a covariance value is proposed in the literature (Shuge Zhao, Pini Guifil, Jingrui Zhang, Initial Costates for Low-thread Minimum-Time Station Change of Geostationary Satellites, Journal of guiding Control and Dynamics,39(12),2016, pp.2745-2754), but the related method is based on specific assumptions and has better convergence only in some cases. The current position keeping strategy emphasizes counteracting the influence of the perturbation of the stationary orbit on the motion of the satellite, and does not fully utilize the long-term characteristic of the perturbation motion of the satellite; on the other hand, a set of collaborative initial value guessing method with strong universality is lacked to improve the convergence stability and the convergence speed of the position maintenance optimization algorithm.
Disclosure of Invention
The invention aims to provide a method for maintaining the low-thrust long-term position of a geostationary orbit satellite. On the basis, a design method of a low-thrust position holding control law is provided, so that the position holding period of the geostationary orbit satellite can be prolonged, the labor cost and equipment loss of a ground measurement and control station for implementing satellite control are reduced, the fuel consumption is reduced, the in-orbit service life of the satellite is prolonged, and the method is generally suitable for geostationary orbit electrically propelled satellites. The method not only can effectively utilize the periodic characteristic of the long-term uncontrolled motion of the satellite to plan the controlled motion, but also can provide a universal collaborative initial value guessing method to ensure the convergence performance of the position keeping optimization algorithm, thereby realizing long-term position keeping.
The purpose of the invention is realized by the following technical scheme.
The invention discloses a method for keeping the low-thrust long-term position of a stationary orbit satellite, which comprises the steps of establishing an average orbit motion model in the orbit plane and out of the orbit plane of the satellite through a spherical coordinate; giving out the long-period motion rule of the geostationary orbit satellite in the plane and out of the plane by a phase plane analysis method; on the basis, a fixed point position holding window of the geostationary orbit satellite is selected to obtain a periodic motion track of the satellite in an uncontrolled state in the fixed point window, namely a drift section track; and designing a low thrust control law, obtaining a motion track of the satellite in a controlled state, namely a thrust section track, and enabling the thrust section track and the drift section track to jointly form a closed-loop track, thereby finishing the low thrust long-term position maintenance of the geostationary orbit satellite.
The invention discloses a method for keeping a low-thrust long-term position of a geostationary orbit satellite, which comprises the following steps of:
the method comprises the following steps: establishing a motion model of the geostationary orbit satellite in a space containing the inside and outside of an orbit plane under the action of environmental perturbation through a spherical coordinate, and analyzing a periodic motion rule of the geostationary orbit satellite;
step 1.1: establishing a two-dimensional model which is described by spherical coordinates and only contains the motion in the satellite orbit plane, and analyzing the periodic motion rule of the geostationary orbit satellite;
and defining a celestial coordinate system, wherein the origin of coordinates is in the center of the earth, the x axis points to the spring equinox, the z axis points to the celestial north pole, and the direction of the y axis accords with the right-hand rule. The motion of the geostationary orbit satellite in and out of the orbital plane is described by two-dimensional spherical coordinates r, λ and three-dimensional spherical coordinates r, λ, φ, respectively. Wherein r is the distance from the geostationary orbit satellite to the geocenter, λ is the geostationary orbit satellite right ascension, and φ is the geostationary orbit satellite declination.
According to the laws of kinematics, the acceleration of a geostationary orbit satellite in the orbital plane is represented in the form:
Figure BDA0002659087810000031
wherein Ω is the ascension of the stationary orbit satellite at the ascending intersection.
According to the satellite perturbation law, the in-plane motion of a geostationary orbit satellite is mainly influenced by the earth's non-spherical perturbation, and the perturbation acceleration has the following form:
Figure BDA0002659087810000032
Wherein the subscript ENP represents the global perturbation, g0Is the constant of gravity of the earth, R0Is the radius of the earth; j. the design is a squarenCoefficient of harmonic terms, J, of the earth's non-spherical perturbationnmThe coefficient is a field harmonic coefficient of the earth non-spherical perturbation, and n and m respectively represent the order and the times of the coefficient; gamma is an included angle between a connecting line of the geostationary orbit satellite and the geocenter and the equatorial plane short axis; xn,Ynm,ZnmRepresenting the earth's non-spherical perturbation constant.
And (3) establishing a two-dimensional model of the satellite moving in the orbital plane by combining the formula (1) with the formula (2), wherein the form is as follows:
Figure BDA0002659087810000033
in general, ignoring the higher order terms in equation (3), only the 2 nd order 2 terms are retained, i.e., n is 2 and m is 2, expressed as:
Figure BDA0002659087810000034
the coordinates (r, lambda, phi) of the satellite sphere are set at the reference point (r)0,λ0,φ0) And (c) is represented as:
Figure BDA0002659087810000035
where Δ r, Δ γ are small amounts.
The formula (4) and the formula (5) are combined to establish a two-dimensional average model of the satellite moving in the orbital plane
Figure BDA0002659087810000041
Equation (6) has a balanced solution, revealing that under long-term evolution the geostationary orbit satellite has a phase plane (lambda,
Figure BDA0002659087810000042
) The motion trajectory periodically moving around the two stable equilibrium points and the two unstable equilibrium points.
Step 1.2: establishing a three-dimensional model of the movement outside the orbital plane described by the spherical coordinates and analyzing the periodic movement rule of the geostationary orbit satellite
According to the kinematic law, the acceleration of a geostationary orbit satellite moving out of orbit is expressed in the form:
Figure BDA0002659087810000043
according to the perturbation law of the satellite, the motion of the static orbit satellite out of the orbit plane is also influenced by the third body gravity in the sun and the moon, and the perturbation acceleration of the third body gravity in the sun and the moon has the following form:
Figure BDA0002659087810000044
Figure BDA0002659087810000045
Figure BDA0002659087810000046
wherein, the subscript LSP represents the third body gravity perturbation of day and month,
Figure BDA0002659087810000047
is the angular velocity of the earth-moon system center rotating around the sun,
Figure BDA0002659087810000048
is the angular velocity of the earth-moon system moving around its center; μ is the gravitational constant; a isx,ay,bx,by,cx,cyIs the third body attraction coefficient of sun and moon.
The motion of the static orbit satellite outside the orbit plane is simultaneously influenced by the non-spherical perturbation of the earth and the third gravitational force of the sun and the moon; simultaneous formulas (2), (7) and (8) establish a three-dimensional motion model of the geostationary orbit satellite out of the orbit plane.
Considering the mean hamiltonian under non-spherical perturbation and gravity perturbation in the sun and moon, it has the following form:
Figure BDA0002659087810000051
wherein n iseIs the angular velocity of rotation of the earth, aeIs the radius of the earth, a is the semi-major axis of the satellite orbit, nbodyIs the rotation angular velocity of the third gravitational body, the main bagIncluding the angular velocity n of the sunsAnd moon self-rotation angular velocity nm(ii) a E is the angle between the berkoff plane and the equatorial plane, s∈＝sin∈，c∈＝cos∈。
And analyzing the periodic rule of the out-of-plane motion of the geostationary orbit satellite by means of the averaged Hamiltonian, and calculating the balance solution of the formula (9) to obtain the periodic motion rule, namely, under the long-term evolution of the orbit, the motion track of the geostationary orbit satellite in a phase plane (omega, phi) carries out periodic motion around a balance point.
Step two: a fixed point position holding window of the geostationary orbit satellite is selected to obtain a long-period motion track, namely a drift section track, of the satellite in the fixed point window in an uncontrolled state;
internationally, the fixed-point windows for geostationary orbit satellites are defined as: a rectangular neighborhood centered at the ideal spot position within the declination phase plane (λ, φ) of the right ascension. The mathematical model of the fixed-point window has the form:
-Δλ≤λ-λd≤Δλ (10)
-Δφ≤φ≤Δφ (11)
wherein subscript d represents the desired pointing position; Δ λ, Δ φ represents the distance of the fixed-point window boundary from the fixed-point window midpoint.
Substituting the constraint of the fixed-point window to the right ascension lambda of the satellite, namely the formula (10) into the formula (6); the constraint of the fixed point window to the declination phi of the satellite, namely the formula (11), is substituted into the formula (9) to obtain the declination phi of the satellite on the phase plane (lambda,
Figure BDA0002659087810000052
) And the phase plane (phi,
Figure BDA0002659087810000053
) Inner drift section trajectory.
Step three: and establishing a low-thrust position keeping control model of the satellite in the orbital plane and out of the orbital plane.
Step 3.1, establishing a low-thrust position keeping control model of the satellite in the orbital plane
The matrix form of the satellite dynamics model considering the small thrust and the environmental perturbation is as follows
Figure BDA0002659087810000054
Wherein, subscript 2D represents a two-dimensional model of the in-plane motion of the satellite, u represents a switching constant of the thruster, T represents the magnitude of the thrust provided by the thruster, and I spIs the specific impulse of the thruster.
In response to the in-plane motion,
Figure BDA0002659087810000055
is a state variable of the two-dimensional model, d2D＝(dr,dλ)TIs the thrust direction vector of the two-dimensional model,
Figure BDA0002659087810000061
is a matrix of thrust coefficients of a two-dimensional model,
Figure BDA0002659087810000062
is a perturbation coefficient matrix of the two-dimensional model.
Step 3.2, establishing a low-thrust position keeping control model of the satellite out of the orbital plane
The matrix form of the satellite dynamics model considering the small thrust and the environmental perturbation is as follows
Figure BDA0002659087810000063
Wherein the subscript 3D represents the three-dimensional model needle of satellite out-of-plane motion,
Figure BDA0002659087810000064
is a state variable of the three-dimensional model, d3D＝(dr,dλ,dφ)TThe thrust direction vector of the three-dimensional model,
Figure BDA0002659087810000065
is a matrix of thrust coefficients for the three-dimensional model,
Figure BDA0002659087810000066
is a perturbation coefficient matrix of the three-dimensional model.
Step four: and establishing a time optimal and fuel optimal solving model for maintaining the low thrust position based on the satellite drift section track obtained in the second step and the satellite low thrust position maintaining control model obtained in the third step.
4.1 establishing initial and final state constraints of thrust segment trajectory
In order to enable the static orbit satellite to be always positioned in a fixed point window, a drift section track and a thrust section track need to form a closed loop track; therefore, the initial state point of the track of the thrust section is superposed with the final state point of the track of the drift section, and the final state point of the track of the thrust section is superposed with the initial state point of the track of the drift section; the initial and final state constraints for the time, fuel optimum model are then expressed as follows:
Figure BDA0002659087810000071
Step 4.2, establishing a time optimal model for maintaining a low-thrust position
The objective function to achieve time optimization has the form:
Figure BDA0002659087810000072
where the subscript MT denotes the time-optimal model.
The Hamiltonian that achieves time optimization has the form:
Figure BDA0002659087810000073
defining the covariance variable as partial derivatives of the state variables x and m of the Hamiltonian H, and the form is as follows:
Figure BDA0002659087810000074
wherein the content of the first and second substances,
Figure BDA0002659087810000075
is a matrix of thrust coefficients for the covariates,
Figure BDA0002659087810000076
is a perturbation coefficient matrix of the covariate.
Obtaining optimal thrust vector d by utilizing Pontryagin maximum principleMT *The matrix expression of (c):
Figure BDA0002659087810000081
therefore, the time optimal model expression for the low thrust position hold is as follows:
Figure BDA0002659087810000082
the constraint that equation (19) needs to satisfy is as follows:
Figure BDA0002659087810000083
step 4.3 based on the time optimal model of step 4.2, establishing a fuel optimal model maintained at a low thrust position
The objective function to achieve fuel optimization has the form:
Figure BDA0002659087810000084
where the subscript MF denotes the fuel optimum model and ε is the homotopy coefficient.
The Hamiltonian to achieve fuel optimization has the form:
Figure BDA0002659087810000085
defining the covariance variable as partial derivatives of the state variables x and m of the Hamiltonian H, and the form is as follows:
Figure BDA0002659087810000086
the optimal thrust vector d can be obtained by utilizing the Pontryagin maximum principleMF *The matrix expression of (c):
Figure BDA0002659087810000091
Thus, the fuel optimum problem for low thrust position hold can be described by a mathematical model of the form:
Figure BDA0002659087810000092
optimal transfer time (t) resulting from solving a time-optimal problemf)MTAs one of the constraints for solving the fuel optimum problem, all the constraints are as follows:
Figure BDA0002659087810000093
step 4.4 converts the time/fuel optimum problem into a two-point boundary value problem.
As can be seen from equations (19) and (25), the state variable value and the covariance variable value at any time t are functions of the state variable value and the covariance variable value at the initial time, as follows:
[x(t),px(t)]T＝f([x(ti),px(ti)]T,ti,t) (27)
the time optimal problem established by equations (19) and (20) is described as findingFind [ x (t)i),(px)MT(ti)]TSo that
[x(tf),px(tf)]T＝f([x(ti),(px)MT(ti)]T,ti,t) (28)
Satisfy the requirement of
Figure BDA0002659087810000094
Similarly, the fuel-optimum problem model established by equations (25) and (26) is described as finding [ x (t)i),(px)MF(ti)]TSo that
[x(tf),px(tf)]T＝f([x(ti),(px)MF(ti)]T,tiT) (30) satisfies
Figure BDA0002659087810000101
Thus, the time and fuel optimization problem translates into a two-point boundary value problem.
Step five: directly solving the two-point boundary value problem established in the step 4.4, and substituting the result into the step four to obtain the time optimal and fuel optimal control law of the low thrust position holding;
step six: and D, applying the position maintaining control law in the step five to realize the low-thrust long-term position maintaining of the geostationary orbit satellite.
Solving the two-point boundary value problem by a collaborative initial value guessing method:
Step 1, calculating pulse thrust delta V required by position maintenance through deviation of the number of initial and final tracks*And the action position of impulse thrust
Figure BDA0002659087810000102
Step (ii) of2. Searching for longitude including impulsive thrust action
Figure BDA0002659087810000103
So that the satellite can complete and apply the pulse thrust delta V*Position maintenance with the same effect later, and recording the start and shut-down right ascension with continuous thrust radian
Figure BDA0002659087810000104
And total time of start-up Δ t*(ii) a By means of continuous thrust FconstAnd mid point of arc segment
Figure BDA0002659087810000105
Calculating the corresponding covariates
Figure BDA0002659087810000106
Step 3, the covariates obtained in step 2
Figure BDA0002659087810000107
And total starting time delta t*And substituting the initial value into a formula (30) and a constraint condition (31), and finally obtaining the optimal initial value by continuously updating so as to solve the problem of two-point edge values.
Advantageous effects
1. The invention discloses a method for keeping a low-thrust long-term position of a geostationary orbit satellite, which is characterized in that a complete track of an uncontrolled drift track of the satellite combined with a controlled track of the low thrust is designed and is always positioned in a fixed-point window of the satellite, so that the position keeping period of the geostationary orbit satellite can be prolonged, the labor cost and equipment loss of a ground measurement and control station for implementing satellite control are reduced, the fuel consumption is reduced, the in-orbit service life of the satellite is prolonged, and the method is generally suitable for an electric propulsion satellite of the geostationary orbit.
2. By establishing the uncontrolled average motion models of the geostationary orbit satellite in the orbit plane and out of the orbit plane, the periodic motion rules of in-plane and out-of-plane motion of the geostationary orbit satellite in the long-term orbit evolution process can be respectively obtained, and further theoretical basis and technical support are provided for planning a controlled motion trajectory by utilizing the periodic characteristics of the uncontrolled motion of the satellite.
3. The invention discloses a method for keeping a small-thrust long-term position of a geostationary orbit satellite, which can quickly search a stable convergence solution under the optimal problem of time and fuel by designing a collaborative initial value guessing method based on a pulse thrust solution and a continuous thrust solution, thereby solving the problem of keeping the small-thrust position with the optimal time/fuel.
Drawings
FIG. 1 is a general flow diagram of a method for low thrust long term position maintenance for a geostationary orbit satellite;
FIG. 2 is a flow chart of a low thrust fuel optimum position hold control law design;
FIG. 3 is a periodic law of in-plane motion for a geostationary orbit satellite;
FIG. 4 is a periodic law of out-of-plane motion for a geostationary orbit satellite;
figure 5a shows a geostationary orbit satellite in (lambda,
Figure BDA0002659087810000111
) A position maintaining trajectory in the phase plane;
FIG. 5b is an enlarged view of a portion of FIG. 5a at the circle;
figure 6a shows a geostationary orbit satellite in (phi,
Figure BDA0002659087810000112
) The position in the phase plane remains the trajectory.
Fig. 6b is an enlarged view of a portion of fig. 6a indicated by an arrow.
Detailed Description
To better illustrate the objects and advantages of the present invention, the following detailed description of the embodiments of the present invention is provided in conjunction with the accompanying drawings.
Example 1:
as shown in fig. 1, in order to verify the method for maintaining the long-term position of a geostationary orbit satellite with a small thrust, a satellite operating in a geostationary orbit of the earth is selected as a main research object. The basic parameters of the satellite are shown in the table below.
TABLE 1 satellite parameters
Figure BDA0002659087810000113
Figure BDA0002659087810000121
The method comprises the following steps: establishing a motion model of the geostationary orbit satellite in a space containing the inside and outside of an orbit plane under the action of environmental perturbation through a spherical coordinate, and analyzing a periodic motion rule of the geostationary orbit satellite;
band harmonic and field harmonic coefficients for calculating earth's non-spherical perturbations are shown in the following table:
TABLE 2 principal coefficient of the Earth's gravitational field
Figure BDA0002659087810000122
By substituting the parameters of tables 1 and 2 into an average model of in-plane motion (i.e., equation (6)), two stable equilibrium solutions λ can be obtained1＝75.5°,λ2255.5 ° and two unstable equilibrium solutions λ3＝165.5°,λ4345.5. The simulation results are shown in fig. 2. FIG. 2 shows the periodic law of motion in the plane of a geostationary orbit satellite, i.e. at any point (λ) in the phase plane k,
Figure BDA0002659087810000123
) After the long-term evolution of the orbit, a periodic motion track around a stable equilibrium point (such as a red line and a yellow line in fig. 3) or an unstable equilibrium point (such as a blue line and a green line in fig. 3) is formed. Whereas a geostationary satellite located at 60 deg. longitude lambda in this example is located in the phase plane (lambda,
Figure BDA0002659087810000124
) Will surround the equilibrium point (λ)1＝75.5°，
Figure BDA0002659087810000125
) A periodic movement is performed.
TABLE 3 correlation coefficient of gravity perturbation between day and month
Figure BDA0002659087810000126
By introducing the parameters of table 3 into the average hamiltonian of the out-of-plane motion (i.e., equation (9)), an equilibrium solution (Ω -0 °, Φ -7.435 °) can be obtained, and the simulation results are shown in fig. 3. FIG. 3 reveals the periodic law of out-of-plane motion of a geostationary orbit satellite, i.e. located at any point (λ) of the phase planek≤14.6°,
Figure BDA0002659087810000127
) After a long-term evolution of the orbit, a periodic motion trajectory around a stable equilibrium point (Ω ═ 0 °, Φ ═ 7.435 °) will be formed, which is also true for the stationary orbit satellite located at latitude Φ ═ 0.5 ° in this example.
Step two: obtaining a long-period motion track, namely a drift section track of the satellite in the fixed point window in an uncontrolled state by selecting a fixed point position holding window of the geostationary orbit satellite;
according to the international consensus that a single geostationary orbit satellite occupies a fixed point longitude, the longitude interval contained in a fixed point window of the geostationary orbit satellite is selected to be [0 degrees, 0.1 degrees ]; according to the periodic law of the out-of-plane motion of the geostationary orbit satellite, the latitude interval included in the fixed-point window of the geostationary orbit satellite is selected to be [0 degrees, 0.5 degrees ]. Thus, the fixed-point window is represented as:
Figure BDA0002659087810000131
The formula (32) is combined with the formulas (6) and (9) to obtain the phase plane (lambda,
Figure BDA0002659087810000132
) Within the sum phase plane (omega, phi)The periodic motion profile of (a). The trajectory of the uncontrolled drift section in the phase plane (lambda,
Figure BDA0002659087810000133
) Respectively at the initial and final state values of
Figure BDA0002659087810000134
The trajectory of the uncontrolled drift segment is in the phase plane (phi,
Figure BDA0002659087810000135
) The initial and final state values of are respectively
Figure BDA0002659087810000136
The following three, four and five steps form a design link of the low-thrust optimal time/fuel position holding control law, and the specific flow is shown in fig. 4.
Step three: establishing a low-thrust position keeping control model of the satellite in and out of the orbital plane;
the static orbit satellite in the example carries an electric propulsion system, and an electric thruster can generate constant continuous small thrust to the satellite, wherein the amplitude of the thrust is T200 mN, and the specific impulse is Isp3800 s. And substituting the related parameters into the formula (12) and the formula (13), namely obtaining the low-thrust position keeping control model of the geostationary orbit satellite in the orbit plane and out of the orbit plane.
Step four: and establishing a time optimal and fuel optimal solving model for maintaining the low thrust position based on the satellite drift section track obtained in the second step and the satellite low thrust position maintaining control model obtained in the third step.
In order to enable the static orbit satellite to be always positioned in a fixed point window, a drift section track and a thrust section track need to form a closed loop track; therefore, the initial state point of the track of the thrust section is superposed with the final state point of the track of the drift section, and the final state point of the track of the thrust section is superposed with the initial state point of the track of the drift section; therefore, the initial and final states of the time and fuel optimal model are constrained to be
Figure BDA0002659087810000141
And (3) substituting the initial and final state constraints into constraint conditions of the time and fuel optimal model, namely formula (20) and formula (26), and converting the established time and fuel optimal model into a two-point boundary value problem.
Step five: directly solving the problem of the two-point boundary value established in the step 4.4, substituting the result into the step four, and obtaining the time optimal control law and the fuel optimal control law for maintaining the low-thrust position;
by directly solving the problem of two-point boundary values, the obtained initial value of the coordination state and the total startup duration are as follows:
Figure BDA0002659087810000142
bringing the coordination initial value and the total startup duration of the formula (33) back to the fourth step to obtain a low thrust position holding control law;
step six: and D, applying the position maintaining control law in the step five to realize the low-thrust long-term position maintaining of the geostationary orbit satellite.
Example 2:
the first four steps of example 2 are the same as those of example 1.
Step five: solving the two-point boundary value problem in the step four by a collaborative initial value guessing method, and substituting the result into the time optimal and fuel optimal control law of the low thrust position maintenance:
step 5.1: calculating the pulse thrust delta V required by position maintenance through the deviation of the number of the initial and final orbits*5.987m/s and the position of action of the impulse thrust
Figure BDA0002659087810000143
And step 5.2: searching involving impulsive thrust action longitude
Figure BDA0002659087810000144
So that the satellite can complete and apply the pulse thrust delta V*Position maintenance with same effect after 5.987m/s and recording of continuous thrust arc of right ascension and shutdown
Figure BDA0002659087810000145
And total time duration at of start-up*47649 s; by means of continuous thrust Fconst200mN and mid point of arc segment
Figure BDA0002659087810000151
Calculating the corresponding covariates
Figure BDA0002659087810000152
Figure BDA0002659087810000153
Step 5.3: with the covariates obtained in step 5.2
Figure BDA0002659087810000154
And total starting time delta t*When 47649s is the initial value, equation (30) and constraint conditions (31) are substituted, and the optimal initial value (p) is finally obtained by continuously updatingx **,Δt**)。
Figure BDA0002659087810000155
Thereby solving the problem of two-point boundary values.
Bringing the optimal co-operating initial value and the total starting time length described in the formula (34) back to the step four to obtain the optimal time and optimal fuel control law for maintaining the low-thrust position;
step six: and D, applying the position maintaining control law in the step five to realize the low-thrust long-term position maintaining of the geostationary orbit satellite.
By comparing the formulas (33) and (34), the total starting time using the collaborative initial value guessing method disclosed by the invention is obviously shorter than the total starting time obtained by directly solving the two-point boundary value problem, and the fuel consumption is less if the time is short, so that the collaborative initial value guessing method disclosed by the invention has better performance.
Fig. 5 shows a schematic diagram of a geostationary orbit satellite in a phase plane (λ,
Figure BDA0002659087810000156
) The complete trajectory of (2). The drift section 1 represents a first section of drift trajectory of the satellite from the initial position in an uncontrolled state, when the satellite reaches the upper boundary lambda-60 DEG of the fixed point window, which is 0.1 DEG, if the uncontrolled state is continuously maintained, the satellite drifts out of the fixed point window, which is 0 DEG-60 DEG-0.1 DEG, so that the satellite is maneuvered to the initial point of the drift section 1 by adopting the thrust section 1; then, a second section of drift trajectory can be formed continuously in an uncontrolled state until the satellite reaches the upper limit lambda-60 degrees of the fixed point window again, namely 0.1 degree, and then second small thrust control is implemented to form a thrust section 2; the above steps are repeated in a circulating way to form a long-term position keeping strategy for the in-plane motion of the satellite in the stationary orbit.
Fig. 6 shows a geostationary orbit satellite positioned at phi 0.5 deg. in a phase plane (lambda,
Figure BDA0002659087810000157
) The complete trajectory of (2). The drift section 1 represents a first section of drift trajectory of the satellite in an uncontrolled state from an initial position, and the motion trend is to reduce phi from 0.5 degrees to phi approximately equal to 0 degrees and then increase phi to 0.5 degrees; if the satellite keeps an uncontrolled state, the satellite drifts out a fixed point window with phi being more than or equal to 0 degree and less than or equal to 0.5 degree, so that the thrust section 1 is adopted to change the variation trend of phi; then the satellite can continue to form a second section of drift trajectory in an uncontrolled state until the satellite reaches the upper bound phi of the fixed point window again to be 0.5 degrees, and then second small thrust control is implemented to form a thrust section 2; the above steps are repeated in a circulating way, and a long-term position keeping strategy for out-of-plane motion of the satellite in the stationary orbit is formed.
Table 4 shows the performance index of the position keeping method designed by the present invention. Conventional low thrust position holding methods aim to overcome environmental perturbations, so the control frequency is once a day or even many times a day. In the method for maintaining the low-thrust long-term position of the geostationary orbit satellite disclosed by the embodiment, the control period is greatly prolonged by utilizing the periodic rule of the long-term uncontrolled motion of the satellite, and the prolonging of the control period means that the labor cost and equipment loss of the ground satellite measurement and control station for implementing satellite control are further reduced. Taking the example of a geostationary orbit satellite pointed at 0 ≦ λ -60 ≦ 0.1 °, 0 ≦ φ ≦ 0.5 °, the period of longitude control is one month (31.82 days) and the period of latitude control is even more than one year (433 days). Therefore, the method can realize the long-term position maintenance of the stationary orbit satellite with low thrust and has better control performance compared with the traditional method.
TABLE 4 Low thrust long-term position keeping method performance index of geostationary orbit satellite
Figure BDA0002659087810000161
In summary, the above description is only a preferred embodiment of the present invention, and is not intended to limit the scope of the present invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (2)

1. A low-thrust long-term position keeping method for a stationary orbit satellite is characterized by comprising the following steps: the method comprises the following steps:
the method comprises the following steps: establishing a motion model of the geostationary orbit satellite in a space containing the inside and outside of an orbit plane under the action of environmental perturbation through a spherical coordinate, and analyzing a periodic motion rule of the geostationary orbit satellite;
step 1.1: establishing a two-dimensional model which is described by spherical coordinates and only contains the motion in the satellite orbit plane, and analyzing the periodic motion rule of the geostationary orbit satellite;
defining a celestial coordinate system, wherein the origin of coordinates is in the center of the earth, the x axis points to the spring equinox, the z axis points to the celestial north pole, and the y axis direction accords with the right-hand rule; the motion of the geostationary orbit satellite in the orbit plane and out of the orbit plane is respectively described by two-dimensional spherical coordinates (r, lambda) and three-dimensional spherical coordinates (r, lambda, phi); wherein r is the distance from the geostationary orbit satellite to the geocenter, lambda is the geostationary orbit satellite right ascension, and phi is the geostationary orbit satellite declination;
according to the laws of kinematics, the acceleration of a geostationary orbit satellite in the orbital plane is represented in the form:
Figure FDA0003553000210000011
Figure FDA0003553000210000012
wherein Ω is the ascension of the stationary orbit satellite;
according to the satellite perturbation law, the in-plane motion of a geostationary orbit satellite is mainly influenced by the earth's non-spherical perturbation, and the perturbation acceleration has the following form:
Figure FDA0003553000210000013
Figure FDA0003553000210000014
Wherein the subscript ENP represents the global perturbation, g0Is the constant of gravity of the earth, R0Is the radius of the earth; j is a unit ofnCoefficient of harmonic terms, J, of the earth's non-spherical perturbationnmThe coefficient is a field harmonic coefficient of the earth non-spherical perturbation, and n and m respectively represent the order and the times of the coefficient; gamma is an included angle between a connecting line of the geostationary orbit satellite and the geocenter and the equatorial plane short axis; xn,Ynm,ZnmRepresenting an earth non-spherical perturbation constant;
and (3) establishing a two-dimensional model of the satellite moving in the orbital plane by combining the formula (1) with the formula (2), wherein the form is as follows:
Figure FDA0003553000210000015
Figure FDA0003553000210000016
in general, higher-order terms in formula (3) are ignored, and only 2 nd order 2-order terms are retained, i.e., n is 2 and m is 2, and is expressed as:
Figure FDA0003553000210000017
Figure FDA0003553000210000018
the coordinates (r, lambda, phi) of the satellite sphere are set at the reference point (r)0,λ0,φ0) And (c) is represented as:
r＝r0+Δr
λ＝λ22+γ0+Δγ
γ＝γ0+Δγ (5)
wherein Δ r, Δ γ are small amounts;
the formula (4) and the formula (5) are combined to establish a two-dimensional average model of the satellite moving in the orbital plane
Figure FDA0003553000210000021
Equation (6) has a balanced solution that reveals the geostationary orbit satellite in the phase plane under long-term evolution
Figure FDA0003553000210000022
The motion trajectory ofPerforming periodic motion around the two stable equilibrium points and the two unstable equilibrium points;
step 1.2: establishing a three-dimensional model of the movement outside the orbital plane described by the spherical coordinates and analyzing the periodic movement rule of the geostationary orbit satellite
According to the kinematic law, the acceleration of a geostationary orbit satellite moving out of orbit is expressed in the form:
Figure FDA0003553000210000023
Figure FDA0003553000210000024
Figure FDA0003553000210000025
according to the perturbation law of the satellite, the motion of the static orbit satellite out of the orbit plane is also influenced by the third body gravity in the sun and the moon, and the perturbation acceleration of the third body gravity in the sun and the moon has the following form:
Figure FDA0003553000210000026
Figure FDA0003553000210000027
Figure FDA0003553000210000028
wherein, the subscript LSP represents the third body gravity perturbation of day and month,
Figure FDA0003553000210000029
is the earth-moon systemThe angular velocity of the system's center of rotation about the sun,
Figure FDA00035530002100000210
is the angular velocity of the earth-moon system moving around its center; μ is the gravitational constant; a is ax,ay,bx,by,cx,cyIs the third gravitational coefficient of sun and moon;
the motion of the static orbit satellite outside the orbit plane is simultaneously influenced by the non-spherical perturbation of the earth and the third gravitational force of the sun and the moon; simultaneous formulas (2), (7) and (8) establish a three-dimensional motion model of the geostationary orbit satellite out of the orbit plane;
considering the mean hamiltonian under non-spherical perturbation and gravity perturbation in the sun and moon, it has the following form:
Figure FDA00035530002100000211
wherein n iseIs the angular velocity of rotation of the earth, aeIs the radius of the earth, a is the semi-major axis of the satellite orbit, nbodyIs the rotation angular velocity of the third gravitational body, mainly comprising the rotation angular velocity n of the sunsAnd moon self-rotation angular velocity nm(ii) a E is the angle between the equatorial plane and the berkoff plane,
Figure FDA0003553000210000036
analyzing the periodic law of the out-of-plane motion of the geostationary orbit satellite by means of the averaged Hamiltonian, and obtaining the periodic motion law by calculating the equilibrium solution of a formula (9), namely under the long-term evolution of the orbit, the motion track of the geostationary orbit satellite in a phase plane (omega, phi) carries out periodic motion around a balance point;
Step two: a fixed point position holding window of the geostationary orbit satellite is selected to obtain a long-period motion track, namely a drift section track, of the satellite in the fixed point window in an uncontrolled state;
internationally, the fixed-point windows for geostationary orbit satellites are defined as: a rectangular neighborhood centered at an ideal fixed point position in the right ascension and declination phase plane (λ, Φ); the mathematical model of the fixed-point window has the form:
-Δλ≤λ-λd≤Δλ (10)
-Δφ≤φ≤Δφ (11)
wherein subscript d represents the desired pointing position; Δ λ, Δ φ represents the distance between the fixed point window boundary and the fixed point window midpoint;
substituting the constraint of the fixed-point window to the right ascension lambda of the satellite, namely the formula (10) into the formula (6); the constraint of the fixed point window to the declination phi of the satellite, namely the formula (11) is substituted into the formula (9), and the phase plane of the satellite is obtained
Figure FDA0003553000210000031
And phase plane
Figure FDA0003553000210000032
Inner drift section trajectories;
step three: establishing a low-thrust position keeping control model of the satellite in and out of the orbital plane;
step 3.1, establishing a low-thrust position keeping control model of the satellite in the orbital plane
The matrix form of the satellite dynamics model considering the small thrust and the environmental perturbation is as follows
Figure FDA0003553000210000033
Wherein, subscript 2D represents a two-dimensional model of the in-plane motion of the satellite, u represents a switching constant of the thruster, T represents the magnitude of the thrust provided by the thruster, and I spIs the specific impulse of the thruster;
in response to the in-plane motion,
Figure FDA0003553000210000034
is a state variable of a two-dimensional model, d2D＝(dr,dλ)TIs the thrust direction vector of the two-dimensional model,
Figure FDA0003553000210000035
is a matrix of thrust coefficients of a two-dimensional model,
Figure FDA0003553000210000041
is a perturbation coefficient matrix of the two-dimensional model;
step 3.2, establishing a low-thrust position keeping control model of the satellite out of the orbital plane
The matrix form of the satellite dynamics model considering the small thrust and the environmental perturbation is as follows
Figure FDA0003553000210000042
Wherein the subscript 3D represents the three-dimensional model needle of satellite out-of-plane motion,
Figure FDA0003553000210000043
is a state variable of the three-dimensional model, d3D＝(dr,dλ,dφ)TThe thrust direction vector of the three-dimensional model,
Figure FDA0003553000210000044
is a matrix of thrust coefficients for the three-dimensional model,
Figure FDA0003553000210000045
is a perturbation coefficient matrix of the three-dimensional model;
step four: establishing a time optimal and fuel optimal solution model for maintaining the low thrust position based on the satellite drift section track obtained in the second step and the satellite low thrust position maintaining control model obtained in the third step;
4.1 establishing initial and final state constraints of thrust segment trajectory
In order to enable the static orbit satellite to be always positioned in a fixed point window, a drift section track and a thrust section track need to form a closed loop track; therefore, the initial state point of the track of the thrust section is superposed with the final state point of the track of the drift section, and the final state point of the track of the thrust section is superposed with the initial state point of the track of the drift section; the initial and final state constraints for the time, fuel optimum model are then expressed as follows:
Figure FDA0003553000210000051
Step 4.2, establishing a time optimal model for maintaining a low-thrust position
The objective function to achieve time optimization has the form:
Figure FDA0003553000210000052
wherein the subscript MT denotes a time-optimal model;
the Hamiltonian that achieves time optimality has the following form:
Figure FDA0003553000210000053
defining the covariance variable as partial derivatives of the state variables x and m of the Hamiltonian H, and the form is as follows:
Figure FDA0003553000210000054
wherein the content of the first and second substances,
Figure FDA0003553000210000055
is a matrix of thrust coefficients for the covariates,
Figure FDA0003553000210000056
is a perturbation coefficient matrix of the covariates;
obtaining time optimal thrust vector d by utilizing Pontryagin maximum principleMT *The matrix expression of (c):
Figure FDA0003553000210000057
therefore, the time optimal model expression for the low thrust position hold is as follows:
Figure FDA0003553000210000058
the constraint that equation (19) needs to satisfy is as follows:
Figure FDA0003553000210000061
step 4.3 based on the time optimal model of step 4.2, establishing a fuel optimal model maintained at a low thrust position
The objective function to achieve fuel optimization has the form:
Figure FDA0003553000210000062
wherein, subscript MF represents a fuel optimum model, and ε is a homotopy coefficient;
the Hamiltonian to achieve fuel optimization has the form:
Figure FDA0003553000210000063
defining the covariance variable as partial derivatives of the state variables x and m of the Hamiltonian H, and the form is as follows:
Figure FDA0003553000210000064
using a PontriThe principle of the submetal maximum value can obtain the optimal thrust vector d of the fuel MF *The matrix expression of (c):
Figure FDA0003553000210000065
thus, the fuel optimum problem for low thrust position hold can be described by a mathematical model of the form:
Figure FDA0003553000210000066
optimal transfer time (t) resulting from solving a time-optimal problemf)MTAs one of the constraints for solving the fuel optimum problem, all the constraints are as follows:
Figure FDA0003553000210000071
4.4, converting the time/fuel optimal problem into a two-point boundary value problem;
as can be seen from equations (19) and (25), the state variable value and the covariance variable value at any time t are functions of the state variable value and the covariance variable value at the initial time, as follows:
[x(t),px(t)]T＝f([x(ti),px(ti)]T,ti,t) (27)
the time optimal problem established by equations (19) and (20) is described as finding [ x (t)i),(px)MT(ti)]TSo that
[x(tf),px(tf)]T＝f([x(ti),(px)MT(ti)]T,ti,t) (28)
Satisfy the requirement of
Figure FDA0003553000210000072
Similarly, the fuel-optimum problem model established by equations (25) and (26) is described as finding [ x (t)i),(px)MF(ti)]TSo that
[x(tf),px(tf)]T＝f([x(ti),(px)MF(ti)]T,ti,t) (30)
Satisfy the requirement of
Figure FDA0003553000210000073
Thus, the time and fuel optimization problem translates into a two-point boundary value problem;
step five: directly solving the two-point boundary value problem established in the step 4.4, and substituting the result into the step four to obtain the time optimal and fuel optimal control law of the low thrust position holding;
step six: and D, applying the position maintaining control law in the step five to realize the low-thrust long-term position maintaining of the geostationary orbit satellite.
2. The method of claim 1, wherein: the two-point boundary value problem stated in the step four is solved by a collaborative initial value guessing method, and the specific steps are as follows:
Step 1, calculating pulse thrust delta V required by position maintenance through deviation of the number of initial and final tracks*And the action position of impulse thrust
Figure FDA0003553000210000074
Step 2, searching longitude including impulse thrust action
Figure FDA0003553000210000081
The continuous thrust arc section of the satellite enables the satellite to complete and apply pulsesThrust force Δ V*Position maintenance with the same effect later, and recording the start and shut-down right ascension with continuous thrust radian
Figure FDA0003553000210000082
And total time of start-up Δ t*(ii) a By using continuous thrust FconstAnd mid point of arc segment
Figure FDA0003553000210000083
Calculating the corresponding covariates
Figure FDA0003553000210000084
Step 3, the covariates obtained in step 2
Figure FDA0003553000210000085
And total starting time delta t*And substituting the initial value into a formula (30) and a constraint condition (31), and finally obtaining the optimal initial value by continuously updating so as to solve the problem of two-point edge values.
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Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN101219713A (en) * 2007-12-26 2008-07-16 北京控制工程研究所 Satellitic self-determination orbital transfer method
CN103235598A (en) * 2013-05-14 2013-08-07 北京理工大学 Method for regulating propeller direction to point to combined-body spacecraft centroid
CN105511490A (en) * 2015-12-15 2016-04-20 北京理工大学 Geostationary orbit satellite station keeping-angular momentum unloading combined control method

Patent Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN101219713A (en) * 2007-12-26 2008-07-16 北京控制工程研究所 Satellitic self-determination orbital transfer method
CN103235598A (en) * 2013-05-14 2013-08-07 北京理工大学 Method for regulating propeller direction to point to combined-body spacecraft centroid
CN105511490A (en) * 2015-12-15 2016-04-20 北京理工大学 Geostationary orbit satellite station keeping-angular momentum unloading combined control method

Non-Patent Citations (2)

* Cited by examiner, † Cited by third party
Title
日地系统L2点Halo轨道自主天文导航及精度分析;赵书阁;《北京理工大学学报》;20131115;全文 *
辐射开环绳系卫星编队自旋展开动力学与控制策略;苏飞;《航空学报》;20160202;全文 *

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