CN115130282A - Halo track maintaining method based on double-base invariant manifold - Google Patents
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Abstract
The invention discloses a Halo track maintaining method based on double-base invariant manifold, which adopts an invariant manifold method to obtain a reduced order kinetic equation and obtain a nonlinear polynomial relationship; solving the obtained kinetic equation by adopting a Linstedt-Pincar method; correcting the deviation between the last integral state value and the target state value of the kinetic equation by adopting a differential iteration method; based on the theoretical basis, taking the restriction three-body problem of the earth-moon-shaped as an example, a Halo track surrounding a translation point is constructed; and iteratively correcting the speed according to the used track keeping method through a polynomial constraint condition. The initial value of the orbit is accurate, the dynamic equation is expanded at the translation point to obtain the nonlinear polynomial relation which is used as constraint, and the accurate orbit initial value can be obtained through LP perturbation solution and iterative correction. The orbit keeping method is suitable for Halo periodic orbits in earth-moon systems and sun-earth systems.
Description
Technical Field
The invention relates to a method for realizing Halo track maintenance near a translation point by a double-base invariant manifold method aiming at a moon model and a sun-earth model, belonging to the field of track dynamics.
Background
The translation points are also called Lagrangian points, which are five special solutions in the circular restrictive three-body problem, discovered by mathematicians Euler and Lagrangian and respectively marked as L 1 To L 5 . According to the dynamic characteristics of the translational point, a plurality of available periodic orbits and quasi-periodic orbits exist around the translational point. The orbits have the characteristics of excellent geographic position and low energy consumption, and have important significance for exploring solar systems and farther outer space environments. The spacecraft is placed at the translation point so as to observe the earth, the sun, other celestial bodies and the depth of the universe by working near the designed nominal orbit for a long time. Studies have shown that although there are available periodic orbits and quasi-periodic orbits, these orbits are unstable, have various perturbation factors, and are prone to fly off the panning points due to the unstable divergence of the orbits, and the initial values of these orbits are very demanding and the required accuracy is not easily achieved in the project. Therefore, the correction and maintenance of the orbit of the spacecraft while performing a mission at a predetermined translational point is indispensable. The invention provides a Halo orbit maintaining method based on double-base invariant manifold, which is used for ensuring that a task orbit of a spacecraft at a translation point is kept stable.
Disclosure of Invention
The invention aims to provide a mission orbit keeping strategy under the influence of complex perturbation in order that a spacecraft can work near a designed nominal orbit for a long time to complete a given mission.
The purpose of the invention is realized by the following technical scheme:
a Halo track maintaining method based on double-base invariant manifold comprises the following steps:
step 1: obtaining a reduced kinetic equation and a nonlinear polynomial relation by adopting an invariant manifold method;
step 2: solving the kinetic equation obtained in the step 1 by adopting a Linstedt-Pincre (LP for short);
and step 3: correcting the deviation between the last integral state value and the target state value of the kinetic equation by adopting a differential iteration method;
and 4, step 4: based on the theoretical basis, taking the limited three-body problem of the earth-moon-shaped as an example, a Halo track surrounding a translation point is constructed;
and 5: and iteratively correcting the speed according to the used track keeping method through a polynomial constraint condition.
Compared with the prior art, the Halo track maintaining method based on the double-base invariant manifold has the following beneficial effects:
(1) the initial value is accurate, a kinetic equation is expanded at a translation point to obtain a nonlinear polynomial relation which is used as constraint, and the accurate initial value of the track can be obtained through LP perturbation solution and iterative correction.
(2) The method has good track keeping effect, adopts the polynomial relation as the constraint condition of track keeping, gives speed increment when the speed needs to be corrected, and ensures the space configuration of the spacecraft track.
(3) The method has strong applicability, and the orbit keeping method is suitable for Halo periodic orbits in earth-moon systems and sun-earth systems.
Drawings
FIG. 1 is a graph of amplitude versus frequency for the present invention;
FIG. 2 is a comparison graph of the integrated initial values of the present invention;
FIG. 3 is a four-time modified Halo track of the present invention;
FIG. 4 is a track maintenance schematic of the present invention;
FIG. 5 is a schematic illustration of the two year rail maintenance of the present invention;
FIG. 6 is a second year track maintenance schematic of the present invention; a) a three-dimensional view; b) an xy view; c) a yz view; d) xz view.
FIG. 7 is a schematic illustration of the Halo track maintenance results of the present invention; a) a three-dimensional view; b) xy view.
Detailed Description
The invention will be further described by means of specific examples in conjunction with the accompanying drawings, which are provided for illustration only and are not intended to limit the scope of the invention.
The invention provides a Halo track maintenance strategy based on double-base invariant manifold, which mainly comprises the following steps:
step 1: and obtaining a reduced kinetic equation and a nonlinear polynomial relation by adopting a non-deformation manifold method.
The method is characterized in that a non-variable manifold method is adopted to research periodic motion at a collinear translational point, motion in zeta and zeta directions are selected as main motion of a Halo track, and firstly, a dimensionless motion equation of the collinear translational point under a mass center convergence coordinate system is quoted:
wherein:
x, Y and Z are coordinate axes of a centroid convergence coordinate system;
omega is the mimicry function in the system and is expressed as
Where μ is the lunar mass, R 1 And R 2 Is the distance of the spacecraft to the earth and moon, respectively, and is denoted
Expanding the equation to a cubic nonlinear equation at the translation point
In the formula
The motion in two selected directions is represented by the lead-in coordinates, and the motion in the other direction is found by the nonlinear relationship and is finally also represented by the lead-in coordinates.
Lead-in principal coordinates of
Taking into account the constant non-linear progression of the planar and vertical periodic orbits as a polynomial, the position and velocity in the eta component is related to u by a polynomial 1 ，v 1 ，u 2 And v 2 And (4) correlating. I.e. if u is known 1 ，v 1 ，u 2 And v 2 The position and velocity in the η direction can be directly obtained from the following non-linear asymptotic relationship.
Is shown as
Determining each coefficient a i And b i An approximately non-linear polynomial relationship between the three directions can be derived. In other words, given displacement and velocity in the ξ and ζ directions, displacement and velocity in the η direction can be determined by such a non-linear polynomial relationship. Therefore, the main thing of this method is to find an explicit nonlinear relationship by determining the coefficients in the equation.
Derivation of Q, P with respect to time
Comparing equations to equations can yield
Substituting the above formula into equation, comparing the coefficients to obtain the coefficient of linear part
The linear portions of these three directions are therefore
As can be seen from the above linear relationship, a exists between the displacement in the eta direction and the speed in the xi direction 2 Multiple relation, speed in eta direction and displacement in xi direction exist b 1 And (4) multiple relation. In addition, the motion of the linear part is an ellipse, where the displacement in one direction is 0 when the velocity in the other direction reaches a maximum, and has no relation to the variable in the ζ direction. That is, the plane motion and the vertical direction are independent and have no mutual influence, the motion in the vertical direction is a simple harmonic motion, and the plane motion is an elliptic motion rotating around a translational point.
By bringing the formula into the formula, the linearized frequency of the motion equation can be obtained
The formula is given by a plus sign and linear frequency omega 2 co There is a pair of real feature roots that cannot generate a periodic orbit.
Coefficients of non-linear part
Wherein
From this, the coefficients of displacement and velocity in the η direction of the polynomial expression are obtained as
Expression of other coefficients
m 30 ＝AA+BB (28)
Step 2: and (3) solving the kinetic equation obtained in the step (1) by adopting a Linstedt-Pincre (LP for short).
The three-dimensional space motion equation can be reduced to a two-dimensional system through an equation, and the three-dimensional space motion equation of Halo can be described by motion in zeta and zeta directions. The kinetic equation in two directions can be obtained by substituting the equation into the equation.
After obtaining the reduced kinetic equation, solving the third-order analytic solution of the Halo track by an LP perturbation method
η＝c 31 sin(ωs)+c 32 sin(2ωs)+c 33 sin(3ωs) (34)
In the formula
So as to obtain a third-order approximate analytic solution of the Halo track near the translational point, wherein the nonlinear frequency of the system is
Ω＝ωδ＝ω(1+εδ 1 +ε 2 δ 2 ) (36)
In the formula
And step 3: and correcting the deviation between the final integral state value and the target state value of the kinetic equation by adopting a differential iteration method.
And taking the nonlinear relation of the equation as a polynomial constraint condition, and continuously iterating by adopting a differential correction method. To the initial guess value X 0 Performing numerical integration to obtain a terminal state value by adopting a fixed time t
Then, according to the nonlinear relation obtained in the previous step, the state value X of the terminal target can be calculated fc 0 。
Wherein
The deviation between the end-of-integration state value and the target state value is
And feeding the deviation back to an initial integral value through a state transition matrix, and recalculating the error by carrying out integral again, so as to continuously iterate until the error meets the precision requirement.
And 4, step 4: based on the above theoretical basis, a Halo orbit around a translational point is constructed, taking the limited three-body problem of the earth-moon circle as an example.
The quality parameter of the march system is μ ═ 0.012150568. Integral calculations were performed using the mathematical software MATLAB. The adopted digital integrators are classical eight-order runge-kutta and seven-order automatic step length control, and the tolerance is 1 multiplied by 10 -14 . We pass through the translation point L 1 Point sum L 2 Halo periodic orbits around a point verify the proposed method. Through the translational point L 1 Point sum L 2 Halo periodic orbits around a point verify the proposed method. Based on the equation-coefficient linear solution, the following table is the nonlinear relationship coefficient of the collinear Halo orbit of the third order polynomial.
For the present conservative system, the effect of the non-linear frequency of the periodic orbit of different amplitudes on the periodic orbit can be derived from FIG. 1, which shows the Earth's moon panning point L in FIG. 1 1 And L 2 The relationship between amplitude and nonlinear frequency, as can be seen from the figure, the nonlinear frequency is gradually decreasing as the track amplitude increases.
And respectively integrating initial values obtained by the linear expansion and the third-order nonlinear expansion under a dynamic model. The integration results are shown in fig. 2, where the solid black line is the first order linear solution direct integration and the solid blue line is the plot resulting from the initial integration provided by the first order non-linearity. As can be seen from the integral curve, the integral initial value obtained by the higher-order expansion is more accurate.
Through iterative correction, the translational point L 1 And L 2 The results of the point-value simulation are shown in fig. 3. Wherein, the red point and the blue point represent the initial value points of the integration after the track correction, and for the Halo periodic track near the translation point,and 4-5 times of iteration is carried out by adopting an iteration method of differential correction, so that a complete periodic orbit can be obtained.
Through the comparison, the initial value of the integral provided by the polynomial is accurate enough to ensure the convergence of the track in the iterative targeting process of the integral. Therefore, the numerical simulation verifies the effectiveness of the initial value provided by the invariant manifold method, and ensures the convergence of the circular restrictive three-body model in the differential correction process. And the invariant manifold method is proved to be a powerful way for analyzing the three-dimensional periodic orbit and clearly shows the nonlinear relation between the displacement and the speed in the zeta direction and the eta direction. Therefore, the Halo periodic track near the translation point can be designed by adopting a double-base invariant manifold method.
And 5: and iteratively correcting the speed according to the used track keeping method through a polynomial constraint condition.
The track-keeping scheme described herein performs velocity corrections based on the desired track-keeping method, primarily through polynomial constraints. The correction speed may be fixed time or on a fixed plane. As shown in fig. 4, the position o is an initial integration point, integration is performed under a restrictive three-body model, and the position p is reached after a half period, at this time, a speed amount to be corrected is obtained through a polynomial relationship, after the speed is changed, forward integration is continued to reach the position f, and then the speed is changed through polynomial constraint again, so that a reciprocating cycle is performed, and an effect of track maintenance is achieved.
When the previous orbit design is adopted, an integral initial value X0 of a complete cycle can be formed after iterative correction and is used as an integral initial value for orbit keeping, namely the ideal position of the spacecraft in orbit when the spacecraft is kept in a target orbit.
X o ＝[x o ,y o ,z o ,v xo ,v yo ,v zo ] (41)
The integral is carried out forward under a complete dynamic model, and the end state of the integral at the cut-off of half period is
X p ＝[x p ,y p ,z p ,v xp ,v yp ,v zp ] (42)
Let the state variable after velocity correction be
X p1 ＝[x p1 ,y p1 ,z p1 ,v xp1 ,v xp1 ,v zp1 ] (43)
For Halo periodic orbits, the motion in the Z direction is a simple harmonic motion, so one can choose not to change v zp Changing only v xp And v yp 。
The required speed increment is
Wherein
Integration is performed by a rail-keeping method, using a fixed time t, at a panning point L 1 Track maintenance is performed with time set to two years, the results are shown in fig. 5, and the track maintenance results for the second year are shown in fig. 6. The correction was performed about 1.739 days in two years, and as can be seen from the two-year Halo integration trace, the integration trace in the first year was similar to the orbital maintenance of the pseudo-Halo periodic trace, with an energy consumption of 0.29997 km/s. The orbit of the second year keeps presenting a Lissajous-like periodic trajectory, and the spatial structure of the nonlinear polynomial presents a Lissajous spatial integral trajectory as the number of corrections increases. The energy consumption in the second year was 0.04736km/s, which is much less than the energy consumption in the first year.
With respect to the particularity of the Halo track, the Halo track is symmetrical about the x-z plane through which v passes p1 0. So that the speed correction can be performed when it passes through the plane, and the control speed becomes v xp The other two directions are solved by a non-linear relation 0.
Wherein
The track hold results of this method are shown in fig. 7. And correcting the track speed by adopting a polynomial as a constraint condition. The spacecraft is guaranteed to keep doing quasi-periodic motion near the translational point, and the convergence on the track is guaranteed due to the space nonlinear relation given by the polynomial.
While the foregoing is directed to the preferred embodiment of the present invention, it is not intended that the invention be limited to the embodiment and the drawings disclosed herein. It is intended that all equivalents and modifications which do not depart from the spirit of the invention disclosed herein are deemed to be within the scope of the invention.
Claims (6)
1. A Halo track maintaining method based on double-base invariant manifold is characterized by comprising the following steps: the method comprises the following steps:
step 1: obtaining a reduced kinetic equation and a nonlinear polynomial relation by adopting an invariant manifold method;
step 2: solving the kinetic equation obtained in the step 1 by adopting a Linstedt-Pincre (LP for short);
and step 3: correcting the deviation between the last integral state value and the target state value of the kinetic equation by adopting a differential iteration method;
and 4, step 4: based on the theoretical basis, taking the limited three-body problem of the earth-moon-shaped as an example, a Halo track surrounding a translation point is constructed;
and 5: and iteratively correcting the speed according to the used track keeping method through a polynomial constraint condition.
2. The bistatic invariant manifold-based Halo track maintenance method according to claim 1, wherein: the method is characterized in that a non-variable manifold method is adopted to research periodic motion at a collinear translational point, motion in zeta and zeta directions are selected as main motion of a Halo track, and firstly, a dimensionless motion equation of the collinear translational point under a mass center convergence coordinate system is quoted:
wherein:
x, Y and Z are coordinate axes of a centroid convergence coordinate system;
omega is the mimicry function in the system and is expressed as
Where μ is the lunar mass, R 1 And R 2 Is the distance of the spacecraft to the earth and moon, respectively, and is denoted
Expanding the equation at translation point to a cubic nonlinear equation
In the formula
Selecting the motion in two directions to be represented by the lead-in coordinates, finding the motion in the other direction through a nonlinear relation, and finally representing the motion in the other direction by the lead-in coordinates;
lead-in principal coordinates of
In terms of polynomialsConstant non-linear asymptotic relationship between the surface and the vertical periodic orbit, the position and velocity in the eta component being related to u by a polynomial relationship 1 ，v 1 ，u 2 And v 2 Correlation; i.e. if u is known 1 ，v 1 ，u 2 And v 2 The position and speed in the η direction can be directly obtained from the following nonlinear asymptotic relationship;
is shown as
Calculating each coefficient a i And b i Then approximate nonlinear polynomial relations among the three directions can be obtained; in other words, given displacement and velocity in the ξ and ζ directions, displacement and velocity in the η direction can be determined by such a non-linear polynomial relationship; therefore, the method mainly finds an explicit nonlinear relation by determining coefficients in an equation;
derivation of Q, P with respect to time
Comparing equations to equations can yield
Substituting the above formula into equation, comparing coefficients to obtain linear part coefficient of
The linear portions of these three directions are therefore
The displacement in the eta direction and the speed in the xi direction have a 2 Multiple relation, speed in eta direction and displacement in xi direction exist b 1 A multiple relationship; in addition, the motion of the linear part is an ellipse, when the speed of one direction reaches the maximum value, the displacement of the other direction is 0, and the displacement has no relation with the variable of the zeta direction; that is, the plane motion and the vertical direction are independent and have no mutual influence, the motion in the vertical direction is a simple harmonic motion, and the plane motion is an elliptic motion rotating around a translational point;
substituting the formula into the formula to obtain the linearized frequency of the motion equation
The formula is given by a plus sign and linear frequency omega 2 co A pair of real characteristic roots are provided, and a periodic orbit cannot be generated;
coefficients of non-linear part
Wherein
From this, the coefficients of displacement and velocity in the η direction of the polynomial expression are obtained, the expression being
Expression of other coefficients
m 30 ＝AA+BB (28)
3. The bistatic invariant manifold-based Halo track maintenance method according to claim 1, wherein: the implementation process of the step 2 is as follows, a three-dimensional space motion equation is reduced to a two-dimensional system through an equation, and the three-dimensional space motion equation of Halo is described by motion in zeta and zeta directions; substituting the equation into an equation to obtain a kinetic equation in two directions;
after obtaining the reduced kinetic equation, solving the third-order analytic solution of the Halo track by an LP perturbation method
η＝c 31 sin(ωs)+c 32 sin(2ωs)+c 33 sin(3ωs) (34)
In the formula
So as to obtain a third-order approximate analytic solution of the Halo track near the translational point, wherein the nonlinear frequency of the system is
Ω＝ωδ＝ω(1+εδ 1 +ε 2 δ 2 ) (36)
In the formula
4. The bistatic invariant manifold-based Halo track maintenance method according to claim 1, wherein: the implementation process of the step 3 is that a differential iteration method is adopted to correct the deviation between the last integral state value and the target state value of the kinetic equation;
taking the nonlinear relation of the equation as a polynomial constraint condition, and continuously iterating by adopting a differential correction method; to the initial guess value X 0 Performing numerical integration to obtain a terminal state value by adopting a fixed time t
Then, according to the nonlinear relation obtained in the previous step, the state value X of the terminal target can be calculated fc 0 ；
Wherein
The deviation between the end-of-integration state value and the target state value is
And feeding the deviation back to an integral initial value through a state transition matrix, and recalculating the error by integrating again, so that iteration is continuously performed until the error meets the precision requirement.
5. The bistatic invariant manifold-based Halo track maintenance method according to claim 1, wherein: the implementation of step 4 is as follows, based on the above theoretical basis, constructing Halo orbits around the translational point with the lunar-circular restriction triple problem.
6. The bistatic invariant manifold-based Halo track maintenance method according to claim 1, wherein: the implementation process of the step 5 is as follows, speed correction is carried out according to the required track keeping method; the speed is corrected by adopting fixed time, or the correction is carried out on a fixed plane; the position o is an integration initial value point, integration is carried out under a restrictive three-body model, the position p is reached after a half period, the speed amount needing to be corrected is obtained through a polynomial relation, after the speed is changed, forward integration is continued to reach the position f, then the speed is changed through polynomial constraint again, and the effect of keeping the track is achieved through reciprocating circulation;
when the previous orbit design is adopted, an integral initial value X0 of a complete period is formed after iterative correction and is used as an integral initial value for orbit keeping, namely the ideal position of the spacecraft in the orbit when the target orbit is kept;
X o ＝[x o ,y o ,z o ,v xo ,v yo ,v zo ] (41)
the integral is carried out forward under a complete dynamic model, and the end state of the integral at the cut-off of half period is
X p ＝[x p ,y p ,z p ,v xp ,v yp ,v zp ] (42)
Let the state variable after velocity correction be
For Halo periodic orbits, the motion in the Z direction is simple harmonic motion, so one can choose not to change v zp Changing only v xp And v yp ；
The required speed increment is
Wherein
Integration by orbit preservation method, using fixed time t, at translational point L 1 Keeping the track, and setting the time to be two years; the correction is carried out once in 1.739 days within two years, and the integral track of the first year is similar to the track maintenance of a simulated Halo periodic track and the energy consumption is 0.29997km/s from the Halo integral track of the two years; the orbit in the second year keeps presenting a Lissajous-like periodic trajectory, and with the increase of the correction times, the spatial structure of the nonlinear polynomial presents a Lissajous spatial integral trajectory; the energy consumption in the second year is 0.04736km/s, which is reduced greatly compared with the energy consumption in the first year;
with respect to the particularity of the Halo track, the Halo track is symmetrical about the x-z plane through which v passes p1 0; so that the speed correction can be performed when it passes through the plane, and the control speed becomes v xp When the direction is equal to 0, solving the other two directions through a nonlinear relation;
wherein
Correcting the track speed by using a polynomial as a constraint condition; the spacecraft is guaranteed to do quasi-periodic motion near the translational point, and the convergence on the track is guaranteed through the space nonlinear relation given by the polynomial.
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| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN106682274A (en) * | 2016-12-05 | 2017-05-17 | 北京理工大学 | Halo orbit on-orbit keeping method considering amplitude constraint |
| CN110015445A (en) * | 2019-02-15 | 2019-07-16 | 北京空间飞行器总体设计部 | A kind of ground moon L2 point Halo track maintains method |
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| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN106682274A (en) * | 2016-12-05 | 2017-05-17 | 北京理工大学 | Halo orbit on-orbit keeping method considering amplitude constraint |
| CN110015445A (en) * | 2019-02-15 | 2019-07-16 | 北京空间飞行器总体设计部 | A kind of ground moon L2 point Halo track maintains method |
Non-Patent Citations (1)
| Title |
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| 郭建宇: "基于双基不变流形法的平动点轨道设计及保持策略研究", 《中国优秀硕士学位论文全文数据库工程科技Ⅱ辑》 * |
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