CN109240340B - Lorentz force multi-satellite formation configuration method based on quasi-periodic orbit - Google Patents

Lorentz force multi-satellite formation configuration method based on quasi-periodic orbit Download PDF

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CN109240340B
CN109240340B CN201811422893.3A CN201811422893A CN109240340B CN 109240340 B CN109240340 B CN 109240340B CN 201811422893 A CN201811422893 A CN 201811422893A CN 109240340 B CN109240340 B CN 109240340B
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periodic
star
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periodic orbit
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CN109240340A (en
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程潏
乔栋
李翔宇
贾飞达
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Beijing Institute of Technology BIT
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
    • G05D1/00—Control of position, course or altitude of land, water, air, or space vehicles, e.g. automatic pilot
    • G05D1/10—Simultaneous control of position or course in three dimensions
    • G05D1/101—Simultaneous control of position or course in three dimensions specially adapted for aircraft
    • G05D1/104—Simultaneous control of position or course in three dimensions specially adapted for aircraft involving a plurality of aircrafts, e.g. formation flying

Abstract

The invention discloses a Lorentz force multi-satellite formation configuration method based on a quasi-periodic orbit, and belongs to the field of aerospace. The implementation method comprises the steps of establishing a motion equation of the charged slave satellite under the artificial magnetic field of the master satellite, carrying out dimensionless simplification processing on the motion equation, determining a balance point of the dimensionless motion equation, obtaining a periodic orbit near the balance point by utilizing a differential correction method and a numerical extension method, calculating a quasi-periodic orbit on a manifold at the center of the periodic orbit, and deploying a plurality of charged satellites on an invariant curve under the stroboscopic mapping of the quasi-periodic orbit to realize the Lorentz force multi-satellite formation configuration without working medium consumption. The method does not need to consume chemical fuel and chemical pollution, and has application prospect in close-range space formation, on-orbit operation and long-time space observation tasks. The invention is suitable for the close-range formation of the electrified satellite around the spacecraft with the spinning magnetic field on the earth high orbit.

Description

Lorentz force multi-satellite formation configuration method based on quasi-periodic orbit
Technical Field
The invention relates to the field of aerospace, in particular to a Lorentz force multi-satellite formation configuration method based on a quasi-periodic orbit, which is suitable for a charged satellite to perform close-range formation around a spacecraft with a spinning magnetic field on an earth high orbit.
Background
When the charged spacecraft moves within the magnetic field, it is subject to lorentz forces, which are perpendicular to both the instantaneous velocity of the spacecraft and the direction of the magnetic field. An artificial magnetic field is deployed on a satellite, namely a main satellite, which runs on the earth high orbit, so that nearby electrified satellites, namely electrified auxiliary satellites, can be subjected to Lorentz force generated by the magnetic field on the main satellite, and the electrified auxiliary satellites and the main satellite can form a Lorentz force formation without working medium consumption under the combined action of the earth attraction and the Lorentz force. As the Lorentz force does not consume the propellant, the Lorentz force is used as a novel power mode, and compared with the traditional chemical propulsion mode, the Lorentz force shows more advantages and can be used as a promising technology in future space tasks.
The existing developed prior art on the formation of lorentz force [1] (see soar. lorentz force propelled charged satellite: orbit motion and propulsion concept [ C ]. second academic conference of the space exploration professional committee of china space science, Dalian, 2009) proposes the concept of formation control using inter-satellite lorentz force, with charged slave satellites moving near the master with spinning artificial magnetic fields, the magnetic fields of the master being generated by high temperature superconductors and the master moving circumferentially around the earth, but no dynamic analysis and formation configuration design is performed.
Prior art [2] (C.Peng, Y.Gao. Formation-flying plate periodic bits inter-presentation of interartelite Lorentz force [ J ]. IEEE Transactions on Aerospace and Electronic Systems,53(3):1412 and 1430,2017.) shows that when the direction of the artificial magnetic field is perpendicular to the orbital plane of the primary star, the balance point and partial periodic orbit of the charged secondary star relative to the movement of the primary star are given. However, there are more periodic orbits in the periodic orbital family, and also there are numerous 2D invariant toroids, i.e., quasi-periodic orbits, in the central manifold of the periodic orbits, which are the preferred choices for the formation flight reference orbit. Because the extension of the periodic orbit and the calculation difficulty of the quasi-periodic orbit are larger, the complete extension of the periodic orbit group is not completed in the prior art [2], any quasi-periodic orbit near the periodic orbit is not obtained, and the technology for realizing formation configuration by using the quasi-periodic orbit is not involved.
In summary, in the technology of implementing formation configuration by using lorentz force, the technology of calculating the quasi-periodic orbit of the charged slave star relative to the master star and the technology of implementing formation configuration based on the quasi-periodic orbit are blank.
Disclosure of Invention
The invention discloses a Lorentz force multi-star formation configuration method based on a quasi-periodic orbit, which aims to solve the technical problems that: the method is characterized in that the movement of a plurality of electrified slave stars on a quasi-periodic orbit around a master star is realized based on the Lorentz force among satellites, so that a multi-star Lorentz force natural formation configuration without working medium consumption is realized by utilizing an invariant curve of the quasi-periodic orbit under stroboscopic mapping, the collision risk among the satellites in the natural formation configuration is avoided, the formation configuration is not dispersed, the electrified slave stars are always kept on the same invariant curve, and no additional fuel is needed for maintaining the electrified slave stars. The invention has the advantages of no fuel consumption and no chemical pollution, is suitable for tasks with photosensitive equipment loads, has great application prospect in close-range space formation, on-orbit operation and long-time space observation tasks,
the purpose of the invention is realized by the following technical scheme.
The invention discloses a Lorentz force multi-satellite formation configuration method based on a quasi-periodic orbit, which is characterized in that a motion equation of a charged slave satellite under an artificial magnetic field of a master satellite is established, dimensionless simplification processing is carried out on the motion equation, a balance point of a dimensionless motion equation is determined, a periodic orbit near the balance point is obtained by utilizing a differential correction method and a numerical extension method, a quasi-periodic orbit on a central manifold of the periodic orbit is calculated, and the Lorentz force multi-satellite formation configuration without working medium consumption is realized by deploying a plurality of charged satellites on an invariant curve under the quasi-periodic orbit stroboscopic mapping. The method does not need to consume chemical fuel and chemical pollution, and has application prospect in close-range space formation, on-orbit operation and long-time space observation tasks.
The invention discloses a Lorentz force multi-star formation configuration method based on a quasi-periodic orbit, which comprises the following steps of:
the method comprises the following steps: and establishing a motion equation of the charged slave star under the artificial magnetic field of the master star, and carrying out dimensionless simplification processing on the motion equation to obtain the dimensionless simplified motion equation.
The main star runs on the earth's high orbit, and the lorentz force required for the movement of the charged slave star is generated by the artificial magnetic field spinning on the main star. The equation of motion of the charged slave star in the local horizontal local vertical coordinate system with the origin at the master star is expressed as,
Figure BDA0001880928480000031
wherein r ═ x, y, z and
Figure BDA0001880928480000032
respectively charged from the star position toQuantity and velocity vectors, n being the average angular velocity of the primary star's motion around the earth. f. ofL＝(fx,fy,fz)TThe lorentz force experienced from the star for charging, expressed as,
Figure BDA0001880928480000033
wherein the content of the first and second substances,
Figure BDA0001880928480000034
charge to mass ratio of charged slave star, vrFor the velocity of the charged slave star relative to the master star, ωcThe angular velocity of the spin magnetic field on the main satellite, B is the strength of the artificial magnetic field.
Equation (1) is dimensionless and simplified to,
Figure BDA0001880928480000035
step two: equilibrium points of the equations of motion for the dimensionless reduction process are determined.
Let X ═ Y ═ Z ═ 0, get the position equation (4) of the equilibrium point,
Figure BDA0001880928480000036
the balance point is a balance point at which the charged slave star is static relative to the master star, namely a special solution that the position of the charged slave star relative to the master star is unchanged and the speed is zero. At the equilibrium point, the attractive earth force on the charged slave satellites cancels the lorentz force.
Step three: a periodic trajectory near an equilibrium point of the equation of motion for the dimension reduction process is determined.
The linearized equation around the equilibrium point may be expressed in the form:
Figure BDA0001880928480000041
wherein v isi1,2, 6 is a feature vector，ciI 1,2, 6 is an arbitrary constant. The characteristic root corresponding to the central manifold is
Figure BDA0001880928480000042
Let c2＝c5＝c3＝c＝60 is obtained as the characteristic root lambda of the central manifold1,4The corresponding periodic solution is:
Figure BDA0001880928480000043
wherein the content of the first and second substances,
Figure BDA0001880928480000044
is v is1Conjugation of (1).
Using the linear approximation represented by equation (6), the initial value of the periodic orbit is obtained as:
Figure BDA0001880928480000045
wherein, XeqIs the position coordinate of the equilibrium point, ∈ is the small perturbation parameter in the central manifold direction, | x (t) | is the euclidean norm of x (t).
Iteratively correcting the initial value of the periodic orbit in the formula (7) by a differential correction method to obtain the accurate value X of the continuous periodic orbitf(t) then obtaining a complete periodic orbital group by numerical continuation.
Step four: and determining a quasi-periodic orbit on the manifold at the center of the periodic orbit based on the periodic orbit near the balance point obtained in the third step.
Selecting one elliptical periodic orbit in the periodic orbit family in the step three, and solving a quasi-periodic orbit nearby the elliptical periodic orbit, wherein the quasi-periodic orbit is a 2D invariable ring surface, the 2D invariable ring surface is represented as a double-parameter function psi (ξ) by parameterization, and the solution of the quasi-periodic orbit is simplified into an invariable curve represented by a single-parameter function through fixing any parameter in a stroboscopic mapping (ξ)
Figure BDA00018809284800000416
Representing invariant curves by truncated Fourier expansion
Figure BDA00018809284800000417
Figure BDA0001880928480000046
Wherein, M is the number of truncation terms,
Figure BDA0001880928480000047
and is
Figure BDA0001880928480000048
Are the fourier coefficients that need to be solved.
Invariant curve
Figure BDA0001880928480000049
After a period of deduction, the points still return to the same invariant curve
Figure BDA00018809284800000410
The phase angle is changed by a rotation number rho, i.e. a constant curve
Figure BDA00018809284800000411
The following invariant equations are satisfied,
Figure BDA00018809284800000412
where ρ is the number of revolutions and the return time T is the period of the periodic orbit.
Invariant curve to be obtained by equation (10)
Figure BDA00018809284800000413
Globally, as an invariant torus ψ (ξ), i.e. a quasi-periodic track on a periodic track center manifold,
Figure BDA0001880928480000051
the fourth specific implementation method comprises the following steps:
step 4.1: determining invariant curves for quasi-periodic orbits
Figure BDA0001880928480000052
Is started.
Using linear flow of periodic orbits as invariant curve of quasi-periodic orbits
Figure BDA0001880928480000053
Is expressed as
Figure BDA0001880928480000054
Wherein, χ0Is any point on the periodic orbit,
Figure BDA0001880928480000055
is the eigenvector corresponding to the unit eigenvalue of the single-valued matrix,
Figure BDA0001880928480000056
is a constant curve
Figure BDA0001880928480000057
Distance to periodic orbit, ξ ∈ [0,2 π ∈]. The initial values of the fourier coefficients are then: c0＝χ0,C1＝∈vr,S1＝-∈viWhen k > 2, Ck＝Sk＝0。
Step 4.2: solving an invariant curve of a quasi-periodic orbit
Figure BDA00018809284800000511
The fourier coefficients of (a).
In the interval [0,2 π]Inner pair invariant curve
Figure BDA00018809284800000512
Discretizing to obtain all discrete points
Figure BDA0001880928480000058
The above satisfies the invariant equation (9), the following equation set is obtained,
Figure BDA0001880928480000059
wherein the constant curve
Figure BDA00018809284800000510
Expressed as a Fourier series expansion shown in equation (8), determining Fourier coefficient C by iteration of the above equation set (12) using Newton's method until the error is less than a given precisionkAnd Sk。
In the iteration process, the frequency number N of the truncated Fourier expansion is adjusted according to the required precisionfThe adjustment rule is as follows: after guarantee NfThe maximum norm of/4 coefficients is an order of magnitude smaller than the required accuracy. If the above regulation rule is not satisfied, the number of the frequencies is increased to 2NfAnd k is equal to Nf+1,…,2NfThe corresponding item is set to Ck＝Sk＝0。
Step 4.3: invariant curve based on quasi-periodic orbit under stroboscopic mapping
Figure BDA00018809284800000515
Curve transformation is carried out by equation (10)
Figure BDA00018809284800000516
The global invariant torus ψ (ξ) is a pseudo-periodic track on a periodic track center manifold.
Step five: and based on the quasi-periodic orbit on the periodic orbit central manifold solved in the step four, realizing the Lorentz force multi-star formation configuration without working medium consumption by utilizing an invariant curve of the quasi-periodic orbit under the stroboscopic mapping.
And based on the quasi-periodic orbit on the central manifold solved in the fourth step, selecting ξ ═ i pi/N, i ═ 1,2, and N equant points on an invariant curve under the quasi-periodic orbit stroboscopic mapping, and deploying one electrified slave star respectively, so that the multi-star Lorentz force natural multi-star formation configuration without working medium consumption is realized.
Further comprises the following steps: and utilizing the step five to obtain a multi-star Lorentz force formation configuration to carry out multi-star formation flying, wherein the electrified slave stars in the formation configuration are positioned on the natural quasi-periodic orbit of the motion equation (3) without additional fuel to maintain, all the electrified slave stars are always kept on the same invariant curve, and the formation configuration is not dispersed.
The method also comprises the seventh step: and D, analyzing the relative distance between the charged slave stars in the multi-star formation configuration obtained in the step five to determine the evolution property of the formation configuration, wherein the charged slave stars in the formation configuration have no collision risk.
Has the advantages that:
1. the invention discloses a Lorentz force multi-satellite formation configuration method based on a quasi-periodic orbit, which is characterized in that a plurality of charged slave satellites move on the quasi-periodic orbit around a master satellite based on the Lorentz force among satellites, and then the Lorentz force multi-satellite formation configuration without working medium consumption is realized by utilizing an invariant curve of the quasi-periodic orbit on a central manifold under stroboscopic mapping, so that the formation configuration is maintained without additional fuel, the charged slave satellites are always on the same invariant curve, and the formation configuration is not dispersed.
2. The invention discloses a quasi-periodic orbit-based Lorentz force multi-satellite formation configuration method, which realizes formation configuration by utilizing inter-satellite Lorentz force, namely, maintains the relative motion between a main satellite and an electrified auxiliary satellite through the Lorentz force, so that the traditional chemical fuel consumption is not needed, the chemical pollution is not generated, and the method is suitable for satellites with photosensitive equipment loads.
3. The invention discloses a Lorentz force multi-satellite formation configuration method based on a quasi-periodic orbit, and the relative distance between electrified slave satellites in the multi-satellite formation configuration obtained in the step five is analyzed to determine that no collision risk exists between electrified satellites.
4. The Lorentz force multi-star formation configuration method based on the quasi-periodic orbit has the beneficial effects, so that the Lorentz force multi-star formation configuration method has a great application prospect in close-range space formation, on-orbit operation and long-time space observation tasks.
Drawings
FIG. 1 is a flow chart of a Lorentz force multi-star formation configuration method of a quasi-periodic orbit of the present invention;
FIG. 2 is the balance point 3 of the present inventionN
Figure BDA0001880928480000061
A nearby periodic family of orbitals;
fig. 3 is a quasi-periodic orbit with a rotation number ρ 0.6625849644 and a return time T2.1505284937 and its invariant curve, wherein the complex pipeline represented by a thin line is the quasi-periodic orbit, and the black thick line is the invariant curve;
FIG. 4 is a Lorentz force three-star formation configuration of the present invention on an invariant curve under a pseudo-periodic orbital strobe mapping, wherein a solid triangle represents three charged slaves, and a black bold line is a connecting line between two adjacent charged slaves;
FIG. 5 is the evolution of the Lorentz force Samsung configuration of the invention over a return time T, wherein: FIGS. 5a) to h) are schematic representations of the formation configuration at times T/8, T/4, …, T, respectively;
FIG. 6 is a rule of the evolution of the relative distance between the charged satellites in the Lorentz force Samsung configuration of the invention.
Detailed Description
The invention is further described with reference to the following drawings and detailed description.
Example 1:
the method for designing the lorentz force multi-star formation configuration based on the quasi-periodic orbit disclosed by the embodiment comprises the following specific steps:
the method comprises the following steps: and establishing a motion equation of the charged slave star under the artificial magnetic field of the master star. The main star runs on the earth high orbit, the Lorentz force required by the movement of the charged auxiliary star is generated by the spinning artificial magnetic field on the main star, the motion equation of the charged auxiliary star in the local horizontal local vertical coordinate system with the main star as the origin is expressed as,
Figure BDA0001880928480000071
wherein the content of the first and second substances,
Figure BDA0001880928480000072
charge to mass ratio of charged slave star, vrFor the velocity of the charged slave star relative to the master star, ωcThe angular velocity of the spin magnetic field on the main satellite, B is the strength of the artificial magnetic field.
Equation (1) is dimensionless and simplified to,
Figure BDA0001880928480000073
step two: equilibrium points of the equations of motion for the dimensionless reduction process are determined.
Let X ═ Y ═ Z ═ 0, get the position equation (4) of the equilibrium point,
Figure BDA0001880928480000081
the balance point is a balance point at which the charged slave star is static relative to the master star, namely a special solution that the position of the charged slave star relative to the master star is unchanged and the speed is zero. At the equilibrium point, the attractive earth force on the charged slave satellites cancels the lorentz force. The positions of the balance points are:
Figure BDA0001880928480000082
step three: determining a periodic trajectory around a balance point of a dimension-reduced equation of motion, the invention uses the balance point 3N
Figure BDA0001880928480000083
The description is given for the sake of example.
The linearized equation around the equilibrium point may be expressed in the form:
Figure BDA0001880928480000084
wherein v isi1,2, 6 is a feature vector, ciI 1,2, 6 is an arbitrary constant. The characteristic root corresponding to the central manifold is
Figure BDA0001880928480000085
Let c2＝c5＝c3＝c 60 can obtain the characteristic root lambda of the central manifold1,4The corresponding periodic solution is:
Figure BDA0001880928480000086
wherein the content of the first and second substances,
Figure BDA0001880928480000087
is v is1Conjugation of (1).
Using the linear approximation represented by equation (6), the initial value of the periodic orbit is obtained as:
Figure BDA0001880928480000088
wherein, XeqIs the position coordinate of the equilibrium point, ∈ is the small perturbation parameter in the central manifold direction, | x (t) | is the euclidean norm of x (t).
Iteratively correcting the initial value of the periodic orbit in the formula (7) by a differential correction method to obtain the accurate value X of the continuous periodic orbitf(t) then using numerical prolongation to obtain the complete periodic orbital family, as shown in FIG. 2.
Step four: and determining a quasi-periodic orbit on the manifold at the center of the periodic orbit based on the periodic orbit near the balance point obtained in the third step.
Selecting one elliptical periodic orbit in the periodic orbit family in the step three to solve the quasi-periodic orbit nearby, wherein the quasi-periodic orbit is a 2D invariant ring surface, the 2D invariant ring surface is parameterized as a biparametric function psi (ξ),the solution of the quasi-periodic orbit is simplified into an invariant curve represented by a single parameter function through any parameter fixed (ξ) by stroboscopic mapping
Figure BDA0001880928480000091
Representing invariant curves by truncated Fourier expansion
Figure BDA0001880928480000092
Figure BDA0001880928480000093
Wherein, M is the number of truncation terms,
Figure BDA0001880928480000094
and is
Figure BDA0001880928480000095
Are the fourier coefficients that need to be solved.
Invariant curve
Figure BDA0001880928480000096
After a period of deduction, the points still return to the same invariant curve
Figure BDA0001880928480000097
The phase angle is changed by a rotation number rho, i.e. a constant curve
Figure BDA0001880928480000098
The following invariant equations are satisfied,
Figure BDA0001880928480000099
where ρ is the number of revolutions and the return time T is the period of the periodic orbit.
Invariant curve to be obtained by equation (10)
Figure BDA00018809284800000910
Globally, as an invariant torus ψ (ξ), i.e. a quasi-periodic track around a periodic track,
Figure BDA00018809284800000911
the fourth step is realized by the following steps:
step 4.1: determining invariant curves for quasi-periodic orbits
Figure BDA00018809284800000912
Is started.
Using linear flow of periodic orbits as invariant curve of quasi-periodic orbits
Figure BDA00018809284800000913
Is expressed as
Figure BDA00018809284800000914
Wherein, χ0V is any point on the periodic orbitr±iviIs the eigenvector corresponding to the unit eigenvalue of the single-valued matrix,
Figure BDA00018809284800000917
is a constant curve
Figure BDA00018809284800000915
Distance to periodic orbit, ξ ∈ [0,2 π ∈]. The initial values of the fourier coefficients are then:
Figure BDA00018809284800000916
when k > 2, Ck＝Sk＝0。
Step 4.2: solving an invariant curve of a quasi-periodic orbit
Figure BDA00018809284800000918
The fourier coefficients of (a).
In the interval [0,2 π]Inner pair invariant curve
Figure BDA0001880928480000101
Discretizing to obtain all discrete points
Figure BDA0001880928480000102
The above satisfy the invariant equation (9), the equation set is obtained,
Figure BDA0001880928480000103
wherein the constant curve
Figure BDA0001880928480000104
Expressed as Fourier series expansion shown in equation (8), the above equation set (12) is iterated by Newton method until the error is less than the given precision, and Fourier coefficient C is determinedk,Sk. In the iteration process, the frequency number N of the truncated Fourier expansion (19) is adjusted according to the required precisionfThe adjustment rule is as follows: after guarantee NfThe maximum norm of/4 coefficients is an order of magnitude smaller than the required accuracy. If the above regulation rule is not satisfied, the number of the frequencies is increased to 2NfAnd k is equal to Nf+1,…,2NfThe corresponding item is set to Ck＝S k0. The invention takes an initial value Nf16 and NfThe upper limit is 128.
Step 4.3: invariant curve based on quasi-periodic orbit under stroboscopic mapping
Figure BDA0001880928480000105
Curve invariant is plotted by equation (10)
Figure BDA0001880928480000106
Globalized as an invariant torus
Figure BDA0001880928480000107
I.e. a quasi-periodic track on the central manifold of the periodic track. Fig. 3 shows a quasi-periodic orbit with a rotation number p of 0.6625849644 and a return time T of 2.1505284937 and under strobe mappingIs constant.
Step five: and based on the quasi-periodic orbit on the periodic orbit central manifold solved in the step four, realizing the multi-star Lorentz force natural formation configuration without working medium consumption by utilizing the invariant curve of the quasi-periodic orbit under the stroboscopic mapping.
Based on the quasi-periodic orbit on the central manifold solved in the fourth step, 3 equally divided points of ξ ═ i pi/4, i ═ 1,2 and 3 are respectively deployed with one charged slave star on the invariant curve under the quasi-periodic orbit stroboscopic mapping, so that the multi-star lorentz force three-star formation configuration without working medium consumption is realized, and the result is shown in fig. 4.
Further comprises the following steps: and utilizing the step five to obtain a multi-star Lorentz force formation configuration to carry out multi-star formation flying, wherein the electrified slave stars in the formation configuration are positioned on a natural quasi-periodic orbit of the motion equation (3) without additional fuel to maintain, all the electrified slave stars are always positioned on the same invariant curve, the formation configuration is not dispersed, and the evolution law is shown in figure 5.
The method also comprises the seventh step: by analyzing the relative distance between the charged slaves in the multi-satellite formation configuration obtained in the step five, the charged slaves in the formation configuration have no collision risk with each other as shown in fig. 6.
The above detailed description is intended to illustrate the objects, aspects and advantages of the present invention, and it should be understood that the above detailed description is only exemplary of the present invention and is not intended to limit the scope of the present invention, and any modifications, equivalents, improvements and the like made within the spirit and principle of the present invention should be included in the scope of the present invention.

Claims (4)

1. A Lorentz force multi-star formation configuration method based on quasi-periodic orbits is characterized by comprising the following steps: comprises the following steps of (a) carrying out,
the method comprises the following steps: establishing a motion equation of the charged slave star under the artificial magnetic field of the master star, and carrying out dimensionless simplification processing on the motion equation to obtain a dimensionless simplified motion equation;
step two: determining a balance point of a motion equation of dimensionless simplification processing;
step three: determining a periodic orbit around a balance point of a motion equation of dimension simplification processing;
step four: determining a quasi-periodic orbit on the manifold at the center of the periodic orbit based on the periodic orbit near the balance point obtained in the third step;
step five: based on the quasi-periodic orbit on the periodic orbit central manifold solved in the step four, realizing Lorentz force multi-star formation configuration without working medium consumption by utilizing an invariant curve of the quasi-periodic orbit under stroboscopic mapping;
step six, the multi-star lorentz force formation configuration obtained in the step five is utilized to carry out multi-star formation flying, the electrified slave stars in the formation configuration are located on a natural quasi-periodic orbit and do not need additional fuel to maintain, all the electrified slave stars are always kept on the same invariant curve, and the formation configuration is not dispersed;
step seven, analyzing the relative distance between the charged slave stars in the multi-star formation configuration obtained in the step five to determine the evolution property of the formation configuration, wherein the charged slave stars in the formation configuration have no collision risk;
wherein, the specific implementation method of the step one is,
the main star runs on the earth high orbit, and the Lorentz force required by the movement of the charged auxiliary star is generated by the artificial magnetic field spinning on the main star; the equation of motion of the charged slave star in the local horizontal local vertical coordinate system with the origin at the master star is expressed as,
Figure FDA0002359230920000011
wherein r ═ x, y, z and
Figure FDA0002359230920000012
respectively charging the position vector and the velocity vector of the slave star, wherein n is the average angular velocity of the master star moving around the earth; f. ofL＝(fx,fy,fz)TThe lorentz force experienced from the star for charging, expressed as,
Figure FDA0002359230920000013
wherein the content of the first and second substances,
Figure FDA0002359230920000014
charge to mass ratio of charged slave star, vrFor the velocity of the charged slave star relative to the master star, ωcThe angular velocity of the spin magnetic field on the main satellite, B is the intensity of the artificial magnetic field;
equation (1) is dimensionless and simplified to,
Figure FDA0002359230920000021
the concrete implementation method of the second step is that,
let X ═ Y ═ Z ═ 0, get the position equation (4) of the equilibrium point,
Figure FDA0002359230920000022
the balance point is a balance point at which the charged slave star is static relative to the master star, namely a special solution that the position of the charged slave star is unchanged relative to the master star and the speed is zero; at the balance point, the earth attraction borne by the charged slave star offsets the Lorentz force;
the third step is realized by the concrete method that,
the linearized equation around the equilibrium point may be expressed in the form:
Figure FDA0002359230920000023
wherein v isi1,2, 6 is a feature vector, ci1,2, 6 is an arbitrary constant; the characteristic root corresponding to the central manifold is
Figure FDA0002359230920000024
Let c2＝c5＝c3＝c60 is obtained as the characteristic root lambda of the central manifold1,4The corresponding periodic solution is:
Figure FDA0002359230920000025
wherein the content of the first and second substances,
Figure FDA0002359230920000026
is v is1Conjugation of (1);
using the linear approximation represented by equation (6), the initial value of the periodic orbit is obtained as:
Figure FDA0002359230920000027
wherein, XeqIs the position coordinate of the equilibrium point, ∈ is the small perturbation parameter in the central manifold direction, | x (t) | is the euclidean norm of x (t);
iteratively correcting the initial value of the periodic orbit in the formula (7) by a differential correction method to obtain the accurate value X of the continuous periodic orbitf(t) then obtaining the complete periodic orbital family by numerical prolongation.
2. The lorentz force multi-star formation configuration method based on quasi-periodic orbits as claimed in claim 1, characterized in that: the concrete implementation method of the step four is that,
selecting one elliptic periodic orbit in the periodic orbit group in the step III, solving a quasi-periodic orbit nearby the elliptic periodic orbit, wherein the quasi-periodic orbit is a 2D invariable ring surface, the 2D invariable ring surface is represented as a double-parameter function psi (ξ) by parameterization, and the solving of the quasi-periodic orbit is simplified into an invariable curve represented by a single-parameter function through any parameter in stroboscopic mapping fixation (ξ)
Figure FDA0002359230920000031
Representing invariant curves by truncated Fourier expansion
Figure FDA0002359230920000032
Figure FDA0002359230920000033
Wherein, M is the number of truncation terms,
Figure FDA0002359230920000034
and is
Figure FDA0002359230920000035
Fourier coefficients to be solved;
invariant curve
Figure FDA0002359230920000036
After a period of deduction, the points still return to the same invariant curve
Figure FDA0002359230920000037
The phase angle is changed by a rotation number rho, i.e. a constant curve
Figure FDA0002359230920000038
The following invariant equations are satisfied,
Figure FDA0002359230920000039
wherein rho is the rotation number, and the return time T is the period of the periodic orbit;
invariant curve to be obtained by equation (10)
Figure FDA00023592309200000310
Globally, as an invariant torus ψ (ξ), i.e. a quasi-periodic track on a periodic track center manifold,
Figure FDA00023592309200000311
3. the lorentz force multi-star formation configuration method based on quasi-periodic orbits as claimed in claim 2, characterized in that: the concrete implementation method of the step four comprises the following steps,
step 4.1: determining invariant curves for quasi-periodic orbits
Figure FDA00023592309200000312
An initial value of (1);
using linear flow of periodic orbits as invariant curve of quasi-periodic orbits
Figure FDA00023592309200000313
Is expressed as
Figure FDA00023592309200000314
Wherein, χ0V is any point on the periodic orbitr±iviIs the eigenvector corresponding to the unit eigenvalue of the single-valued matrix, and belongs to the invariant curve
Figure FDA00023592309200000315
Distance to periodic orbit, ξ ∈ [0,2 π ∈](ii) a The initial values of the fourier coefficients are then: c0＝χ0,C1＝∈vr,S1＝-∈viWhen k is>2 hour, Ck＝Sk＝0；
Step 4.2: solving an invariant curve of a quasi-periodic orbit
Figure FDA0002359230920000046
The Fourier coefficients of (1);
in the interval [0,2 π]Inner pair invariant curve
Figure FDA0002359230920000047
Discretizing to obtain all discrete points
Figure FDA0002359230920000041
The above satisfies the invariant equation (9), the following equation set is obtained,
Figure FDA0002359230920000042
wherein the constant curve
Figure FDA0002359230920000043
Expressed as a Fourier series expansion shown in equation (8), determining Fourier coefficient C by iteration of the above equation set (12) using Newton's method until the error is less than a given precisionkAnd Sk；
In the iteration process, the frequency number N of the truncated Fourier expansion is adjusted according to the required precisionfThe adjustment rule is as follows: after guarantee NfThe maximum norm of the/4 coefficients is one order of magnitude smaller than the required accuracy; if the above regulation rule is not satisfied, the number of the frequencies is increased to 2NfAnd k is equal to Nf+1,…,2NfThe corresponding item is set to Ck＝Sk＝0；
Step 4.3: invariant curve based on quasi-periodic orbit under stroboscopic mapping
Figure FDA0002359230920000044
Curve transformation is carried out by equation (10)
Figure FDA0002359230920000045
The global invariant torus ψ (ξ) is a pseudo-periodic track on a periodic track center manifold.
4. The method of claim 3, wherein the method comprises the following steps: the concrete implementation method of the step five is that,
and based on the quasi-periodic orbit on the central manifold solved in the fourth step, selecting ξ ═ i pi/N, i ═ 1,2, and N equant points on an invariant curve under the quasi-periodic orbit stroboscopic mapping, and deploying one electrified slave star respectively, so that the multi-star Lorentz force natural multi-star formation configuration without working medium consumption is realized.
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