CN112307615A - Rapid evolution method of space debris orbit - Google Patents

Rapid evolution method of space debris orbit Download PDF

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CN112307615A
CN112307615A CN202011164513.8A CN202011164513A CN112307615A CN 112307615 A CN112307615 A CN 112307615A CN 202011164513 A CN202011164513 A CN 202011164513A CN 112307615 A CN112307615 A CN 112307615A
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孙冲
陈建林
张震
袁源
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Northwestern Polytechnical University
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Abstract

The invention discloses a space debris orbit rapid evolution method, and belongs to the field of spacecraft orbit prediction. States of a plurality of fragments are described as a nominal state and a corresponding state deviation distribution form through Taylor polynomials, and polynomial integration is used for replacing a large number of fragment repetitive orbit integration operations. A large amount of fragment evolution is changed into a numerical value introduction process with extremely high speed, the overall calculation efficiency is greatly reduced, and the acceleration effect of several orders of magnitude can be achieved. By adopting high-order Taylor expansion, the operation precision of the technical scheme provided by the invention can theoretically reach the same precision as that of the traditional Monte Carlo targeting method. By balancing the calculation precision and the calculation efficiency, the error of the fifth-order approximation method is less than 0.01 m, and the sacrifice of high precision due to accelerated operation is effectively avoided. The method can rapidly evolve a large number of space fragment orbits, greatly accelerate the orbital evolution speed of a large number of fragments, and ensure the orbital precision in the whole evolution process.

Description

Rapid evolution method of space debris orbit
Technical Field
The invention belongs to the field of spacecraft orbit prediction, and relates to a space debris orbit rapid evolution method.
Background
With the continuous development of aerospace technology, the number of artificial satellites, space debris and the like is remarkably increased. The risk of in-orbit collisions and self-disassembly of space objects has severely impacted normal space activities. If an explosion or collision is accidentally generated, more space debris is generated, thereby further aggravating the deterioration of the space environment. Therefore, experts in the aerospace field have recently recognized that: merely slowing down the formation of space debris is far from enough that even if future space activities no longer produce new space debris, the existing debris and satellites of the space will face a tremendous threat of collision. Therefore, the existing space debris needs to be removed, and the accurate prediction of the motion track of the space debris is a precondition for removing the space debris and reducing the space collision accidents.
Z.A. Antassssi and T.E.Simos in the document "optimized run-Kutta method for the solution of orbital evolution" use the Longge library tower integral method to evolve the orbital motion equation, thereby predicting the future motion trend of the orbit, but along with the increase of the orbital evolution time, the operation time will be obviously increased; in order to reduce the time cost consumed in the process of evolving a large number of orbits, an introduction and an introduction of an et al adopt a longge tower method to solve a large number of fragments according to the distribution of fragment clouds in the long-term evolution modeling and analysis of fragmenting the fragment clouds by a spatial object, which reduces the operation time but greatly reduces the operation precision.
In order to predict an accurate space debris orbit and reduce collision risk, the accuracy of orbit evolution needs to be ensured in the process of orbit evolution of the space debris. The current mainstream space debris trajectory calculation method is to solve by integrating the nonlinear dynamical equation system. However, space collision and space breakdown accidents, when they occur, tend to produce a large amount of debris. At the same time, the amount of existing space debris is very large. Therefore, the long-term evolution of the spatial debris trajectory is computationally expensive. Besides the gravity of the earth, the movement of the space debris is also influenced by many perturbations such as complex lunar gravity, solar gravity, sunlight pressure and thin atmosphere, and the complexity of calculation is further increased.
In view of the above, it is desirable to develop a new fast algorithm for the motion trajectory of the space debris.
Disclosure of Invention
The invention aims to overcome the defects of long operation time and low operation precision of the space debris motion orbit calculation method in the prior art, and provides a space debris orbit rapid evolution method.
In order to achieve the purpose, the invention adopts the following technical scheme to realize the purpose:
a method for rapidly evolving space debris orbits comprises the following steps:
step 1, performing space decomposition simulation on a space decomposition model to obtain a series of fragments with different sizes and surface-to-quality ratio, quality and speed increment of the fragments;
step 2, modeling the speed increment of all fragments by adopting a polynomial to form an initial orbit state set;
step 3, carrying out Taylor polynomial approximation on the initial orbit state set in the step 2, setting evolution time, integrating the dynamic model under a polynomial framework to obtain a polynomial with position increment and speed increment as variables;
and 4, obtaining the state of each fragment at the final moment on the basis of the initial velocity increment of each fragment and the polynomial obtained in the step 3, and finishing the polynomial orbit evolution of the fragments and the statistical information evolution of all the fragments.
Preferably, the number of fragments N (L) obtained after the spatial decomposition simulation in step 1c) Calculating according to the formula (1):
N(Lc)＝S6Lc -1.6 (1)
in the formula (1), LcFor feature length, S is a parameter determined from the spatial solution model.
Preferably, the area-to-mass ratio p (r, θ) of the fragments in step 1 is calculated according to the formula (2):
Figure BDA0002745353460000031
in the formula (2), the reaction mixture is,
Figure BDA0002745353460000032
η＝A/M；θAlg(Lc) (ii) a ε (θ) are all weight coefficients, μ, determined by a spatial solution modeliAnd σiRespectively, the mean and variance of the distribution function, determined according to the weight coefficient epsilon (theta);
the mass M of the fragments is calculated according to equation (3):
M＝AX/η (3)
in the formula (3), AXDetermining the characteristic length of the space solution model and the fragment, wherein eta represents the surface-to-mass ratio of the fragment;
the velocity increment p (v) of the debris is calculated according to equation (4):
Figure BDA0002745353460000033
in the formula (4), ν Alg (Δ ν); mean value μν0.9 γ + 2.90; variance σν0.4, the velocity increments are randomly evenly distributed in three directions.
Preferably, the initial orbit state set in step 2 is:
Ω＝{Δνi|i＝1,…N} (5)
in the formula (5), Ω is a vector set composed of all space debris velocity state increment vectors; Δ νiIs the velocity increment vector of the ith fragment; n represents the total number of space fragments.
Preferably, the modeling in step 2 specifically includes establishing an earth central gravity field model and a perturbation vector field dynamics model of the space debris:
Figure BDA0002745353460000041
in the formula (6), the reaction mixture is,
Figure BDA0002745353460000042
is the position vector of the space debris;
Figure BDA0002745353460000043
is the velocity vector of the space debris; r is the distance from the space debris to the center of the track;
Figure BDA0002745353460000044
acceleration vectors resulting from non-spherical oblate perturbation of the earth;
Figure BDA0002745353460000045
acceleration vectors caused by perturbation of solar attraction;
Figure BDA0002745353460000046
acceleration vectors resulting from perturbation of lunar gravity;
Figure BDA0002745353460000047
the resulting acceleration vector is perturbed by the solar pressure.
Preferably, the specific operation of step 3 is to perform polynomial integration on the kinetic equation under a polynomial framework and obtain an approximate solution in a polynomial form, starting from the initial orbit state set.
Preferably, the state solution in polynomial form is:
Figure BDA0002745353460000048
in the formula (7), n is determined by the polynomial expansion order; j is more than or equal to 1 and less than or equal to 6, and j represents the position component and the speed component of three axes in a Cartesian coordinate system; c. CjiIs a polynomial coefficient; beta is ajiIs a polynomial expansion; f denotes the final time tf。
Preferably, the taylor polynomial approximation is to describe the initial velocity increment of the space debris in a polynomial form, obtain an ordinary differential equation describing the motion law of the space debris, and perform taylor polynomial approximation on a vector field of the ordinary differential equation.
Preferably, the polynomial integration is polynomial integration of the ordinary differential equation along time to obtain a state solution in a polynomial form at a certain time.
Preferably, the initial orbit state set in step 2 further comprises earth non-spherical oblate perturbation, sun and moon gravitational perturbation and sunlight pressure perturbation.
Compared with the prior art, the invention has the following beneficial effects:
the invention discloses a space debris orbit fast evolution method, which expresses the states of a plurality of debris as a polynomial form by a Taylor polynomial approximation method and uses a polynomial integral to replace the orbit integral operation of a large number of debris repeatability. Although the integral process in the form of the polynomial has lower calculation efficiency compared with the numerical evolution of a single track, once the result in the form of the polynomial is obtained, a large amount of fragment evolution becomes a numerical value introduction process with extremely high speed, the overall calculation efficiency is greatly reduced, and the acceleration effect of several orders of magnitude can be achieved. By adopting high-order Taylor expansion, the operation precision of the technical scheme provided by the invention can theoretically reach the same precision as that of the traditional Monte Carlo targeting method. By balancing the calculation accuracy and the calculation efficiency, the error of the fifth-order approximation method is less than 0.01 m, and the sacrifice of high accuracy caused by accelerated operation can be effectively avoided. Aiming at different space collision and disintegration activities, the only different places are the number of generated fragments and the respective physical characteristics are different when the fragments are formed, and the integral algorithm execution cannot be influenced. Therefore, the method of the invention can rapidly evolve a large amount of space fragment orbits, greatly accelerate the orbital evolution speed of a large amount of fragments and ensure the orbital precision in the whole evolution process.
Further, in the application process of the method, a user can simulate different forms of space collision or space disintegration by modifying the initial velocity increment of the track fragment; in addition, by setting the perturbation force condition and the polynomial expansion order, fragment evolution results with different precisions and different calculation efficiencies can be obtained, and the whole evolution process of the space fragments under different perturbation conditions can be simulated and considered.
Further, the differential equation in polynomial form is integrated along time to obtain a state solution in polynomial form at a certain time. The approximation error of the state solution depends on the order of the taylor expansion order, the size of the initial velocity increment, and the time length of the orbital evolution.
Further, the implementation of the method of the present invention allows for the consideration of perturbation models with high accuracy for different spatial perturbations, the introduction of which can simulate a more realistic spatial debris evolution process, but at the same time slightly reduces the computational efficiency of the method.
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FIG. 1 is a schematic diagram of space debris in an orbital altitude interval outside the earth;
FIG. 2 is a diagram illustrating the effect of the polynomial algorithm proposed by the present invention;
FIG. 3 is a graph comparing the operation time consumed by the polynomial algorithm of the present invention and the conventional Monte Carlo method.
Detailed Description
In order to make the technical solutions of the present invention better understood, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
It is to be understood that the terms in the description and in the claims, and in the drawings, are used for distinguishing between similar elements and not necessarily for describing a particular sequential or chronological order. It is to be understood that the data so used is interchangeable under appropriate circumstances such that the embodiments of the invention described herein are capable of operation in sequences other than those illustrated or described herein. Furthermore, the terms "comprises," "comprising," and "having," and any variations thereof, are intended to cover a non-exclusive inclusion, such that a process, method, system, article, or apparatus that comprises a list of steps or elements is not necessarily limited to those steps or elements expressly listed, but may include other steps or elements not expressly listed or inherent to such process, method, article, or apparatus.
The invention is described in further detail below with reference to the accompanying drawings:
example 1
A high-efficiency space debris orbit fast evolution method is divided into the following five steps by performing orbit evolution:
step 1, a space solution model of the American national aerospace agency is adopted, space solution simulation is carried out on a certain spacecraft, and a series of fragments with different sizes and corresponding surface-to-mass ratio, mass and speed increment (solution generation) are obtained.
The number of fragments produced by the split model is determined by equation (1):
N(Lc)＝S6Lc -1.6 (1)
wherein L iscAnd S is a parameter and is determined according to the split model.
The surface-to-mass ratio of each fragment was then determined from the bi-normal distribution (2):
Figure BDA0002745353460000071
wherein, gamma Alg (eta), eta is A/M; theta Alg (L)c) (ii) a ε (θ) is the weight coefficient, determined by the different types of decompositions. Mu.siAnd σiRespectively, the mean and variance of the distribution function, determined from the weight coefficient epsilon (theta).
The mass of each fragment is determined by equation (3):
M＝AX/η (3)
wherein A isXAnd (3) determined by the solution model and the characteristic length, wherein eta represents the surface-to-quality ratio of the fragments.
The speed increment is determined by equation (4):
Figure BDA0002745353460000072
wherein vAlg (Δ ν); mean value μν＝0.9γ+2.90；Variance σνUsually 0.4. The velocity increments are randomly and evenly distributed in three directions.
Step 2, modeling the speed increment sets of all fragments by adopting a polynomial to form an initial orbit state set:
Ω＝{Δνi|i＝1,…N} (5)
in the formula (5), Ω is a vector set composed of all space debris velocity state increment vectors; Δ νiIs the velocity increment vector of the ith fragment; n represents the total number of space fragments.
The space debris kinetic equation is:
Figure BDA0002745353460000081
in the formula (6), the reaction mixture is,
Figure BDA0002745353460000082
is the position vector of the space debris;
Figure BDA0002745353460000083
is the velocity vector of the space debris; r is the distance from the space debris to the center of the track;
Figure BDA0002745353460000084
acceleration vectors resulting from non-spherical oblate perturbation of the earth;
Figure BDA0002745353460000085
acceleration vectors caused by perturbation of solar attraction;
Figure BDA0002745353460000086
acceleration vectors resulting from perturbation of lunar gravity;
Figure BDA0002745353460000087
the resulting acceleration vector is perturbed by the solar pressure.
Step 3, carrying out Taylor polynomial description on the space debris initial state set (5) and the orbit dynamics equation and (6), and carrying out polynomial integration along a time axis to obtain the orbit state in a polynomial form
Figure BDA0002745353460000088
In the formula (7), n is determined by the polynomial expansion order; j represents the position component and the velocity component of three axes in a Cartesian coordinate system from 1 to 6 respectively; c. CjiIs a polynomial coefficient; beta is ajiIs a polynomial expansion; f denotes the final time tf. The process can select whether conditions such as earth non-spherical flat perturbation, day and month gravitation perturbation, sunlight pressure perturbation and the like are added.
Step 4, setting evolution time to obtain a space debris state in a polynomial form with position increment and speed increment as variables at final time
Figure BDA0002745353460000089
Where i is the fragment number, j is the index value of the state component, and N is the fragment number. Since the disassembly model does not result in the generation of a position increment at the moment of disassembly, the position increment generally takes 0.
And 5, substituting the speed increment of each fragment into the polynomial in the step 4, and solving to obtain the state of each fragment at the final time.
In step 1, fragments generated by satellite disintegration have different surface-to-mass ratios, masses and velocity increments, and the step is mainly used for simulating the satellite disintegration process and generating corresponding fragment initial states and parameters; in step 2, a polynomial is used for carrying out set form modeling on the initial speeds of all the fragments, and the speed increments of all the fragments form a probability distribution; in step 3, a high-order Taylor polynomial is used for approximating the earth spherical vector field or the perturbation vector field, and a polynomial integration method is adopted for integrating the approximated polynomial vector field; in step 4, setting an orbit evolution time, obtaining a final time by using a polynomial integration method in step 3, and solving in a polynomial form by taking the initial fragment speed increment as an independent variable; in step 5, the velocity increment of each fragment is substituted into a state solution in a polynomial form to obtain the final state of each fragment, and the single fragment orbit evolution and the statistical information evolution of all fragments are completed.
Example 2
Step 1, establishing an earth center gravitational field model of space debris and a dynamic model under a perturbation condition;
step 2, constructing a computer algorithm and a polynomial integral algorithm of Taylor polynomial automatic approximation;
step 3, carrying out polynomial description on the initial velocity increment of the space debris, and carrying out Taylor polynomial approximation on a vector field of an ordinary differential equation for describing the motion rule of the space debris;
and 4, integrating the differential equation in the polynomial form obtained in the step 3 along time by adopting a polynomial integration method to obtain a state solution in the polynomial form at a certain moment. Notably, the approximation error of the solution depends on the order of the taylor expansion order, the size of the initial velocity increment and the time length of the orbital evolution;
and 5, substituting the respective speed increment of all the fragments into a state polynomial to obtain a state solution of the space fragment. The deconstructed distribution of all fragments is the statistical information of the fragments at that moment.
And 6, aiming at different space collision and disintegration activities, the only different places are that when the fragments are formed, the number of the generated fragments is different from the respective physical characteristics, and the integral algorithm execution is not influenced.
The implementation of the method allows to consider perturbation models of high precision for different spatial perturbations, step 7. The introduction of the perturbation model can simulate a more real space debris evolution process, but at the same time slightly reduces the computational efficiency of the method.
A schematic diagram of space debris in a certain orbit altitude interval outside the earth is shown in fig. 1.
The method of the invention is utilized to carry out polynomial orbit evolution on the orbit of 1581 fragments, the effect schematic diagram is shown in figure 2, and as can be seen from figure 2, the orbit undergoes 96-hour evolution. The figure shows the evolution accuracy (maximum error) and the required computation time (CPU time) that can be obtained with polynomials of different expansion orders. The result shows that the precision of the algorithm is improved along with the improvement of the expansion order of the polynomial, but the corresponding operation time is slightly prolonged. When the five-order operation is used, the operation error is reduced to be within 0.01 meter, and the error of all fragment evolution can be controlled to be in millimeter level by adopting the six-order and the above methods.
The time spent comparing the method of the present invention with the conventional Monte Carlo method is shown in FIG. 3. In the same system and the same operation environment of the same computer, the orbit of 1581 fragments is evolved by adopting a millimeter-grade precision six-order approximate polynomial expansion algorithm and a traditional Monte Carlo method. The result shows that the method provided by the invention can greatly reduce the operation time, and the effect is more and more obvious along with the increase of the evolution time.
In summary, the invention discloses a rapid evolution method of a space debris orbit, which adopts a polynomial expansion algorithm to accelerate the orbit evolution speed of the space debris, utilizes polynomial expansion to approximate a space vector field of the debris, and adopts a polynomial integration method to perform collective evolution on the orbit of the space debris. On the premise of ensuring the precision, the overall evolution speed of the space debris can be increased by several orders of magnitude. In the algorithm application process, a user can simulate different forms of space collision or space disintegration by modifying the initial velocity increment of the track fragment; in addition, fragment evolution results with different accuracies and different calculation efficiencies can be obtained by setting the shooting force condition and the polynomial expansion order. According to the method, Taylor polynomials of different expansion orders are adopted, after the space is decomposed to form fragments, polynomial approximation and polynomial integration are carried out on a vector field of a fragment kinetic equation to obtain a polynomial of a final state, speed increment is substituted into the polynomial to obtain the final state after solving, the traditional Monte Carlo targeting method for researching the whole evolution of the fragments is improved, and the deceleration efficiency is obviously improved under the condition of not losing calculation accuracy.
The above-mentioned contents are only for illustrating the technical idea of the present invention, and the protection scope of the present invention is not limited thereby, and any modification made on the basis of the technical idea of the present invention falls within the protection scope of the claims of the present invention.

Claims (10)

1. A method for rapidly evolving space debris orbit is characterized by comprising the following steps:
step 1, performing space decomposition simulation on a space decomposition model to obtain a series of fragments with different sizes and surface-to-quality ratio, quality and speed increment of the fragments;
step 2, modeling the speed increment of all fragments by adopting a polynomial to form an initial orbit state set;
step 3, carrying out Taylor polynomial approximation on the initial orbit state set in the step 2, setting evolution time, integrating the dynamic model under a polynomial framework to obtain a polynomial with position increment and speed increment as variables;
and 4, obtaining the state of each fragment at the final moment on the basis of the initial velocity increment of each fragment and the polynomial obtained in the step 3, and finishing the polynomial orbit evolution of the fragments and the statistical information evolution of all the fragments.
2. The method for fast evolving spatial debris trajectory according to claim 1, wherein the number of debris N (L) obtained after the spatial decomposition simulation in step 1c) Calculating according to the formula (1):
N(Lc)＝S6Lc -1.6 (1)
in the formula (1), LcFor feature length, S is a parameter determined from the spatial solution model.
3. The method for rapidly evolving space debris trajectory according to claim 1, wherein the surface-to-mass ratio p (r, θ) of the debris in step 1 is calculated according to formula (2):
Figure FDA0002745353450000011
in the formula (2), the reaction mixture is,
Figure FDA0002745353450000012
η＝A/M；θAlg(Lc) (ii) a ε (θ) are all weight coefficients, μ, determined by a spatial solution modeliAnd σiRespectively, the mean and variance of the distribution function, determined according to the weight coefficient epsilon (theta);
the mass M of the fragments is calculated according to equation (3):
M＝AX/η (3)
in the formula (3), AXDetermining the characteristic length of the space solution model and the fragment, wherein eta represents the surface-to-mass ratio of the fragment;
the velocity increment p (v) of the debris is calculated according to equation (4):
Figure FDA0002745353450000021
in the formula (4), ν Alg (Δ ν); mean value μν0.9 γ + 2.90; variance σν0.4, the velocity increments are randomly evenly distributed in three directions.
4. The method for fast evolving space debris trajectory according to claim 1, wherein the initial set of trajectory states in step 2 is:
Ω＝{Δνi|i＝1,…N} (5)
in the formula (5), Ω is a vector set composed of all space debris velocity state increment vectors; Δ νiIs the velocity increment vector of the ith fragment; n represents the total number of space fragments.
5. The method for fast orbital evolution of space debris according to claim 1, wherein the modeling in step 2 specifically includes establishing an earth central gravitational field model and a perturbation vector field dynamics model of the space debris:
Figure FDA0002745353450000022
in the formula (6), the reaction mixture is,
Figure FDA0002745353450000023
is the position vector of the space debris;
Figure FDA0002745353450000024
is the velocity vector of the space debris; r is the distance from the space debris to the center of the track;
Figure FDA0002745353450000025
acceleration vectors resulting from non-spherical oblate perturbation of the earth;
Figure FDA0002745353450000026
acceleration vectors caused by perturbation of solar attraction;
Figure FDA0002745353450000027
acceleration vectors resulting from perturbation of lunar gravity;
Figure FDA0002745353450000028
the resulting acceleration vector is perturbed by the solar pressure.
6. The method for fast evolving space debris trajectory according to claim 1, wherein the specific operation of step 3 is to perform polynomial integration on the kinetic equation under a polynomial framework and obtain an approximate solution in a polynomial form, starting from the initial trajectory state set.
7. The method for fast evolving space debris trajectory according to claim 6, wherein the state solution in polynomial form is:
Figure FDA0002745353450000031
in the formula (7), n is determined by the polynomial expansion order; j is more than or equal to 1 and less than or equal to 6, and j represents the position component and the speed component of three axes in a Cartesian coordinate system; c. CjiIs a polynomial coefficient; beta is ajiIs a polynomial expansion; f denotes the final time tf。
8. The method for rapidly evolving a space debris orbit according to claim 6, wherein the Taylor polynomial approximation is a polynomial description of the initial velocity increment of the space debris, so as to obtain a ordinary differential equation describing the law of motion of the space debris, and the Taylor polynomial approximation is performed on a vector field of the ordinary differential equation.
9. The method for fast evolving space debris trajectory according to claim 8, wherein the polynomial integration is a polynomial integration of ordinary differential equations along time to obtain a state solution in a polynomial form at a certain time.
10. The method for fast evolution of space debris orbit according to claim 1, wherein the initial orbit state set of step 2 further comprises earth aspheric oblate perturbation, solar and lunar gravitational perturbation and solar pressure perturbation.
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