A method and system for analyzing the evolution of orbital deviations of space objects

Abstract

The present application belongs to the technical field of spacecraft orbit dynamics, and relates to a space target orbit deviation evolution analysis method and system. N The space target orbit deviation evolution analysis method adopts a probability space subdivision technique to obtain one initial orbit state deviation representative point of a space target containing deviation and a corresponding assigned probability of each initial orbit state deviation representative point of the space target; based on the principle of probability conservation, a probability density integral equation of a space target terminal time orbit state response is obtained, which is a high-dimensional integral containing a Dirac function; the Dirac function is smoothed, the probability density integral equation of the space target terminal time orbit state response is solved by using a direct probability integral method, and the probability density function of the space target terminal time orbit state response is obtained. The present application is suitable for space target orbit deviation evolution analysis caused by various uncertain factors, and has strong applicability, high calculation efficiency and high accuracy.

Description

Technical Field

[0001] This invention relates to the field of spacecraft orbital dynamics technology, and in particular to a method and system for analyzing the evolution of orbital deviations of space targets. Background Technology

[0002] In actual space missions, spacecraft orbits are typically affected by various uncertainties (also known as deviation factors, such as model parameter errors, navigation errors, and control errors), causing the spacecraft's actual orbit to deviate from its designed nominal orbit, thus affecting the normal completion of the space mission. Therefore, conducting orbital deviation evolution analysis on specific space targets is a crucial foundation for achieving space target collision early warning, spacecraft tracking, orbit determination and cataloging maintenance, orbital maneuver detection, and spacecraft trajectory planning, and is of great significance for the safety protection of spacecraft in orbit and the safety of astronauts.

[0003] Due to the complexity of the on-orbit space target dynamics model and the mutual influence of various deviation factors during actual flight control, the spacecraft orbital dynamic system is essentially a nonlinear stochastic dynamic system when considering the impact of deviation factors. In this case, the spacecraft trajectory deviation evolution problem becomes a nonlinear system state deviation prediction problem. By solving the Fokker-Planck equation or the stochastic Liouville equation without process noise, the evolution process of the probability density function of the state deviation can be obtained, thus acquiring complete probability distribution information of the state deviation at any given time. However, due to the complete coupling between the physical state and the probability density evolution in the Fokker-Planck or Liouville equations, solving high-dimensional Fokker-Planck or Liouville equations remains a challenge. Existing methods for orbit deviation evolution analysis mainly include Monte Carlo simulation, linear methods, and nonlinear methods. The Monte Carlo simulation method is the most direct solution method for orbit deviation evolution analysis. While simple to implement, it has high computational costs and requires a sufficiently large number of sample points to guarantee accuracy, resulting in low computational efficiency. Furthermore, the calculation results exhibit stochastic convergence, making it difficult to reveal the intrinsic evolution mechanism of orbit deviation. The main idea of ​​the linear method is to linearize the nonlinear orbital dynamic system and assume that the initial orbital deviation follows a Gaussian distribution, calculating the mean and covariance matrix of the output deviation. This method is computationally efficient. However, because the propagated orbital deviation exhibits non-Gaussian properties, the mean and covariance matrix alone cannot fully describe the probability distribution characteristics of the orbital deviation. Therefore, this method suffers from significant computational errors in strongly nonlinear systems and non-Gaussian orbital deviation propagation problems.

[0004] To improve the computational accuracy of linear methods, various analytical or semi-analytical nonlinear orbit deviation evolution analysis methods have been proposed, including: the unscented variation method, the polynomial chaotic expansion method, the state transition tensor method, the differential algebra method, the Gaussian sum model, and the hybrid deviation propagation method. The state transition tensor method expands the deviation trajectory along the design trajectory using Taylor series to higher-order nonlinear terms, derives and calculates the state transition tensors of each order, and finally performs deviation evolution analysis only through analytical algebraic operations. This method requires the orbital dynamics equations to be continuously differentiable and necessitates manual derivation of higher-order partial derivatives of the dynamics equations. Due to the difficulty in deriving higher-order partial derivatives and the complexity of their expressions, as well as the complexity of calculating the state transition tensor, the state transition tensor method is difficult to apply to strongly nonlinear orbital dynamics systems. The differential algebra method automatically calculates higher-order partial derivatives, achieving deviation prediction with arbitrary-order accuracy. It avoids the tedious work of manually deriving higher-order partial derivatives of the dynamics model, and exhibits high adaptability and computational efficiency. However, this method requires the orbital dynamics equations to be continuously differentiable, making it unsuitable for dynamic systems with discontinuities. It necessitates operating the model within a differential-algebraic framework, and the expansion order is difficult to determine a priori. Gaussian sum models approximate the probability density function of any arbitrary distribution by weighted summing of a finite number of Gaussian distribution probability density functions, reducing the influence of nonlinear terms in the dynamic system. However, the number of sub-Gaussian distributions, their weights, and the selection of their update and partitioning directions are key issues affecting the computational accuracy of this method. In recent years, many scholars have proposed hybrid deviation propagation methods by fully combining the advantages of different deviation propagation methods. However, existing methods and their improvements still require calculating higher-order partial derivatives or deriving state transition tensors of various orders, resulting in large computational loads and complexity for strongly nonlinear orbital dynamic systems.

[0005] In summary, existing methods for analyzing orbital deviations suffer from technical bottlenecks in terms of computational accuracy, efficiency, and applicability. Therefore, there is an urgent need to propose a high-precision, high-efficiency, and highly versatile method for analyzing the orbital deviations of space targets, in order to facilitate the prediction of orbital deviations. Summary of the Invention

[0006] The purpose of this invention is to address the shortcomings of existing methods for analyzing orbital deviations in terms of computational efficiency, accuracy, and applicability, and to propose a new method and system for analyzing the orbital deviations of space targets. This method introduces probability space partitioning and Dirac function smoothing techniques, and employs the direct probability integral method to numerically solve the probability density integral equation. This efficiently and accurately obtains the probability density function of the orbital state response at the terminal moment of the space target, achieving decoupling between the physical evolution of the structural system and the probability density evolution. This method boasts advantages such as high computational accuracy, high computational efficiency, and strong versatility.

[0007] To achieve the above objectives, the present invention adopts the following technical solution: In a first aspect, the present invention provides a method for analyzing the evolution of orbital deviations of a space target, comprising the following steps: S1. Configure the space target and its initial state nominal values. The initial orbital state deviation of the space target forms an input random vector space. Using the generalized F-bias point selection method, the input random vector space is probabilistically partitioned to obtain the initial orbital state of the space target. N Each deviation represents a point And the assigned probability corresponding to each deviation point. , ; S2. Will N Each deviation represents a point Each value is added to the initial state nominal value of the space target to obtain the corresponding result. N The initial orbital state deviation of a space target with bias is represented by a point. And the assigned probability corresponding to the point representing the initial orbital state deviation of each space target. , Among them, the first The initial orbital state deviation of each space target represents a point With the Each deviation represents a point The corresponding assignment probabilities are the same, that is , ; S3. Solve the orbital dynamics equations of the space target, and calculate the orbital state response at the terminal moment corresponding to the initial orbital state deviation representative point of each space target based on the orbital dynamics equations, thereby obtaining the physical mapping between the initial orbital state deviation representative point of each space target and the corresponding orbital state response; S4. Configure the probability density function of the initial orbital state deviation of the space target. Based on the principle of probability conservation, utilize the physical mapping between the representative point of each initial orbital state deviation of the space target and the corresponding orbital state response to establish the relationship between the probability density function of the orbital state response at the terminal moment of the space target and the probability density function of the initial orbital state deviation. Obtain the probability density integral equation of the orbital state response at the terminal moment of the space target. This probability density integral equation is a high-dimensional integral containing the Dirac function. S5. Smooth the Dirac function, use the assigned probability corresponding to the point represented by the initial orbital state deviation of the space target, and use the direct probability integral method to solve the probability density integral equation of the orbital state response at the terminal moment of the space target, so as to obtain the probability density function of the orbital state response at the terminal moment of the space target, thus completing the evolution analysis of the orbital deviation of the space target.

[0008] As one possible implementation, the input random vector space is partitioned into probability spaces using a generalized F-bias point selection method, including: S10. Using the Sobol point set as the initial representative point set, the input random vector space is divided into multiple non-overlapping representative regions using the Voronoi unit partitioning method. This yields a uniformly distributed set of low-bias points. S11. Perform generalized F-bias minimization renormalization on a uniformly distributed set of low-bias points to obtain multiple representative bias points of the initial orbital state of the space target. S12. Calculate the probability of assigning a point representing the initial orbital state deviation of each space target.

[0009] As one possible implementation, the probability density integral equation of the orbital state response of the space target at the terminal moment is obtained by the following method: in, The probability density function representing the orbital state response of the space target at the terminal moment. The independent variable represents the probability density function of the orbital state response of the space target at the terminal moment. The probability density function representing the initial orbital state deviation of a space target. This indicates the spatial target orbital state response at the terminal moment. For the Dirac function, This represents the input random vector space.

[0010] As one possible approach, the orbital dynamics equations can be solved using any one of the following numerical integration methods: Runge-Kutta method, Nermark method, or central difference method.

[0011] As one possible implementation, the Dirac function can be smoothed using any one of the following functions: Gaussian function, trigonometric function, Poisson function, Cauchy function, and rectangular wave function.

[0012] As one possible implementation, the probability density integral equation of the orbital state response at the terminal moment of the space target is solved using the direct probability integral method, specifically: In the formula, This represents the number of points indicating the initial orbital state deviation of a space target. Indicates the first The initial orbital state deviation of each space target represents a point. , Indicates the first The initial orbital state deviation of each space target represents the orbital state response of that point at the terminal moment. Indicates the first The initial orbital state deviation of each space target represents the assigned probability of that point. This represents the smoothing parameter of the Dirac function.

[0013] As one possible implementation, the following method is used to calculate the first... The initial orbital state deviation of each space target represents the assigned probability of the point: In the formula, The probability density function representing the initial orbital state deviation of a space target. Represents the i-th element in the input random vector space The initial orbital state deviation of each space target represents the representative region corresponding to the point.

[0014] As one possible implementation, the smoothing parameters of the Dirac function are selected using the following method. : In the formula, Represents the smoothing factor. ; express N The initial orbital state deviation of each space target represents the standard deviation of the orbital state response at the terminal moment. express N The initial orbital state deviation of a space target represents the interquartile range of the orbital state response at the terminal moment.

[0015] Secondly, the present invention provides a space target orbit deviation evolution analysis system, comprising: The deviation representative point acquisition unit configures the space target and its initial state nominal value. The input random vector space is constructed from the initial orbital state deviation of the space target. The input random vector space is then probabilistically partitioned using the generalized F-deviation point selection method to obtain the initial orbital state of the space target. N Each deviation represents a point And the assigned probability corresponding to each deviation point. , ; The initial orbital state deviation of the space target represents the acquisition unit, which will... N Each deviation represents a point Each value is added to the initial state nominal value of the space target to obtain the corresponding result. N The initial orbital state deviation of a space target with bias is represented by a point. And the assigned probability corresponding to the point representing the initial orbital state deviation of each space target. , Among them, the first The initial orbital state deviation of each space target represents a point With the Each deviation represents a point The corresponding assignment probabilities are the same, that is , ; The unit obtains the probability density integral equation of the orbital state response at the terminal moment of a space target, solves the orbital dynamics equation of the space target, calculates the orbital state response at the terminal moment corresponding to the representative point of the initial orbital state deviation of each space target based on the orbital dynamics equation, and then obtains the physical mapping between the representative point of the initial orbital state deviation of each space target and the corresponding orbital state response; configures the probability density function of the orbital state deviation at the initial moment of the space target, and based on the principle of probability conservation, uses the physical mapping between the representative point of the initial orbital state deviation of each space target and the corresponding orbital state response to establish the relationship between the probability density function of the orbital state response at the terminal moment of the space target and the probability density function of the orbital state deviation at the initial moment, and obtains the probability density integral equation of the orbital state response at the terminal moment of the space target. The probability density integral equation is a high-dimensional integral containing the Dirac function. The probability density function solving unit for the orbital state response at the terminal moment of the space target smooths the Dirac function and uses the assigned probability corresponding to the representative point of the initial orbital state deviation of the space target. It then uses the direct probability integral method to solve the probability density integral equation of the orbital state response at the terminal moment of the space target, thereby obtaining the probability density function of the orbital state response at the terminal moment of the space target, thus completing the evolution analysis of the orbital deviation of the space target.

[0016] Compared with the prior art, the beneficial effects of the present invention are as follows: 1. The space target orbit deviation evolution analysis method and system proposed in this invention employs probabilistic space partitioning and Dirac function smoothing techniques. It is applicable to the analysis of space target orbit deviation caused by various uncertainties, such as model parameter errors, navigation errors, and control errors. Furthermore, it does not restrict the distribution type of the input random vector (i.e., the input random vector can follow any type of distribution, such as Gaussian distribution or non-Gaussian distribution). This provides new research ideas for the design of aerospace missions such as space target collision warning, orbital maneuver detection, and spacecraft trajectory planning. It has strong applicability and high computational efficiency and accuracy.

[0017] 2. The space target orbit deviation evolution analysis method and system proposed in this invention, through probability space partitioning and Dirac function smoothing techniques, uses the direct probability integral method to solve the probability density integral equation of the space target's orbital state response at the terminal moment. This enables accurate and efficient prediction of the probability density function and cumulative distribution function of the space target's orbital state at the terminal moment, thereby obtaining the statistical characteristics of the uncertainty of the space target's state at the terminal moment. Furthermore, compared with the Monte Carlo simulation method, the calculation results of this invention are deterministic, while the calculation results of the Monte Carlo simulation method exhibit stochastic convergence.

[0018] 3. The space target orbit deviation evolution analysis method and system proposed in this invention are based on the direct probability integral method for space target orbit deviation evolution analysis, realizing the decoupling of the physical evolution of the structural system and the probability density evolution. It can be integrated into existing aerospace mission design software and is convenient for engineering applications. Attached Figure Description

[0019] The accompanying drawings, which are included to provide a further understanding of the invention and form part of this invention, illustrate exemplary embodiments of the invention and are used to explain the invention, but do not constitute an undue limitation of the invention. In the drawings: Figure 1 This is a flowchart of the space target orbit deviation evolution analysis method in an embodiment of the present invention; Figure 2 This is the distribution of 1300 deviation representative points generated after probability space partitioning of the input random vector space in this embodiment of the invention; Figure 3 In this embodiment of the invention, 1300 representative deviation points are used. The deviation in direction represents the distribution of the point set; Figure 4 In this embodiment of the invention, 1300 representative deviation points are used. The deviation in direction represents the distribution of the point set; Figure 5 In this embodiment of the invention, 1300 representative deviation points are used. The deviation in direction represents the distribution of the point set; Figure 6 For each deviation representative point in the embodiments of the present invention, The position of the three-dimensional plane and in The position of the plane; Figures 7 to 8 This is a screenshot of some of the calculated data for 1300 representative deviation points and their assigned probabilities in an embodiment of the present invention; Figure 9 In this embodiment of the invention, the direct probability integral method and Monte Carlo simulation method are used to predict five orbital periods. The probability density function curve and cumulative distribution function curve of the spacecraft's position in the direction; Figure 10 In this embodiment of the invention, the direct probability integral method and Monte Carlo simulation method are used to predict five orbital periods. The probability density function curve and cumulative distribution function curve of the spacecraft's position in the direction; Figure 11 In this embodiment of the invention, the direct probability integral method and Monte Carlo simulation method are used to predict five orbital periods. The probability density function curve and cumulative distribution function curve of the spacecraft's position in the direction; Figure 12 In this embodiment of the invention, the direct probability integral method and Monte Carlo simulation method are used to predict five orbital periods. The probability density function curve and cumulative distribution function curve of the spacecraft velocity in the direction; Figure 13 In this embodiment of the invention, the direct probability integral method and Monte Carlo simulation method are used to predict five orbital periods. The probability density function curve and cumulative distribution function curve of the spacecraft velocity in the direction; Figure 14 In this embodiment of the invention, the direct probability integral method and Monte Carlo simulation method are used to predict five orbital periods. The probability density function curve and cumulative distribution function curve of the spacecraft velocity in the direction; Figure 15 In this embodiment of the invention, the direct probability integral method and Monte Carlo simulation method are used to predict the spacecraft's terminal time after 5 orbital periods. x Screenshot of the probability density function calculation data for direction and position; Figure 16 In this embodiment of the invention, the direct probability integral method and Monte Carlo simulation method are used to predict the spacecraft's terminal time after 5 orbital periods. x Screenshot of the calculation data for the cumulative distribution function of direction and position; Figure 17 In this embodiment of the invention, the direct probability integral method and Monte Carlo simulation method are used to predict the spacecraft's terminal time after 5 orbital periods. y Screenshot of the probability density function calculation data for direction and position; Figure 18 In this embodiment of the invention, the direct probability integral method and Monte Carlo simulation method are used to predict the spacecraft's terminal time after 5 orbital periods. y Screenshot of the calculated data for the cumulative distribution function of direction and position. Detailed Implementation

[0020] To facilitate a clear description of the technical solutions in the embodiments of the present invention, the terms "first" and "second" are used to distinguish identical or similar items with essentially the same function and effect. For example, the first threshold and the second threshold are merely used to distinguish different thresholds and do not limit their order. Those skilled in the art will understand that the terms "first" and "second" do not limit the quantity or execution order, and that the terms "first" and "second" are not necessarily different.

[0021] It should be noted that in this invention, the terms "exemplary" or "for example" are used to indicate examples, illustrations, or descriptions. Any embodiment or design described as "exemplary" or "for example" in this invention should not be construed as being more preferred or advantageous than other embodiments or designs. Specifically, the use of terms such as "exemplary" or "for example" is intended to present the relevant concepts in a concrete manner.

[0022] In this invention, "at least one" refers to one or more, and "more than one" refers to two or more. "And / or" describes the relationship between related objects, indicating that three relationships can exist. For example, A and / or B can represent: A alone, A and B simultaneously, or B alone, where A and B can be singular or plural. The character " / " generally indicates that the preceding and following related objects are in an "or" relationship. "At least one" or similar expressions refer to any combination of these items, including any combination of singular or plural items. For example, "at least one of a, b, or c" can represent: a, b, c, a combination of a and b, a combination of a and c, a combination of b and c, or a, b, and c, where a, b, and c can be single or multiple.

[0023] This invention aims to provide a method and system for analyzing the evolution of orbital deviations of space targets. It employs a direct probability integral method to accurately and efficiently solve for the probability density function and cumulative distribution function of the orbital state response at the terminal moment of the space target, thereby obtaining the statistical characteristics of the uncertainty in the orbital state of the space target at the terminal moment. This method boasts high computational accuracy and efficiency, strong applicability, and can be integrated into existing aerospace mission design software, facilitating engineering applications.

[0024] In a first aspect, embodiments of the present invention provide a method for analyzing the evolution of orbital deviations of space targets, see [link to previous document]. Figure 1 , Figure 1 The implementation steps of this method are shown. Specifically, the implementation process of this method will be explained in detail below using a spacecraft as an example.

[0025] Besides the gravitational pull of the central celestial body, the orbital motion of spacecraft is typically influenced by perturbations such as Earth's non-spherical gravity, atmospheric drag, solar radiation pressure, and third-body gravity. Among these, Earth's non-spherical gravity has the greatest impact. The Earth's oblateness... The term is a major factor in the non-spherical gravitational perturbation of the Earth. This embodiment considers... A perturbed two-body orbit model of a spacecraft is used to derive the probability density integral equation of the spacecraft's orbital state response. A spacecraft orbital deviation evolution analysis method based on the direct probability integral method is proposed, which can accurately and efficiently predict the evolution of the spacecraft's initial state deviation.

[0026] The space target orbit deviation evolution analysis method proposed in this embodiment includes the following steps: S1. Configure the spacecraft and its initial state nominal values. The initial orbital state deviation of the spacecraft forms the input random vector space. Using a generalized F-bias point selection method, the input random vector space is probabilistically partitioned to obtain the initial orbital state of the spacecraft. N Each deviation represents a point , And the assigned probability corresponding to each deviation point. .

[0027] In practice, given the initial moment, the spacecraft has six classical orbital elements (i.e., semi-major axes) in the geocentric inertial coordinate system. eccentricity Track inclination Right ascension of ascending node Perigee latitude arc And true near point angle The nominal values ​​of ) are respectively =6778.138km, =0.2, =0.001rad, =0 rad, =0 rad, =0 rad. The nominal values ​​of the spacecraft's initial nominal orbital state (i.e., position and velocity) can be obtained by converting the nominal values ​​of the initial nominal orbital elements. Assume that in the geocentric inertial coordinate system, the deviation of the spacecraft's initial nominal orbital state follows a normal distribution, and the initial nominal orbit is in... , and The standard deviation of the position distribution in all three directions is 1. =100m, the standard deviation of the velocity distribution is... =10 m / s. The spacecraft orbital deviation prediction time is 5 orbital periods, that is... =5 =27768s, where The nominal orbital period of a spacecraft is represented by its nominal semi-major axis. Decision, that is , This represents the Earth's gravitational constant.

[0028] Spacecraft initial orbital state It can be represented as .in, This represents the nominal value of the spacecraft's initial orbital state. This represents the initial orbital state deviation of the spacecraft. Using probabilistic space partitioning techniques and the generalized F-bias point selection method, the input random vector space composed of the initial orbital state deviation of the spacecraft is probabilistically partitioned to generate the initial orbital state (including position and velocity) of the spacecraft. N Each deviation represents a point And calculate the assigned probability for each deviation point. , .

[0029] As one possible implementation, the input random vector space is partitioned into probability spaces using a generalized F-bias point selection method, including: S10. Using the Sobol point set as the initial representative point set, the Voronoi unit partitioning method is used to divide the input random vector space. Divided into multiple non-overlapping representative regions This yields a uniformly distributed set of low-bias points. For example, non-overlapping representative regions It can be represented as: (1) In the formula, This represents the number of low-bias point sets obtained from the probability space partitioning. and Representing non-overlapping representative regions and The representative point in the middle.

[0030] S11. Perform generalized F-bias minimization renormalization on a uniformly distributed set of low-bias points to obtain multiple representative deviation points of the spacecraft's initial orbital state; S12. Calculate the probability of assigning a point representing the orbital state deviation at the initial moment for each spacecraft.

[0031] As an example, the first q The initial orbital state deviation of each spacecraft represents the probability of assignment at that point. It can be calculated using the following formula: (2) In the formula, Indicates the input random vector The joint probability density function.

[0032] See Figure 2 These are the 1300 representative deviation points generated after performing probability space partitioning on the input random vector space in this embodiment. Figures 3 to 5 The deviations in each direction represent the distribution of the point set. Figure 6 The calculation considerations for using the direct probability integral method and Monte Carlo simulation method in this embodiment are given. The position of each representative point at the terminal moment in the perturbed two-body orbit model of the spacecraft.

[0033] S2. Will N Each deviation represents a point Each one is compared with the nominal value of the spacecraft's initial orbital state. Add them together to get the corresponding result. N Representative points of the initial orbital state deviation of a spacecraft with deviation. , And the assigned probability corresponding to each spacecraft's initial orbital state deviation point. Among them, the first Representative points of the initial orbital state deviation of each spacecraft The corresponding assigned probability With the Each deviation represents a point The corresponding assigned probability Same, that is , .

[0034] See Figures 7 to 8 The image shows a screenshot of the calculated data of 1300 representative deviation points selected using the direct probability integration method in this embodiment, along with their assigned probabilities. Figure 7 Each column represents a deviation point. Figure 8 Each column represents the assigned probability corresponding to a deviation point.

[0035] This method employs probability space partitioning and Dirac function smoothing techniques, making it suitable for analyzing the evolution of orbital deviations of space targets caused by various uncertainties, such as model parameter errors, navigation errors, and control errors. Furthermore, it does not restrict the distribution type of the input random vector and has the advantages of high computational efficiency, high computational accuracy, and strong applicability.

[0036] S3. Solve the orbital dynamics equations of the spacecraft, and calculate the orbital state response at the terminal moment corresponding to each initial orbital state deviation representative point based on the orbital dynamics equations, thereby obtaining the physical mapping between each initial orbital state deviation representative point and the corresponding orbital state response.

[0037] In a geocentric inertial coordinate system, consider The orbital dynamics equations of the perturbed two-body orbit model of the spacecraft are expressed as follows: (3) Among them, the coefficients of the Earth's second-order zone harmonics are =1.08263×10 -3 The Earth's equatorial radius is =6378.137km; the distance between the spacecraft and the Earth's center (i.e., the distance from the Earth's center) is The Earth's gravitational constant is =3.9860×10 5 km 3 s -2 .

[0038] Representative points of the initial orbital state deviation of each spacecraft ( Substituting the above considerations In the orbital dynamics equation (3) of the perturbed two-body orbital model of the spacecraft, any one of the numerical integration methods among Runge-Kutta method, Nermark method, and central difference method is used to solve the orbital dynamics equation. In this embodiment, the Runge-Kutta method (i.e., the variable step size RKF89 integral scheme) is used to solve the equation to obtain the representative points of each spacecraft's initial orbital state deviation. Track state response at the terminal moment The above solution process constitutes a physical mapping of the orbital dynamics system, that is, a physical mapping between the initial orbital state deviation representative point and the corresponding orbital state response: .

[0039] S4. Configure the probability density function of the orbital state deviation at the initial moment of the spacecraft. Based on the principle of probability conservation, utilize the physical mapping between each point represented by the initial orbital state deviation and the corresponding orbital state response to establish the relationship between the probability density function of the orbital state response at the terminal moment of the spacecraft and the probability density function of the orbital state deviation at the initial moment. Obtain the probability density integral equation of the orbital state response at the terminal moment of the spacecraft. The probability density integral equation is a high-dimensional integral containing the Dirac function.

[0040] As one possible implementation, the probability density integral equation of the spacecraft's orbital state response at the terminal moment is obtained using the following method: (4) in, The probability density function representing the orbital state response of the spacecraft at the terminal moment. The independent variable represents the probability density function of the spacecraft's orbital state response at the terminal moment. The probability density function representing the initial orbital state deviation of the spacecraft. Indicates consideration The orbital state response of the perturbed two-body orbital model of the spacecraft at the terminal moment. For the Dirac function, This represents the input random vector space.

[0041] As can be seen from equation (4), the probability density integral equation of the orbital state response at the terminal moment is a high-dimensional integral containing the Dirac function, which is usually difficult to solve directly analytically. It is necessary to smooth the Dirac function to improve the calculation accuracy of the numerical integral.

[0042] S5. Smooth the Dirac function, use the assigned probability corresponding to the point represented by the initial orbital state deviation of the spacecraft, and use the direct probability integral method to solve the probability density integral equation of the orbital state response at the spacecraft's terminal moment to obtain the probability density function of the orbital state response at the spacecraft's terminal moment, thus completing the orbital deviation evolution analysis.

[0043] The space target orbit deviation evolution analysis method proposed in this invention is based on the direct probability integral method to analyze the orbit deviation evolution of space targets. It achieves the decoupling of the physical evolution of the structural system and the probability density evolution, and is easy to integrate into existing aerospace mission design software, which is convenient for engineering applications.

[0044] As one possible implementation, the Dirac function can be smoothed using any one of the following functions: Gaussian function, trigonometric function, Poisson function, Cauchy function, and rectangular wave function.

[0045] In this embodiment, the Dirac function is smoothed using a Gaussian function.

[0046] As one possible implementation, the probability density integral equation of the spacecraft's orbital state response at the terminal moment is solved using the direct probability integral method, specifically: (5) In the formula, This represents the number of representative points of the initial orbital state deviation of the spacecraft. In this embodiment, a total of 1300 representative points of the initial orbital state deviation were generated. Indicates the first Each initial orbital state deviation represents a point. ; Indicates the first Each initial orbital state deviation represents the orbital state response of a point at the terminal time. Indicates the first Each initial orbital state deviation represents the probability assigned to a point. This represents the smoothing parameter of the Dirac function.

[0047] As one possible implementation, this embodiment is based on kernel density estimation and uses the following method to select the smoothing parameters of the Dirac function. : (6) In the formula, Represents the smoothing factor. ; express N The initial orbital state deviation of a spacecraft represents the standard deviation of the orbital state response at the terminal time. express N The initial orbital state deviation of a spacecraft represents the interquartile range of the orbital state response at the terminal moment.

[0048] Secondly, embodiments of the present invention provide a space target orbit deviation evolution analysis system, comprising: The deviation representative point acquisition unit configures the space target and its initial state nominal value. The input random vector space is constructed from the initial orbital state deviation of the space target. The input random vector space is then probabilistically partitioned using the generalized F-deviation point selection method to obtain the initial orbital state of the space target. N Each deviation represents a point And the assigned probability corresponding to each deviation point. , ; The initial orbital state deviation of the space target represents the acquisition unit, which will... N Each deviation represents a point Each value is added to the initial state nominal value of the space target to obtain the corresponding result. N The initial orbital state deviation of a space target with bias is represented by a point. And the assigned probability corresponding to the point representing the initial orbital state deviation of each space target. , Among them, the first The initial orbital state deviation of each space target represents a point With the Each deviation represents a point The corresponding assignment probabilities are the same, that is , ; The unit obtains the probability density integral equation of the orbital state response at the terminal moment of a space target, solves the orbital dynamics equation of the space target, calculates the orbital state response at the terminal moment corresponding to the representative point of the initial orbital state deviation of each space target based on the orbital dynamics equation, and then obtains the physical mapping between the representative point of the initial orbital state deviation of each space target and the corresponding orbital state response; configures the probability density function of the orbital state deviation at the initial moment of the space target, and based on the principle of probability conservation, uses the physical mapping between the representative point of the initial orbital state deviation of each space target and the corresponding orbital state response to establish the relationship between the probability density function of the orbital state response at the terminal moment of the space target and the probability density function of the orbital state deviation at the initial moment, and obtains the probability density integral equation of the orbital state response at the terminal moment of the space target. The probability density integral equation is a high-dimensional integral containing the Dirac function. The probability density function solving unit for the orbital state response at the terminal moment of the space target smooths the Dirac function and uses the assigned probability corresponding to the representative point of the initial orbital state deviation of the space target. It then uses the direct probability integral method to solve the probability density integral equation of the orbital state response at the terminal moment of the space target, thereby obtaining the probability density function of the orbital state response at the terminal moment of the space target, thus completing the evolution analysis of the orbital deviation of the space target.

[0049] This invention is a versatile and easy-to-operate methodology that can be applied to the problem of analyzing the orbital deviation evolution of space targets in non-Gaussian and strongly nonlinear systems. The obtained deviation evolution information can be further applied to the design of space missions such as space target collision early warning, orbital maneuver detection, and spacecraft trajectory planning. It has strong applicability and high computational efficiency and accuracy.

[0050] To verify the calculation accuracy of this method, this embodiment also uses 10 5 Monte Carlo simulation method of subsampling for considering Deviation evolution analysis was performed on a perturbed two-body orbital model of a spacecraft. Figures 9 to 14 Two methods were used to predict five orbital periods: the direct probability integral method and the Monte Carlo simulation method. , and The probability density function curves and cumulative distribution function curves of the spacecraft's position and velocity in three directions. Figures 15 to 18This document presents partial calculations of the probability density function and cumulative distribution function of the spacecraft at its terminal moment after predicting five orbital periods using both the direct probability integral method and the Monte Carlo simulation method. It shows that the results calculated by the direct probability integral method are largely consistent with those of the Monte Carlo simulation method, and the probability density function curve of the spacecraft's terminal state is no longer a Gaussian distribution, indicating that the distribution of the spacecraft's terminal state follows a non-Gaussian distribution. Furthermore, to quantitatively verify the accuracy of the probability density function of the spacecraft's orbital state at the terminal moment calculated by the direct probability integral method, this embodiment calculates the Kullback-Leibler relative entropies of the spacecraft's position and velocity probability density functions in each direction calculated by both methods. These entropies are 0.069004, 0.00077054, 0.011610, 0.0004765, 0.029059, and 0.032631, respectively. The very small relative entropies in each direction indicate that the calculation results of the two methods agree well. Therefore, the direct probability integral method can accurately predict the probability density function of the spacecraft's orbital state at the terminal moment in space.

[0051] Table 1 below shows the direct probability integral method and Monte Carlo simulation method for considering... The time required for deviation evolution analysis of the perturbed two-body orbit model of the spacecraft shows that the Monte Carlo simulation method takes approximately 34.06 times longer than the direct probability integral method. Therefore, the direct probability integral method can efficiently capture the non-Gaussian distribution characteristics of the spacecraft's terminal state deviation distribution.

[0052] Table 1. Direct probability integral method and Monte Carlo simulation method for considering Comparison of the time required for deviation evolution analysis of perturbed two-body orbit models of spacecraft In Table 1 above, T 1 represents the time required to generate representative points (generated by the direct probability integral method) or sample points (generated by the Monte Carlo simulation method). T 2 indicates that the calculation takes into account The time of the state response of the perturbed spacecraft two-body orbital dynamics equations. T 3 represents the time interval for calculating the probability density function curves and cumulative distribution function curves of the spacecraft's position and velocity at the terminal moment. T total for T 1. T 2 and T The sum of 3.

[0053] The proposed method for analyzing the evolution of orbital deviations of a space target utilizes probability space partitioning and Dirac function smoothing techniques, employing the direct probability integral method to solve the probability density integral equation of the orbital state response at the terminal moment, thereby obtaining the probability density function of the orbital state response at the terminal moment. Compared to Monte Carlo simulation methods, the proposed method based on the direct probability integral method can accurately and efficiently predict the probability density function and cumulative distribution function of the orbital state of the space target at the terminal moment, thus obtaining the statistical characteristics of the uncertainty of the space target's state at the terminal moment.

[0054] Although the invention has been described herein in conjunction with various embodiments, those skilled in the art will understand and implement other variations of the disclosed embodiments by reviewing the accompanying drawings, the disclosure, and the description of the drawings, in carrying out the claimed invention. In this specification, the word "comprising" does not exclude other components or steps, and "a" or "an" does not exclude multiple components. A single processor or other unit can implement several of the functions listed in the specification. While certain measures are described in different embodiments, this does not mean that these measures cannot be combined to produce good results.

[0055] Although the invention has been described in conjunction with specific features and embodiments, it is obvious that various modifications and combinations can be made therein without departing from the spirit and scope of the invention. Accordingly, this specification and drawings are merely illustrative of the invention and are considered to cover any and all modifications, variations, combinations, or equivalents within the scope of the invention. Clearly, those skilled in the art can make various alterations and modifications to the invention without departing from its spirit and scope. Thus, if such modifications and modifications fall within the scope of the invention and its equivalents, the invention is also intended to include such modifications and modifications.

Claims

1. A method for analyzing the evolution of orbital deviations of space targets, characterized in that, Includes the following steps: S1. Configure the space target and its initial state nominal values. The initial orbital state deviation of the space target forms an input random vector space. Using the generalized F-bias point selection method, the input random vector space is probabilistically partitioned to obtain the initial orbital state of the space target. N Each deviation represents a point And the assigned probability corresponding to each deviation point. ; S2. Will N Each deviation represents a point Each value is added to the initial state nominal value of the space target to obtain the corresponding result. N The initial orbital state deviation of a space target with bias is represented by a point. And the assigned probability corresponding to the point representing the initial orbital state deviation of each space target. Among them, the first The initial orbital state deviation of each space target represents a point With the Each deviation represents a point The corresponding assignment probabilities are the same, that is , ; S3. Solve the orbital dynamics equations of the space target, and calculate the orbital state response at the terminal moment corresponding to the initial orbital state deviation representative point of each space target based on the orbital dynamics equations, thereby obtaining the physical mapping between the initial orbital state deviation representative point of each space target and the corresponding orbital state response; S4. Configure the probability density function of the initial orbital state deviation of the space target. Based on the principle of probability conservation, utilize the physical mapping between the representative point of each initial orbital state deviation of the space target and the corresponding orbital state response to establish the relationship between the probability density function of the orbital state response at the terminal moment of the space target and the probability density function of the initial orbital state deviation. Obtain the probability density integral equation of the orbital state response at the terminal moment of the space target. The probability density integral equation is a high-dimensional integral containing the Dirac function. S5. Smooth the Dirac function, use the assigned probability corresponding to the point represented by the initial orbital state deviation of the space target, and use the direct probability integral method to solve the probability density integral equation of the orbital state response at the terminal moment of the space target, so as to obtain the probability density function of the orbital state response at the terminal moment of the space target, thus completing the evolution analysis of the orbital deviation of the space target. The probability density integral equation of the orbital state response of the space target at the terminal moment is obtained by the following method: in, The probability density function representing the orbital state response of the space target at the terminal moment. The independent variable represents the probability density function of the orbital state response of the space target at the terminal moment. The probability density function representing the initial orbital state deviation of a space target. This indicates the spatial target orbital state response at the terminal moment. For the Dirac function, Represents the input random vector space; When smoothing the Dirac function using a Gaussian function, the probability density integral equation of the orbital state response at the terminal moment of the space target is solved using the direct probability integral method, specifically: In the formula, It is a Gaussian function. This represents the number of points indicating the initial orbital state deviation of a space target. Indicates the first The initial orbital state deviation of each space target represents a point. , Indicates the first The initial orbital state deviation of each space target represents the orbital state response of that point at the terminal moment. Indicates the first The initial orbital state deviation of each space target represents the assigned probability of that point. This represents the smoothing parameter of the Dirac function.

2. The method for analyzing the evolution of orbital deviations of a space target according to claim 1, characterized in that, The probability space partitioning of the input random vector space is performed using the generalized F-bias point selection method, including: S10. Using the Sobol point set as the initial representative point set, the input random vector space is divided into multiple non-overlapping representative regions using the Voronoi unit partitioning method. This yields a uniformly distributed set of low-bias points. S11. Perform generalized F-bias minimization renormalization on a uniformly distributed set of low-bias points to obtain multiple representative bias points of the initial orbital state of the space target. S12. Calculate the probability of assigning a point representing the initial orbital state deviation of each space target.

3. The method for analyzing the evolution of orbital deviations of a space target according to claim 1, characterized in that, The orbital dynamics equations can be solved using any one of the following numerical integration methods: Runge-Kutta method, Nermark method, or central difference method.

4. The method for analyzing the evolution of orbital deviations of a space target according to claim 1, characterized in that, The Dirac function is smoothed using any one of the following functions: Gaussian function, trigonometric function, Poisson function, Cauchy function, and rectangular wave function.

5. The method for analyzing the evolution of orbital deviations of a space target according to claim 1, characterized in that, The following method is used to calculate the first... The initial orbital state deviation of each space target represents the assigned probability of the point: In the formula, The probability density function representing the initial orbital state deviation of a space target. Represents the i-th element in the input random vector space The initial orbital state deviation of each space target represents the representative region corresponding to the point.

6. The method for analyzing the evolution of orbital deviations of a space target according to claim 1, characterized in that, The smoothing parameters of the Dirac function are selected using the following method. : In the formula, Represents the smoothing factor. ; express N The initial orbital state deviation of each space target represents the standard deviation of the orbital state response at the terminal moment. express N The initial orbital state deviation of a space target represents the interquartile range of the orbital state response at the terminal moment.

7. A space target trajectory deviation evolution analysis system, characterized in that, include: The deviation representative point acquisition unit configures the space target and its initial state nominal value. The input random vector space is constructed from the initial orbital state deviation of the space target. The input random vector space is then probabilistically partitioned using the generalized F-deviation point selection method to obtain the initial orbital state of the space target. N Each deviation represents a point And the assigned probability corresponding to each deviation point. ; The initial orbital state deviation of the space target represents the acquisition unit, which will... N Each deviation represents a point Each value is added to the initial state nominal value of the space target to obtain the corresponding result. N The initial orbital state deviation of a space target with bias is represented by a point. And the assigned probability corresponding to the point representing the initial orbital state deviation of each space target. , Among them, the first The initial orbital state deviation of each space target represents a point With the Each deviation represents a point The corresponding assignment probabilities are the same, that is , ; The unit obtains the probability density integral equation of the orbital state response at the terminal moment of a space target, solves the orbital dynamics equation of the space target, calculates the orbital state response at the terminal moment corresponding to the representative point of the initial orbital state deviation of each space target based on the orbital dynamics equation, and then obtains the physical mapping between the representative point of the initial orbital state deviation of each space target and the corresponding orbital state response; configures the probability density function of the orbital state deviation at the initial moment of the space target, and based on the principle of probability conservation, uses the physical mapping between the representative point of the initial orbital state deviation of each space target and the corresponding orbital state response to establish the relationship between the probability density function of the orbital state response at the terminal moment of the space target and the probability density function of the orbital state deviation at the initial moment, and obtains the probability density integral equation of the orbital state response at the terminal moment of the space target. The probability density integral equation is a high-dimensional integral containing the Dirac function. The probability density function solving unit for the orbital state response at the terminal moment of the space target smooths the Dirac function and uses the assigned probability corresponding to the representative point of the initial orbital state deviation of the space target. It then uses the direct probability integral method to solve the probability density integral equation of the orbital state response at the terminal moment of the space target, thereby obtaining the probability density function of the orbital state response at the terminal moment of the space target, thus completing the evolution analysis of the orbital deviation of the space target.

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