Optical measurement segmental arc correlation method based on probability expression

Abstract

The invention discloses an optical measurement segmental arc correlation method based on probability expression, and the method comprises the steps: obtaining an initial orbit feasible region of a segmental arc through an admissible domain, carrying out the uniform distribution discrete sampling of the admissible domain, and describing the uncertainty distribution of the initial orbit of the segmental arc through a Gaussian mixture model. The uncertainty distribution and propagation of the orbit are described in a corrected spring equinox orbital element space, and the state transition tensor of a gravity potential model of any order is applied, so that the precision and speed of orbit uncertainty propagation are improved. In combination with a Gaussian mixture model and a state transition tensor, sparse sampling points are used as component mean propagation dense sampling points of the Gaussian mixture model, and uncertainty distribution of the initial orbit of the arc section is propagated to a common moment. And giving a definition of association probability measurement, calculating the association probability measurement by utilizing a probability density function of track distribution of the segmental arc at the public moment, and carrying out association judgment on the optical measurement segmental arc based on probability expression.

Description

Technical Field

[0001] The present invention discloses an optical measurement arc segment association method based on probability expression. It involves using a Gaussian mixture model to describe the uncertainty distribution of the arc segment, then propagating orbit uncertainty in the orbit element space, and using a probability density function to associate the optical measurement arc segment. It belongs to the field of aerospace technology. Background Art

[0002] As of September 2023, the number of human spacecraft in orbit has reached 8,800, and 1,893 payloads have been launched in 2023 alone. Satellites in the geosynchronous orbit belt are an important space asset because they can continuously monitor and communicate with the ground. In order to solve the orbital drift caused by disturbance forces, GEO satellites usually implement orbital maneuvers to maintain orbital stability. The congestion of the geosynchronous orbit belt and frequent orbital maneuvers increase the risk of close approaches and collisions. In addition, orbital transfers and military confrontation exercises by uncooperative satellites pose a huge threat to the safety of geosynchronous orbit satellites.

[0003] With the rapid growth of the number of space objects, the space environment has become increasingly crowded and complex. The number of newly launched satellites continues to increase year by year. By the end of 2023, there will be more than 34,000 space objects orbiting the Earth in space, and 4,460 new space targets will be added in 2023. With the continuous entry of manned spacecraft, space stations and various satellites into orbit, ensuring the safe operation of these satellites and spacecraft and more effectively utilizing the precious and limited space orbital resources requires a more complete and accurate understanding of the space environment. Therefore, the continuous improvement of the optical monitoring capability of space objects is crucial for current and future space missions, and one of the key links is the correlation of optical measurement arcs, which is an important part of the field of space situational awareness.

[0004] Space surveillance combines the detection, tracking, identification and cataloging of satellites. It is an important issue in space activities. Space surveillance is mainly driven by military needs, but it is becoming increasingly important for civilian needs. The surveillance of geosynchronous orbit (GEO) targets is mainly carried out through ground-based or space-based optical observations. At present, China's detection and surveillance of geosynchronous orbit targets mainly rely on ground-based optical surveillance systems, which are affected by atmospheric and meteorological conditions and cannot observe all day long. Due to the advancement of optical sensors and the deployment of space-based platforms, the optical detection capability of space-based space surveillance (SBSS) for space targets has been further enhanced. Moreover, compared with ground-based observations, space-based surveillance systems are not restricted by the atmosphere and geographical location. They have the advantages of strong orbital observation capabilities, long observation distances, short repeated observation cycles, low equipment complexity, low power consumption, and all-weather observation. The space-based space surveillance system uses visible light sensors on satellites to obtain optical measurement data of space objects, and then determines the orbit of space objects based on these measurement data, thereby providing support for cataloging space objects, spacecraft collision avoidance, and enhancing space situational awareness capabilities.

[0005] The initial orbit obtained by the single arc segment initial orbit determination method under space-based optical observation generally has a large error, and the arc segment association also involves complex orbit perturbation problems and a huge number of initial orbits. It is not easy and unrealistic to use only a single arc segment independent calculation to achieve the cataloging of space objects. It has gradually become an important research content in the field of space situational awareness to gradually determine new space objects and catalog them through the association solution of non-associated arc segments of uncataloged objects that are not in the space object catalog. In terms of association based on admissible domain, in 2012, K. Fujimoto et al. proposed a method to associate multiple optical observations using the probability distribution defined by the admissible domain in the Poincare orbital element space. The method reduces the computational cost by dividing the admissible domain into smaller sub-intervals and linearly mapping them, and its accuracy will not be significantly reduced. In 2014, K. Fujimoto et al. proposed a direct Bayesian admissible domain method, which can effectively deal with observation data errors by combining the least squares method. Finally, this hybrid method can effectively reduce the proportion of false positives in the association results. In 2016, Marcus J. Holzinger et al. introduced and discussed a method for probabilistic classification and prioritization of non-associated arcs using admissible domain and Bayesian theory without prior information. This method can also be further extended to the associated classification process with prior information. In 2020, Jiang Ping et al. optimized the orbits of two observation arcs based on the admissible domain method to determine their association. At the same time, they proposed a method for false association identification, which effectively improved the association success rate by analyzing the characteristics of angle errors. In terms of initial value and boundary value problems, in 2014, J. A. Siminski et al. restricted the four-dimensional state of the observed target by the angle and angle change rate of an observation arc, and used two arcs to construct initial value problem and boundary value problem methods respectively to associate the two arcs. In the initial value problem, the observation data of the first arc segment is used as the initial value, propagated to the moment of the second arc segment, and then the observation covariance matrix at the same moment is used to construct the loss function to determine whether the two arc segments are associated; in the boundary value problem, the loss function under the boundary value problem is constructed by solving the Lambert problem to determine whether they are associated. In 2018, H. Cai et al. improved the solution to the traditional initial value problem in the association of short arc segments. This scheme determines whether the arc segments are associated by searching and defining a new loss function in the non-singular canonical space and finding the global minimum. In 2021, Huang Qiushi et al. used the joint positioning method of multi-star sensors to convert the optical observation quantity into the position vector of the observed target, took a point from each arc segment to be associated as the hypothetical boundary and determined the orbit, and determined whether the arc segments are associated by calculating the Mahalanobis distance.Correlating arc segments based on covariance propagation is a classic correlation method. In 2008, K. Hill et al. proposed an algorithm for processing radar arc segment correlation. By propagating two initial orbits and their covariances to a common time, the Mahalanobis distance between the two initial orbit state quantities after propagation is calculated using an equiprobable ellipsoid under a Gaussian distribution to determine the correlation between the initial orbits. In 2012, K. Hill et al. found that in covariance correlation, the choice of parameters used to represent the orbit state has a great influence on the results. By using many different orbital parameters and propagation methods, a 10-day simulation of 1,000 space targets within a range of 20 kilometers on the semi-major axis was conducted, and compared with the correlation method using a fixed position threshold. The results show that the use of UT transformation and elliptic curve coordinate system has good results. In addition, the UT transformation propagation effect using the average number of points of orbital roots is the best. In 2013, C. Sabol et al. proposed that if the measurement error is a zero-mean Gaussian error, the orbit error represented by the covariance statistics is also a Gaussian error. When the orbit error becomes too large and is no longer Gaussian, the orbit error cannot be represented by the covariance, which will have a negative impact on the arc association process. In 2018, Yan Ruidong et al. established a covariance generation and evolution algorithm based on linear and unscented Kalman filtering, which was applied to arc association. Through simulation of 20 satellites, their orbital states can be fully associated.

[0006] Regarding the orbit uncertainty propagation method when arc segments are associated, in 2006, Ryan S. Park et al. discussed the high-order state transition tensors (STTs) that describe the local nonlinear motion of the nominal orbit. Through the analytical representation of nonlinear uncertainty propagation, the results showed that the high-order state transition tensor can be used for semi-analytical methods to replace Monte Carlo simulation. In 2012, K. Fujimoto et al. used the Poincare orbital elements and state transition tensor method based on two-body dynamics to describe the nonlinear propagation of uncertainty, in which the probability density function (PDF) and mean covariance can be expressed analytically. In 2013, Brandon A. Jones et al. used a non-intrusive polynomial chaos expansion method, that is, a sampling-based method. In the scenario of sun-synchronous orbit and lightning orbit propagation for up to 10 days, this method has fewer samples than the MC method and can achieve the expected accuracy. In 2013, Kyle J. DeMars et al. proposed a Gaussian mixture method based on adaptive entropy, which can more accurately approximate the probability density function. In addition, this method is different from the standard Gaussian mixture model. It can replace the components that produce nonlinear effects with smaller Gaussian components when necessary through nonlinear detection. In 2015, Fujimoto et al. used a Gaussian mixture model to describe the orbital state distribution of space objects under the two-body hypothesis, and used the state transfer tensor of the reference orbit to propagate uncertainty for all Gaussian components. In 2021, Yan et al. used weighted Gaussian mixture components to fit the initial Gaussian distribution when predicting the collision of space debris, and used an appropriate number of Gaussian mixture components to approximate the initial orbital covariance. In 2024, Acciarini et al. converted the Fokker–Planck equation into a set of linear ordinary differential equations in the context of orbital dynamics, decoupled the time- and state-related elements, and maintained the overall probability density function of the state throughout time, and propagated the uncertainty distribution nonlinearly.

[0007] Based on the research idea of ​​extending the covariance correlation to the arc segment correlation under the non-Gaussian distribution assumption, this paper designs an optical measurement arc segment correlation method based on probability representation. This method samples the initial orbit distribution of the optical measurement arc segment and describes its initial probability density function with a Gaussian mixture model. The initial distribution of the arc segment is propagated to the same epoch time using the state transfer tensor or the separate propagation method, and the probability density function of the two arc segments is used to calculate their correlation probability metric. According to the pre-given metric threshold, the correlation between the optical measurement arc segments is judged. Summary of the invention

[0008] (I) Purpose of the invention

[0009] In order to provide correlation technical support for the detection, tracking, identification and cataloging of space targets in space surveillance missions, and to achieve the description of the correlation between optical measurement arc segments from a probabilistic level in non-Gaussian situations, thereby improving the accuracy of arc segment correlation, the present invention proposes an optical measurement arc segment correlation method based on probabilistic representation.

[0010] (II) Technical solution

[0011] The present invention proposes an optical measurement arc segment association method based on probability expression, which comprises the following steps:

[0012] Step 1: Obtain the initial uncertainty distribution of the optical measurement arc

[0013] The admissible region method is used to obtain the feasible region of the initial orbit of the optical measurement arc segment and perform uniform distribution sampling. Sparse sampling points and dense sampling points are obtained by sampling at sparse intervals and dense intervals respectively. The Gaussian mixture model is used to describe the uncertainty distribution of the admissible region of the arc segment or the initial orbit.

[0014]

[0015] The vector γ corresponding to the admissible domain includes the distance, angle and its rate of change. is the Gaussian probability density function. Fitting the optical measurement arc segment can obtain the vector γ

[0016]

[0017] It includes distance, distance change rate, right ascension, right ascension change rate, declination, and declination change rate. Its probability density function can be approximated by the Gaussian mixture model as follows:

[0018]

[0019] Step 2: Description of the initial uncertainty distribution of the arc segment in the root number space

[0020] Introducing the modified vernal equinox orbital element X MEE

[0021]

[0022] The initial uncertainty distribution of the arc segment is transformed from the "range-range change rate" space to the "corrected equinox orbit element" space. Each dense sampling point represents the mean of a Gaussian mixture model component, whose corrected equinox orbit element X MEE The corresponding covariance matrix is

[0023]

[0024] Step 3: Build a state transition tensor model

[0025] Applying the Gravitational Potential Model

[0026]

[0027] Using the gravitational potential of any order to calculate the position and velocity as the state quantity

[0028]

[0029] Thus, the third-order state transfer tensor of position and velocity is calculated as the state quantity The state transition tensor needs to be calculated using its ordinary differential equation

[0030]

[0031]

[0032] The differential model is further used to calculate the third-order state transfer tensor with the number of orbital elements of the corrected vernal equinox as the state quantity.

[0033] Step 4: Propagation of initial uncertainty distribution of arc segments

[0034] Calculate the third-order state transfer tensor of each sparse sampling point at the corrected vernal equinox orbital root, and use the sparse sampling points to propagate the mean and covariance matrix of all dense sampling points, propagate the Gaussian mixture model of the initial orbit of the arc segment to the common time, and calculate the probability density function. The state deviation at any epoch can be expressed as

[0035]

[0036] The state transfer matrix required to calculate the deviation point covariance matrix can be approximated as

[0037]

[0038] Step 5: Calculate the association probability metric and perform optical measurement arc segment association judgment

[0039] At the common moment, the two admissible domains "intersect" at the true orbit state point. Based on this idea, the likelihood score of correlation (LSC) can be defined as

[0040]

[0041] Wherein, subscripts a and b represent two optical observation arcs. The association probability measure under non-Gaussian distribution is expressed as

[0042]

[0043] Using a pre-given threshold of the association probability measure, if the joint probability measure of two arcs is higher than the threshold, it is considered that they are associated or belong to the same spatial target.

[0044] (III) Advantages

[0045] The advantages of the optical measurement arc segment association method based on probability expression provided by the present invention are:

[0046] ① The method proposed in the present invention introduces the modified vernal equinox orbital roots, which have small nonlinearity in the orbital equation, are suitable for orbital uncertainty propagation, and clarify the conversion relationship between them and the position and velocity state quantities.

[0047] ② The method proposed in this invention combines the orbital dynamics model, proposes a state transfer tensor using the gravity potential model, and provides a calculation method for the state transfer tensor with position velocity as the state quantity and the number of orbital elements of the modified vernal equinox as the state quantity. When the third-order state transfer tensor is used to propagate the Gaussian component of the Gaussian mixture model, the propagation speed can be greatly improved, and the rapid propagation of the orbital uncertainty distribution can be achieved.

[0048] ③ The method proposed in the present invention defines an associated probability metric. Under the Gaussian distribution assumption, the associated probability metric is related to the Mahalanobis distance and can be converted into each other. Under the non-Gaussian distribution assumption, the orbit uncertainty propagation of the arc segment is carried out with the help of the Gaussian mixture model and the state transfer tensor, and the associated probability metric is calculated, which overcomes the problem that traditional methods are difficult to deal with non-Gaussian nonlinear uncertainty propagation. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] Figure 1 The present invention is a schematic diagram of optical observation of space targets.

[0050] Figure 2 It is a schematic diagram of arc segment allowable domain sampling at different sampling intervals involved in the present invention.

[0051] Figure 3 It is a schematic diagram of the probability density function of the admissible domain approximated by the Gaussian mixture model involved in the present invention.

[0052] Figure 4 It is a schematic diagram of mapping the sampling points of the admissible domain involved in the present invention to the orbital elements of the modified vernal equinox.

[0053] Figure 5 It is a schematic diagram of the distribution results of Gaussian mixture components of the separate propagation at the reference epoch and the propagation based on the state transfer tensor involved in the present invention.

[0054] Figure 6It is a schematic diagram of the calculation result of the association probability measurement involved in the present invention.

[0055] Figure 7 It is a schematic flow chart of the method of the present invention. DETAILED DESCRIPTION

[0056] The technical solution and algorithm flow of the present invention will be described in detail and completely below with reference to the accompanying drawings.

[0057] The present invention discloses an optical measurement arc segment association method based on probability expression. Figure 1 As shown in Figure 1, the position coordinates of the space-based optical sensor located on the LEO satellite in the ECI coordinate system are considered as known information, which can be provided by the GPS precision positioning system in practical applications. The optical measurement information of the space target is obtained through the optical sensor, including right ascension and declination and their corresponding time series.

[0058] Step 1: Obtain the initial uncertainty distribution of the optical measurement arc

[0059] Based on the optical observation arc, the admissible region method is used to obtain the feasible region of the initial orbit of the optical measurement arc and perform uniform distribution sampling. Sparse sampling points and dense sampling points are obtained by sampling with sparse intervals and dense intervals respectively. The Gaussian mixture model is used to describe the uncertainty distribution of the arc admissible region or the initial orbit. For example, the sampling interval Δρ and When the speeds are 20 km and 20 m / s respectively, the sampling results are as follows: Figure 2 As shown; the probability density function of the admissible domain approximated by the Gaussian mixture model is as follows Figure 3 shown.

[0060] Step 2: Description of the initial uncertainty distribution of the arc segment in the root number space

[0061] Introducing the modified vernal equinox orbital element X MEE After that, the initial uncertainty distribution of the arc segment is converted from the "distance-distance change rate" space to the "corrected equinox orbit element" space. Each dense sampling point represents the mean of a Gaussian mixture model component. In the corrected equinox orbit element space, the distribution of dense and sparse sampling points of the optical measurement arc segment is as follows: Figure 4 shown.

[0062] Step 3: Build a state transition tensor model

[0063] Introduce and apply the gravity potential model to calculate the position and velocity as the state quantity using the gravity potential

[0064]

[0065] Thus, the third-order state transfer tensor of position and velocity is calculated as the state quantity The differential model is further used to calculate the third-order state transfer tensor with the number of orbital elements of the corrected vernal equinox as the state quantity. That is, the orbital elements X can be given as follows MEE With position speed X RV The p-order partial derivative relationship between

[0066]

[0067] These two equations can be calculated by difference. From position velocity X RV The state transition tensor can be converted to the track element X MEE The state transition tensor

[0068]

[0069]

[0070] Step 4: Propagation of initial uncertainty distribution of arc segments

[0071] The third-order state transfer tensor of each sparse sampling point at the corrected vernal equinox orbital root is calculated, and the mean and covariance matrix of all dense sampling points are propagated using the sparse sampling points to propagate the Gaussian mixture model of the initial orbit of the arc segment to the common time. Examples of the mean distribution of dense sampling points based on individual propagation and state transfer tensor propagation are shown below: Figure 5 As shown, and the probability density function is calculated.

[0072] Step 5: Calculate the association probability metric and perform optical measurement arc segment association judgment

[0073] At the common moment, the two admissible domains "intersect" at the real orbit state point. Based on this idea, the association probability metric is defined. The association probability metric between all optical measurement arc segments is calculated under the assumption of non-Gaussian distribution, such as Figure 6 This is an example of the association probability measurement between the first arc and other arcs based on the state transfer tensor propagation, where the horizontal axis represents the relative epoch time of each arc within a day. Taking the space target with NORAD number 53148 as an example, the circle pattern represents the arc belonging to the target. When the threshold of the given association probability measurement is 10 6 When , all arcs above the threshold are considered to be associated with the first arc, and erroneous arcs are excluded.

Claims

1. An optical measurement arc segment association method based on probability expression, characterized by: Firstly, the admissible region method is used to obtain the feasible region of the initial orbit of the optical measurement arc segment and perform uniform distribution sampling. Sparse sampling points and dense sampling points are obtained by sampling with sparse intervals and dense intervals respectively. The Gaussian mixture model is used to describe the uncertainty distribution of the arc segment admissible region or the initial orbit. Secondly, the modified vernal equinox orbital elements are introduced to transform the initial uncertainty distribution of the arc segment from the "distance-distance change rate" space to the "modified vernal equinox orbital elements" space, and an arbitrary order gravity potential model is introduced to calculate the state transfer tensor with position and velocity as the state quantity, and further calculate the state transfer tensor with the modified vernal equinox orbital elements as the state quantity; Thirdly, the third-order state transfer tensor of each sparse sampling point at the corrected vernal equinox orbital root is calculated, and the mean and covariance matrices of all dense sampling points are propagated using the sparse sampling points, the Gaussian mixture model of the initial orbit of the arc segment is propagated to the common moment, and the probability density function is calculated; Finally, the association probability metric is calculated according to the probability density function of the arc segment's initial orbit propagating to the common time, and the arc segment is associated using the pre-given threshold of the association probability metric.

2. The optical measurement arc segment association method based on probability expression according to claim 1 uses the state transition tensor of the orbital elements at the corrected vernal equinox point, characterized in that: Introducing and applying the gravity potential model Using gravitational potential to calculate position and velocity as state quantity Thus, the third-order state transfer tensor of position and velocity is calculated as the state quantity The state transition tensor needs to be calculated using its ordinary differential equation Use the difference to calculate the orbital elements X as follows MEE With position speed X RV The p-order partial derivative relationship between Further calculation of the third-order state transfer tensor of the modified vernal equinox orbital elements as state quantity 3. The optical measurement arc segment association method based on probability expression according to claim 1 uses association probability measurement, characterized in that: At the common time, the probability measure of the association between arc segments a and b in the modified equinox orbital element space is The association probability measure using the Gaussian mixture model under the non-Gaussian distribution assumption is expressed as By using a pre-given threshold of the association probability measure, if the joint probability measure of two arcs is higher than the threshold, it is determined that they are associated or belong to the same spatial target.

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