Space target detectable probability prediction method considering state uncertainty
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- BEIJING INST OF TECH
- Filing Date
- 2023-10-24
- Publication Date
- 2026-06-09
Smart Images
Figure CN117408057B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to a method for predicting the detectability probability of a space target considering state uncertainty, belonging to the field of space target state estimation. Background Technology
[0002] With the increasing frequency of space activities and the large-scale deployment of low-Earth orbit (LEO) mega-constellations, the number of near-Earth space targets is rapidly increasing, leading to a more crowded space environment. The large number and high speed of LEO targets pose a serious threat to human space activities if they are not accurately and efficiently tracked to avoid potential space traffic accidents. Compared to traditional ground-based systems, space-based space target tracking systems offer significant advantages in accuracy and timeliness due to their all-weather, 24 / 7 operation. However, in large-scale space target tracking, limitations such as the rapid changes in the relative positions of sensors and space targets, and the time intervals between sensor observations, necessitate estimation and prediction of the space target's state within a filtering framework. During this state transfer process, the position, velocity, and other states of the space target are uncertain, and the currently widely used geometrically based space-based observation models are insufficient to accurately characterize the space target's state. Therefore, it is necessary to consider the uncertainty of the space target's state and establish a space-based observation model based on probabilistic relationships to improve the accuracy of space target state estimation. Summary of the Invention
[0003] The purpose of this invention is to provide a method for predicting the detectability probability of space targets that considers state uncertainty. This method incorporates the state uncertainty of space targets into a space-based observation model, establishes a dynamic model of the space-based tracking system and a space target state estimation model, and, based on a geometrically based space-based observation model, obtains a probabilistically based space-based observation model by combining the Monte Carlo random sampling concept. This yields a numerically continuously distributed detectability probability of space targets, thereby improving the accuracy of space target state estimation.
[0004] The objective of this invention is achieved through the following technical solutions.
[0005] This invention discloses a method for predicting the detectability probability of a spatial target considering state uncertainty, comprising the following steps:
[0006] Step 1: Large-scale space target tracking is carried out using a space-based multi-sensor tracking system. Cataloged targets are selected as space targets to be observed, and the Walker constellation configuration, which is uniformly distributed in space, is used as the basic configuration of the space-based observation constellation. Dynamic models of the space target state and the observation satellite state are constructed, and the state description of the space-based tracking system is realized based on the dynamic model.
[0007] A space-based tracking system consists of multiple satellites, which are distributed in a specific configuration in space to ensure coverage and stability. The Walker constellation configuration consists of several circular orbits, with satellites evenly distributed within each orbital plane, enabling global coverage and uniform revisit. The space-based tracking system adopts the Walker configuration.
[0008] The two-body elliptical orbit of a single satellite in a space-based tracking system is described by a set of Keplerian orbital parameters (a,e,i,Ω,ω,θ), namely, the semi-major axis a, eccentricity e, inclination i, right ascension of the ascending node Ω, argument of perigee ω, and true anomaly θ. The semi-major axis a and eccentricity e together determine the size and shape of the orbit; the orbital inclination i is the angle between the satellite's orbital plane and the Earth's equatorial plane; the right ascension of the ascending node Ω is the angle between the vernal equinox of the Earth's equatorial plane and the ascending node of the satellite's orbit with respect to the Earth's center O, indicating the azimuth of the satellite's orbital plane within the Earth's equatorial plane; simultaneously, the orbital inclination i and the right ascension of the ascending node Ω together determine the position of the satellite's orbital plane; the argument of perigee ω is the angle between the perigee and the ascending node within the satellite's orbital plane, indicating the azimuth of the major and minor axes of the elliptical orbit; and the true anomaly θ is the angle between the satellite's position within the orbital plane and the perigee, indicating the satellite's position within its orbit.
[0009] The configuration of all satellites in the space-based tracking system is described by a set of Walker constellation parameters (N, P, F), namely the total number of satellites N, the number of orbital planes P, and the phase factor F. The total number of satellites N and the number of orbital planes P together determine the number of satellites S = N / P in each orbital plane; the phase factor F represents the phase relationship between corresponding satellites on adjacent orbital planes, and the phase difference Δu between corresponding satellites on adjacent orbital planes is...
[0010]
[0011] For satellite i (i = 1, 2, ..., N) in the space-based tracking system s Its right ascension of the ascending node and argument of the perigee are expressed as follows:
[0012]
[0013] Since the satellites in the Walker constellation have the same orbital semi-major axis a and inclination i, and the eccentricity e = 0, and the initial true anomaly angle θ = 0, therefore, the right ascension Ω of the ascending node of satellite i given by equation (2) is... i and perigee argument ω i That is, the position and velocity of each satellite in the orbital plane coordinate system are obtained from the Kepler orbital parameters of each satellite.
[0014] The origin of the satellite's orbital plane coordinate system is located at the Earth's center, Xp The axis points to the perigee of the orbital plane, Y p The axis lies within the orbital plane and is perpendicular to X. p Axis, Z p The axis is perpendicular to the orbital plane and conforms to a right-handed coordinate system. The elliptical orbit of the satellite is approximated as a circular orbit; the angular velocity of the satellite in this circular orbit is the same as its average angular velocity in the elliptical orbit, and the period of motion of the satellite in the circular orbit is the same as its period in the elliptical orbit. If the true anomaly angle θ of the satellite at time k is ∠NOS, then the corresponding deviated anomaly angle E of the satellite in the circular orbit is ∠NCQ.
[0015] At the initial time k0, the real satellite and the satellite in the circular orbit pass through the perigee simultaneously and in the same direction. Therefore, at time t, the mean perigee angle M of the real satellite is defined as the angular distance of the satellite in the circular orbit, i.e.
[0016] M=n(t-k0) (3)
[0017] Where n is the average angular velocity of the actual satellite moving in an elliptical orbit. The mean anomaly angle M is a hypothetical auxiliary quantity and does not correspond to an angle in the actual physical sense. According to Kepler's third law of orbital motion, the average angular velocity n of the satellite in orbit has the following relationship with its semi-major axis a.
[0018]
[0019] Where G is the gravitational constant; M e For Earth's mass; μ e is the gravitational constant.
[0020] The relationship between the anomalous angle E and the level anomalous angle M is as follows:
[0021] M = Ee sin E (5)
[0022] The position r of the satellite in the orbital plane coordinate system in the space-based tracking system p Represented as
[0023]
[0024] Differentiating equation (6) yields the velocity of the satellite in the orbital plane coordinate system within the space-based tracking system.
[0025]
[0026] According to equations (6) and (7), the satellite's position and velocity in the orbital plane coordinate system are... Based on the transformation relationship between the orbital plane coordinate system and the geocentric inertial coordinate system, the satellite's position and velocity in the geocentric inertial coordinate system are obtained.
[0027] Coordinate transformation matrix from orbital plane coordinate system to geocentric inertial coordinate system Represented as
[0028]
[0029] in,
[0030]
[0031] The position and velocity of the satellite in the geocentric inertial coordinate system in the space-based tracking system They are respectively represented as
[0032]
[0033] If a satellite in a space-based tracking system is only subject to Earth's gravity and not to other perturbations, then all satellites in the space-based tracking system satisfy the following dynamic equations.
[0034]
[0035] Among them, r0 and The initial position and velocity of the satellite are obtained based on the parameters of the space-based tracking system and equations (6), (7), and (10). Equation (11) is integrated to obtain the dynamic model of any satellite in the space-based tracking system, thus realizing the state description of the space-based tracking system.
[0036] Other perturbation forces include atmospheric drag and solar radiation pressure.
[0037] Step 2: Model the space target tracking problem as a correspondence between the observation space and the state space of the space target. Since only part of the space target state is observable, the Bayesian recursive method is used to estimate the space target state, obtain a set of space target state information in the time series, and establish a space target state estimation model.
[0038] In a space target tracking system, the intrinsic relationship between a space target and time is the state change of the space target over time. The space target tracking problem can be modeled as a correspondence between the observation space and the state space of the space target. In the state space, the state of the space target transitions from the previous time k-1 to the next time k. However, in the observation space, since only a portion of the space target's state can be observed, a Bayesian recursive method is used to estimate the space target's state to obtain a set of space target state information over time.
[0039] For a discrete-time state-space system, the state at time k is determined by the random variable X. kThis indicates that the state vector propagates in the state space along with the Markov transfer function. At each time k, a noisy observation process produces an observation Z. k If the state information is correlated with the prediction process of the Bayesian filter through a likelihood function, then the prediction process is expressed as follows:
[0040] P k|k-1 (X k |Z 1:k-1 )=∫f k|k-1 (X k |X)P k-1 (X|Z 1:k-1 )dX (12)
[0041] Among them, f k|k-1 (X k |X) is the state transition function; P k|k-1 Let Z be the prior probability density distribution at time k, i.e., without merging the observations at time k. k Estimation of the probability density distribution of information. The update process of the Bayesian filter is expressed as follows:
[0042]
[0043] Among them, g k Let P be the likelihood function. At time k-1, all information is governed by a probability density distribution P. k-1 This means that at time k, a new observation value Z is obtained. k The purpose of Bayesian estimation is to estimate P... k-1 With Z k The information contained herein is fused together to obtain the posterior probability density distribution P at time k. k .
[0044] The prediction part of the Kalman filter covers the dynamics model of the linear system, expressed as:
[0045] X k|k-1 =F k X k-1 +B k u k +w k (14)
[0046] Among them, F k B is the state transition matrix; k To control the input model; u k For control variables; w k It is process noise, and
[0047] The update section covers the linear observation model, denoted as
[0048]
[0049] Among them, H k For observation model; v k To observe noise, and
[0050] The prediction and update process is summarized as follows:
[0051]
[0052] If the Kalman gain K k The larger the value, the larger the change in the posterior estimate. To address nonlinear system problems, the extended Kalman filter linearizes the nonlinear system by performing a first-order Taylor expansion of the dynamic and observation models, ensuring that the Gaussian distribution remains Gaussian after the linear transformation, thus processing the observation information of the nonlinear system.
[0053] The nonlinear system dynamics model of the extended Kalman filter is expressed as follows:
[0054] X k|k-1 =f(X) k-1 ,u k )+w k (17)
[0055] Where f is the dynamic model; w k It is process noise, and
[0056] The nonlinear observation model of the extended Kalman filter is expressed as:
[0057]
[0058] Where h is the observation model; v k To observe noise, and
[0059] For any spatial target to be tracked, the prediction and update process of the extended Kalman filter can be expressed as follows:
[0060]
[0061] Here, f cannot be directly used for covariance estimation; the Jacobian matrix needs to be calculated.
[0062]
[0063] The advantage of the Extended Kalman Filter (EKF) lies in its ability to locally linearize the nonlinear equations through Taylor expansion, allowing the Kalman filter framework to be applied to nonlinear systems and achieving good filtering results when the system's nonlinearity is not very strong. However, for systems with strong nonlinearity, the linear function cannot approximate the error propagation function well, leading to divergence in the state estimation of the spatial target.
[0064] Therefore, an unscented Kalman filter is used for the space target, approximating a Gaussian distribution with a fixed number of parameters, making the nonlinear system equations applicable to the standard Kalman system under the linear assumption. A set of Sigma sampling points is taken from the original distribution of the space target, where the mean and covariance of the Sigma points are equal to those of the original distribution. The Sigma points are substituted into the nonlinear function to obtain the corresponding set of nonlinear function values. The mean and covariance of the transformed space target are then calculated using the Sigma point set.
[0065] For an L-dimensional variable x, take 2L+1 Sigma points {χ i}, i = 0, ..., 2L, where each Sigma point has first-order and second-order weights, such that
[0066]
[0067] The transformed mean and covariance are
[0068]
[0069] Among them, y i =f(χ) i ), i = 0, ..., 2L.
[0070] The prediction process of the unscented Kalman filter for any spatial target to be tracked is as follows:
[0071] ① Generate Sigma points based on the posterior estimate at time k-1.
[0072] (X k-1 ,P k-1 )→{χ i} (twenty three)
[0073] ② Perform nonlinear state transitions
[0074] X i =f(χ) i ) (twenty four)
[0075] ③ Obtain the prior state estimate and the prior estimate covariance.
[0076]
[0077] Then the update process begins:
[0078] ① Generate Sigma points based on prior estimates
[0079] (X k|k-1 ,P k|k-1 )→{χ i} (26)
[0080] ② Obtain the nonlinear observation model
[0081] Z i =h(χ i (27)
[0082] ③ Obtain the simulated observations, the covariance of the simulated observations, and the cross covariance.
[0083]
[0084] ④ Calculate the Kalman gain
[0085]
[0086] By using equations (12) to (29), the spatial target tracking problem is modeled as a correspondence between the observation space and the state space of the spatial target, a set of spatial target state information in the time series is obtained, and a spatial target state estimation model is established.
[0087] Step 3: Establish a space-based observation model based on geometric relationships and determine the detectability of space targets relative to sensors.
[0088] Whether a space target can be observed by the sensor within a time window depends on the visibility p of the space target relative to the sensor. v,k The judgment is given by the following expression:
[0089] p v,k =(1-p s,k (1-p) b,k (30)
[0090] Where, p s,k Whether a space target is covered by Earth's shadow is determined by the geometric relationship between the Sun, the space target, and Earth. When the space target is not in the shadow cast by the Sun on Earth, p s,k p is 0 if it is not 1 otherwise b,k Whether a space target is obscured by the Earth is determined by the geometric relationship between the sensor, the space target, and the Earth. If the space target is not obscured by the Earth within the sensor's field of view, then p... b,k It is 0 if it is not 0, otherwise it is 1.
[0091] p s,k and p b,k The calculation is defined as a "0-1 model". Calculate ps,k It consists of two steps. The first step defines the length of the perpendicular line from the Earth's center to the line connecting the Sun and the space target as d1, and the Earth's radius as R. e Then p s,k Represented as
[0092]
[0093] When d1>R e The space target is not in the shadow cast by the sun on Earth, i.e., p s,k = 0. However, when d1 ≤ R e Whether a space target lies in the shadow depends on its position relative to the Sun and Earth. Let d0 be the distance from the center of the Sun to a straight line d1, and d2 be the distance from the center of the Sun to the space target. By comparing the lengths of d2 and d0, we can determine whether the space target is obscured by the Earth's shadow.
[0094]
[0095] Calculate p b,k It consists of two steps. The first step defines the length of the perpendicular line from the Earth's center to the line connecting the sensor and the space target as d1, and the Earth's radius as R. e Then p b,k Represented as
[0096]
[0097] When d1>R e The space target is not in an area that may be obscured by the Earth, i.e., p s,k = 0. However, when d1 ≤ R e Whether a space target is located within the Earth's occlusion zone depends on its position relative to the sensor and the Earth. Let d0 be the distance from the sensor to line d1, and d2 be the distance from the sensor to the space target. By comparing the lengths of d2 and d0, we can determine whether the space target is occluded by the Earth.
[0098]
[0099] All parameters in equations (30)-(34) are determined by the optimization variables (h,i,Ω,N). orb N spo It is obtained through geometric transformation.
[0100] Step 4: Based on the space-based observation model established in Step 3, and combined with the space-based tracking system dynamic model and space target state estimation model established in Steps 1 and 2, considering the uncertainty of space target state, establish a space-based observation model based on probability. Based on the space-based observation model based on probability, obtain the numerically continuously distributed probability of space target detection, that is, realize the prediction of the probability of space target detection.
[0101] Because the large-scale space target tracking problem uses a limited number of space-based optical sensors to track a large number of space targets, a large number of space targets frequently enter and exit the sensor's field of view. Therefore, at time k, correctly calculating the detectability p of the space targets is crucial. D,k Measuring the presence of space targets is crucial for accurately estimating their number. However, due to variations in lighting conditions and the relative distance between the sensor and the space target, the p-value of the space target... D,k It is time-varying. Therefore, it is necessary to consider p. D,k Develop more accurate prediction models.
[0102] The detectability of a space target is determined solely by its visibility relative to the sensor. In practical missions, the apparent magnitude of a space target relative to the sensor significantly impacts the quality of observation. The detectability p of a space target... D,k It consists of three parts: the probability of successful detection of a space target, p D The apparent magnitude p of a space target relative to a sensor am,k And the visibility p of space targets relative to sensors v,k Its calculation expression is:
[0103] p D,k =p D p am,k p v,k (35)
[0104] Where, p D The constant value used to measure the sensor's performance is [0,1]; p am,k Calculated based on the relative positions of space targets, sensors, and the sun; p v,k =(1-p s,k (1-p) b,k ), and p s,k and p b,k Determined by the "0-1 model".
[0105] Based on the uncertainty and probability theory of space target state, p s,k and p b,k The "0-1 model" is refined into and The "0-1 model" is used to obtain the detectability p relative to space targets. D,k More accurate detectable probability in, and , representing the probability that a space target is in Earth's shadow and is obscured by Earth, respectively, with values ranging from [0,1].
[0106] calculate This requires two steps. The first step is to define σ as the maximum two-dimensional standard deviation of the three-dimensional covariance of the space target on the plane of the Sun, Earth, and space target. Since the distance between the Earth and the Sun is much greater than the distance between the Earth and the space target, this is determined by comparing |d1-R... e The relationship between |σ| and |σ| can be used to approximate whether a space target is within the critical region. Represented as
[0107]
[0108] in, This represents the probability that a space target in the critical region is in the Earth's shadow. It can be observed that if |d1-R e If |≤σ, then the space target may be located in the shadowed or unshadowed region. Therefore, further calculations of the space target in the critical region are needed.
[0109] Considering both the uncertainty of the space target's state and computational complexity, the probability of a space target being located in the Earth's shadow is calculated using the Monte Carlo sampling method. For space target j, N Gaussian distributions are randomly generated within its state distribution space. The sample points represent a spatial target, and each spatial target may be located in the shaded area or the unshaded area. N is obtained from equations (31) and (32). o One located in the shaded area and N u N sample points located in the non-shaded area. o Calculation of the ratio of N Right now
[0110]
[0111] Equation (35) is transformed into
[0112]
[0113] Among them, the visibility probability of a space target relative to the sensor
[0114] Thus, equations (36)-(38) yield the space target observation model of the space-based tracking system based on probability relationships, obtaining the detectability probability of space targets that is continuously distributed in numerical terms, thereby realizing the prediction of the detectability probability of space targets. Compared with the existing discrete distribution method for judging the detectability of space targets, this improves the accuracy of space target state estimation.
[0115] Beneficial effects:
[0116] 1. The present invention discloses a method for large-scale space target tracking using a space-based multi-sensor tracking system. The method adopts the Walker constellation configuration that is uniformly distributed in space as the basic configuration of the space-based observation constellation, constructs a dynamic model of the space target state and the observation satellite state, and realizes the state description of the space-based tracking system based on the dynamic model, thereby saving computational resources.
[0117] 2. The present invention discloses a method for establishing a space target state estimation model, which models the space target tracking problem as a correspondence between the observation space and the state space of the space target. Since only part of the space target state is observable, the Bayesian recursive method is used to estimate the space target state, effectively obtaining a set of space target state information in the time series.
[0118] 3. The present invention discloses a method for predicting the detectability probability of space targets considering state uncertainty. Based on a space-based observation model based on geometric relationships, the method considers the state uncertainty of space targets and establishes a space-based observation model based on probability relationships. The detectability probability of space targets is obtained by numerically continuously distributed according to the space-based observation model based on probability relationships, thereby improving the accuracy of space target state estimation. Attached Figure Description
[0119] Figure 1 This is a flowchart of a spatial target detectability probability prediction method considering state uncertainty according to the present invention;
[0120] Figure 2 This is a schematic diagram of the satellite orbital plane coordinate system in this invention;
[0121] Figure 3 This is a flowchart of the spatial target state estimation process in this invention;
[0122] Figure 4 This is a flowchart illustrating the process of calculating whether a space target is covered by the Earth's shadow in this invention.
[0123] Figure 5 This is a flowchart illustrating the process of calculating the probability of a space target being covered by the Earth's shadow, as described in this invention. Detailed Implementation
[0124] To better illustrate the purpose, technical solution, and advantages of the present invention, the following description, in conjunction with the accompanying drawings and embodiments, will further explain the invention.
[0125] like Figure 1 As shown in the figure, this embodiment discloses a method for predicting the detectability probability of spatial targets considering state uncertainty. The specific implementation steps are as follows:
[0126] Step 1: Large-scale space target tracking is carried out using a space-based multi-sensor tracking system. 1000 cataloged targets in Starlink are selected as the space targets to be observed. The Walker constellation configuration, which is uniformly distributed in space, is used as the basic configuration of the space-based observation constellation. Dynamic models of the space target state and the observation satellite state are constructed. Based on the dynamic model, the state description of the space-based tracking system is realized.
[0127] A space-based tracking system consists of multiple satellites, which are distributed in a specific configuration in space to ensure coverage and stability. The Walker constellation configuration consists of several circular orbits, with satellites evenly distributed within each orbital plane, enabling global coverage and uniform revisit. The space-based tracking system adopts the Walker configuration.
[0128] The two-body elliptical orbit of a single satellite in a space-based tracking system is described by a set of Keplerian orbital parameters (a,e,i,Ω,ω,θ), namely, the semi-major axis a, eccentricity e, inclination i, right ascension of the ascending node Ω, argument of perigee ω, and true anomaly θ. The semi-major axis a and eccentricity e together determine the size and shape of the orbit; the orbital inclination i is the angle between the satellite's orbital plane and the Earth's equatorial plane; the right ascension of the ascending node Ω is the angle between the vernal equinox of the Earth's equatorial plane and the ascending node of the satellite's orbit with respect to the Earth's center O, indicating the azimuth of the satellite's orbital plane within the Earth's equatorial plane; simultaneously, the orbital inclination i and the right ascension of the ascending node Ω together determine the position of the satellite's orbital plane; the argument of perigee ω is the angle between the perigee and the ascending node within the satellite's orbital plane, indicating the azimuth of the major and minor axes of the elliptical orbit; and the true anomaly θ is the angle between the satellite's position within the orbital plane and the perigee, indicating the satellite's position within its orbit.
[0129] The configuration of all satellites in the space-based tracking system is described by a set of Walker constellation parameters (N, P, F), namely the total number of satellites N, the number of orbital planes P, and the phase factor F. The total number of satellites N and the number of orbital planes P together determine the number of satellites S = N / P in each orbital plane; the phase factor F represents the phase relationship between corresponding satellites on adjacent orbital planes, and the phase difference Δu between corresponding satellites on adjacent orbital planes is...
[0130]
[0131] For satellite i (i = 1, 2, ..., N) in the space-based tracking system s Its right ascension of the ascending node and argument of the perigee are expressed as follows:
[0132]
[0133] Since the satellites in the Walker constellation have the same orbital semi-major axis a and inclination i, and the eccentricity e = 0, and the initial true anomaly angle θ = 0, therefore, the right ascension Ω of the ascending node of satellite i given by equation (2) is... i and perigee argument ω i That is, the position and velocity of each satellite in the orbital plane coordinate system are obtained from the Kepler orbital parameters of each satellite.
[0134] The origin of the satellite's orbital plane coordinate system is located at the Earth's center, X p The axis points to the perigee of the orbital plane, Y p The axis lies within the orbital plane and is perpendicular to X. p Axis, Z p The axis is perpendicular to the orbital plane and conforms to a right-handed coordinate system. For example... Figure 2 As shown, the satellite's elliptical orbit is approximated as a circular orbit. The angular velocity of the satellite in this circular orbit is the same as its average angular velocity in the elliptical orbit, and the period of the satellite's motion in the circular orbit is the same as its period in the elliptical orbit. If the true anomaly angle θ of the satellite at time k is ∠NOS, then the corresponding deviated anomaly angle E of the satellite in the circular orbit is ∠NCQ.
[0135] At the initial time k0, the real satellite and the satellite in the circular orbit pass through the perigee simultaneously and in the same direction. Therefore, at time t, the mean perigee angle M of the real satellite is defined as the angular distance of the satellite in the circular orbit, i.e.
[0136] M=n(t-k0) (3)
[0137] Where n is the average angular velocity of the actual satellite moving in an elliptical orbit. The mean anomaly angle M is a hypothetical auxiliary quantity and does not correspond to an angle in the actual physical sense. According to Kepler's third law of orbital motion, the average angular velocity n of the satellite in orbit has the following relationship with its semi-major axis a.
[0138]
[0139] Where G is the gravitational constant; M e For Earth's mass; μ e is the gravitational constant.
[0140] The relationship between the anomalous angle E and the level anomalous angle M is as follows:
[0141] M = Ee sin E (5)
[0142] The position r of the satellite in the orbital plane coordinate system in the space-based tracking system p Represented as
[0143]
[0144] Differentiating equation (6) yields the velocity of the satellite in the orbital plane coordinate system within the space-based tracking system.
[0145]
[0146] According to equations (6) and (7), the satellite's position and velocity in the orbital plane coordinate system are... Based on the transformation relationship between the orbital plane coordinate system and the geocentric inertial coordinate system, the satellite's position and velocity in the geocentric inertial coordinate system are obtained.
[0147] Coordinate transformation matrix from orbital plane coordinate system to geocentric inertial coordinate system Represented as
[0148]
[0149] in,
[0150]
[0151] The position and velocity of the satellite in the geocentric inertial coordinate system in the space-based tracking system They are respectively represented as
[0152]
[0153] If a satellite in a space-based tracking system is only subject to Earth's gravity and not to other perturbations, then all satellites in the space-based tracking system satisfy the following dynamic equations.
[0154]
[0155] Among them, r0 and The initial position and velocity of the satellite are obtained based on the parameters of the space-based tracking system and equations (6), (7), and (10). Equation (11) is integrated to obtain the dynamic model of any satellite in the space-based tracking system, thus realizing the state description of the space-based tracking system.
[0156] Other perturbation forces include atmospheric drag and solar radiation pressure.
[0157] Step 2: Model the space target tracking problem as a correspondence between the observation space and the state space of the space target. Since only part of the space target state is observable, the Bayesian recursive method is used to estimate the space target state, obtain a set of space target state information in the time series, and establish a space target state estimation model.
[0158] In a space target tracking system, the intrinsic relationship between a space target and time is the state change of the space target over time. The space target tracking problem can be modeled as a correspondence between the observation space and the state space of the space target. In the state space, the state of the space target transitions from the previous time k-1 to the next time k. However, in the observation space, since only a portion of the space target's state can be observed, a Bayesian recursive method is used to estimate the space target's state to obtain a set of space target state information over time.
[0159] For a discrete-time state-space system, the state at time k is determined by the random variable X. k This indicates that the state vector propagates in the state space along with the Markov transition function. For example... Figure 3 As shown, at each time k, the noisy observation process produces the observation value Z. k If the state information is correlated with the prediction process of the Bayesian filter through a likelihood function, then the prediction process is expressed as follows:
[0160] P k|k-1 (X k |Z 1:k-1 )=∫f k|k-1 (X k |X)P k-1 (X|Z 1:k-1 )dX (12)
[0161] Among them, f k|k-1 (X k |X) is the state transition function; P k|k-1 Let Z be the prior probability density distribution at time k, i.e., without merging the observations at time k. k Estimation of the probability density distribution of information. The update process of the Bayesian filter is expressed as follows:
[0162]
[0163] Among them, g k Let P be the likelihood function. At time k-1, all information is governed by a probability density distribution P. k-1 This means that at time k, a new observation value Z is obtained. k The purpose of Bayesian estimation is to estimate P... k-1 With Z k The information contained herein is fused together to obtain the posterior probability density distribution P at time k.k .
[0164] The prediction part of the Kalman filter covers the dynamics model of the linear system, expressed as:
[0165] X k|k-1 =F k X k-1 +B k u k +w k (14)
[0166] Among them, F k B is the state transition matrix; k To control the input model; u k For control variables; w k It is process noise, and
[0167] The update section covers the linear observation model, denoted as
[0168]
[0169] Among them, H k For observation model; v k To observe noise, and
[0170] The prediction and update process is summarized as follows:
[0171]
[0172] If the Kalman gain K k The larger the value, the larger the change in the posterior estimate. To address nonlinear system problems, the extended Kalman filter linearizes the nonlinear system by performing a first-order Taylor expansion of the dynamic and observation models, ensuring that the Gaussian distribution remains Gaussian after the linear transformation, thus processing the observation information of the nonlinear system.
[0173] The nonlinear system dynamics model of the extended Kalman filter is expressed as follows:
[0174] X k|k-1 =f(X) k-1 ,u k )+w k (17)
[0175] Where f is the dynamic model; w k It is process noise, and
[0176] The nonlinear observation model of the extended Kalman filter is expressed as:
[0177]
[0178] Where h is the observation model; v k To observe noise, and
[0179] For any spatial target to be tracked, the prediction and update process of the extended Kalman filter can be expressed as follows:
[0180]
[0181] Here, f cannot be directly used for covariance estimation; the Jacobian matrix needs to be calculated.
[0182]
[0183] The advantage of the Extended Kalman Filter (EKF) lies in its ability to locally linearize the nonlinear equations through Taylor expansion, allowing the Kalman filter framework to be applied to nonlinear systems and achieving good filtering results when the system's nonlinearity is not very strong. However, for systems with strong nonlinearity, the linear function cannot approximate the error propagation function well, leading to divergence in the state estimation of the spatial target.
[0184] Therefore, an unscented Kalman filter is used for the space target, approximating a Gaussian distribution with a fixed number of parameters, making the nonlinear system equations applicable to the standard Kalman system under the linear assumption. A set of Sigma sampling points is taken from the original distribution of the space target, where the mean and covariance of the Sigma points are equal to those of the original distribution. The Sigma points are substituted into the nonlinear function to obtain the corresponding set of nonlinear function values. The mean and covariance of the transformed space target are then calculated using the Sigma point set.
[0185] For an L-dimensional variable x, take 2L+1 Sigma points {χ i}, i = 0, ..., 2L, where each Sigma point has first-order and second-order weights, such that
[0186]
[0187] The transformed mean and covariance are
[0188]
[0189] Among them, y i =f(χ) i ), i = 0, ..., 2L.
[0190] The prediction process of the unscented Kalman filter for any spatial target to be tracked is as follows:
[0191] ① Generate Sigma points based on the posterior estimate at time k-1.
[0192] (Xk-1 ,P k-1 )→{χ i} (twenty three)
[0193] ② Perform nonlinear state transitions
[0194] X i =f(χ) i ) (twenty four)
[0195] ③ Obtain the prior state estimate and the prior estimate covariance.
[0196]
[0197] Then the update process begins:
[0198] ① Generate Sigma points based on prior estimates
[0199] (X k|k-1 ,P k|k-1 )→{χ i} (26)
[0200] ② Obtain the nonlinear observation model
[0201] Z i =h(χ i (27)
[0202] ③ Obtain the simulated observations, the covariance of the simulated observations, and the cross covariance.
[0203]
[0204] ④ Calculate the Kalman gain
[0205]
[0206] By using equations (12) to (29), the spatial target tracking problem is modeled as a correspondence between the observation space and the state space of the spatial target, a set of spatial target state information in the time series is obtained, and a spatial target state estimation model is established.
[0207] Step 3: Establish a space-based observation model based on geometric relationships and determine the detectability of space targets relative to sensors.
[0208] like Figure 4 As shown, whether a space target can be observed by the sensor within a time window depends on the visibility p of the space target relative to the sensor. v,k The judgment is given by the following expression:
[0209] p v,k =(1-p s,k (1-p) b,k(30)
[0210] Where, p s,k Whether a space target is covered by Earth's shadow is determined by the geometric relationship between the Sun, the space target, and Earth. When the space target is not in the shadow cast by the Sun on Earth, p s,k p is 0 if it is not 1 otherwise b,k Whether a space target is obscured by the Earth is determined by the geometric relationship between the sensor, the space target, and the Earth. If the space target is not obscured by the Earth within the sensor's field of view, then p... b,k It is 0 if it is not 0, otherwise it is 1.
[0211] p s,k and p b,k The calculation is defined as a "0-1 model". Calculate p s,k It consists of two steps. The first step defines the length of the perpendicular line from the Earth's center to the line connecting the Sun and the space target as d1, and the Earth's radius as R. e Then p s,k Represented as
[0212]
[0213] When d1>R e The space target is not in the shadow cast by the sun on Earth, i.e., p s,k = 0. However, when d1 ≤ R e Whether a space target lies in the shadow depends on its position relative to the Sun and Earth. Let d0 be the distance from the center of the Sun to a straight line d1, and d2 be the distance from the center of the Sun to the space target. By comparing the lengths of d2 and d0, we can determine whether the space target is obscured by the Earth's shadow.
[0214]
[0215] Calculate p b,k It consists of two steps. The first step defines the length of the perpendicular line from the Earth's center to the line connecting the sensor and the space target as d1, and the Earth's radius as R. e Then p b,k Represented as
[0216]
[0217] When d1>R e The space target is not in an area that may be obscured by the Earth, i.e., p s,k = 0. However, when d1 ≤ R eWhether a space target is located within the Earth's occlusion zone depends on its position relative to the sensor and the Earth. Let d0 be the distance from the sensor to line d1, and d2 be the distance from the sensor to the space target. By comparing the lengths of d2 and d0, we can determine whether the space target is occluded by the Earth.
[0218]
[0219] All parameters in equations (30)-(34) are determined by the optimization variables (h,i,Ω,N). orb N spo It is obtained through geometric transformation.
[0220] Step 4: Based on the space-based observation model established in Step 3, and combined with the space-based tracking system dynamic model and space target state estimation model established in Steps 1 and 2, and considering the uncertainty of the space target state, establish a space-based observation model based on probability.
[0221] Because the large-scale space target tracking problem uses a limited number of space-based optical sensors to track a large number of space targets, a large number of space targets frequently enter and exit the sensor's field of view. Therefore, at time k, correctly calculating the detectability p of the space targets is crucial. D,k Measuring the presence of space targets is crucial for accurately estimating their number. However, due to variations in lighting conditions and the relative distance between the sensor and the space target, the p-value of the space target... D,k It is time-varying. Therefore, it is necessary to consider p. D,k Develop more accurate prediction models.
[0222] The detectability of a space target is determined solely by its visibility relative to the sensor. In practical missions, the apparent magnitude of a space target relative to the sensor significantly impacts the quality of observation. The detectability p of a space target... D,k It consists of three parts: the probability of successful detection of a space target, p D The apparent magnitude p of a space target relative to a sensor am,k And the visibility p of space targets relative to sensors v,k Its calculation expression is:
[0223] p D,k =p D p am,k p v,k (35)
[0224] Where, p D The constant value used to measure the sensor's performance is [0,1]; p am,k Calculated based on the relative positions of space targets, sensors, and the sun; p v,k =(1-ps,k (1-p) b,k ), and p s,k and p b,k Determined by the "0-1 model".
[0225] like Figure 5 As shown, based on the uncertainty and probability theory of the space target state, p s,k and p b,k The "0-1 model" is refined into and The "0-1 model" is used to obtain the detectability p relative to space targets. D,k More accurate detectable probability in, and , representing the probability that a space target is in Earth's shadow and is obscured by Earth, respectively, with values ranging from [0,1].
[0226] calculate This requires two steps. The first step is to define σ as the maximum two-dimensional standard deviation of the three-dimensional covariance of the space target on the plane of the Sun, Earth, and space target. Since the distance between the Earth and the Sun is much greater than the distance between the Earth and the space target, this is determined by comparing |d1-R... e The relationship between |σ| and |σ| can be used to approximate whether a space target is within the critical region. Represented as
[0227]
[0228] in, This represents the probability that a space target in the critical region is in the Earth's shadow. It can be observed that if |d1-R e If |≤σ, then the space target may be located in the shadowed or unshadowed region. Therefore, further calculations of the space target in the critical region are needed.
[0229] Considering both the uncertainty of the space target's state and computational complexity, the probability of a space target being located in the Earth's shadow is calculated using the Monte Carlo sampling method. For space target j, N Gaussian distributions are randomly generated within its state distribution space. The sample points represent a spatial target, and each spatial target may be located in the shaded area or the unshaded area. N is obtained from equations (31) and (32). o One located in the shaded area and N u N sample points located in the non-shaded area. o Calculation of the ratio of N Right now
[0230]
[0231] Equation (35) is transformed into
[0232]
[0233] Among them, the visibility probability of a space target relative to the sensor
[0234] Thus, equations (36)-(38) yield the space target observation model of the space-based tracking system based on probability relationships, obtaining the detectability probability of space targets that is continuously distributed in numerical terms, thereby realizing the prediction of the detectability probability of space targets. Compared with the existing discrete distribution method for judging the detectability of space targets, this improves the accuracy of space target state estimation.
[0235] The above detailed description further illustrates the purpose, technical solution, and beneficial effects of the invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.
Claims
1. A method for predicting the detectability probability of a spatial target considering state uncertainty, characterized in that: Includes the following steps, Step 1: Large-scale space target tracking is carried out using a space-based multi-sensor tracking system. Cataloged targets are selected as space targets to be observed, and the Walker constellation configuration, which is uniformly distributed in space, is used as the basic configuration of the space-based observation constellation. Dynamic models of the space target state and the observation satellite state are constructed, and the state description of the space-based tracking system is realized based on the dynamic model. Step 2: Model the space target tracking problem as a correspondence between the observation space and the state space of the space target. Since only part of the space target state is observable, the Bayesian recursive method is used to estimate the space target state, obtain a set of space target state information in the time series, and establish a space target state estimation model. Step 3: Establish a space-based observation model based on geometric relationships and determine the detectability of space targets relative to sensors; Step 4: Based on the space-based observation model established in Step 3, and combined with the space-based tracking system dynamic model and space target state estimation model established in Steps 1 and 2, considering the uncertainty of space target state, establish a space-based observation model based on probability. Based on the space-based observation model based on probability, obtain the numerically continuous probability of space target detection, that is, realize the prediction of space target detection probability. Step four is implemented as follows: Because the problem of large-scale space target tracking uses a limited number of space-based optical sensors to track large-scale space targets, a large number of space targets frequently enter and exit the sensor's field of view; therefore, in At any given moment, accurately calculate the detectability of space targets. Measuring the presence of space targets is crucial for accurately estimating their number; however, variations in lighting conditions and the relative distance between the sensor and the space target can affect the accuracy of space target measurements. It is time-varying; therefore, it is necessary to... Develop more accurate prediction models; The detectability of a space target is determined solely by its visibility relative to the sensor; the apparent magnitude of a space target relative to the sensor significantly impacts the quality of observation; the detectability of space targets... It consists of three parts: the probability of successful detection of space targets. The apparent magnitude of a space target relative to a sensor. And the visibility of space targets relative to sensors Its calculation expression is: in, A constant value used to measure the sensor's own performance, with a range of values. ; Calculated based on the relative positions of space targets, sensors, and the sun; ,and and Determined by the "0-1 model", Whether a space target is covered by the Earth's shadow is determined by the geometric relationship between the Sun, the space target, and the Earth. Whether a space target is obscured by the Earth is determined by the geometric relationship between the sensor, the space target, and the Earth. Based on the uncertainty and probability theory of space target state, and The "0-1 model" is refined into and The "0~1 model" is used to obtain detectability relative to space targets. More accurate detectable probability ;in, and These represent the probabilities of a space target being in Earth's shadow and being obscured by Earth, respectively, with values ranging from [value range missing]. ; calculate This requires two steps; the first step is to define the maximum two-dimensional standard deviation of the three-dimensional covariance of the space target on the plane of the Sun, Earth, and space target. Because the distance between Earth and the Sun is much greater than the distance between Earth and a space target, by comparison... and The relative sizes of the targets can be used to approximate whether a space target is within the critical region; then... Represented as in, The length of the perpendicular line from the Earth's center to the line connecting the sensor and the space target. For the Earth's radius, It is the probability that a space target in the critical zone is in the Earth's shadow; if If the target is located in the shadowed or unshadowed region, then further calculations of the critical zone target are needed. ; Taking into account the uncertainty of the space target's state and computational complexity, the probability of the space target being located in the Earth's shadow is calculated using the Monte Carlo sampling method; for space targets Randomly generated within its state distribution space Each follows a Gaussian distribution. The sample points represent a spatial target, and each spatial target may be located in the shaded area or in the unshaded area; by equation... Japanese style Together One located in the shaded area and A sample point located in the non-shaded area; through and Ratio calculation ,Right now ,Mode Transform into Among them, the visibility probability of a space target relative to the sensor ; Mode -Mode That is, a space target observation model based on probability relationship is obtained for the space-based tracking system. Based on the space target observation model of the space-based tracking system, the detectable probability of space targets is obtained in a numerically continuous distribution, that is, the detectable probability of space targets is predicted.
2. The method for predicting the detectability probability of a spatial target considering state uncertainty as described in claim 1, characterized in that: The implementation method for step one is as follows: The space-based tracking system consists of multiple satellites; the Walker constellation configuration consists of several circular orbits, with satellites evenly distributed within each orbital plane; the space-based tracking system adopts the Walker configuration. The two-body elliptical orbit of a single satellite in a space-based tracking system is determined by a set of Kepler orbital parameters. Description, i.e., the semi-major axis of the track eccentricity ,inclination Right ascension of ascending node Perimeter Argument and true near point angle Among them, the semi-major axis of the track and eccentricity Together they determine the size and shape of the track; the track inclination angle The angle between the satellite's orbital plane and the Earth's equatorial plane; the right ascension of the ascending node. The vernal equinox at the Earth's equatorial plane and the ascending node of the satellite's orbit relative to the Earth's center. The angle between the two points indicates the azimuth of the satellite's orbital plane within the Earth's equatorial plane; simultaneously, the orbital inclination angle... Right ascension of ascending node Together they determine the position of the satellite's orbital plane; perigee argument The true anomaly is the angle between the perigee and the ascending node within the satellite's orbital plane, representing the azimuth of the major and minor axes of the satellite's elliptical orbit; The angle between the satellite's position within its orbital plane and its perigee represents the satellite's position within its orbit. The configuration of all satellites in the space-based tracking system is determined by a set of Walker constellation parameters. Description, i.e., the total number of satellites in the constellation. The number of orbital planes of constellations and phase factor Among them, the total number of satellites And the number of orbital planes of constellations The number of satellites in each orbital plane is determined jointly. Phase factor This refers to the phase relationship between corresponding satellites on adjacent orbital planes, and the phase difference between corresponding satellites on adjacent orbital planes. for For satellites in space-based tracking systems Its right ascension of the ascending node and argument of the perigee are expressed as follows: Because the satellites in the Walker constellation have the same orbital semi-major axis and tilt angle And eccentricity Initial true anterior angle Therefore, the combined The given satellite Right ascension of the ascending node and perigee argument That is, the position and velocity of each satellite in the orbital plane coordinate system are obtained from the Kepler orbital parameters of each satellite. ; The origin of the satellite's orbital plane coordinate system is located at the Earth's center. The axis points towards the perigee of the orbital plane. The axis is located in the track plane and perpendicular to the plane. axis, The axis is perpendicular to the orbital plane and conforms to the right-hand coordinate system; the elliptical orbit of the satellite is approximated as a circular orbit, and the angular velocity of the satellite moving in the circular orbit is the same as the average angular velocity moving in the elliptical orbit, and the period of the satellite's motion in the circular orbit is the same as the period of its motion in the elliptical orbit. True perimeter of the satellite at any given time for Then the corresponding apogee angle of the satellite in the circular orbit for ; At the initial moment If a real satellite and a satellite in a circular orbit pass through perigee simultaneously and in the same direction, then... At any given moment, the actual mean angle of the satellite Defined as the angular distance of a satellite in a circular orbit, i.e. in, The average angular velocity of a real satellite moving in an elliptical orbit; the angle of approach. It is a fictitious auxiliary quantity that does not correspond to an angle in the real physical sense; according to Kepler's third law of orbital motion, the average angular velocity of a satellite in orbit... Its orbital semi-major axis The following relationships exist in, It is the gravitational constant; Earth mass; It is the gravitational constant; Nearest point angle Angle of near point The relationship is The position of a satellite in the orbital plane coordinate system in a space-based tracking system Represented as Differentiating equation (6), we obtain the velocity of the satellite in the orbital plane coordinate system within the space-based tracking system. , According to equations (6) and (7), the satellite's position and velocity in the orbital plane coordinate system are... By determining the transformation relationship between the orbital plane coordinate system and the geocentric inertial coordinate system, the position and velocity of the satellite in the geocentric inertial coordinate system can be obtained. ; Coordinate transformation matrix from orbital plane coordinate system to geocentric inertial coordinate system Represented as in, The position and velocity of the satellite in the geocentric inertial coordinate system in the space-based tracking system They are respectively represented as If a satellite in a space-based tracking system is only subject to Earth's gravity and not to other perturbations, then all satellites in the space-based tracking system satisfy the following dynamic equations. in, and To be based on the parameters of the space-based tracking system and the formula ,Mode and style The obtained initial position and velocity of the satellite; for equation By performing integration, a dynamic model of any satellite in the space-based tracking system is obtained, thus realizing the state description of the space-based tracking system; Other perturbation forces include atmospheric drag and solar radiation pressure.
3. The method for predicting the detectability probability of a spatial target considering state uncertainty as described in claim 2, characterized in that: The second step is implemented as follows: In a space target tracking system, the intrinsic relationship between a space target and time is the state change of the space target over time; the space target tracking problem is modeled as the correspondence between the observation space and the state space of the space target. In the state space, the spatial target state changes from the previous time step. Next moment However, in the observation space, since only a portion of the spatial target state can be observed, a Bayesian recursive method is used to estimate the spatial target state in order to obtain a set of spatial target state information in the time series. For a discrete-time state-space system, time 1 The state is determined by random variables This means that the state vector propagates in the state space along with the Markov transition function; at each time step... Noisy observation processes produce observation values. If the state information is correlated with the prediction process of the Bayesian filter through a likelihood function, then the prediction process is expressed as follows: in, This is the state transition function; for The prior probability density distribution at time t, i.e., without fusion Time observation value Information probability density distribution estimation; the Bayesian filter update process is expressed as... in, Let be the likelihood function; at time All the information is represented by a probability density distribution. It indicates that at any time To obtain new observations The purpose of Bayesian estimation is to... and The information contained is integrated to obtain Posterior probability density distribution at time 1 ; The prediction part of the Kalman filter covers the dynamics model of the linear system, expressed as: in, This is the state transition matrix; To control the input model; For control variables; It is process noise, and ; The update section covers the linear observation model, denoted as in, For observation models; To observe noise, and ; The prediction and update process is summarized as follows: If Kalman gain The larger the value, the larger the change in the posterior estimate; the extended Kalman filter linearizes the nonlinear system by performing a first-order Taylor expansion of the dynamic model and the observation model, so that the Gaussian distribution remains Gaussian after the linear transformation, thus processing the observation information of the nonlinear system. The nonlinear system dynamics model of the extended Kalman filter is expressed as follows: in, For dynamic models; It is process noise, and ; The nonlinear observation model of the extended Kalman filter is expressed as: in, For observation models; To observe noise, and ; For any spatial target to be tracked, the prediction and update process of the extended Kalman filter can be expressed as follows: in, It cannot be directly used for covariance estimation; the Jacobian matrix needs to be calculated. An unscented Kalman filter is applied to the space target, approximating a Gaussian distribution with a fixed number of parameters, so that the nonlinear system equations are applicable to the standard Kalman system under the linear assumption. A set of Sigma sampling points is taken from the original distribution of the space target, and the mean and covariance of the Sigma points are equal to those of the original distribution. The Sigma points are substituted into the nonlinear function to obtain the corresponding set of nonlinear function values. The mean and covariance of the transformed space target are calculated using the set of Sigma points. for Dimensional variables ,Pick Sigma points Each Sigma point has first-order and second-order weights, such that The transformed mean and covariance are in, ; The prediction process of the unscented Kalman filter for any spatial target to be tracked is as follows: ①According to The posterior estimate at time 1 generates the Sigma point. ② Perform nonlinear state transitions ③ Obtain the prior state estimate and the prior estimate covariance. Then the update process begins: ① Generate Sigma points based on prior estimates ② Obtain the nonlinear observation model ③ Obtain the simulated observations, the covariance of the simulated observations, and the cross covariance. ④ Calculate the Kalman gain Through -Mode The problem of space target tracking is modeled as a correspondence between the observation space and the state space of the space target, and a set of space target state information in time series is obtained to establish a space target state estimation model.
4. The method for predicting the detectability probability of a spatial target considering state uncertainty as described in claim 3, characterized in that: The method for implementing step three is as follows: Whether a space target can be observed by the sensor within a time window depends on the visibility of the space target relative to the sensor. The judgment is given by the following expression: in, Whether a space target is covered by Earth's shadow is determined by the geometric relationship between the Sun, the space target, and Earth. When the space target is not in the Sun's shadow cast by Earth, it indicates whether it is covered by Earth's shadow. It is 0 if it is not 1 otherwise; Whether a space target is obscured by the Earth is determined by the geometric relationship between the sensor, the space target, and the Earth. If the space target is not obscured by the Earth within the sensor's field of view, then... It is 0 if it is not 1 otherwise; and The calculation is defined as a "0-1 model"; the calculation It includes two steps; the first step is to define the length of the perpendicular line from the Earth's center to the line connecting the Sun and the space target as... Earth's radius is ,but Represented as when The space target is not in the shadow cast by the sun on the earth, that is... However, when Whether a space target is located in the shadow region will depend on its position relative to the Sun and Earth; the definition is a straight line from the center of the Sun. The distance is The distance from the center of the sun to the space target is By comparison and The length of the object determines whether it is obscured by Earth's shadow. calculate It includes two steps; the first step is to define the length of the perpendicular line from the Earth's center to the line connecting the sensor and the space target as... Earth's radius is ,but Represented as when The space target is not in an area that might be obscured by Earth, that is... However, when Whether a space target is located in the Earth's obstruction zone will depend on its position relative to the sensor and the Earth; the definition is from the sensor to the line of sight. The distance is The distance from the sensor to the space target is By comparison and The length of the object determines whether it is obscured by Earth. Mode -Mode All parameters are determined by the optimization variables. It is obtained through geometric transformation.