Optical formation satellite injection error propagation analysis method and system

Abstract

The invention provides an optical formation satellite injection error propagation analysis method and system. The method comprises the following steps: S1, establishing a kinetic model of a formation satellite; s2, establishing an expression for the configuration key indexes of the arm length, the respiration angle and the arm length change rate in the space gravitational wave detection task formation problem; s3, expressing a relationship between a mean value and a covariance of the arm length, the respiratory angle and the arm length change rate in the formation configuration by using a UT sampling method; and S4, correcting the sampling points and the weights by using a ratio UT method. According to the method, the proportion correction sampling UT method is adopted to carry out injection error analysis on the space gravitational wave detection task, and compared with a traditional targeting method, the calculation amount is greatly reduced, and the calculation speed is effectively improved.

Description

Technical Field

[0001] This invention relates to the field of aerospace optical satellites, specifically to a method and system for analyzing the propagation of orbit insertion errors in optical formation satellites. Background Technology

[0002] For space-based gravitational wave detection missions—a current global research hotspot—there are two main approaches: heliocentric orbit and geocentric orbit. Both approaches plan to launch three identical satellites, forming a formation in either a heliocentric-Earth-lagging orbit or a high Earth orbit. Gravitational wave signals are weak, and detection requires the formation satellites to maintain high stability in space for more than five years. However, due to launch errors and orbital evolution errors, the formation configuration will gradually diverge during the mission, potentially even failing to meet the formation requirements and causing mission interruption. Because of the complexity of the dynamic models and the long mission duration, the traditional Monte Carlo method is time-consuming in analyzing the impact of insertion errors on the formation configuration. Therefore, there is an urgent need to find a fast and accurate method for analyzing the propagation of insertion errors in space-based gravitational wave detection missions.

[0003] Previous studies have addressed the problem of error propagation analysis. Wu Weiren et al. [Wu Weiren, Tang Yuhua, Qiao Dong, Li Xiangyu. "Optimal Error Correction Point Selection Method for Earth-Moon L2 Point Transfer Orbit" Application No. 202010386041.4] proposed a method for selecting the optimal error correction point for the Earth-Moon L2 point transfer orbit. This method calculates the error change over time based on the state transition matrix, offering advantages such as minimal velocity increment, high correction efficiency, and low fuel consumption. Huang Jian et al. [Huang Jian, Wang Dongya, Zhang Bing, Wu Wentang. "Determination Method of Satellite Orbiting Aircraft Dynamic Parameters Based on Orbit Error Analysis" Application No. 202110585731.7] calculated the initial relative velocity parameters of the satellite based on the real-time position difference between the monitoring satellite and the target, utilizing the relationship between orbital position error and velocity error. This method has the advantages of clear principle and simple calculation. Wu Weiren et al. [Wu Weiren, Tang Yuhua, Qiao Dong, Li Xiangyu. "A Method for Reconstructing Equilibrium Point Transfer Tracks Based on State Transition Matrix" Application No. 202010448584.4] derived the functional relationship between the reconstruction maneuver and the change in the orbital state based on the state transition matrix, which can enable transfer tracks with errors to enter the same mission track. Li Dawei et al. [Li Dawei, Yang Xu, Cheng Haowen, Jin Yong. "A Method and Device for Calculating Initial Orbit Error of Space Targets" Application No. 202010263472.1] obtained the prediction error in the U-axis, N-axis and W-axis directions at a specified time based on the initial orbit covariance at a specified time in the J2000 coordinate system. Zhang Gang et al. [Zhang Gang, Ma Huidong, Cao Xibin, Geng Yunhai, Wu Baolin, Kong Xianren, Dong Limin. A method and device for correcting the Earth-Moon transfer orbit of a probe. Application No. 201910108507.1] calculated the difference between the terminal time state quantity and the terminal time standard quantity, and corrected the corrected time state quantity based on this value. Summary of the Invention

[0004] To address the shortcomings of existing technologies, the purpose of this invention is to provide a method and system for analyzing the propagation of optical formation satellite orbit insertion errors.

[0005] A method for analyzing the propagation of optical formation satellite orbit insertion errors according to the present invention includes:

[0006] Step S1: Establish a dynamic model of the formation satellites;

[0007] Step S2: Establish expressions for the key configuration indicators arm length, breathing angle, and arm length change rate in the formation problem of space gravitational wave detection missions;

[0008] Step S3: Use the UT sampling method to express the relationship between the mean and covariance of arm length, breathing angle and arm length change rate in the formation configuration;

[0009] Step S4: Correct the sampling points and weights using the proportional UT method.

[0010] Preferably, in step S1:

[0011] A dynamic model of the satellite formation is established. The dynamic model of the heliocentric orbit space gravitational wave detection mission is represented in the heliocentric J2000 coordinate system as follows:

[0012]

[0013] Where, r i =[x i y i z i ] T , r k This represents the position vector of the three satellites, k = 1, 2, 3; v k Represents the velocity vectors of the three satellites, k = 1, 2, 3; R i The vectors representing the positions of Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, and the Moon, i = 1, 2, ..., 8, 9; μ0 represents the solar gravitational constant, μ i Let i represent the gravitational constants of the eight planets and the Moon, i = 1, 2, ..., 8, 9.

[0014] Preferably, in step S2:

[0015] The arm length of the formation determines the range of the detected wave source, the change in the breathing angle affects the pointing accuracy of the onboard telescope, and the rate of change of arm length affects the Doppler frequency shift. These three indicators can be expressed in formula form as follows:

[0016] L ij (t)=||R ij (t)|| (3)

[0017]

[0018] Among them, R ij Let L represent the arm length vector from satellite i to satellite j, where L represents the arm length, θ represents the breathing angle, and V represents the rate of change of arm length; t is the orbital recursion time.

[0019] Preferably, in step S3:

[0020] The Utahedron method is used for sampling calculation to obtain the mean and initial covariance matrix of arm length, breathing angle and arm length change rate in the formation configuration; sampling Sigma points are generated according to the formula, where the covariance matrix is ​​obtained by Cholesky decomposition; sampling weights are calculated using the formula.

[0021] The dynamic model of a space-based gravitational wave detection mission is expressed as follows:

[0022]

[0023] Among them, the state variable x has a total of 27 terms, x1 to x 18 These are the positions and velocities of the three satellites; x 19 To x 21 It is the arm length of the formation; x 22 To x 24 It is the breathing angle of the formation; x 25 To x 27 is the rate of change of the arm length of the formation configuration; f(x,t) is the dynamic expression;

[0024] Generate Sigma point χ i The method is as follows, where the mean and covariance of the state vector x are respectively and P x

[0025]

[0026] Where χ0 is the sampling center point, χ i These are other sampling points, which together constitute the sampling point set. i represents the subscript of the matrix column, n represents the dimension of x, and λ is the parameter of the sampling distribution.

[0027] in Obtained through Cholesky decomposition, i.e.

[0028]

[0029] Let represent the i-th column of the matrix, and λ be the parameter of the sampling distribution;

[0030] The sampling weights are calculated using the following formula.

[0031]

[0032] Among them, W i (m) and W i (m) These are the weights of the mean and covariance, respectively.

[0033] Preferably, in step S4:

[0034] The proportional UT transformation method is used to correct the UT method sampling. The values ​​of parameters α and β in the proportional UT transformation method are adjusted to keep the prediction covariance matrix positive semidefinite.

[0035] By integrating the dynamic equations, the transformed point set is obtained. The weighted average is then used to obtain the mean and standard deviation of the transformed points. Finally, the average and standard deviation of the formation arm length, breathing angle, and arm length change rate at any time during the mission cycle are obtained by integrating the equations.

[0036] {χ} obtained from UT transformation i W i The proportional UT transformation formula is:

[0037] χ′ i =χ0+α(χ i -χ0) (10)

[0038] y′ i =f(χ′) i (11)

[0039]

[0040] Where, χ′ i These are the transformed sampling points, y′ i This is the result of the transformation and integration, where W0 is the corresponding initial weight in equation (9), and P... yy It is the variance obtained after weighting;

[0041] For each χ′ i Integrating the dynamic equations, we obtain the transformed y′ i The point set {Y′} i}, based on the point set {Y′ i The different weight coefficients W′ for each element in} i The weighted average yields the transformed mean and standard deviation. and Integrating formulas (13) and (14) yields the mean and standard deviation of the formation arm length, breathing angle, and arm length change rate at any time during the mission cycle.

[0042] An optical formation satellite orbit insertion error propagation analysis system provided by the present invention includes:

[0043] Module M1: Establish a dynamic model of the formation satellites;

[0044] Module M2: Establishes expressions for key configuration indicators such as arm length, breathing angle, and arm length change rate in the formation problem of space gravitational wave detection missions;

[0045] Module M3: Uses UT sampling to express the relationship between the mean and covariance of arm length, breathing angle, and arm length change rate in formation configuration;

[0046] Module M4: Corrects sampling points and weights using the proportional UT method.

[0047] Preferably, in module M1:

[0048] A dynamic model of the satellite formation is established. The dynamic model of the heliocentric orbit space gravitational wave detection mission is represented in the heliocentric J2000 coordinate system as follows:

[0049]

[0050] Where, r i =[x i y i z i ] T , r k This represents the position vector of the three satellites, k = 1, 2, 3; v k Represents the velocity vectors of the three satellites, k = 1, 2, 3; R i The vectors representing the positions of Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, and the Moon, i = 1, 2, ..., 8, 9; μ0 represents the solar gravitational constant, μ i Let i represent the gravitational constants of the eight planets and the Moon, i = 1, 2, ..., 8, 9.

[0051] Preferably, in module M2:

[0052] The arm length of the formation determines the range of the detected wave source, the change in the breathing angle affects the pointing accuracy of the onboard telescope, and the rate of change of arm length affects the Doppler frequency shift. These three indicators can be expressed in formula form as follows:

[0053] L ij (t)=||R ij (t)|| (3)

[0054]

[0055] Among them, R ij Let L represent the arm length vector from satellite i to satellite j, where L represents the arm length, θ represents the breathing angle, and V represents the rate of change of arm length; t is the orbital recursion time.

[0056] Preferably, in module M3:

[0057] The Utahedron method is used for sampling calculation to obtain the mean and initial covariance matrix of arm length, breathing angle and arm length change rate in the formation configuration; sampling Sigma points are generated according to the formula, where the covariance matrix is ​​obtained by Cholesky decomposition; sampling weights are calculated using the formula.

[0058] The dynamic model of a space-based gravitational wave detection mission is expressed as follows:

[0059]

[0060] Among them, the state variable x has a total of 27 terms, x1 to x 18 These are the positions and velocities of the three satellites; x 19 To x 21It is the arm length of the formation; x 22 To x 24 It is the breathing angle of the formation; x 25 To x 27 is the rate of change of the arm length of the formation configuration; f(x,t) is the dynamic expression;

[0061] Generate Sigma point χ i The method is as follows, where the mean and covariance of the state vector x are x and P, respectively. x

[0062]

[0063] Where χ0 is the sampling center point, χ i These are other sampling points, which together constitute the sampling point set. i represents the subscript of the matrix column, n represents the dimension of x, and λ is the parameter of the sampling distribution.

[0064] in Obtained through Cholesky decomposition, i.e.

[0065]

[0066] Let represent the i-th column of the matrix, and λ be the parameter of the sampling distribution;

[0067] The sampling weights are calculated using the following formula.

[0068]

[0069] Among them, W i (m) and W i (m) These are the weights of the mean and covariance, respectively.

[0070] Preferably, in module M4:

[0071] The proportional UT transformation method is used to correct the UT method sampling. The values ​​of parameters α and β in the proportional UT transformation method are adjusted to keep the prediction covariance matrix positive semidefinite.

[0072] By integrating the dynamic equations, the transformed point set is obtained. The weighted average is then used to obtain the mean and standard deviation of the transformed points. Finally, the average and standard deviation of the formation arm length, breathing angle, and arm length change rate at any time during the mission cycle are obtained by integrating the equations.

[0073] {χ} obtained from UT transformation i W i The proportional UT transformation formula is:

[0074] χ′ i =χ0+α(χi -χ0) (10)

[0075] y′ i =f(χ′) i (11)

[0076]

[0077]

[0078] Where, χ′ i These are the transformed sampling points, y′ i This is the result of the transformation and integration, where W0 is the corresponding initial weight in equation (9), and P... yy It is the variance obtained after weighting;

[0079] For each χ′ i Integrating the dynamic equations, we obtain the transformed y′ i The point set {Y′} i}, based on the point set {Y′ i The different weight coefficients W′ for each element in} i The weighted average yields the transformed mean and standard deviation. and Integrating formulas (13) and (14) yields the mean and standard deviation of the formation arm length, breathing angle, and arm length change rate at any time during the mission cycle.

[0080] Compared with the prior art, the present invention has the following beneficial effects:

[0081] 1. This invention uses the proportionally corrected sampling UT method to analyze the orbital error of space gravitational wave detection missions. Compared with the traditional target shooting method, it greatly reduces the amount of calculation and effectively improves the calculation speed.

[0082] 2. This invention does not require finding the Jacobian matrix of the dynamic equations, which has the advantage of simplifying calculations for problems with complex dynamic models;

[0083] 3. This invention can solve the nonlocal effects caused by sampling asymmetry;

[0084] 4. The mean matrix and covariance matrix of this invention are flexible in their construction and can be used to add or remove state vectors. Attached Figure Description

[0085] Other features, objects, and advantages of the present invention will become more apparent from the following detailed description of non-limiting embodiments with reference to the accompanying drawings:

[0086] Figure 1 This is a flowchart of an optical formation satellite orbit insertion error propagation analysis method according to the present invention;

[0087] Figure 2 This is a schematic diagram of the formation arm length. Detailed Implementation

[0088] The present invention will now be described in detail with reference to specific embodiments. These embodiments will help those skilled in the art to further understand the present invention, but do not limit the invention in any way. It should be noted that those skilled in the art can make several changes and improvements without departing from the concept of the present invention. These all fall within the scope of protection of the present invention.

[0089] Example 1:

[0090] This invention relates to a method for analyzing the propagation of orbital insertion errors in optical formation satellites. The invention includes the following steps: First, establishing a dynamic model of the formation satellites; second, establishing expressions for key configuration indicators—arm length, breathing angle, and arm length change rate—in the formation problem of space gravitational wave detection missions; then, using the UT sampling method to express the relationship between the mean and covariance of these three parameters; finally, using the proportional UT method to correct the sampling points and weights. This invention effectively solves the problem of orbital insertion error analysis for formation satellites, and is fast and accurate, making it suitable for the field of spacecraft formation flight.

[0091] According to the present invention, a method for analyzing the propagation of optical formation satellite orbit insertion errors is provided, such as... Figure 1 As shown, it includes:

[0092] Step S1: Establish a dynamic model of the formation satellites;

[0093] Specifically, in step S1:

[0094] A dynamic model of the satellite formation is established. The dynamic model of the heliocentric orbit space gravitational wave detection mission is represented in the heliocentric J2000 coordinate system as follows:

[0095]

[0096] Where, r i =[x i y i z i ] T , r k This represents the position vector of the three satellites, k = 1, 2, 3; v k Represents the velocity vectors of the three satellites, k = 1, 2, 3; R i The vectors representing the positions of Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, and the Moon, i = 1, 2, ..., 8, 9; μ0 represents the solar gravitational constant, μ iLet i represent the gravitational constants of the eight planets and the Moon, i = 1, 2, ..., 8, 9.

[0097] Step S2: Establish expressions for the key configuration indicators arm length, breathing angle, and arm length change rate in the formation problem of space gravitational wave detection missions;

[0098] Specifically, in step S2:

[0099] The arm length of the formation determines the range of the detected wave source, the change in the breathing angle affects the pointing accuracy of the onboard telescope, and the rate of change of arm length affects the Doppler frequency shift. These three indicators can be expressed in formula form as follows:

[0100] L ij (t)=||R ij (t)|| (3)

[0101]

[0102] Among them, R ij Let L represent the arm length vector from satellite i to satellite j, where L represents the arm length, θ represents the breathing angle, and V represents the rate of change of arm length; t is the orbital recursion time.

[0103] Step S3: Use the UT sampling method to express the relationship between the mean and covariance of arm length, breathing angle and arm length change rate in the formation configuration;

[0104] Specifically, in step S3:

[0105] The Utahedron method is used for sampling calculation to obtain the mean and initial covariance matrix of arm length, breathing angle and arm length change rate in the formation configuration; sampling Sigma points are generated according to the formula, where the covariance matrix is ​​obtained by Cholesky decomposition; sampling weights are calculated using the formula.

[0106] The dynamic model of a space-based gravitational wave detection mission is expressed as follows:

[0107]

[0108] Among them, the state variable x has a total of 27 terms, x1 to x 18 These are the positions and velocities of the three satellites; x 19 To x 21 It is the arm length of the formation; x 22 To x 24 It is the breathing angle of the formation; x 25 To x 27 is the rate of change of the arm length of the formation configuration; f(x,t) is the dynamic expression;

[0109] Generate Sigma point χ iThe method is as follows, where the mean and covariance of the state vector x are x and P, respectively. x

[0110]

[0111] Where χ0 is the sampling center point, χ i These are other sampling points, which together constitute the sampling point set. i represents the subscript of the matrix column, n represents the dimension of x, and λ is the parameter of the sampling distribution.

[0112] in Obtained through Cholesky decomposition, i.e.

[0113]

[0114] Let represent the i-th column of the matrix, and λ be the parameter of the sampling distribution;

[0115] The sampling weights are calculated using the following formula.

[0116]

[0117] Among them, W i (m) and W i (m) These are the weights of the mean and covariance, respectively.

[0118] Step S4: Correct the sampling points and weights using the proportional UT method.

[0119] Specifically, in step S4:

[0120] The proportional UT transformation method is used to correct the UT method sampling. The values ​​of parameters α and β in the proportional UT transformation method are adjusted to keep the prediction covariance matrix positive semidefinite.

[0121] By integrating the dynamic equations, the transformed point set is obtained. The weighted average is then used to obtain the mean and standard deviation of the transformed points. Finally, the average and standard deviation of the formation arm length, breathing angle, and arm length change rate at any time during the mission cycle are obtained by integrating the equations.

[0122] {χ} obtained from UT transformation i W i The proportional UT transformation formula is:

[0123] χ′ i =χ0+α(χ i -χ0) (10)

[0124] y′ i =f(χ′) i (11)

[0125]

[0126]

[0127] Where, χ′ i These are the transformed sampling points, y′ i This is the result of the transformation and integration, where W0 is the corresponding initial weight in equation (9), and P... yy It is the variance obtained after weighting;

[0128] For each χ′ i Integrating the dynamic equations, we obtain the transformed y′ i The point set {Y′} i}, based on the point set {Y′ i The different weight coefficients W′ for each element in} i The weighted average yields the transformed mean and standard deviation. and Integrating formulas (13) and (14) yields the mean and standard deviation of the formation arm length, breathing angle, and arm length change rate at any time during the mission cycle.

[0129] Example 2:

[0130] Example 2 is a preferred embodiment of Example 1, and is used to illustrate the present invention in more detail.

[0131] The present invention also provides an optical formation satellite orbit insertion error propagation analysis system. The optical formation satellite orbit insertion error propagation analysis system can be implemented by executing the process steps of the optical formation satellite orbit insertion error propagation analysis method. That is, those skilled in the art can understand the optical formation satellite orbit insertion error propagation analysis method as a preferred embodiment of the optical formation satellite orbit insertion error propagation analysis system.

[0132] An optical formation satellite orbit insertion error propagation analysis system provided by the present invention includes:

[0133] Module M1: Establish a dynamic model of the formation satellites;

[0134] Specifically, in module M1:

[0135] A dynamic model of the satellite formation is established. The dynamic model of the heliocentric orbit space gravitational wave detection mission is represented in the heliocentric J2000 coordinate system as follows:

[0136]

[0137] Where, r i =[x i y i zi ] T , r k This represents the position vector of the three satellites, k = 1, 2, 3; v k Represents the velocity vectors of the three satellites, k = 1, 2, 3; R i The vectors representing the positions of Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, and the Moon, i = 1, 2, ..., 8, 9; μ0 represents the solar gravitational constant, μ i Let i represent the gravitational constants of the eight planets and the Moon, i = 1, 2, ..., 8, 9.

[0138] Module M2: Establishes expressions for key configuration indicators such as arm length, breathing angle, and arm length change rate in the formation problem of space gravitational wave detection missions;

[0139] Specifically, in module M2:

[0140] The arm length of the formation determines the range of the detected wave source, the change in the breathing angle affects the pointing accuracy of the onboard telescope, and the rate of change of arm length affects the Doppler frequency shift. These three indicators can be expressed in formula form as follows:

[0141] L ij (t)=||R ij (t)|| (3)

[0142]

[0143] Among them, R ij Let L represent the arm length vector from satellite i to satellite j, where L represents the arm length, θ represents the breathing angle, and V represents the rate of change of arm length; t is the orbital recursion time.

[0144] Module M3: Uses UT sampling to express the relationship between the mean and covariance of arm length, breathing angle, and arm length change rate in formation configuration;

[0145] Specifically, in module M3:

[0146] The Utahedron method is used for sampling calculation to obtain the mean and initial covariance matrix of arm length, breathing angle and arm length change rate in the formation configuration; sampling Sigma points are generated according to the formula, where the covariance matrix is ​​obtained by Cholesky decomposition; sampling weights are calculated using the formula.

[0147] The dynamic model of a space-based gravitational wave detection mission is expressed as follows:

[0148]

[0149] Among them, the state variable x has a total of 27 terms, x1 to x 18These are the positions and velocities of the three satellites; x 19 To x 21 It is the arm length of the formation; x 22 To x 24 It is the breathing angle of the formation; x 25 To x 27 is the rate of change of the arm length of the formation configuration; f(x,t) is the dynamic expression;

[0150] Generate Sigma point χ i The method is as follows, where the mean and covariance of the state vector x are respectively and P x

[0151]

[0152] Where χ0 is the sampling center point, χ i These are other sampling points, which together constitute the sampling point set. i represents the subscript of the matrix column, n represents the dimension of x, and λ is the parameter of the sampling distribution.

[0153] in Obtained through Cholesky decomposition, i.e.

[0154]

[0155] Let represent the i-th column of the matrix, and λ be the parameter of the sampling distribution;

[0156] The sampling weights are calculated using the following formula.

[0157]

[0158] Among them, W i (m) and W i (m) These are the weights of the mean and covariance, respectively.

[0159] Module M4: Corrects sampling points and weights using the proportional UT method.

[0160] Specifically, in module M4:

[0161] The proportional UT transformation method is used to correct the UT method sampling. The values ​​of parameters α and β in the proportional UT transformation method are adjusted to keep the prediction covariance matrix positive semidefinite.

[0162] By integrating the dynamic equations, the transformed point set is obtained. The weighted average is then used to obtain the mean and standard deviation of the transformed points. Finally, the average and standard deviation of the formation arm length, breathing angle, and arm length change rate at any time during the mission cycle are obtained by integrating the equations.

[0163] {χ} obtained from UT transformation i W i The proportional UT transformation formula is:

[0164] χ′ i =χ0+α(χ i -χ0) (10)

[0165] y′ i =f(χ′) i (11)

[0166]

[0167]

[0168] Where, χ′ i These are the transformed sampling points, y′ i This is the result of the transformation and integration, where W0 is the corresponding initial weight in equation (9), and P... yy It is the variance obtained after weighting;

[0169] For each χ′ i Integrating the dynamic equations, we obtain the transformed y′ i The point set {Y′} i}, based on the point set {Y′ i The different weight coefficients W′ for each element in} i The weighted average yields the transformed mean and standard deviation. and Integrating formulas (13) and (14) yields the mean and standard deviation of the formation arm length, breathing angle, and arm length change rate at any time during the mission cycle.

[0170] Example 3:

[0171] Example 3 is a preferred example of Example 1, and is used to illustrate the present invention in more detail.

[0172] To address the problem of orbit insertion error analysis for formation satellites, the purpose of this invention is to provide a method for rapidly and accurately analyzing the impact of orbit insertion error on formation configuration. This method overcomes the computational speed limitations caused by the computational complexity of the traditional Monte Carlo method, meets the requirements for orbit insertion error analysis, and provides a foundation for orbit insertion control of formation satellites.

[0173] According to one aspect of the present invention, a method for analyzing the propagation of orbital insertion errors in formation missions is provided, characterized in that the method includes the following steps:

[0174] Step 1: Establish a dynamic model of the formation satellites. Taking the heliocentric orbit space gravitational wave detection formation flight mission as an example, the formation satellites are subject to the gravitational pull of the Sun, Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, and the Moon.

[0175] Step 2: In space-based gravitational wave detection missions, three key indicators need to be monitored: formation arm length, breathing angle, and arm length variation rate. The formation arm length determines the range of the detected wave source, the breathing angle affects the pointing accuracy of the onboard telescope, and the arm length variation rate affects the Doppler frequency shift. Express these three indicators using formulas.

[0176] Step 3: Use the basic UT method to perform sampling calculations to obtain the mean and initial covariance matrix of arm length, breathing angle and arm length change rate in the formation configuration.

[0177] Step 4: Generate sampling Sigma points according to a specific formula, where the covariance matrix is ​​obtained through Cholesky decomposition.

[0178] Step 5: Calculate the sampling weights using a specific formula.

[0179] Step 6: Correct the UT method sampling described above using the proportional UT transformation method. Adjusting the values ​​of parameters α and β in the proportional UT transformation method can keep the prediction covariance matrix positive semi-definite, which is applicable to all sampling strategies.

[0180] Step 7: Integrate the dynamic equations to obtain the transformed point set, and then weight the transformed mean and standard deviation. The integral yields the mean and standard deviation of the formation arm length, breathing angle, and arm length change rate at any time within the mission cycle.

[0181] The specific steps are as follows:

[0182] Step 1: Establish a dynamic model for the satellite formation. Taking a heliocentric orbit space gravitational wave detection mission as an example, its dynamic model in the heliocentric J2000 coordinate system can be expressed as follows:

[0183]

[0184] Where r i =[x i y i z i ] T , r k This represents the position vector of the three satellites, k = 1, 2, 3; v k Let R represent the velocity vectors of the three satellites, where k = 1, 2, 3. iRepresents the position vectors of Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, and the Moon, i = 1, 2, ..., 8, 9. μ0 represents the solar gravitational constant, μ i Let i represent the gravitational constants of the eight planets and the Moon, i = 1, 2, ..., 8, 9.

[0185] Step 2: List the expressions for the three key indicators of the space gravitational wave detection mission formation mission: arm length, breathing angle, and arm length change rate.

[0186] L ij (t)=||R ij (t)|| (3)

[0187]

[0188] Among them, R ij Let L represent the arm length vector from satellite i to satellite j, where L represents the arm length, θ represents the breathing angle, and V represents the rate of change of arm length.

[0189] The subscripts for the arm length vector R and arm length L are 12 (arm length between satellite 1 and satellite 2), 13 (arm length between satellite 1 and satellite 3), and 23 (arm length between satellite 2 and satellite 3), respectively. The subscripts for the breathing angle are 1 (angle between L12 and L13), 2 (angle between L12 and L23), and 3 (angle between L31 and L32). A schematic diagram is shown below. Figure 2 .

[0190] Step 3: Use the basic UT method to perform sampling calculations to obtain the mean and initial covariance matrix of arm length, breathing angle and arm length change rate in the formation configuration.

[0191] The dynamic model of the space gravitational wave detection mission in this invention can be expressed as follows:

[0192]

[0193] The state variable x has 27 terms, from x1 to x... 18 These are the positions and velocities of the three satellites; x 19 To x 21 It is the arm length of the formation; x 22 To x 24 It is the breathing angle of the formation; x 25 To x 27 It is the rate of change of the arm length of the formation configuration. f(x,t) is the dynamic expression, namely formula (1) and formula (2) above.

[0194] Step 4: Generate Sigma points χ i The method is as follows, where the mean and covariance of the state vector x are respectively and P x

[0195]

[0196] in It can be obtained through Cholesky decomposition, i.e.

[0197]

[0198] Let λ represent the i-th column of the matrix, and λ be the parameter of the sampling distribution, which is usually taken as n+λ=3.

[0199] Step 5: Calculate the sampling weights, which can be obtained using the following formula.

[0200]

[0201] Step 6: Correct the UT method sampling described above using the proportional UT transformation method. Adjusting the values ​​of parameters α and β in the proportional UT transformation method can keep the prediction covariance matrix positive semi-definite, which is applicable to all sampling strategies.

[0202] Regarding {χ} obtained from the UT transformation above... i W i The proportional UT transformation formula is:

[0203] χ′ i =χ0+α(χ i -χ0) (10)

[0204] y′ i =f(x′) i (11)

[0205]

[0206] Step 7: For each χ′ i Integrating the dynamic equations, we obtain the transformed y′ i The point set {Y′} i}. Based on the point set {Y′ i The different weight coefficients W′ for each element in} i The weighted average yields the transformed mean and standard deviation. and Integrating formulas (13) and (14) yields the mean and standard deviation of the formation arm length, breathing angle, and arm length change rate at any time during the mission cycle.

[0207] Those skilled in the art will understand that, besides implementing the system and its various devices, modules, and units provided by this invention in the form of purely computer-readable program code, the same functions can be achieved entirely through logical programming of the method steps, making the system and its various devices, modules, and units of this invention function in the form of logic gates, switches, application-specific integrated circuits, programmable logic controllers, and embedded microcontrollers. Therefore, the system and its various devices, modules, and units provided by this invention can be considered as a hardware component, and the devices, modules, and units included therein for implementing various functions can also be considered as structures within the hardware component; alternatively, the devices, modules, and units for implementing various functions can be considered as both software modules implementing the method and structures within the hardware component.

[0208] Specific embodiments of the present invention have been described above. It should be understood that the present invention is not limited to the specific embodiments described above, and those skilled in the art can make various changes or modifications within the scope of the claims, which do not affect the essence of the present invention. Unless otherwise specified, the embodiments and features described in this application can be arbitrarily combined with each other.

Claims

1. A method for analyzing orbital error propagation of optical formation satellites, characterized in that: include: Step S1: Establishing a dynamic model of formation satellites; Step S2: establishing expressions for the key configuration indicators arm length, breathing angle and arm length change rate in the space gravitational wave detection mission formation problem; Step S3: using the UT sampling method to express the relationship between the mean and covariance of the arm length, breathing angle and arm length change rate in the formation configuration; Step S4: Correct the sampling points and weights using the proportional UT method.

2. The optical formation satellite orbit entry error propagation analysis method according to claim 1, characterized in that: In step S1: The dynamic model of the formation satellites is established. The dynamic model of the heliocentric orbit space gravitational wave detection mission is expressed in the heliocentric J2000 coordinate system as follows: Among them, r i =[x i y i z i ] T , r k represents the position vectors of three satellites, k = 1, 2, 3; v k Represents the velocity vectors of the three satellites, k = 1, 2, 3; R i represents the position vectors of Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, and the Moon, i = 1, 2, ..., 8, 9; μ0 represents the solar gravitational constant, μ i They represent the gravitational constants of the eight planets and the moon respectively, i = 1, 2, ..., 8, 9.

3. The optical formation satellite orbit entry error propagation analysis method according to claim 1, characterized in that: In step S2: The arm length of the formation configuration determines the range of the detection wave source, the change of the breathing angle affects the pointing accuracy of the on-board telescope, and the arm length change rate affects the Doppler frequency shift. The above three indicators are expressed in the form of formula: L ij (t)=||R ij (t)|| (3) Among them, R ij represents the arm length vector from satellite i to satellite j, L represents the arm length, θ represents the breathing angle, V represents the arm length change rate; t is the orbit recursion time.

4. The optical formation satellite orbit entry error propagation analysis method according to claim 1, characterized in that: In step S3: The UT method is used to perform sampling calculations to obtain the mean and covariance initial matrix of the arm length, breathing angle and arm length change rate in the formation configuration; Generate sampling Sigma points according to the formula, where the covariance matrix is ​​obtained by Cholesky decomposition; Calculate the sampling weight using the formula; The dynamic model of the space gravitational wave detection mission is expressed as Among them, the state variable x has 27 items, x1 to x 18 are the position velocities of the three satellites; x 19 to x 21 is the arm length of the formation configuration; x 22 to x 24 is the breathing angle of the formation configuration; x 25 to x 27 is the arm length change rate of the formation configuration; f(x,t) is the dynamic expression; Generate Sigma Point χ i The method is as follows, where the mean and covariance of the state vector x are and P x Among them, χ0 is the sampling center point, χ i are other sampling points, and the two together constitute the sampling point set, i represents the subscript of the matrix column, n represents the dimension of x, and λ is the parameter of the sampling distribution; in Through Cholesky decomposition, we get represents the i-th column of the matrix, λ is the parameter of the sampling distribution; For the calculation of sampling weights, the following formula is used to obtain Among them, W i (m) and W i (m) are the weights of the mean and covariance respectively.

5. The optical formation satellite orbit entry error propagation analysis method according to claim 1, characterized in that: In step S4: The proportional UT transformation method is used to correct the UT method sampling, and the values ​​of parameters α and β in the proportional UT transformation method are adjusted to keep the prediction covariance matrix semi-positive definite; According to the integration of the dynamic equation, the transformed point set is obtained, and the mean and standard deviation of the transformed point set are obtained by weighting. The mean and standard deviation of the formation configuration arm length, breathing angle and arm length change rate at any time during the mission cycle are obtained by integration. {χ obtained by UT transformation i ,W i }, the proportional UT transformation formula is x i ′=χ0+α(χ i -x0) (10) y i ′=f(x i ′) (11) Among them, χ i ′ is the transformed sampling point, y i ′ is the transformed integral result, W0 is the corresponding initial weight in equation (9), P yy is the variance obtained after weighting; For each χ i ′ According to the dynamic equation, the transformation is obtained by integrating y i ′ is composed of a set of points {Y i ′}, according to the point set {Y i Different weight coefficients W for each element in ′} i ′, weighted to obtain the transformed mean and standard deviation and By integrating formulas (13) and (14), we can obtain the mean and standard deviation of the formation configuration arm length, breathing angle and arm length change rate at any time during the mission cycle.

6. An optical formation satellite orbit entry error propagation analysis system, characterized in that: include: Module M1: Establish the dynamic model of formation satellites; Module M2: Establish expressions for the key configuration indicators arm length, breathing angle and arm length change rate in the space gravitational wave detection mission formation problem; Module M3: Use UT sampling method to express the relationship between the mean and covariance of arm length, breathing angle and arm length change rate in formation configuration; Module M4: Correct sampling points and weights using proportional UT method.

7. The optical formation satellite orbit entry error propagation analysis system according to claim 6, characterized in that: In the module M1: The dynamic model of the formation satellites is established. The dynamic model of the heliocentric orbit space gravitational wave detection mission is expressed in the heliocentric J2000 coordinate system as follows: Among them, r i =[x i y i z i ] T , r k represents the position vectors of three satellites, k = 1, 2, 3; v k Represents the velocity vectors of the three satellites, k = 1, 2, 3; R i represents the position vectors of Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, and the Moon, i = 1, 2, ..., 8, 9; μ0 represents the solar gravitational constant, μ i They represent the gravitational constants of the eight planets and the moon respectively, i = 1, 2, ..., 8, 9.

8. The optical formation satellite orbit entry error propagation analysis system according to claim 6, characterized in that: In the module M2: The arm length of the formation configuration determines the range of the detection wave source, the change of the breathing angle affects the pointing accuracy of the on-board telescope, and the arm length change rate affects the Doppler frequency shift. The above three indicators are expressed in the form of formula: L ij (t)=||R ij (t)|| (3) Among them, R ij represents the arm length vector from satellite i to satellite j, L represents the arm length, θ represents the breathing angle, V represents the arm length change rate; t is the orbit recursion time.

9. The optical formation satellite orbit entry error propagation analysis system according to claim 6, characterized in that: In the module M3: The UT method is used to perform sampling calculations to obtain the mean and covariance initial matrix of the arm length, breathing angle and arm length change rate in the formation configuration; the sampling Sigma points are generated according to the formula, where the covariance matrix is ​​obtained by Cholesky decomposition; Calculate the sampling weight using the formula; The dynamic model of the space gravitational wave detection mission is expressed as Among them, the state variable x has 27 items, x1 to x 18 are the position velocities of the three satellites; x 19 to x 21 is the arm length of the formation configuration; x 22 to x 24 is the breathing angle of the formation configuration; x 25 to x 27 is the arm length change rate of the formation configuration; f(x,t) is the dynamic expression; Generate Sigma Point χ i The method is as follows, where the mean and covariance of the state vector x are and P x Among them, χ0 is the sampling center point, χ i are other sampling points, and the two together constitute the sampling point set, i represents the subscript of the matrix column, n represents the dimension of x, and λ is the parameter of the sampling distribution; in Through Cholesky decomposition, we get represents the i-th column of the matrix, λ is the parameter of the sampling distribution; For the calculation of sampling weights, the following formula is used to obtain Among them, W i (m) and W i (m) are the weights of the mean and covariance respectively.

10. The optical formation satellite orbit entry error propagation analysis system according to claim 6, characterized in that: In the module M4: The proportional UT transformation method is used to correct the UT method sampling, and the values ​​of parameters α and β in the proportional UT transformation method are adjusted to keep the prediction covariance matrix semi-positive definite; According to the integration of the dynamic equation, the transformed point set is obtained, and the mean and standard deviation of the transformed point set are obtained by weighting. The mean and standard deviation of the formation configuration arm length, breathing angle and arm length change rate at any time during the mission cycle are obtained by integration. {χ obtained by UT transformation i ,W i }, the proportional UT transformation formula is x i ′=χ0+α(χ i -x0) (10) y i ′=f(x i ′) (11) Among them, χ i ′ is the transformed sampling point, y i ′ is the transformed integral result, W0 is the corresponding initial weight in equation (9), P yy is the variance obtained after weighting; For each χ i ′ According to the dynamic equation, the transformation is obtained by integrating y i ′ is composed of a set of points {Y i ′}, according to the point set {Y i Different weight coefficients W for each element in ′} i ′, weighted to obtain the transformed mean and standard deviation and By integrating formulas (13) and (14), we can obtain the mean and standard deviation of the formation configuration arm length, breathing angle and arm length change rate at any time during the mission cycle.

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