Stability Analysis Method of Interferometric Constellations Based on Decoupled State Transition Tensors

Abstract

This invention discloses a method for analyzing the configuration stability of a high-precision space measurement constellation based on decoupled state transition tensors, belonging to the field of space technology. The method involves: decoupling the influence of different perturbations on configuration stability; establishing a mapping relationship between the spacecraft's orbital state and configuration stability indices; substituting the obtained orbital state transition tensor expression for perturbation decoupling into the method to obtain the state transition tensor expression for the configuration stability indices; analyzing and obtaining the influence of different perturbations on configuration stability, thereby supporting the optimization of the high-precision space measurement constellation configuration, improving the performance of the high-precision space measurement constellation system, and increasing detection accuracy. This invention has the advantages of high accuracy and high efficiency in analyzing the configuration stability of high-precision space measurement constellations, which is beneficial for the rapid iterative optimization of high-precision space measurement constellation configurations and for improving the stability and detection performance of high-precision space measurement constellation configurations.

Description

Technical Field

[0001] This invention relates to a method for high-precision space measurement and constellation configuration stability analysis, belonging to the field of space technology. Background Technology

[0002] Observing outer space is crucial for understanding the origin of the universe, exploring space, and discovering extraterrestrial life and Earth-like planets. It is of great significance for the development of space science and technology, planetary physics, and the exploration of the universe. Due to the vast distances and weak signals from targets in outer space, interferometry is typically used to improve observation sensitivity. However, ground-based interferometry methods have limited applicability due to atmospheric obstruction, ground vibration noise, and the influence of surface gravity gradients. Therefore, deploying multiple interferometric spacecraft to form a space interferometric constellation has become a feasible solution for future high-precision interferometry. For a high-precision space-based constellation, it is necessary to control the stability of the verification quality and ensure the measurement accuracy of laser interferometry and space inertial sensing. This places high demands on the stability of the probe configuration. The dynamic environment of a high-precision space-based constellation is complex, with multiple perturbations coupling with each other, affecting the stability of the configuration. Therefore, it is necessary to analyze the impact of different perturbations on the configuration to provide a basis for selecting the configuration design area. However, due to the coupling of perturbations, conventional methods are insufficient to analyze the influence mechanism of individual perturbations. Therefore, it is necessary to study the stability analysis method of decoupled high-precision space measurement constellation configuration, analyze the influence of different perturbations on the stability of the configuration, and provide a reference for the selection and design of high-precision space measurement constellations. Summary of the Invention

[0003] The main objective of this invention is to provide a stability analysis method for high-precision space measurement constellations based on decoupled state transition tensors. Addressing the stability of high-precision space measurement constellation configurations under complex dynamic environments, this method decouples the influence of different perturbations on configuration stability, establishes a mapping relationship between spacecraft orbital states and configuration stability indices, substitutes the obtained decoupled orbital state transition tensor expressions into the method, and categorizes perturbations according to their types to obtain state transition tensor expressions for perturbation-decoupled configuration stability indices. Based on these expressions, the influence of different perturbations on configuration stability is analyzed, thereby supporting the optimization of high-precision space measurement constellation configurations, improving system performance, enhancing detection accuracy, and ultimately solving related technical problems in the field of high-precision space measurement constellations. This invention offers advantages such as high accuracy and efficiency in analyzing the stability of high-precision space measurement constellation configurations, facilitating rapid iterative optimization and improving the stability and detection performance of high-precision space measurement constellation configurations.

[0004] The objective of this invention is achieved through the following technical solutions.

[0005] This invention discloses a stability analysis method for interferometric constellations based on decoupled state transition tensors. The method decomposes the spacecraft's dynamic equations in a high-precision space measurement constellation configuration according to perturbations, obtaining decoupled dynamic equations. Then, based on the type of perturbation, a decoupled state transition tensor is defined and initialized. Next, the decoupled state transition tensor is integrated based on the decomposed dynamic equations to obtain the orbital state transition tensor expression for perturbation decoupling. A mapping relationship is established between the spacecraft's orbital state and configuration stability indices. Substituting the obtained perturbation-decoupled orbital state transition tensor expression into this method, and classifying it according to different perturbations, a state transition tensor expression for the configuration stability index is obtained. Finally, based on the obtained state transition tensor expression for the decoupled configuration stability index, the influence of different perturbations on configuration stability is analyzed, thus providing support for optimizing the configuration of a high-precision space measurement constellation, improving the system performance, increasing detection accuracy, and ultimately solving related technical problems in the field of high-precision space measurement constellations. This invention has the advantages of high accuracy and high efficiency in the stability analysis of high-precision space measurement constellations, which is conducive to the rapid iterative optimization of the configuration of high-precision space measurement constellations and to improving the stability and detection performance of the configuration of high-precision space measurement constellations.

[0006] This invention discloses a method for analyzing the stability of interferometric constellations based on decoupled state transition tensors, comprising the following steps:

[0007] Step 1: Establish the dynamic equations of the spacecraft in the high-precision space measurement constellation configuration, decompose them according to the perturbation force, and obtain the decoupled dynamic equations.

[0008] The dynamic equations of the spacecraft in the high-precision space measurement constellation configuration are as follows:

[0009]

[0010] Where x(t) = [r(t)] T ,v(t) T ] T Let f(x,t) represent the spacecraft's state (position and velocity) at time t, and let f(x,t) represent the spacecraft's dynamic equations. x0 is the initial state given at time t0.

[0011] The dynamics in equation (1) are decoupled according to different perturbation forces as follows:

[0012]

[0013] Where f (E)(x,t) f (M) (x,t) and f (S) (x,t) represent the dynamic equations for the Earth's central gravity, the Earth's J2 term perturbation, the Moon's gravitational perturbation, and the Sun's gravitational perturbation, respectively.

[0014] Step 2: Define the decoupled state transition tensor according to the type of perturbation force, and initialize it.

[0015] The decoupled state transition tensor is not limited to second order and can be extended to any order according to actual needs. Preferably, when the decoupled state transition tensor is decomposed into a second-order tensor, the implementation method is as follows:

[0016] Consider the first-order and second-order state transition tensors as follows:

[0017]

[0018]

[0019] In formulas (3) and (4), Φ i,a With Φ i,ab These are the first-order state transition tensor and the second-order state transition tensor, respectively. and For the influence of a single perturbation force, and This is the coupling effect term of the two perturbation forces. and The coupling effect term of the three perturbation forces, and This represents the coupling effect term of the four perturbation forces. In the above state transition tensor, and Initialize to the identity matrix. and Initialize it as a negative identity matrix, and the rest as zero matrices.

[0020] Step 3: Based on the dynamic equations decomposed in Step 1, integrate the decoupled state transition tensor to obtain the expression for the orbital state transition tensor of the perturbation decoupling. By integrating the decoupled state transition tensor defined in Step 2, the decoupled orbital state transition tensor Φ at any time is obtained. i,a Φ i,ab .

[0021] Based on the decomposed dynamic equations, integrating the decoupled state transition tensor yields the expression for the orbital state transition tensor of the perturbation decoupling. By integrating the decoupled state transition tensor defined in step 2, the decoupled orbital state transition tensor Φ at any given time can be obtained. i,a Φ i,abThe differential equations for the integral of the state transition tensor are shown in equations (5) and (6).

[0022]

[0023]

[0024] The subscripts in formulas (5) and (6) represent Einstein's summation notation. In formulas (5) and (6) and Representing the first-order and second-order Jacobian matrices:

[0025]

[0026]

[0027] in and The first- and second-order Jacobian matrices are used for decoupling based on the perturbation force, with superscripts corresponding to different perturbation forces. The differential equations for the state transition tensors of each decoupling are obtained from the superscripts as follows:

[0028]

[0029]

[0030]

[0031] Step 4: Establish the mapping relationship between the spacecraft's orbital state and configuration stability index, and express the obtained orbital state transition tensor expression Φ after decoupling from the perturbation force. i,a Φ i,ab Substituting the mapping relationship, and classifying according to different perturbation forces, we obtain the state transition tensor expression of the configuration stability index for perturbation force decoupling.

[0032] The state transition tensor model of the configuration stability index with respect to the orbital state is established as follows:

[0033]

[0034] Where y is the configuration stability index, x is the orbital state; G i,a With H i,ab These are the first- and second-order state transition tensors of the configuration stability index with respect to the orbital state.

[0035] The obtained orbital state transition tensor expression Φ is decoupled from the perturbation force. i,a Φ i,ab Substituting into formula (12), and classifying according to different perturbation forces, the state transition tensor expression of the configuration stability index for perturbation force decoupling is obtained as follows:

[0036]

[0037]

[0038] Where P i,p Q i,pl Let be the state transition tensor of the configuration stability index with respect to the initial state. and For the influence of a single perturbation force, and This is the coupling effect term of the two perturbation forces. and The coupling effect term of the three perturbation forces, and The coupling effects of the four perturbation forces are as follows:

[0039]

[0040]

[0041]

[0042]

[0043] Step 5: Based on the state transition tensor expression of the decoupled configuration stability index, analyze the influence of different perturbation forces on configuration stability, and then obtain the influence of different perturbation forces on configuration stability.

[0044] The invention also includes step 6: based on the impact of different perturbation forces on configuration stability analyzed in step 5, the most influential factor is determined, thereby supporting the optimization of the space high-precision measurement constellation configuration, improving the performance of the space high-precision measurement constellation system, increasing detection accuracy, and ultimately solving related technical problems in the field of space high-precision measurement constellations. This invention has the advantages of high accuracy and high efficiency in analyzing the stability of space high-precision measurement constellation configurations, which is conducive to the rapid iterative optimization of space high-precision measurement constellation configurations and helps improve the stability and detection performance of space high-precision measurement constellation configurations.

[0045] The decoupled state transition tensor is not limited to second order, but can be extended to any order according to actual needs.

[0046] Beneficial effects:

[0047] 1. This invention discloses a stability analysis method for interferometric constellations based on decoupled state transition tensors. The method decomposes the dynamic equations of spacecraft in a high-precision space measurement constellation configuration according to perturbation forces, obtaining decoupled dynamic equations. Then, based on the type of perturbation force, a decoupled state transition tensor is defined and initialized. Next, the decoupled state transition tensor is integrated based on the decomposed dynamic equations to obtain the orbital state transition tensor expression for perturbation decoupling. A mapping relationship is established between the spacecraft's orbital state and the configuration stability index. Substituting the obtained perturbation decoupling orbital state transition tensor expression into this relationship, and classifying it according to different perturbations, the method obtains the state transition tensor expression for the configuration stability index of perturbation decoupling. Finally, based on the obtained state transition tensor expression for the decoupled configuration stability index, the influence of different perturbations on configuration stability is analyzed, thereby supporting the optimization of the high-precision space measurement constellation configuration, improving the performance of the high-precision space measurement constellation system, and solving related technical problems in the field of high-precision space measurement constellations.

[0048] 2. The present invention discloses an interferometric constellation stability analysis method based on decoupled state transition tensor, which requires only one integration, has a small computational load, and can be reused for different problems, which helps to reduce the computation time of gravitational wave detection configuration stability analysis.

[0049] 3. The present invention discloses an interferometric constellation stability analysis method based on decoupled state transition tensor. By decomposing the influence of different perturbation forces on the configuration stability, it is possible to decouple and analyze the influence of a single perturbation force on the configuration stability of a high-precision space measurement constellation. This method is helpful for the selection and design of high-precision space measurement constellation configurations, improves the performance of high-precision space measurement constellation systems, and solves related technical problems in the field of high-precision space measurement constellations.

[0050] 4. The present invention discloses an interferometric constellation stability analysis method based on decoupled state transition tensor. On the basis of achieving the above three beneficial effects, it has the advantages of high accuracy and high efficiency in the stability analysis of high-precision space measurement constellation configuration. It is conducive to the rapid iterative optimization of high-precision space measurement constellation configuration and to improving the stability and detection performance of high-precision space measurement constellation configuration.

[0051] 5. This invention discloses an interferometric constellation stability analysis method based on decoupled state transition tensors. It employs decoupled state transition tensors to analyze the stability of high-precision space measurement constellation configurations. By replacing the high-precision space measurement constellation configuration with near-Earth spacecraft dynamics, deep-space spacecraft dynamics, and small celestial body appendage spacecraft dynamics, it extends the application to near-Earth spacecraft orbit stability analysis, deep-space transfer orbit stability analysis, and small celestial body near-object orbital stability analysis, and solves related technical problems. Attached Figure Description

[0052] Figure 1 This is a flowchart of an interferometric constellation stability analysis method based on decoupled state transition tensor disclosed in this invention;

[0053] Figure 2 A comparative diagram showing the impact of different perturbation forces on the stability of the "Tianqin" space high-precision interferometry constellation configuration is provided for analysis. Figure 2 (a1) is a projection diagram showing the influence of the arm lengths of spacecraft 1 and 2 on the arm lengths of spacecraft 1 and 3. Figure 2 (a2) is a projection diagram showing the influence of the arm lengths of spacecraft 1 and spacecraft 2 on the arm lengths of spacecraft 2 and spacecraft 3. Figure 2 (a3) is a projection diagram showing the influence of the arm lengths of spacecraft 1 and spacecraft 3 on the arm lengths of spacecraft 2 and spacecraft 3. Figure 2 (b1) is a projection diagram showing the influence of the breathing angles of spacecraft 1 and spacecraft 2. Figure 2 (b2) is a projection diagram showing the influence of the breathing angles of spacecraft 1 and spacecraft 3. Figure 2 (b3) is a projection diagram showing the influence of the breathing angles of spacecraft 2 and spacecraft 3. Figure 2 (c1) is a projection diagram showing the influence of the rate of change of the arm lengths of spacecraft 1 and spacecraft 2 on the rate of change of the arm lengths of spacecraft 1 and spacecraft 3. Figure 2 (c2) is a projection diagram showing the influence of the rate of change of the arm lengths of spacecraft 1 and spacecraft 2 on the rate of change of the arm lengths of spacecraft 2 and spacecraft 3. Figure 2 (c3) is a projection diagram showing the influence of the change rate of the arm length of spacecraft 1 and spacecraft 3 on the change rate of the arm length of spacecraft 2 and spacecraft 3. Figure 2 (d) is a projection diagram showing the effect of the configuration pointing angle.

[0054] Figure 3 A comparison of the effects of different perturbation forces on the arm length of the "Tianqin" space high-precision interferometry constellation configuration under different semi-major axis designs. Detailed Implementation

[0055] To better illustrate the purpose and advantages of the present invention, the invention will be further described below in conjunction with the accompanying drawings and embodiments.

[0056] According to the method proposed in this invention and Figure 1 As shown, this example uses the geocentric gravitational wave detection configuration as an example to analyze the stability of the "Tianqin" space high-precision interferometry constellation configuration, and uses the method proposed in this invention for analysis.

[0057] This example discloses a method for stability analysis of interferometric constellations based on decoupled state transition tensors. The specific implementation method is as follows:

[0058] Step 1: Establish the dynamic equations of the spacecraft in the configuration of the "Tianqin" space high-precision interferometry constellation, decompose them according to the perturbation force, and obtain the decoupled dynamic equations.

[0059] The "Tianqin" space-based high-precision interferometry constellation consists of three spacecraft, and their corresponding orbital parameters are shown in Table 2. In this example, the high-precision dynamic model used includes the Earth's J2-order gravitational field, the Moon's gravitational pull, and the Sun's central gravity. The stability parameters considered include the arm length l. ij (t), breathing angle θ i The arm length change rate and pointing angle were measured. The stability analysis lasted for 3 months. The arm length expression is as follows:

[0060] l ij (t)=||r ij || (19)

[0061] The expression for the breathing angle is:

[0062]

[0063] The expression for the rate of change of arm length is:

[0064]

[0065] Where r ij =r i -r j This represents the relative position vector from spacecraft i to spacecraft j.

[0066] Table 1. Orbital parameters of the Lyra constellation

[0067]

[0068] Step 2: Define and initialize the decoupled state transition tensor according to the type of perturbation force. The initialization method for the decoupled state transition tensor is as follows:

[0069]

[0070]

[0071]

[0072] Where t0 represents the initial time, I 6×6 Represents the identity matrix, 0 6×6×6 It is a zero matrix.

[0073] Step 3: Based on the decomposed dynamic equations, integrate the decoupled state transition tensor to obtain the orbital state transition tensor expression for perturbation decoupling.

[0074] Step 4: Establish the mapping relationship between the spacecraft's orbital state and configuration stability index. Substitute the obtained orbital state transition tensor expression for perturbation decoupling into the perturbation force and classify it according to different perturbations to obtain the state transition tensor expression for the configuration stability index for perturbation decoupling.

[0075] Step 5: Based on the state transition tensor expression of the decoupled configuration stability index, analyze the influence of different perturbation forces on configuration stability, and then obtain the influence of different perturbation forces on configuration stability.

[0076] Step 6: Based on the influence of different perturbation forces on configuration stability analyzed in Step 5, determine the most influential factor, thereby supporting the optimization of the configuration of the "Tianqin" space high-precision interferometry constellation, improving the performance of the "Tianqin" space high-precision interferometry constellation system, increasing detection accuracy, and ultimately solving related technical problems in the field of space high-precision interferometry constellations.

[0077] Figure 2 The influence of different perturbation forces on the arm length, breathing angle, rate of change of arm length, and pointing angle of the "Tianqin" space high-precision interferometry constellation configuration is presented. As can be seen from the figure, the gravitational perturbation of Earth (J2 term) and the Moon has the greatest impact on the stability of the "Tianqin" space high-precision interferometry constellation configuration.

[0078] Figure 3 The effects of different perturbations under varying semi-major axes on the arm length, breathing angle, rate of change of arm length, and pointing angle of the Tianqin space high-precision interferometry constellation configuration are presented. When the semi-major axis is less than 50,000 km, the influence of the Earth's J2 term is relatively significant. When the semi-major axis is greater than 160,000 km, the lunar gravitational perturbation has a significant impact on the stability of the Tianqin space high-precision interferometry constellation configuration. Therefore, these two regions should be avoided when designing the configuration.

[0079] 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 analyzing the stability of interferometric constellations based on decoupled state transition tensors, characterized in that: Includes the following steps, Step 1: Establish the dynamic equations of the spacecraft in the high-precision space measurement constellation configuration, decompose them according to the perturbation force, and obtain the decoupled dynamic equations; Step 2: Define the decoupled state transition tensor according to the type of perturbation force, and initialize it; Step 3: Based on the dynamic equations decomposed in Step 1, integrate the decoupled state transition tensor to obtain the expression for the orbital state transition tensor of the perturbation decoupling; by integrating the decoupled state transition tensor defined in Step 2, obtain the decoupled orbital state transition tensor at any time. , ; Step 4: Establish the mapping relationship between the spacecraft's orbital state and configuration stability index, and express the obtained orbital state transition tensor expression decoupled from the perturbation force. , Substituting the mapping relationship, we classify the data according to different perturbation forces and obtain the state transition tensor expression of the configuration stability index for perturbation force decoupling; Step 5: Based on the state transition tensor expression of the decoupled configuration stability index, analyze the influence of different perturbation forces on configuration stability, and then obtain the influence of different perturbation forces on configuration stability.

2. The method for analyzing the stability of an interferometric constellation based on a decoupled state transition tensor as described in claim 1, characterized in that: It also includes step 6, Based on the influence of different perturbation forces on configuration stability analyzed in step 5, the most influential factor is determined, thereby supporting the optimization of the configuration of the high-precision space measurement constellation, improving the performance of the high-precision space measurement constellation system, increasing detection accuracy, and solving related technical problems in the field of high-precision space measurement constellations. This invention has the advantages of high accuracy and high efficiency in the stability analysis of the configuration of high-precision space measurement constellations, which is conducive to the rapid iterative optimization of the configuration of high-precision space measurement constellations and to improving the stability and detection performance of the configuration of high-precision space measurement constellations.

3. The method for analyzing the stability of an interferometric constellation based on a decoupled state transition tensor as described in claim 2, characterized in that: The decoupled state transition tensor is not limited to second order, but can be extended to any order according to actual needs.

4. A method for analyzing the stability of an interferometric constellation based on a decoupled state transition tensor as described in claim 1, 2, or 3, characterized in that: Step 1 is implemented as follows: The dynamic equations of the spacecraft in the high-precision space measurement constellation configuration are as follows: in Indicates the spacecraft is in The state at any given moment (position and velocity). The equations representing the dynamics of a spacecraft; for The initial state given at any given time; The formula The dynamics in the middle are decoupled according to different perturbation forces as follows: in , , and These represent the dynamic equations for the Earth's central gravity, the Earth's J2 term perturbation, the Moon's gravitational perturbation, and the Sun's gravitational perturbation, respectively.

5. The method for analyzing the stability of an interferometric constellation based on a decoupled state transition tensor as described in claim 4, characterized in that: Step 2 is implemented as follows: The decoupled state transition tensor is not limited to second order and can be extended to any order according to actual needs. When the decoupled state transition tensor is decomposed into a second-order tensor, the implementation method is as follows: Consider the first-order and second-order state transition tensors as follows: ， formula and middle and These are the first-order state transition tensor and the second-order state transition tensor, respectively. and For the influence of a single perturbation force, and This is the coupling effect term of the two perturbation forces. and The coupling effect term of the three perturbation forces, and The coupling effects of the four perturbation forces are the terms; in the above state transition tensor, and Initialize to the identity matrix. and Initialize it as a negative identity matrix, and the rest as zero matrices.

6. The method for analyzing the stability of an interferometric constellation based on a decoupled state transition tensor as described in claim 5, characterized in that: Step 3 is implemented as follows: Based on the decomposed dynamic equations, the decoupled state transition tensor is integrated to obtain the expression for the orbital state transition tensor of the perturbation decoupling; by integrating the decoupled state transition tensor defined in step 2, the decoupled orbital state transition tensor at any time can be obtained. , The differential equation for the integral of the state transition tensor is as follows: and As shown; , formula and The subscript indicates Einstein's summation notation; the formula and middle and Representing the first-order and second-order Jacobian matrices: , in , , , , , , , and The first- and second-order Jacobian matrices are used for decoupling based on the perturbation force, with the superscripts corresponding to different perturbation forces. The differential equations for the state transition tensors of each decoupling are obtained based on the superscripts as follows: ， ， 7. The method for analyzing the stability of an interferometric constellation based on a decoupled state transition tensor as described in claim 6, characterized in that: Step 4 is implemented as follows: The state transition tensor model of the configuration stability index with respect to the orbital state is established as follows: in As an indicator of configurational stability, It refers to the orbital state; and These are the first- and second-order state transition tensors of the configuration stability index with respect to the orbital state; The obtained orbital state transition tensor expression decoupled from the perturbation force. , Substitute into the formula Based on different perturbation forces, the state transition tensor expression for the configuration stability index of perturbation force decoupling is obtained as follows: ， in , Let be the state transition tensor of the configuration stability index with respect to the initial state. and For the influence of a single perturbation force, and This is the coupling effect term of the two perturbation forces. and The coupling effect term of the three perturbation forces, and The coupling effects of the four perturbation forces are as follows: ， ， ， 。

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