Spatial high-precision interferometry geocentric configuration injection stability domain expansion method

Abstract

The invention discloses a space high-precision interference measurement geocentric configuration injection stability domain expansion method, and belongs to the field of aerospace. The implementation method comprises the following steps: establishing a spatial high-precision interference measurement geocentric configuration spacecraft kinetic equation; establishing a spatial high-precision interference measurement geocentric configuration stability characterization model, and establishing a mapping relation between a spatial high-precision interference measurement geocentric configuration stability index and a spacecraft state vector in the spatial high-precision interference measurement geocentric configuration. And establishing a semi-analytic expression of the orbit deviation propagation of the geocentric configuration spacecraft in spatial high-precision interference measurement. And establishing a semi-analytical expression of the geocentric configuration orbit deviation propagation of spatial high-precision interferometry. And establishing a semi-analytical expression of deviation propagation of the stability index of the geocentric configuration in spatial high-precision interferometry. And increasing the weight of the stability index at the moment in each Newton iteration, and performing iterative optimization on the initial state of the spacecraft according to the optimal solution of the Newton method to obtain an expanded space high-precision interference measurement geocentric configuration injection stability domain.

Description

Technical Field

[0001] The invention relates to a method for expanding the orbital stability domain of a space high-precision interference measurement of a geocentric configuration, and belongs to the technical field of aerospace. Background Art

[0002] In recent years, high-precision interferometry in space has gradually become an important means for space gravitational wave detection, observation of terrestrial planets and high-precision imaging of the universe. By deploying interferometric sensors in space, it has achieved precise observation of cosmic phenomena, and is not limited by seismic waves and interferometric arm length. High-precision interferometry in space is one of the high-precision interferometry configurations in space. The main gravitational force on the spacecraft is the earth's gravity, such as the "Tianqin" program of China's space gravitational wave detection. Configuration stability is an important constraint in the design of high-precision interferometry in space, and has an important impact on the accuracy of interferometry. Although the nominal orbit of high-precision interferometry in space can usually meet the configuration stability requirements through effective optimization design, due to the influence of orbital deviation, when the orbital deviation exceeds the allowable range of the orbital deviation of high-precision interferometry in space, the spacecraft in the high-precision interferometry in space will deviate from its pre-set nominal orbit, thereby destroying the stability of the high-precision interferometry in space. Therefore, it is necessary to expand the allowable range of orbital deviation of space high-precision interferometry measurement of geocentric configuration, and provide a larger allowable range for orbital deviation of space high-precision interferometry measurement of geocentric configuration. The boundary of the allowable range of orbital deviation of space high-precision interferometry measurement of geocentric configuration is the stable domain of space high-precision interferometry measurement of geocentric configuration. Summary of the invention

[0003] In order to solve the problem that the orbital deviation exceeds the original stable domain of the geocentric configuration measured by high-precision space interference, the purpose of the present invention is to provide a method for expanding the orbital stability domain of the geocentric configuration measured by high-precision space interference. The method establishes a semi-analytical expression for the propagation of the deviation of the stability index of the geocentric configuration measured by high-precision space interference, and uses Newton iteration and variable weight method to iteratively optimize the initial configuration of the geocentric configuration measured by high-precision space interference to obtain the optimized configuration, thereby expanding the orbital stability domain of the geocentric configuration measured by high-precision space interference, meeting the orbital requirements of the geocentric configuration measured by high-precision space interference when there is a large orbital deviation, and improving the orbital stability of the geocentric configuration measured by high-precision space interference.

[0004] The purpose of the present invention is achieved through the following technical solutions.

[0005] The invention discloses a method for expanding the stable domain of orbital entry of high-precision interferometric measurement of geocentric configuration, sets the orbital entry time, mission end time and initial states of three spacecraft for high-precision interferometric measurement of geocentric configuration, and establishes the dynamic equation of spacecraft for high-precision interferometric measurement of geocentric configuration. A stability characterization model for high-precision interferometric measurement of geocentric configuration is established, and a mapping relationship between the stability index of high-precision interferometric measurement of geocentric configuration and the state vector of spacecraft in high-precision interferometric measurement of geocentric configuration is further established. The state transfer tensor of spacecraft for high-precision interferometric measurement of geocentric configuration is calculated, and a semi-analytical expression for the propagation of orbital deviation of spacecraft for high-precision interferometric measurement of geocentric configuration is established. A semi-analytical expression for the propagation of orbital deviation of high-precision interferometric measurement of geocentric configuration is established. A semi-analytical expression for the propagation of deviation of stability index of high-precision interferometric measurement of geocentric configuration is established. The expected value of stability index is set, a cost function is constructed, the expected value of position stability domain is set, the time exceeding the expected value is calculated, the weight of stability index at the time is increased in each Newton iteration, and the initial state of spacecraft is iteratively optimized according to the optimal solution of Newton method. Based on the expanded stable domain of high-precision interferometry of the geocentric configuration into orbit obtained after iterative optimization, a control strategy for high-precision interferometry of the geocentric configuration into orbit is constructed to achieve high-precision interferometry of the geocentric configuration into orbit, enhance the stability of high-precision interferometry of the geocentric configuration into orbit, and improve the observation efficiency of high-precision interferometry of the geocentric configuration into orbit.

[0006] The present invention discloses a method for expanding the stable domain of geocentric configuration orbital entry by high-precision interferometry in space, comprising the following steps:

[0007] Step 1: Set the orbit entry time t0 and mission end time t0 for high-precision interferometric measurement of the geocentric configuration f and the initial states x of the three spacecraft 1,0 、x 2,0 、x 3,0 , establish the dynamic equation f of spacecraft for high-precision interferometric measurement of geocentric configuration;

[0008] Set the time t0 for high-precision interferometric measurement of the geocentric configuration to enter orbit and the time t0 for mission completion. f , the initial state of the first spacecraft is x 1,0 , the initial state of the second spacecraft is x 2,0 , the initial state of the third spacecraft is x 3,0 .

[0009] The dynamic equation f of the spacecraft in the high-precision interferometric measurement of the geocentric configuration is established as follows:

[0010]

[0011] where t∈[t0,t f] is any time, x i =[r i ;v i ] represents the state vector of the i-th spacecraft at time t in the high-precision interferometric measurement of the geocentric configuration, where i belongs to {1, 2, 3}, r i represents the position vector of the i-th spacecraft at time t in the high-precision interferometric measurement of the geocentric configuration, v i represents the velocity vector of the i-th spacecraft at time t in the space high-precision interferometry measurement of the geocentric configuration, and f is the dynamic equation of the spacecraft in the space high-precision interferometry measurement of the geocentric configuration.

[0012] Step 2: Based on the equation f established in step 1, a characterization model for the stability of the geocentric configuration measured by spatial high-precision interferometry is established; further, a mapping relationship between the stability index and the state vector in the geocentric configuration measured by spatial high-precision interferometry is established.

[0013] The stability index of the space high-precision interferometry geocentric configuration includes: the arm length l between the i-th spacecraft and the j-th spacecraft in the space high-precision interferometry geocentric configuration ij , space high-precision interferometry measurement of the breathing angle θ corresponding to the i-th spacecraft in the geocentric configuration i , Space high-precision interferometry measurement of the arm length change rate between the i-th spacecraft and the j-th spacecraft in the geocentric configuration The characterization model of the stability of the geocentric configuration measured by high-precision interferometry in space is as follows:

[0014]

[0015] in:

[0016] r ij =r i -r j (3)

[0017] v ij =v i -v j (4)

[0018] Wherein, i, j, k represent the subscripts of the spacecraft, and i, j, k all belong to {1, 2, 3}.

[0019] According to formula (2), the mapping relationship between the stability index and the state vector in the space high-precision interferometry measurement of the geocentric configuration is further established as follows:

[0020]

[0021] in represents the stability index vector of the geocentric configuration measured by space high-precision interferometry, X=[x1; x2; x3] represents the three spacecraft state vectors in the geocentric configuration measured by space high-precision interferometry; x1=[r1; v1], x2=[r2; v2], x3=[r3; v3] represent the first, second and third spacecraft state vectors respectively.

[0022] Step 3: Based on the orbit entry time t0 and mission end time t set in step 1 f 、The initial states of the three spacecraft x 1,0 、x 2,0 、x 3,0 , and the equation f established in step 1, calculate the state transfer tensor of the spacecraft with high-precision interferometry of the geocentric configuration, and establish a semi-analytical expression for the propagation of the orbit deviation of the spacecraft with high-precision interferometry of the geocentric configuration.

[0023] The state transfer tensor of the spacecraft for high-precision interferometric measurement of the geocentric configuration is as follows:

[0024]

[0025] where t∈[t0,t f ] is any time, Represents the first-order state transfer tensor of a high-precision interferometric measurement of a geocentric configuration spacecraft. represents the second-order state transfer tensor of the spacecraft for high-precision interferometric measurement of the geocentric configuration; superscript x i represents the i-th element of the state vector x, with a superscript represents the k1th element of the initial state vector x0, with the superscript Represents the k2th element of the initial state vector x0

[0026] The semi-analytical expression for the propagation of orbit deviation of spacecraft in geocentric configuration with high-precision interferometry is established as follows:

[0027]

[0028] Where superscript δx i represents the i-th element of the spacecraft state vector deviation δx; represents the k1th element of the spacecraft initial state vector deviation δx0, Represents the k2th element of the initial state vector deviation δx0.

[0029] Formula (8) is valid for different spacecraft in the high-precision interferometric measurement of the geocentric configuration.

[0030] Step 4: Based on the semi-analytical expression established in step 3, establish a semi-analytical expression for the propagation of the orbital deviation of the geocentric configuration measured by high-precision interferometry in space.

[0031] The semi-analytical expression for the propagation of orbit deviation of geocentric configuration measured by high-precision interferometry in space is as follows:

[0032]

[0033] where δX i represents the i-th element of the deviation vector δX, represents the k1th element of the deviation vector δX0, Represents the k2th element of the deviation vector δX0. δx1 represents the deviation of the first spacecraft into orbit using high-precision interferometry of the geocentric configuration, δx2 represents the deviation of the second spacecraft into orbit using high-precision interferometry of the geocentric configuration, δx3 represents the deviation of the third spacecraft into orbit using high-precision interferometry of the geocentric configuration, and δx 1,0 represents the initial orbital deviation of the first spacecraft in high-precision interferometric measurement of the geocentric configuration, δx 2,0 represents the initial orbital deviation of the second spacecraft in the high-precision interferometric measurement of the geocentric configuration, δx 3,0 It represents the initial orbital deviation of the third spacecraft using high-precision interferometric measurement of the geocentric configuration.

[0034] In formula (9), and The state transfer tensor of the orbital state of the geocentric configuration is measured by high-precision interferometry in space, and its expression is as follows:

[0035]

[0036] Among them, the superscripts 1, 2, and 3 on the right side of the equal sign represent the state transfer tensors of the first spacecraft, the second spacecraft, and the third spacecraft corresponding to the spatial high-precision interferometric measurement of the geocentric configuration, and the superscripts i, i′, i″ and k1, k1″, k1″ and k2, k2′, k2″ represent the ordinal numbers of the corresponding elements.

[0037] Step 5: Based on the semi-analytical expression of the propagation of the orbit deviation of the geocentric configuration measured by high-precision interferometry in space established in step 4 and the mapping relationship established in step 2, a semi-analytical expression of the propagation of the deviation of the stability index of the geocentric configuration measured by high-precision interferometry in space is established.

[0038] The semi-analytical expression for the propagation of the deviation of the geocentric configuration stability index measured by space high-precision interferometry is as follows:

[0039]

[0040] where z j is the jth element of the vector deviation δz of the geocentric configuration stability index measured by high-precision spatial interferometry, It is the first-order improved state transfer tensor for the stability index of the geocentric configuration in space high-precision interferometry. It is the second-order improved state transfer tensor of the stability index of the geocentric configuration measured by high-precision interferometry in space, and its expression is as follows:

[0041]

[0042] in and represents the space high-precision interferometry measurement of the orbital state transfer tensor of the geocentric configuration. The upper right subscripts i, k1, and k2 represent the order of the elements. j,i and H j,ii is the correlation coefficient matrix, and its expression is as follows:

[0043]

[0044] where h j (X) represents the jth element of the geocentric configuration stability index h(X) of high-precision spatial interferometry. i Represents the i-th element of the geocentric configuration orbit X measured by high-precision interferometry in space.

[0045] Step 6: According to the orbit entry time t0 and mission end time t set in step 1 f 、The initial states of the three spacecraft x 1,0 、x 2,0 、x 3,0 , and the semi-analytical expression of the deviation propagation of the geocentric configuration stability index of spatial high-precision interferometry established in step 5, setting the expected value of the stability index and Constructing the cost function Set the expected value of the position stability region Calculation exceeds expectations The time t m In each Newton iteration, the weight of the stability index at that moment is increased, and according to the optimal solution of Newton's method, the initial state of the spacecraft is iteratively optimized to achieve the expansion of the stability domain of the geocentric configuration in space high-precision interferometric measurement.

[0046] Cost function

[0047]

[0048] Where Nt k The weight vector of the moment stability index is λ l (t k ) is t k The weight corresponding to the arm length at the moment, λ θ(t k ) is t k The weight corresponding to the breathing angle at the moment, λ l (t k ) is t k The weight corresponding to the arm length change rate at the moment t k The stability index vector at time t k The expected stability index vector at each moment is y i (t k ) indicates t k The stability index vector y(t k ), Indicates t k The stability index vector of the expected moment The i-th element of i (t k ) indicates t k The weight vector λ(t k ). N represents the time interval [t0,t f ]Discrete number.

[0049] Set the expected value of the position stability region Calculation exceeds expectations The time t m , in each Newton iteration, the weight λ of the stability index corresponding to the moment l (t m ), λ θ (t m ) and λ l (t m ) increases to twice the previous iteration;

[0050] According to Newton's iteration method, the optimal solution of equation (17) is

[0051] δX0≈-B -1 A(18) is a two-dimensional coefficient matrix, is a three-dimensional coefficient matrix, and the calculation formula is:

[0052]

[0053] in Represents the k1th element in the 1st row of the two-dimensional coefficient matrix A. Represents the 1st row, k1th column, k2th element of the three-dimensional coefficient matrix B. And in formula (13) Both represent the first-order improved state transfer tensor of the stability index of the geocentric configuration measured by spatial high-precision interferometry. For the convenience of formula writing, the subscript symbols are different, but the meanings are the same. And in formula (14) Both represent the second-order improved state transfer tensor of the stability index of the geocentric configuration measured by spatial high-precision interferometry. For the convenience of formula writing, the subscript symbols are different, but the meanings are the same.

[0054] According to formula (20), the initial state of the spacecraft is iteratively optimized to obtain the optimized initial state of the spacecraft. The stable domain of the high-precision interferometric measurement of the geocentric configuration obtained by integration of the initial state is expanded compared with the stable domain of the high-precision interferometric measurement of the geocentric configuration before optimization.

[0055] This has achieved the expansion of the stable domain of high-precision interferometric measurement of the geocentric configuration in space.

[0056] It also includes step seven, based on the expanded space high-precision interferometry measurement of the geocentric configuration orbital integration stability domain obtained after iterative optimization, constructing a space high-precision interferometry measurement of the geocentric configuration orbital integration control strategy, realizing space high-precision interferometry measurement of the geocentric configuration orbital integration, enhancing the space high-precision interferometry measurement of the geocentric configuration orbital integration stability, and improving the space high-precision interferometry measurement of the geocentric configuration observation efficiency; the space high-precision interferometry measurement of the geocentric configuration observation includes space gravitational wave observation, terrestrial planet observation, and high-precision cosmic observation.

[0057] Step 3 and It is obtained by integrating equations (21) and (22),

[0058]

[0059] in and Respectively and The derivative of and The meanings are the same, both representing the first-order state transfer tensor of a spacecraft for high-precision interferometric measurement of geocentric configuration; and They have the same meaning and both represent the second-order state transfer tensor of the spacecraft with high-precision interferometric measurement of the geocentric configuration; different subscripts α, β and i are used for ease of writing.

[0060] A in formula (21) and (22) i,α and A i,αβ is the local dynamics tensor, and its calculation formula is as follows:

[0061]

[0062] where fi (x, t) represents the i-th element of the dynamic equation f of the spacecraft in the geocentric configuration of high-precision interferometric measurement. α represents the αth element of the spacecraft state vector x, x β represents the βth element of the spacecraft state vector x.

[0063] At time t in step 6 m The calculation method is as follows:

[0064] The position stability domain σ at each time t is obtained by traversal and dichotomy R (t), compare the position stability domain σ at each time t R (t) and the expected value of the position stability domain The size of σ R (t) is greater than the expected value The moment is recorded as t m .

[0065] Beneficial effects:

[0066] 1. The present invention discloses a method for expanding the stable domain of high-precision interferometry of geocentric configuration for space into orbit. Based on the expanded stable domain of high-precision interferometry of geocentric configuration for space into orbit obtained after iterative optimization, a control strategy for high-precision interferometry of geocentric configuration for space into orbit is constructed, which can realize high-precision interferometry of geocentric configuration for space into orbit, and further realize high-precision interferometry of space gravitational wave observation, terrestrial planet observation, high-precision cosmic observation and other phenomena.

[0067] 2. The present invention discloses a method for expanding the stable domain of the geocentric configuration when entering orbit using high-precision interferometry in space. By establishing a semi-analytical expression for the propagation of the deviation of the stability index of the geocentric configuration when entering orbit using high-precision interferometry in space, the calculation accuracy of the stable domain of the geocentric configuration when entering orbit using high-precision interferometry in space can be improved.

[0068] 3. The present invention discloses a method for expanding the stable domain of space high-precision interferometry measurement of geocentric configuration into orbit. Compared with the traditional configuration optimization method, the proposed iterative optimization algorithm for space high-precision interferometry measurement of geocentric configuration greatly improves the optimization efficiency of the stable domain, reduces the calculation time, and can effectively expand the stable domain of space high-precision interferometry measurement of geocentric configuration into orbit.

[0069] 4. The method for expanding the stable domain of high-precision interferometry of geocentric configuration into orbit disclosed by the present invention has the advantages of fast calculation speed and high calculation efficiency for expanding the stable domain of high-precision interferometry of constellation into orbit, based on achieving the above three beneficial effects. It is beneficial to enhance the stability of high-precision interferometry of geocentric configuration into orbit and improve the observation efficiency of high-precision interferometry of geocentric configuration into orbit. BRIEF DESCRIPTION OF THE DRAWINGS

[0070] Figure 1 This is a flow chart of a method for expanding the orbital stability domain of high-precision interferometric measurement of geocentric configurations disclosed in the present invention;

[0071] Figure 2 The image of the stable domain before optimization is provided in an embodiment of a method for expanding the stable domain of orbiting the geocentric configuration by high-precision interferometric measurement of space disclosed by the present invention;

[0072] Figure 3 The invention discloses a method for expanding the stable domain of a space high-precision interferometric measurement of a geocentric configuration for orbital entry, and optimizes the stable domain image. DETAILED DESCRIPTION

[0073] In order to better illustrate the purpose and advantages of the present invention, the present invention is explained in detail below in conjunction with specific implementation examples.

[0074] Embodiment 1:

[0075] In the context of this embodiment, the present invention is applied to the task of high-precision interferometric measurement of geocentric configuration in space. Interferometric measurement is an important means of detecting gravitational waves, observing terrestrial planets and conducting precise cosmic observations. High-precision interferometric measurement of geocentric configuration in space is a way to achieve interferometric measurement. Although through optimized design, the configuration must remain stable for a long time to ensure high-precision interferometric measurement. However, when the orbital deviation exceeds the allowable range of the orbital deviation of the high-precision interferometric measurement of geocentric configuration in space, the spacecraft in the high-precision interferometric measurement of geocentric configuration in space will deviate from its pre-set nominal orbit, thereby destroying the stability of the high-precision interferometric measurement of geocentric configuration in space.

[0076] like Figure 1 As shown, the present embodiment discloses a method for expanding the stable domain of geocentric configuration orbital entry using high-precision interferometric measurement, and the specific implementation steps are as follows:

[0077] Step 1: Set the orbit entry time t0 and mission end time t0 for high-precision interferometric measurement of the geocentric configuration f and the initial states x of the three spacecraft 1,0 、x 2,0 、x 3,0 , establish the dynamic equation f of spacecraft for high-precision interferometric measurement of geocentric configuration;

[0078] The time for the space high-precision interferometry measurement of the geocentric configuration to enter orbit is set to UTC 2031 / 11 / 22 12:00:00, and the time for the space high-precision interferometry measurement of the geocentric configuration mission to end is set to UTC 2032 / 03 / 22 12:00:00.

[0079] Set the initial state of the first spacecraft in the high-precision interferometric measurement of the geocentric configuration to x 1,0, set the initial state of the second spacecraft in the high-precision interferometric measurement of the geocentric configuration to x 2,0 , set the initial state of the third spacecraft in the high-precision interferometric measurement of the geocentric configuration to x 3,0 , the corresponding initial state is shown in Table 1.

[0080] Table 1 Initial state of the constellation spacecraft for high-precision interferometry measurement (Earth-centered inertial system)

[0081]

[0082] The dynamic equation f of the spacecraft in the high-precision interferometric measurement of the geocentric configuration is established as follows:

[0083]

[0084] where t∈[t0,t f ] is any time, x i =[r i ;v i ] represents the state vector of the i-th spacecraft at time t in the high-precision interferometric measurement of the geocentric configuration, where i belongs to {1, 2, 3}, r i represents the position vector of the i-th spacecraft at time t in the high-precision interferometric measurement of the geocentric configuration, v i μ represents the velocity vector of the i-th spacecraft at time t in the high-precision interferometric measurement of the geocentric configuration, and f is the dynamic equation of the spacecraft in the high-precision interferometric measurement of the geocentric configuration. E ,μ S and μ M are the gravitational constants of the Earth, the Sun, and the Moon, respectively, r S and r M are the position vectors of the sun and the moon in the geocentric inertial system, a NE is the non-spherical perturbation acceleration of the Earth.

[0085] Step 2: Based on the equation f established in step 1, a characterization model for the stability of the geocentric configuration measured by spatial high-precision interferometry is established; further, a mapping relationship between the stability index and the state vector in the geocentric configuration measured by spatial high-precision interferometry is established.

[0086] The stability index of the space high-precision interferometry geocentric configuration includes: the arm length l between the i-th spacecraft and the j-th spacecraft in the space high-precision interferometry geocentric configuration ij , space high-precision interferometry measurement of the breathing angle θ corresponding to the i-th spacecraft in the geocentric configuration i , Space high-precision interferometry measurement of the arm length change rate between the i-th spacecraft and the j-th spacecraft in the geocentric configuration The characterization model of the stability of the geocentric configuration measured by high-precision interferometry in space is as follows:

[0087]

[0088] in:

[0089] r ij =r i -r j (27)

[0090] v ij =v i -v j (28)

[0091] Wherein, i, j, k represent the subscripts of the spacecraft, and i, j, k all belong to {1, 2, 3}.

[0092] According to formula (2), the mapping relationship between the stability index and the state vector in the space high-precision interferometry measurement of the geocentric configuration is further established as follows:

[0093]

[0094] in represents the stability index vector of the geocentric configuration measured by space high-precision interferometry, X=[x1; x2; x3] represents the three spacecraft state vectors in the geocentric configuration measured by space high-precision interferometry; x1=[r1; v1], x2=[r2; v2], x3=[r3; v3] represent the first, second and third spacecraft state vectors respectively.

[0095] Step 3: Based on the orbit entry time t0 and mission end time t set in step 1 f 、The initial states of the three spacecraft x 1,0 、x 2,0 、x 3,0 , and the equation f established in step 1, calculate the state transfer tensor of the spacecraft with high-precision interferometry of the geocentric configuration, and establish a semi-analytical expression for the propagation of the orbit deviation of the spacecraft with high-precision interferometry of the geocentric configuration.

[0096] The state transfer tensor of the spacecraft for high-precision interferometric measurement of the geocentric configuration is as follows:

[0097]

[0098] where t∈[t0,t f ] is any time, Represents the first-order state transfer tensor of a high-precision interferometric measurement of a geocentric configuration spacecraft. represents the second-order state transfer tensor of the spacecraft for high-precision interferometric measurement of the geocentric configuration; superscript x i represents the i-th element of the state vector x, with a superscript represents the k1th element of the initial state vector x0, with the superscript Represents the k2th element of the initial state vector x0

[0099] and It is obtained by integrating equations (32) and (33)

[0100]

[0101] in and Respectively and The derivative of and The meanings are the same, both representing the first-order state transfer tensor of a spacecraft for high-precision interferometric measurement of geocentric configuration; and They have the same meaning and both represent the second-order state transfer tensor of the spacecraft with high-precision interferometric measurement of the geocentric configuration; different subscripts α, β and i are used for ease of writing.

[0102] A in formula (32) and (33) i,α and A i,αβ is the local dynamics tensor, and its calculation formula is as follows:

[0103]

[0104] where f i (x, t) represents the i-th element of the dynamic equation f of the spacecraft in the geocentric configuration of high-precision interferometric measurement. α represents the αth element of the spacecraft state vector x, x β represents the βth element of the spacecraft state vector x.

[0105] The semi-analytical expression for the propagation of orbit deviation of spacecraft in geocentric configuration with high-precision interferometry is established as follows:

[0106]

[0107] Where superscript δx i represents the i-th element of the spacecraft state vector deviation δx; represents the k1th element of the spacecraft initial state vector deviation δx0, Represents the k2th element of the initial state vector deviation δx0.

[0108] Formula (8) is valid for different spacecraft in the high-precision interferometric measurement of the geocentric configuration.

[0109] Step 4: Based on the semi-analytical expression established in step 3, establish a semi-analytical expression for the propagation of the orbital deviation of the geocentric configuration measured by high-precision interferometry in space.

[0110] The semi-analytical expression for the propagation of orbit deviation of geocentric configuration measured by high-precision interferometry in space is as follows:

[0111]

[0112] where δX i represents the i-th element of the deviation vector δX, represents the k1th element of the deviation vector δX0, Represents the k2th element of the deviation vector δX0. δx1 represents the deviation of the first spacecraft into orbit using high-precision interferometry of the geocentric configuration, δx2 represents the deviation of the second spacecraft into orbit using high-precision interferometry of the geocentric configuration, δx3 represents the deviation of the third spacecraft into orbit using high-precision interferometry of the geocentric configuration, and δx 1,0 represents the initial orbital deviation of the first spacecraft in high-precision interferometric measurement of the geocentric configuration, δx 2,0 represents the initial orbital deviation of the second spacecraft in the high-precision interferometric measurement of the geocentric configuration, δx 3,0 It represents the initial orbital deviation of the third spacecraft using high-precision interferometric measurement of the geocentric configuration.

[0113] In formula (9), and The state transfer tensor of the orbital state of the geocentric configuration is measured by high-precision interferometry in space, and its expression is as follows:

[0114]

[0115]

[0116] Among them, the superscripts 1, 2, and 3 on the right side of the equal sign represent the state transfer tensors of the first spacecraft, the second spacecraft, and the third spacecraft corresponding to the spatial high-precision interferometric measurement of the geocentric configuration, and the superscripts i, i′, i″ and k1, k1″, k1″ and k2, k2′, k2″ represent the ordinal numbers of the corresponding elements.

[0117] Step 5: Based on the semi-analytical expression of the propagation of the orbit deviation of the geocentric configuration measured by high-precision interferometry in space established in step 4 and the mapping relationship established in step 2, a semi-analytical expression of the propagation of the deviation of the stability index of the geocentric configuration measured by high-precision interferometry in space is established.

[0118] The semi-analytical expression for the propagation of the deviation of the geocentric configuration stability index measured by space high-precision interferometry is as follows:

[0119]

[0120] where z j is the jth element of the vector deviation δz of the geocentric configuration stability index measured by high-precision spatial interferometry, It is the first-order improved state transfer tensor for the stability index of the geocentric configuration in space high-precision interferometry. It is the second-order improved state transfer tensor of the stability index of the geocentric configuration measured by high-precision interferometry in space, and its expression is as follows:

[0121]

[0122] in and represents the space high-precision interferometry measurement of the orbital state transfer tensor of the geocentric configuration. The upper right subscripts i, k1, and k2 represent the order of the elements. j,i and H j,ii is the correlation coefficient matrix, and its expression is as follows:

[0123]

[0124] where h j (X) represents the jth element of the geocentric configuration stability index h(X) of high-precision spatial interferometry. i Represents the i-th element of the geocentric configuration orbit X measured by high-precision interferometry in space.

[0125] Step 6: According to the orbit entry time t0 and mission end time t set in step 1 f 、The initial states of the three spacecraft x 1,0 、x 2,0 、x 3,0 , and the semi-analytical expression of the deviation propagation of the geocentric configuration stability index of spatial high-precision interferometry established in step 5, setting the expected value of the stability index and Constructing the cost function Set the expected value of the position stability region Calculation exceeds expectations The time t m In each Newton iteration, the weight of the stability index at that moment is increased, and according to the optimal solution of Newton's method, the initial state of the spacecraft is iteratively optimized to achieve the expansion of the stability domain of the geocentric configuration in space high-precision interferometric measurement.

[0126] Cost function

[0127]

[0128] Where Nt kThe weight vector of the moment stability index is λ l (t k ) is t k The weight corresponding to the arm length at the moment, λ θ (t k ) is t k The weight corresponding to the breathing angle at the moment, λ l (t k ) is t k The weight corresponding to the arm length change rate at the moment t k The stability index vector at time t k The expected stability index vector at each moment is y i (t k ) indicates t k The stability index vector y(t k ), Indicates t k The stability index vector of the expected moment The i-th element of i (t k ) indicates t k The weight vector λ(t k ). N represents the time interval [t0,t f ]Discrete number.

[0129] Set the expected value of the position stability region The position stability domain σ at each time t is obtained by traversal and dichotomy R (t), compare the position stability domain σ at each time t R (t) and the expected value of the position stability domain The size of σ R (t) is greater than the expected value The moment is recorded as t m In each Newton iteration, the weight λ of the stability index corresponding to the moment is l (t m ), λ θ (t m ) and λ l (t m ) increases to twice that of the previous iteration.

[0130] According to Newton's iteration method, the optimal solution of equation (17) is

[0131] δX0≈-B -1 A(46) is a two-dimensional coefficient matrix, is a three-dimensional coefficient matrix, and the calculation formula is:

[0132]

[0133] in Represents the k1th element in the 1st row of the two-dimensional coefficient matrix A. Represents the 1st row, k1th column, k2th element of the three-dimensional coefficient matrix B. And in formula (13) Both represent the first-order improved state transfer tensor of the stability index of the geocentric configuration measured by spatial high-precision interferometry. For the convenience of formula writing, the subscript symbols are different, but the meanings are the same. And in formula (14) Both represent the second-order improved state transfer tensor of the stability index of the geocentric configuration measured by spatial high-precision interferometry. For the convenience of formula writing, the subscript symbols are different, but the meanings are the same.

[0134] According to formula (20), the initial state of the spacecraft is iteratively optimized to obtain the optimized initial state of the spacecraft. The stable domain of the high-precision interferometric measurement of the geocentric configuration obtained by integration of the initial state is expanded compared with the stable domain of the high-precision interferometric measurement of the geocentric configuration before optimization.

[0135] According to the optimal solution of Newton’s method (46), the initial state of the spacecraft is iteratively optimized, and the final iterative optimization solution is shown in Table 2.

[0136] Table 2. State of the spacecraft after iterative optimization of the space high-precision interferometry constellation (Earth-centered inertial system)

[0137]

[0138] This has achieved the expansion of the stable domain of high-precision interferometric measurement of the geocentric configuration in space.

[0139] Step seven, based on the expanded stable domain of space high-precision interferometry of geocentric configuration into orbit obtained after iterative optimization, construct a control strategy for space high-precision interferometry of geocentric configuration into orbit, achieve high-precision space high-precision interferometry of geocentric configuration into orbit, enhance the stability of space high-precision interferometry of geocentric configuration into orbit, and improve the observation efficiency of space high-precision interferometry of geocentric configuration; the space high-precision interferometry of geocentric configuration observation includes space gravitational wave observation, terrestrial planet observation, and high-precision cosmic observation.

[0140] The above-mentioned specific description further describes the purpose, technical scheme and advantages of the invention in detail. It should be understood that the above description is only a specific implementation example of the present invention, which is used to explain the present invention and is not intended to limit the scope of protection of the present invention. Any modification, equivalent substitution, improvement, etc. made within the spirit and principle of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for expanding the stable domain of geocentric configuration in space by high-precision interferometry, characterized by: The following steps are included: Step 1: Set the orbit entry time t0 and mission end time t0 for high-precision interferometric measurement of the geocentric configuration f and the initial states x of the three spacecraft 1,0 、x 2,0 、x 3,0 , establish the dynamic equation f of spacecraft for high-precision interferometric measurement of geocentric configuration; Step 2: Based on the equation f established in step 1, a characterization model for the stability of the geocentric configuration measured by spatial high-precision interferometry is established; and a mapping relationship between the stability index and the state vector in the geocentric configuration measured by spatial high-precision interferometry is further established; Step 3: Based on the orbit entry time t0 and mission end time t set in step 1 f 、The initial states of the three spacecraft x 1,0 、x 2,0 、x 3,0 , and the equation f established in step 1, calculate the state transfer tensor of the spacecraft for high-precision interferometric measurement of the geocentric configuration, and establish a semi-analytical expression for the propagation of the orbit deviation of the spacecraft for high-precision interferometric measurement of the geocentric configuration; Step 4: Based on the semi-analytical expression established in step 3, establish a semi-analytical expression for the propagation of the orbit deviation of the geocentric configuration measured by high-precision interferometry in space; Step 5: Based on the semi-analytical expression of the propagation of the orbit deviation of the geocentric configuration measured by spatial high-precision interferometry established in step 4 and the mapping relationship established in step 2, a semi-analytical expression of the propagation of the deviation of the stability index of the geocentric configuration measured by spatial high-precision interferometry is established; Step 6: According to the orbit entry time t0 and mission end time t set in step 1 f 、The initial states of the three spacecraft x 1,0 、x 2,0 、x 3,0 , and the semi-analytical expression of the deviation propagation of the geocentric configuration stability index of spatial high-precision interferometry established in step 5, setting the expected value of the stability index and Constructing the cost function Set the expected value of the position stability region Calculation exceeds expectations The time t m In each Newton iteration, the weight of the stability index at that moment is increased, and according to the optimal solution of Newton's method, the initial state of the spacecraft is iteratively optimized to achieve the expansion of the stability domain of the geocentric configuration in space high-precision interferometric measurement.

2. A method for expanding the stable region of orbital entry by high-precision interferometric measurement of geocentric configuration as claimed in claim 1, characterized in that: The implementation method of step one is: Set the time t0 for high-precision interferometric measurement of the geocentric configuration to enter orbit and the time t0 for mission completion. f , the initial state of the first spacecraft is x 1,0 , the initial state of the second spacecraft is x 2,0 , the initial state of the third spacecraft is x 3,0 ; The dynamic equation f of the spacecraft in the high-precision interferometric measurement of the geocentric configuration is established as follows: where t∈[t0,t f ] is any time, x i =[r i ;v i ] represents the state vector of the i-th spacecraft at time t in the high-precision interferometric measurement of the geocentric configuration, where i belongs to {1, 2, 3}, r i represents the position vector of the i-th spacecraft at time t in the high-precision interferometric measurement of the geocentric configuration, v i represents the velocity vector of the i-th spacecraft at time t in the space high-precision interferometry measurement of the geocentric configuration, and f is the dynamic equation of the spacecraft in the space high-precision interferometry measurement of the geocentric configuration.

3. A method for expanding the stable region of orbital entry by high-precision interferometric measurement of geocentric configuration as claimed in claim 2, characterized in that: In step 2, The stability index of the space high-precision interferometry geocentric configuration includes: the arm length l between the i-th spacecraft and the j-th spacecraft in the space high-precision interferometry geocentric configuration ij , space high-precision interferometry measurement of the breathing angle θ corresponding to the i-th spacecraft in the geocentric configuration i , Space high-precision interferometry measurement of the arm length change rate between the i-th spacecraft and the j-th spacecraft in the geocentric configuration The characterization model of the stability of the geocentric configuration measured by high-precision interferometry in space is as follows: in: r ij ＝r i -r j (3) v ij =v i -v j (4) Wherein, i, j, k represent the subscripts of the spacecraft, and i, j, k all belong to {1, 2, 3}; According to formula (2), the mapping relationship between the stability index and the state vector in the space high-precision interferometry measurement of the geocentric configuration is further established as follows: in represents the stability index vector of the geocentric configuration measured by space high-precision interferometry, X=[x1; x2; x3] represents the three spacecraft state vectors in the geocentric configuration measured by space high-precision interferometry; x1=[r1; v1], x2=[r2; v2], x3=[r3; v3] represent the first, second and third spacecraft state vectors respectively.

4. A method for expanding the stable region of orbital entry by high-precision interferometric measurement of geocentric configuration as claimed in claim 3, characterized in that: The implementation method of step three is: The state transfer tensor of the spacecraft for high-precision interferometric measurement of the geocentric configuration is as follows: where t∈[t0,t f ] is any time, Represents the first-order state transfer tensor of a high-precision interferometric measurement of a geocentric configuration spacecraft. represents the second-order state transfer tensor of the spacecraft for high-precision interferometric measurement of the geocentric configuration; superscript x i represents the i-th element of the state vector x, with a superscript represents the k1th element of the initial state vector x0, with the superscript Represents the k2th element of the initial state vector x0 The semi-analytical expression for the propagation of orbit deviation of spacecraft in geocentric configuration with high-precision interferometry is established as follows: Where superscript δx i represents the i-th element of the spacecraft state vector deviation δx; represents the k1th element of the spacecraft initial state vector deviation δx0, represents the k2th element of the initial state vector deviation δx0; Formula (8) is valid for different spacecraft in the high-precision interferometric measurement of the geocentric configuration.

5. A method for expanding the stable region of orbital entry by high-precision interferometric measurement of geocentric configuration as claimed in claim 4, characterized in that: The implementation method of step 4 is: The semi-analytical expression for the propagation of orbit deviation of geocentric configuration measured by high-precision interferometry in space is as follows: where δX i represents the i-th element of the deviation vector δX, represents the k1th element of the deviation vector δX0, represents the k2th element of the deviation vector δX0; δx1 represents the deviation of the first spacecraft into orbit using high-precision interferometry of the geocentric configuration, δx2 represents the deviation of the second spacecraft into orbit using high-precision interferometry of the geocentric configuration, δx3 represents the deviation of the third spacecraft into orbit using high-precision interferometry of the geocentric configuration, and δx 1,0 represents the initial orbital deviation of the first spacecraft in high-precision interferometric measurement of the geocentric configuration, δx 2,0 represents the initial orbital deviation of the second spacecraft in high-precision interferometric measurement of the geocentric configuration, δx 3,0 It represents the initial orbital deviation of the third spacecraft in the high-precision interferometric measurement of the geocentric configuration; In formula (9) and The state transfer tensor of the orbital state of the geocentric configuration is measured by high-precision interferometry in space, and its expression is as follows: Among them, the superscripts 1, 2, and 3 on the right side of the equal sign represent the state transfer tensors of the first spacecraft, the second spacecraft, and the third spacecraft corresponding to the spatial high-precision interferometric measurement of the geocentric configuration, and the superscripts i, i′, i″ and k1, k1″, k1″ and k2, k2′, k2″ represent the ordinal numbers of the corresponding elements.

6. A method for expanding the stable region of orbital entry by high-precision interferometric measurement of geocentric configuration as claimed in claim 5, characterized in that: The implementation method of step five is: The semi-analytical expression for the propagation of the deviation of the geocentric configuration stability index measured by space high-precision interferometry is as follows: where z j is the jth element of the vector deviation δz of the geocentric configuration stability index measured by high-precision spatial interferometry, It is the first-order improved state transfer tensor for the stability index of the geocentric configuration in space high-precision interferometry. It is the second-order improved state transfer tensor of the stability index of the geocentric configuration measured by high-precision interferometry in space, and its expression is as follows: in and represents the space high-precision interferometry measurement of the orbital state transfer tensor of the geocentric configuration. The upper right subscripts i, k1, and k2 represent the order of the elements. j,i and H j,ii is the correlation coefficient matrix, and its expression is as follows: where h j (X) represents the jth element of the geocentric configuration stability index h(X) of high-precision interferometric measurement in space, X i Represents the i-th element of the geocentric configuration orbit X measured by high-precision interferometry in space.

7. A method for expanding the stable region of orbital entry by high-precision interferometric measurement of geocentric configuration as claimed in claim 6, characterized in that: The implementation method of step six is: Cost function Where Nt k The weight vector of the moment stability index is λ l (t k ) is t k The weight corresponding to the arm length at the moment, λ θ (t k ) is t k The weight corresponding to the breathing angle at each moment, t k The weight corresponding to the arm length change rate at each moment; t k The stability index vector at time t k The expected stability index vector at each moment is y i (t k ) indicates t k The stability index vector y(t k ), Indicates t k The stability index vector of the expected moment The i-th element of i (t k ) indicates t k The weight vector λ(t k )’s ith element; N represents the time interval [t0,t f ]Discrete number; Set the expected value of the position stability region Calculation exceeds expectations The time t m , in each Newton iteration, the weight λ of the stability index corresponding to the moment l (t m ), λ θ (t m )and Increase to twice the previous iteration; According to Newton's iteration method, the optimal solution of equation (17) is δX0≈-B -1 A(18) is a two-dimensional coefficient matrix, is a three-dimensional coefficient matrix, and the calculation formula is: in represents the k1th element in the first row of the two-dimensional coefficient matrix A; represents the 1st row, k1th column, k2th element of the three-dimensional coefficient matrix B; And in formula (13) Both represent the first-order improved state transfer tensor of the stability index of the geocentric configuration measured by spatial high-precision interferometry. For the convenience of formula writing, the subscript symbols are different, but the meanings are the same; And in formula (14) Both represent the second-order improved state transfer tensor of the stability index of the geocentric configuration measured by spatial high-precision interferometry. For the convenience of formula writing, the subscript symbols are different, but the meanings are the same; According to formula (20), the initial state of the spacecraft is iteratively optimized to obtain the optimized initial state of the spacecraft. The stable domain of the high-precision interferometric measurement of the geocentric configuration obtained by integrating the initial state is expanded compared with the stable domain of the high-precision interferometric measurement of the geocentric configuration before optimization. Thus, the expansion of the stable domain of geocentric configuration in space high-precision interferometric measurement has been achieved; Step 6, time t m The calculation method is as follows: The position stability domain σ at each time t is obtained by traversal and dichotomy R (t), compare the position stability domain σ at each time t R (t) and the expected value of the position stability domain The size of σ R (t) is greater than the expected value The moment is recorded as t m .

8. A method for expanding the stable region of orbital entry by high-precision interferometric measurement of geocentric configuration as claimed in claim 1, 2, 3, 4, 5, 6 or 7, characterized in that: It also includes step seven, which is to construct a control strategy for the orbital insertion of high-precision interferometry of the geocentric configuration in space based on the expanded stable domain of the orbital insertion of high-precision interferometry of the geocentric configuration obtained after iterative optimization, to achieve high-precision orbital insertion of high-precision interferometry of the geocentric configuration in space, to enhance the stability of the orbital insertion of high-precision interferometry of the geocentric configuration in space, and to improve the observation efficiency of the geocentric configuration in space.

9. A method for expanding the stable region of orbital entry by high-precision interferometric measurement of geocentric configuration as claimed in claim 8, characterized in that: Space high-precision interferometry observations of the geocentric configuration include space gravitational wave observations, Earth-like planet observations, and high-precision cosmic observations.

10. A method for expanding the stable region of orbital entry by high-precision interferometric measurement of geocentric configuration as claimed in claim 8, characterized in that: The implementation method of step seven is: Step 3 and It is obtained by integrating equations (21) and (22), in and Respectively and The derivative of and The meanings are the same, both representing the first-order state transfer tensor of a spacecraft for high-precision interferometric measurement of geocentric configuration; and They have the same meaning, and both represent the second-order state transfer tensor of a spacecraft for high-precision interferometric measurement of geocentric configuration; different subscripts α, β and i are used for ease of writing; A in formula (21) and (22) i,α and A i,αβ is the local dynamics tensor, and its calculation formula is as follows: where f i (x, t) represents the i-th element of the dynamic equation f of the spacecraft in the geocentric configuration of high-precision interferometric measurement. α represents the αth element of the spacecraft state vector x, x β represents the βth element of the spacecraft state vector x.

Images