A method for extending the orbital stability region of geocentric configuration in high-precision space interferometry

Abstract

A method for expanding the orbital stability domain of a geocentric configuration spacecraft using high-precision interferometry, belonging to the aerospace field, is disclosed. The method involves: establishing the dynamic equations of the geocentric configuration spacecraft; establishing a stability characterization model for the geocentric configuration and a mapping relationship between the stability index and the spacecraft's state vector within the geocentric configuration; establishing a semi-analytical expression for the propagation of orbital deviations in the geocentric configuration; establishing a semi-analytical expression for the propagation of orbital deviations in the geocentric configuration; and establishing a semi-analytical expression for the propagation of deviations in the stability index of the geocentric configuration. In each Newton iteration, the weight of the stability index at that moment is increased, and the initial state of the spacecraft is iteratively optimized based on the optimal solution of Newton's method to obtain the expanded orbital stability domain of the geocentric configuration.

Description

Technical Field

[0001] This invention relates to a method for expanding the orbital stability domain of a geocentric configuration in high-precision space interferometry, belonging to the field of aerospace technology. Background Technology

[0002] In recent years, high-precision space interferometry has gradually become an important means of space-based gravitational wave detection, Earth-like planet observation, and high-precision cosmological imaging. By deploying interferometric sensors in space, it enables precise observation of cosmic phenomena, unaffected by seismic waves or the length of the interferometric arms. The geocentric configuration of high-precision space interferometry is one type of high-precision space interferometry configuration, where the spacecraft is subjected to Earth's gravity, such as China's "Tianqin" space gravitational wave detection program. Configuration stability is a crucial constraint in the design of geocentric configurations for high-precision space interferometry, significantly impacting interferometric accuracy. Although effective optimization can usually ensure that the nominal orbit of a geocentric configuration meets stability requirements, orbital insertion deviations can cause spacecraft to deviate from their pre-set nominal orbits, thus compromising the stability of the geocentric configuration. Therefore, it is necessary to expand the allowable range of orbital deviation for geocentric configurations in high-precision space interferometry, and to provide a larger allowable range for orbital deviation for geocentric configurations in high-precision space interferometry. The boundary of the allowable range of orbital deviation for geocentric configurations in high-precision space interferometry is the stability domain of geocentric configurations in high-precision space interferometry. Summary of the Invention

[0003] To address the issue of spacecraft orbital stability being affected by orbital insertion deviations exceeding the original stability domain of the geocentric configuration in high-precision interferometry (HCI) spacecraft, this invention aims to provide a method for expanding the orbital stability domain of the geocentric configuration in high-precision interferometry. This method establishes a semi-analytical expression for the propagation of deviation in the stability index of the geocentric configuration in high-precision interferometry, and uses Newton's iteration and variable weighting methods to iteratively optimize the initial configuration of the geocentric configuration in high-precision interferometry, obtaining an optimized configuration. This expands the orbital stability domain of the geocentric configuration in high-precision interferometry, ensuring orbital insertion stability even with large orbital insertion deviations, thereby improving the orbital stability of the geocentric configuration in high-precision interferometry.

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

[0005] This invention discloses a method for expanding the orbital stability domain of a geocentric configuration for high-precision space interferometry. It sets the orbital insertion time, mission termination time, and initial states of three spacecraft within the geocentric configuration, and establishes the spacecraft dynamic equations. A stability characterization model for the geocentric configuration is established, further defining the mapping relationship between the stability index and the spacecraft state vector within the geocentric configuration. The state transition tensor of the geocentric configuration is calculated, and a semi-analytical expression for the propagation of orbital deviations is established. A semi-analytical expression for the propagation of stability index deviations is also established. An expected value for the stability index is set, a cost function is constructed, and an expected value for the position stability domain is set. The time exceeding the expected value is calculated, and the weight of the stability index at each Newton iteration is increased. Based on the optimal solution of Newton's method, the initial state of the spacecraft is iteratively optimized. Based on the expanded orbital stability domain of the geocentric configuration for high-precision space interferometry obtained through iterative optimization, an orbital control strategy for the geocentric configuration for high-precision space interferometry is constructed to achieve high-precision orbital insertion of the geocentric configuration for high-precision space interferometry, enhance the orbital stability of the geocentric configuration for high-precision space interferometry, and improve the observation efficiency of the geocentric configuration for high-precision space interferometry.

[0006] This invention discloses a method for extending the orbital stability region of a geocentric configuration in high-precision space interferometry, comprising the following steps:

[0007] Step 1: Set the orbit insertion time t0 and mission termination time t0 for the high-precision interferometry geocentric configuration in space. f And the initial states x of the three spacecraft 1,0 x 2,0 x 3,0 Establish the dynamic equation f of geocentric configuration spacecraft for high-precision interferometry in space;

[0008] Set the orbit insertion time t0 and mission end time t for high-precision interferometry geocentric configuration. 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 geocentric configuration of high-precision space interferometry is established as follows:

[0010]

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

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

[0013] The stability index of the geocentric configuration in high-precision space interferometry includes: the arm length l between the i-th spacecraft and the j-th spacecraft in the geocentric configuration in high-precision space interferometry. ij The breathing angle θ of the i-th spacecraft in the geocentric configuration of high-precision space interferometry. i The rate of change of arm length between the i-th and j-th spacecraft in a high-precision space interferometry configuration. The characterization model for the stability of the geocentric configuration in high-precision space interferometry is as follows:

[0014]

[0015] in:

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

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

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

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

[0020]

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

[0022] Step 3: Based on the orbit insertion time t0 and mission end time t0 set in Step 1 f Initial states of the three spacecraft x 1,0 x 2,0 x 3,0 And, based on the equation f established in step one, calculate the state transition tensor of the geocentric configuration spacecraft for high-precision interferometry in space, and establish a semi-analytical expression for the propagation of orbital deviations of the geocentric configuration spacecraft for high-precision interferometry in space.

[0023] The state transition tensor for a geocentric spacecraft using high-precision interferometry is as follows:

[0024]

[0025] Where t∈[t0,t f [This refers to any time.] This represents the first-order state transition tensor of a geocentric configuration spacecraft used for high-precision interferometry in space. This represents the second-order state transition tensor of a geocentric configuration spacecraft used for high-precision interferometry in space; the superscript x i The superscript represents the i-th element of the state vector x. The superscript represents the k1-th element of the initial state vector x0. This represents the k2th element of the initial state vector x0.

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

[0027]

[0028] Where the superscript δx i The i-th element represents the spacecraft state vector deviation δx; This represents the k1th element of the spacecraft's initial state vector deviation δx0. This represents the k2th element of the initial state vector deviation δx0.

[0029] Equation (8) holds true for different spacecraft in geocentric configurations for high-precision interferometry in space.

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

[0031] The semi-analytical expression for the propagation of orbital deviations in geocentric configurations using high-precision space interferometry is as follows:

[0032]

[0033] Where δX i This represents the i-th element of the deviation vector δX. This represents the k1-th element of the deviation vector δX0. This represents the k2th element of the deviation vector δX0. δx1 represents the orbital insertion deviation of the first spacecraft in the high-precision interferometry geocentric configuration, δx2 represents the orbital insertion deviation of the second spacecraft in the high-precision interferometry geocentric configuration, and δx3 represents the orbital insertion deviation of the third spacecraft in the high-precision interferometry geocentric configuration. 1,0 δx represents the initial orbital deviation of the first spacecraft in a high-precision space interferometric measurement of the geocentric configuration. 2,0 δx represents the initial orbital deviation of the second spacecraft in a high-precision interferometric measurement of the geocentric configuration. 3,0 This indicates the initial orbital deviation of the third spacecraft in the geocentric configuration, measured using high-precision interferometry in space.

[0034] In equation (9) and The orbital state transition tensor for high-precision space interferometry geocentric configuration is expressed as follows:

[0035]

[0036] The superscripts 1, 2, and 3 on the right side of the equal sign represent the state transition tensors of the first, second, and third spacecraft in the geocentric configuration of the high-precision interferometry spacecraft, respectively. 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 for the propagation of orbital deviation in geocentric configuration of high-precision interferometry established in Step 4, and the mapping relationship established in Step 2, establish a semi-analytical expression for the propagation of deviation in stability index of geocentric configuration in high-precision interferometry.

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

[0039]

[0040] Where z j Let be the j-th element of the vector deviation δz, which is the stability index of the geocentric configuration in high-precision space interferometry. This is a first-order improved state transition tensor for the stability index of the geocentric configuration in high-precision space interferometry. The second-order improved state transition tensor for the stability index of the geocentric configuration in high-precision space interferometry is expressed as follows:

[0041]

[0042] in and This represents the orbital state transition tensor for high-precision interferometry in space with geocentric configuration. The superscripts i, k1, and k2 indicate the ordinal numbers of the elements. H j,i and H j,ii The correlation coefficient matrix is ​​expressed as follows:

[0043]

[0044] Where h j (X) represents the j-th element of the geocentric configuration stability index h(X) for high-precision interferometry in space. i The i-th element represents the geocentric configuration orbit X in high-precision spatial interferometry.

[0045] Step Six: Based on the orbit insertion time t0 and mission end time t0 set in Step One... f Initial states of the three spacecraft x 1,0 x 2,0 x 3,0 And the semi-analytical expression for the propagation of the deviation of the geocentric configuration stability index established in step five, and the expected value of the stability index are set. and Construct cost function Determine the expected value of the location stability region. Calculation exceeds expected value The moment t m In each Newton iteration, the weight of the stability index at that moment is increased. Based on the optimal solution of Newton's method, the initial state of the spacecraft is iteratively optimized to expand the stability domain of the geocentric configuration for high-precision interferometry in space.

[0046] Cost function

[0047]

[0048] Where N t k The weight vector of the time-stability index is λ l (t k ) for t k The weight λ corresponding to the arm length at time step θ(t k ) for t k The weight λ corresponding to the breathing angle at any given time l (t k ) for t k The weight corresponding to the rate of change of arm length at time t. k The stability index vector at time t is t k The expected stability index vector at time step is y i (t k ) represents t k The stability index vector y(t) at time t k The i-th element of ) Indicates t k Stability index vector at time-to-time The i-th element, λ i (t k ) represents t k The weight vector λ(t) of the stability index at time t k The i-th element of ). N represents the time interval [t0, t f The number of discrete elements.

[0049] Determine the expected value of the location stability region. Calculation exceeds expected value The moment t m In each Newton iteration, the weight λ of the stability index corresponding to that moment is... l (t m ), λ θ (t m ) and λ l (t m The value is increased to twice that of the previous iteration.

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

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

[0052]

[0053] in This represents the k1th element in the first row of the two-dimensional coefficient matrix A. This represents the element in the 1st row, k1st column, and k2nd position of the three-dimensional coefficient matrix B. In formula (13) Both represent the first-order improved state transition tensor of the geocentric configuration stability index in high-precision interferometry in space. The subscript symbols are different for ease of formula writing, but the meanings are the same. In formula (14) Both represent the second-order improved state transition tensor of the geocentric configuration stability index for high-precision interferometry in space. The subscript symbols are different for ease of formula writing, but the meanings are the same.

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

[0055] This achieves the expansion of the geocentric configuration stability domain for high-precision space interferometry.

[0056] It also includes step seven, which involves constructing a high-precision space interferometry geocentric configuration orbital stability domain obtained after iterative optimization, thereby achieving high-precision orbital insertion of the high-precision space interferometry geocentric configuration, enhancing the orbital stability of the high-precision space interferometry geocentric configuration, and improving the observation efficiency of the high-precision space interferometry geocentric configuration; the observation of the high-precision space interferometry geocentric configuration includes space gravitational wave observation, Earth-like planet observation, and high-precision cosmological observation.

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

[0058]

[0059] in and They represent and The derivative, and They have the same meaning, both representing the first-order state transition tensor of a geocentric configuration spacecraft used for high-precision interferometry in space. and They have the same meaning, all representing the second-order state transition tensor of a geocentric configuration spacecraft for high-precision interferometry in space; however, different subscripts α, β, and i are used for ease of writing.

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

[0061]

[0062] Where fi (x,t) represents the i-th element of the dynamic equation f of the geocentric configuration spacecraft in high-precision interferometry, where x α Let x represent the α-th element of the spacecraft state vector x. β This represents the β-th element of the spacecraft state vector x.

[0063] Step six, time t m The calculation method is as follows:

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

[0065] Beneficial effects:

[0066] 1. The present invention discloses a method for expanding the orbital stability domain of geocentric configuration in high-precision space interferometry. Based on the expanded orbital stability domain of geocentric configuration in high-precision space interferometry obtained by iterative optimization, a control strategy for orbital insertion of geocentric configuration in high-precision space interferometry is constructed, which can realize high-precision orbital insertion of geocentric configuration in high-precision space interferometry, and thus realize high-precision interferometry of phenomena such as space gravitational wave observation, Earth-like planet observation, and high-precision cosmological observation.

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

[0068] 3. The present invention discloses a method for expanding the orbital stability domain of geocentric configurations in high-precision space interferometry. Compared with traditional configuration optimization methods, the proposed iterative optimization algorithm for geocentric configurations in high-precision space interferometry greatly improves the optimization efficiency of the stability domain, reduces the computation time, and can efficiently expand the orbital stability domain of geocentric configurations in high-precision space interferometry.

[0069] 4. The method for expanding the orbital stability domain of a geocentric configuration in high-precision space interferometry disclosed in this invention, while achieving the above three beneficial effects, has the advantages of fast calculation speed and high calculation efficiency for expanding the orbital stability domain of a high-precision space interferometry constellation. This is beneficial for enhancing the orbital stability of geocentric configurations in high-precision space interferometry and improving the observation efficiency of geocentric configurations in high-precision space interferometry. Attached Figure Description

[0070] Figure 1 This is a flowchart of a method for expanding the orbital stability domain of a geocentric configuration for high-precision interferometry in space, as disclosed in this invention.

[0071] Figure 2 This is an example of a method for expanding the orbital stability domain of a geocentric configuration in high-precision interferometry, as disclosed in this invention, before optimization of the stability domain image.

[0072] Figure 3 This is an optimized stability domain image of an embodiment of a method for expanding the orbital stability domain of a geocentric configuration in high-precision space interferometry disclosed in this invention. Detailed Implementation

[0073] To better illustrate the purpose and advantages of the present invention, the present invention will be explained in detail below with reference to specific implementation examples.

[0074] Example 1:

[0075] In the context of this embodiment, the present invention is applied to a high-precision space interferometric geocentric configuration mission. Interferometry is an important means of detecting gravitational waves, observing Earth-like planets, and conducting precise cosmological observations. A high-precision space interferometric geocentric configuration is one way to achieve interferometry. Although optimized design can ensure the configuration remains stable over a long period, guaranteeing high-precision interferometry, when the orbital insertion deviation exceeds the allowable boundary of the high-precision space interferometric geocentric configuration's orbital insertion deviation, the spacecraft in the high-precision space interferometric geocentric configuration will deviate from its pre-set nominal orbit, thereby compromising the stability of the high-precision space interferometric geocentric configuration.

[0076] like Figure 1 As shown in this embodiment, a method for extending the orbital stability region of a geocentric configuration for high-precision space interferometry is disclosed. The specific implementation steps are as follows:

[0077] Step 1: Set the orbit insertion time t0 and mission termination time t0 for the high-precision interferometry geocentric configuration in space. f And the initial states x of the three spacecraft 1,0 x 2,0 x 3,0 Establish the dynamic equation f of geocentric configuration spacecraft for high-precision interferometry in space;

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

[0079] The initial state of the first spacecraft in the high-precision interferometric geocentric configuration is set as x. 1,0The initial state of the second spacecraft, configured for high-precision interferometry in space with a geocentric configuration, is set as x. 2,0 The initial state of the third spacecraft in the high-precision interferometry geocentric configuration is set as x. 3,0 The corresponding initial states are shown in Table 1.

[0080] Table 1. Initial states of spacecraft in the high-precision interferometry constellation (Geocentric inertial frame)

[0081]

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

[0083]

[0084] Where t∈[t0,t f [x] represents any time interval. i =[r i ;v i ] represents the state vector of the i-th spacecraft at time t in a high-precision interferometric geocentric configuration, where i belongs to {1,2,3}, r i v represents the position vector of the i-th spacecraft at time t in a geocentric configuration of high-precision interferometry in space. i Let μ represent the velocity vector of the i-th spacecraft in the geocentric configuration of high-precision interferometry in space, and f be the spacecraft dynamics equation in the geocentric configuration of high-precision interferometry in space. E ,μ S and μ M The gravitational constants of the Earth, the Sun, and the Moon, respectively, r S and r M Let a be the position vectors of the Sun and Moon in the Earth-centered inertial frame, respectively. NE This refers to the non-spherical perturbation acceleration of the Earth.

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

[0086] The stability index of the geocentric configuration in high-precision space interferometry includes: the arm length l between the i-th spacecraft and the j-th spacecraft in the geocentric configuration in high-precision space interferometry. ij The breathing angle θ of the i-th spacecraft in the geocentric configuration of high-precision space interferometry. i The rate of change of arm length between the i-th and j-th spacecraft in a high-precision space interferometry configuration. The characterization model for the stability of the geocentric configuration in high-precision space interferometry is as follows:

[0087]

[0088] in:

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

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

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

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

[0093]

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

[0095] Step 3: Based on the orbit insertion time t0 and mission end time t0 set in Step 1 f Initial states of the three spacecraft x 1,0 x 2,0 x 3,0 And, based on the equation f established in step one, calculate the state transition tensor of the geocentric configuration spacecraft for high-precision interferometry in space, and establish a semi-analytical expression for the propagation of orbital deviations of the geocentric configuration spacecraft for high-precision interferometry in space.

[0096] The state transition tensor for a geocentric spacecraft using high-precision interferometry is as follows:

[0097]

[0098] Where t∈[t0,t f [This refers to any time.] This represents the first-order state transition tensor of a geocentric configuration spacecraft used for high-precision interferometry in space. This represents the second-order state transition tensor of a geocentric configuration spacecraft used for high-precision interferometry in space; the superscript x i The superscript represents the i-th element of the state vector x. The superscript represents the k1-th element of the initial state vector x0. This 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 They represent and The derivative, and They have the same meaning, both representing the first-order state transition tensor of a geocentric configuration spacecraft used for high-precision interferometry in space. and They have the same meaning, all representing the second-order state transition tensor of a geocentric configuration spacecraft for high-precision interferometry in space; however, different subscripts α, β, and i are used for ease of writing.

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

[0103]

[0104] Where f i (x,t) represents the i-th element of the dynamic equation f of the geocentric configuration spacecraft in high-precision interferometry, where x α Let x represent the α-th element of the spacecraft state vector x. β This represents the β-th element of the spacecraft state vector x.

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

[0106]

[0107] Where the superscript δx i The i-th element represents the spacecraft state vector deviation δx; This represents the k1th element of the spacecraft's initial state vector deviation δx0. This represents the k2th element of the initial state vector deviation δx0.

[0108] Equation (8) holds true for different spacecraft in geocentric configurations for high-precision interferometry in space.

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

[0110] The semi-analytical expression for the propagation of orbital deviations in geocentric configurations using high-precision space interferometry is as follows:

[0111]

[0112] Where δX i This represents the i-th element of the deviation vector δX. This represents the k1-th element of the deviation vector δX0. This represents the k2th element of the deviation vector δX0. δx1 represents the orbital insertion deviation of the first spacecraft in the high-precision interferometry geocentric configuration, δx2 represents the orbital insertion deviation of the second spacecraft in the high-precision interferometry geocentric configuration, and δx3 represents the orbital insertion deviation of the third spacecraft in the high-precision interferometry geocentric configuration. 1,0 δx represents the initial orbital deviation of the first spacecraft in a high-precision space interferometric measurement of the geocentric configuration. 2,0 δx represents the initial orbital deviation of the second spacecraft in a high-precision interferometric measurement of the geocentric configuration. 3,0 This indicates the initial orbital deviation of the third spacecraft in the geocentric configuration, measured using high-precision interferometry in space.

[0113] In equation (9) and The orbital state transition tensor for high-precision space interferometry geocentric configuration is expressed as follows:

[0114]

[0115]

[0116] The superscripts 1, 2, and 3 on the right side of the equal sign represent the state transition tensors of the first, second, and third spacecraft in the geocentric configuration of the high-precision interferometry spacecraft, respectively. 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 for the propagation of orbital deviation in geocentric configuration of high-precision interferometry established in Step 4, and the mapping relationship established in Step 2, establish a semi-analytical expression for the propagation of deviation in stability index of geocentric configuration in high-precision interferometry.

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

[0119]

[0120] Where z j Let be the j-th element of the vector deviation δz, which is the stability index of the geocentric configuration in high-precision space interferometry. This is a first-order improved state transition tensor for the stability index of the geocentric configuration in high-precision space interferometry. The second-order improved state transition tensor for the stability index of the geocentric configuration in high-precision space interferometry is expressed as follows:

[0121]

[0122] in and This represents the orbital state transition tensor for high-precision interferometry in space with geocentric configuration. The superscripts i, k1, and k2 indicate the ordinal numbers of the elements. H j,i and H j,ii The correlation coefficient matrix is ​​expressed as follows:

[0123]

[0124] Where h j (X) represents the j-th element of the geocentric configuration stability index h(X) for high-precision interferometry in space. i The i-th element represents the geocentric configuration orbit X in high-precision spatial interferometry.

[0125] Step Six: Based on the orbit insertion time t0 and mission end time t0 set in Step One... f Initial states of the three spacecraft x 1,0 x 2,0 x 3,0 And the semi-analytical expression for the propagation of the deviation of the geocentric configuration stability index established in step five, and the expected value of the stability index are set. and Construct cost function Determine the expected value of the location stability region. Calculation exceeds expected value The moment t m In each Newton iteration, the weight of the stability index at that moment is increased. Based on the optimal solution of Newton's method, the initial state of the spacecraft is iteratively optimized to expand the stability domain of the geocentric configuration for high-precision interferometry in space.

[0126] Cost function

[0127]

[0128] Where N t kThe weight vector of the time-stability index is λ l (t k ) for t k The weight λ corresponding to the arm length at time step θ (t k ) for t k The weight λ corresponding to the breathing angle at any given time l (t k ) for t k The weight corresponding to the rate of change of arm length at time t. k The stability index vector at time t is t k The expected stability index vector at time step is y i (t k ) represents t k The stability index vector y(t) at time t k The i-th element of ) Indicates t k Stability index vector at time-to-time The i-th element, λ i (t k ) represents t k The weight vector λ(t) of the stability index at time t k The i-th element of ). N represents the time interval [t0, t f The number of discrete elements.

[0129] Determine the expected value of the location stability region. The position stability region σ at each time t is obtained by traversal and binary search. R (t), compare the position stability domain σ at each time t. R (t) and the expected value of the location stability region The size, record σ R (t) is greater than the expected value The moment, and denoted as t. m In each Newton iteration, the weight λ of the stability index corresponding to that moment is... l (t m ), λ θ (t m ) and λ l (t m The value is increased 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) It is a two-dimensional coefficient matrix. The coefficient matrix is ​​a three-dimensional matrix, and the calculation formula is as follows:

[0132]

[0133] in This represents the k1th element in the first row of the two-dimensional coefficient matrix A. This represents the element in the 1st row, k1st column, and k2nd position of the three-dimensional coefficient matrix B. In formula (13) Both represent the first-order improved state transition tensor of the geocentric configuration stability index in high-precision interferometry in space. The subscript symbols are different for ease of formula writing, but the meanings are the same. In formula (14) Both represent the second-order improved state transition tensor of the geocentric configuration stability index for high-precision interferometry in space. The subscript symbols are different for ease of formula writing, but the meanings are the same.

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

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

[0136] Table 2. Iterative optimization status of spacecraft in the high-precision interferometry constellation (Geocentric inertial frame)

[0137]

[0138] This achieves the expansion of the geocentric configuration stability domain for high-precision space interferometry.

[0139] Step 7: Based on the expanded orbital stability domain of the geocentric configuration obtained after iterative optimization, construct the orbital control strategy for the geocentric configuration of ...

[0140] The above detailed description further illustrates the purpose, technical solution, and advantages of the invention. It should be understood that the above description is merely a specific example of the invention's implementation, used to explain the invention, and is not intended to limit the scope of protection of the invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the invention should be included within the scope of protection of the invention.

Claims

1. A method for expanding the orbital stability region of a geocentric configuration in high-precision space interferometry, characterized in that: Includes the following steps, Step 1: Set the orbit insertion time t0 and mission termination time t0 for the high-precision interferometry geocentric configuration in space. f And the initial states x of the three spacecraft 1,0 x 2,0 x 3,0 Establish the dynamic equation f of geocentric configuration spacecraft for high-precision interferometry in space; Step 2: Based on the equation f established in Step 1, establish a characterization model for the stability of the geocentric configuration in high-precision space interferometry; further establish the mapping relationship between the stability index and the state vector in the geocentric configuration of high-precision space interferometry. Step 3: Based on the orbit insertion time t0 and mission end time t0 set in Step 1 f Initial states of the three spacecraft x 1,0 x 2,0 x 3,0 And the equation f established in step one, calculate the state transition tensor of the geocentric configuration spacecraft for high-precision interferometry, and establish a semi-analytical expression for the propagation of orbital deviations of the geocentric configuration spacecraft for high-precision interferometry. Step 4: Based on the semi-analytical expression established in Step 3, establish a semi-analytical expression for the propagation of orbital deviations in geocentric configurations for high-precision space interferometry. Step 5: Based on the semi-analytical expression for the propagation of orbital deviation in geocentric configuration of high-precision interferometry established in Step 4, and the mapping relationship established in Step 2, establish a semi-analytical expression for the propagation of deviation in stability index of geocentric configuration in high-precision interferometry. Step Six: Based on the orbit insertion time t0 and mission end time t0 set in Step One... f Initial states of the three spacecraft x 1,0 x 2,0 x 3,0 And the semi-analytical expression for the propagation of the deviation of the geocentric configuration stability index established in step five, and the expected value of the stability index are set. and Construct cost function Determine the expected value of the location stability region. Calculation exceeds expected value The moment t m In each Newton iteration, the weight of the stability index at that moment is increased. Based on the optimal solution of Newton's method, the initial state of the spacecraft is iteratively optimized to expand the stability domain of the geocentric configuration for high-precision interferometry in space.

2. The method for expanding the orbital stability region of a geocentric configuration in high-precision space interferometry as described in claim 1, characterized in that: The implementation method for step one is as follows: Set the orbit insertion time t0 and mission end time t for high-precision interferometry geocentric configuration. 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 geocentric configuration of high-precision space interferometry is established as follows: Where t∈[t0,t f [x] represents any time interval. i =[r i ;v i ] represents the state vector of the i-th spacecraft at time t in a high-precision interferometric geocentric configuration, where i belongs to {1,2,3}, r i v represents the position vector of the i-th spacecraft at time t in a geocentric configuration of high-precision interferometry in space. i Let f represent the velocity vector of the i-th spacecraft in the geocentric configuration of high-precision interferometry in space, and let f be the spacecraft dynamics equation in the geocentric configuration of high-precision interferometry in space.

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

4. The method for expanding the orbital stability region of a geocentric configuration for high-precision space interferometry as described in claim 3, characterized in that: The method for implementing step three is as follows: The state transition tensor for a geocentric spacecraft using high-precision interferometry is as follows: Where t∈[t0,t f [This refers to any time.] This represents the first-order state transition tensor of a geocentric configuration spacecraft used for high-precision interferometry in space. This represents the second-order state transition tensor of a geocentric configuration spacecraft used for high-precision interferometry in space; the superscript x i The superscript represents the i-th element of the state vector x. The superscript represents the k1-th element of the initial state vector x0. This represents the k2th element of the initial state vector x0. The semi-analytical expression for the propagation of orbital deviations in geocentric configuration spacecraft using high-precision interferometry is established as follows: Where the superscript δx i The i-th element represents the spacecraft state vector deviation δx; This represents the k1th element of the spacecraft's initial state vector deviation δx0. This represents the k2th element of the initial state vector deviation δx0; Equation (8) holds true for different spacecraft in geocentric configurations for high-precision interferometry in space.

5. The method for expanding the orbital stability region of a geocentric configuration for high-precision space interferometry as described in claim 4, characterized in that: Step four is implemented as follows: The semi-analytical expression for the propagation of orbital deviations in geocentric configurations using high-precision space interferometry is as follows: Where δX i This represents the i-th element of the deviation vector δX. This represents the k1-th element of the deviation vector δX0. This represents the k2th element of the deviation vector δX0; δx1 represents the orbital insertion deviation of the first spacecraft in the high-precision interferometry geocentric configuration, δx2 represents the orbital insertion deviation of the second spacecraft in the high-precision interferometry geocentric configuration, and δx3 represents the orbital insertion deviation of the third spacecraft in the high-precision interferometry geocentric configuration. 1,0 δx represents the initial orbital deviation of the first spacecraft in a high-precision space interferometric measurement of the geocentric configuration. 2,0 δx represents the initial orbital deviation of the second spacecraft in a high-precision interferometric measurement of the geocentric configuration. 3,0 This indicates the initial orbital deviation of the third spacecraft in the geocentric configuration, measured using high-precision interferometry in space. In equation (9) and The orbital state transition tensor for high-precision space interferometry geocentric configuration is expressed as follows: The superscripts 1, 2, and 3 on the right side of the equal sign represent the state transition tensors of the first, second, and third spacecraft in the geocentric configuration of the high-precision interferometry spacecraft, respectively. The superscripts i, i′, i″ and k1, k1″, k1″ and k2, k2′, k2″ represent the ordinal numbers of the corresponding elements.

6. The method for expanding the orbital stability region of a geocentric configuration in high-precision space interferometry as described in claim 5, characterized in that: Step five is implemented as follows: The semi-analytical expression for the propagation of the deviation index of geocentric configuration stability in high-precision space interferometry is as follows: Where z j Let be the j-th element of the vector deviation δz, which is the stability index of the geocentric configuration in high-precision space interferometry. This is a first-order improved state transition tensor for the stability index of the geocentric configuration in high-precision space interferometry. The second-order improved state transition tensor for the stability index of the geocentric configuration in high-precision space interferometry is expressed as follows: in and This represents the orbital state transition tensor for high-precision interferometry in space with geocentric configuration. The superscripts i, k1, and k2 indicate the ordinal numbers of the elements. H j,i and H j,ii The correlation coefficient matrix is ​​expressed as follows: Where h j (X) represents the j-th element of the geocentric configuration stability index h(X) for high-precision interferometry in space. i The i-th element represents the geocentric configuration orbit X in high-precision spatial interferometry.

7. The method for expanding the orbital stability region of a geocentric configuration for high-precision space interferometry as described in claim 6, characterized in that: Step six is ​​implemented as follows: Cost function Where N t k The weight vector of the time-stability index is λ l (t k ) for t k The weight λ corresponding to the arm length at time step θ (t k ) for t k The weight corresponding to the breathing angle at any given time. For t k The weight corresponding to the rate of change of arm length at any given time; t k The stability index vector at time t is t k The expected stability index vector at time step is y i (t k ) represents t k The stability index vector y(t) at time t k The i-th element of ) Indicates t k Stability index vector at time-to-time The i-th element, λ i (t k ) represents t k The weight vector λ(t) of the stability index at time t k The i-th element of ); N represents the time interval [t0, t f The number of discrete numbers; Determine the expected value of the location stability region. Calculation exceeds expected value The moment t m In each Newton iteration, the weight λ of the stability index corresponding to that moment is... l (t m ), λ θ (t m )and Increase it to twice the size of the previous iteration; According to Newton's iteration method, the optimal solution of equation (17) is δX0≈-B. -1 A(18) It is a two-dimensional coefficient matrix. The coefficient matrix is ​​a three-dimensional matrix, and the calculation formula is as follows: in This represents the k1-th element in the first row of the two-dimensional coefficient matrix A; This represents the element in the 1st row, k1st column, and k2nd position of the three-dimensional coefficient matrix B; In formula (13) Both represent the first-order improved state transition tensor of the geocentric configuration stability index in high-precision interferometry in space. The subscript symbols are different for ease of formula writing, but the meanings are the same. In formula (14) Both represent the second-order improved state transition tensor of the geocentric configuration stability index in high-precision interferometry in space. The subscript symbols are different for ease of formula writing, but the meanings are the same. According to Equation (20), the initial state of the spacecraft is iteratively optimized to obtain the optimized initial state of the spacecraft. The geocentric configuration stability domain of the high-precision interferometry in space obtained by integration of the initial state is expanded compared with the geocentric configuration stability domain of the high-precision interferometry in space before optimization. This achieves the expansion of the stability domain for high-precision space interferometry geocentric configuration; Step six, time t m The calculation method is as follows: The position stability region σ at each time t is obtained by traversal and binary search. R (t), compare the position stability domain σ at each time t. R (t) and the expected value of the location stability region The size, record σ R (t) is greater than the expected value The moment, and denoted as t. m .

8. A method for expanding the orbital stability region of a geocentric configuration for high-precision space interferometry as described in claims 1, 2, 3, 4, 5, 6, or 7, characterized in that: It also includes step seven, which involves constructing a high-precision space interferometric geocentric configuration orbital insertion control strategy based on the expanded orbital stability domain obtained after iterative optimization, thereby achieving high-precision orbital insertion of the high-precision space interferometric geocentric configuration, enhancing the orbital stability of the high-precision space interferometric geocentric configuration, and improving the observation efficiency of the high-precision space interferometric geocentric configuration.

9. The method for expanding the orbital stability region of a geocentric configuration in high-precision space interferometry as described in claim 8, characterized in that: High-precision space interferometry geocentric configuration observations include space gravitational wave observations, terrestrial planet observations, and high-precision cosmological observations.

10. The method for expanding the orbital stability region of a geocentric configuration for high-precision space interferometry as described in claim 8, characterized in that: Step seven is implemented as follows: Step 3 and It is obtained by integrating equations (21) and (22). in and They represent and The derivative, and They have the same meaning, both representing the first-order state transition tensor of a geocentric configuration spacecraft used for high-precision interferometry in space. and They have the same meaning, all representing the second-order state transition tensor of a geocentric configuration spacecraft used for high-precision interferometry in space; different subscripts α, β, and i are used for ease of writing. A in equations (21) and (22) i,α and A i,αβ For the local dynamic tensor, the calculation formula is as follows: Where f i (x,t) represents the i-th element of the dynamic equation f of the geocentric configuration spacecraft in high-precision interferometry, where x α Let x represent the α-th element of the spacecraft state vector x. β This represents the β-th element of the spacecraft state vector x.

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