A gravitational wave detection equilateral triangle formation design method based on a second-order CW equation
By using an optimization design method based on second-order CW equations, the relative position and velocity of spacecraft were corrected, solving the stability problem of equilateral triangle formation and achieving long-term stability and optimization effect of the formation.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- NORTHWESTERN POLYTECHNICAL UNIV
- Filing Date
- 2023-03-31
- Publication Date
- 2026-05-12
AI Technical Summary
Existing technologies lack stability in designing equilateral triangular formations for space gravitational wave detection missions, are computationally complex, cannot explain the underlying mechanisms of configuration divergence, and have limited optimization results.
A gravitational wave detection equilateral triangle formation design method based on the second-order CW equation is adopted. By constructing an optimized design model, the relative position and relative velocity of the spacecraft are corrected by utilizing the nonlinear information in the second-order CW equation, and the phase angle is optimized to achieve long-term stability of the formation.
The divergence of formation arm length was reduced, the balance of breathing angle was optimized, and the calculation efficiency was improved, ensuring the long-term stability and optimization effect of the formation.
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Figure CN116384102B_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of aerospace technology and relates to a method for designing equilateral triangle formations for gravitational wave detection based on second-order CW equations. Background Technology
[0002] Space-based gravitational wave detection missions utilize a three-spacecraft formation of an equilateral triangle. Based on the Michelson interferometry principle, gravitational wave signals are characterized by measuring the change in arm length between adjacent spacecraft. In practical engineering, achieving this goal requires extremely high precision from the three Michelson interferometers formed between the spacecraft, necessitating a certain level of overall stability for the equilateral triangle formation. Therefore, designing a stable equilateral triangle formation that can be maintained over a long period becomes a crucial issue for space-based gravitational wave detection missions.
[0003] To address the stability issues of the aforementioned formation configurations during long-term operation, Nayak derived the relationship between the formation arm length and the formation plane inclination angle based on the second-order CW equation. He discovered that changing the formation plane inclination angle could reduce the peak fluctuation of the formation arm length, thus partially solving the configuration divergence problem of equilateral triangular formations designed based on the CW equation under a two-body nonlinear gravitational field. However, his method still has two shortcomings: 1. Nayak obtained a periodic solution to the CW equation with second-order accuracy, rather than an analytical solution, and therefore cannot directly explain the deep-seated mechanism of configuration divergence; 2. Nayak's method is computationally complex and not intuitive, failing to explain the physical mechanism of better stable solutions, and the optimization results for configuration stability remain limited. Summary of the Invention
[0004] The purpose of this invention is to solve the problems in the prior art and provide a method for designing equilateral triangle formations for gravitational wave detection based on the second-order CW equation.
[0005] To achieve the above objectives, the present invention employs the following technical solution:
[0006] A method for designing an equilateral triangle formation for gravitational wave detection based on the second-order CW equation includes the following steps:
[0007] Using the CW equations, a nominal configuration of an equilateral triangle formation for gravitational wave detection was designed, and the nominal relative positions and nominal relative velocities of the three spacecraft were obtained.
[0008] An optimization design model for the gravitational wave detection equilateral triangle formation based on the second-order CW equation is constructed. Using nominal relative position and nominal relative velocity as the reference, the correction amount of the gravitational wave detection equilateral triangle formation is obtained.
[0009] The relative positions and relative velocities of the three spacecraft are corrected based on the correction amount of the gravitational wave detection equilateral triangle formation, resulting in a new gravitational wave detection equilateral triangle formation.
[0010] Furthermore, the nominal relative position is
[0011] r0 = [x0, y0, z0] T
[0012] The nominal relative speed is
[0013]
[0014] Where x0 represents the satellite's coordinates relative to the formation center along the orbital radial direction, y0 represents the satellite's coordinates relative to the formation center along the flight direction, and z0 represents the satellite's coordinates relative to the formation center along the orbital plane normal. This represents the first derivative of x0 with respect to time. This represents the first derivative of y0 with respect to time. Let z0 be the first derivative with respect to time.
[0015] Furthermore, the nominal relative position and nominal relative velocity are calculated using the CW equation spatial circular formation formula:
[0016]
[0017] Where n represents the orbital angular velocity of the virtual reference center of the formation, r represents the formation size, α is the phase angle of the spacecraft, and the superscript / subscript i = 1, 2, 3 represents the numbers of the three spacecraft in the equilateral triangle formation.
[0018] Furthermore, the optimized design model for the gravitational wave detection equilateral triangle formation based on the second-order CW equation is as follows:
[0019]
[0020]
[0021] Where, Δl ij (t) represents the difference between the arm length formed by spacecraft i and j in an equilateral triangle formation and the nominal arm length l0 of the formation, t∈[t0,t... f ] represents the optimization variable of the inner-layer optimization problem, represents the running time of the formation, t0 represents the start time, t f Indicates the termination time, Δα i For the outer optimization problem, δα represents the correction amount of the phase angle of each spacecraft in the formation relative to the nominal phase angle, and δα represents the range of values for the outer optimization variable.
[0022] Furthermore, the expression for the difference between the arm length formed by spacecraft i and spacecraft j in the equilateral triangle formation and the nominal arm length l0 of the formation is as follows:
[0023]
[0024] Where x, y, and z are calculated from the approximate analytical solution of the second-order CW equation:
[0025]
[0026] Where ε=n 2 / R is a parameter without specific physical meaning, where α0~α8, β0~β7, and γ0~γ6 are related to x0, y0, z0, ... The relevant algebraic expressions.
[0027] Furthermore, the correction amount for the gravitational wave detection equilateral triangle formation includes the correction amount for the phase angle of each spacecraft in the formation relative to the nominal phase angle and the correction amount for the relative motion period conditions of each spacecraft in the formation.
[0028] Furthermore, the method for calculating the corrected relative positions and relative velocities of the three spacecraft is as follows:
[0029]
[0030] Here, the nonlinear function f represents the nonlinear periodic correction formula based on energy matching.
[0031] Furthermore, the nonlinear periodic correction formula based on energy matching is as follows:
[0032]
[0033] Where R represents the orbital radius of the virtual center of the formation.
[0034] Compared with the prior art, the present invention has the following beneficial effects:
[0035] This invention provides a method for designing equilateral triangular formations for gravitational wave detection based on second-order CW equations. Utilizing the nonlinear information contained in the second-order CW equations, an optimization model for the design of equilateral triangular formations based on second-order CW equations is constructed. This model corrects the equilateral triangular formations designed based on CW equations, reducing the divergence of the formation arm lengths and breathing angle under a two-body nonlinear gravitational field. Compared with existing formation design methods based on second-order CW equations, the method proposed in this invention reduces the maximum divergence of the arm length from 1% to 0.32%. Furthermore, because this invention selects the phase angle of the spacecraft as the optimization variable during the optimization process, it considers the balance of the changes in the three arm lengths. Therefore, while using arm length as the optimization index, it also optimizes the breathing angle. Compared with existing optimization methods based on other models, the performance index of the optimization problem in this invention is expressed analytically, resulting in higher computational efficiency. Simultaneously, it employs nonlinear periodic conditions based on energy matching to ensure long-term stability and non-divergence of the formation, leading to better optimization results. Attached Figure Description
[0036] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the embodiments will be briefly introduced below. It should be understood that the following drawings only show some embodiments of the present invention and should not be regarded as a limitation on the scope. For those skilled in the art, other related drawings can be obtained based on these drawings without creative effort.
[0037] Figure 1 This is a flowchart of the space gravitational wave detection equilateral triangle formation configuration design method based on the second-order CW equation of the present invention.
[0038] Figure 2 This is a schematic diagram illustrating a scenario example of a space gravitational wave detection equilateral triangle formation mission according to the present invention.
[0039] Figure 3 This is a schematic diagram illustrating the stability index of the equilateral triangle formation for space gravitational wave detection according to the present invention.
[0040] Figure 4 This is a schematic diagram illustrating the physical meaning of the optimization variables in the space gravitational wave detection equilateral triangle formation configuration optimization design model based on the second-order CW equation of the present invention;
[0041] Figure 5 This invention only considers the evolution of formation arm length after correction based on nonlinear periodic conditions using energy matching;
[0042] Figure 6 This invention only considers the evolution of the formation breathing angle after correction based on nonlinear periodic conditions of energy matching;
[0043] Figure 7 This describes the evolution of the arm length of the formation designed based on the design method of this invention;
[0044] Figure 8 This describes the evolution of the breathing angle of the formation designed based on the design method of this invention. Detailed Implementation
[0045] The following description, in conjunction with the accompanying drawings, illustrates exemplary embodiments of this application, including various details to aid understanding. These should be considered merely exemplary. Therefore, those skilled in the art will recognize that various changes and modifications can be made to the embodiments described herein without departing from the scope and spirit of this application. Similarly, for clarity and brevity, descriptions of well-known functions and structures are omitted in the following description.
[0046] Obviously, the described embodiments are only some, not all, of the embodiments in this application. All other embodiments obtained by those skilled in the art based on the embodiments in this application without inventive effort are within the scope of protection of this application.
[0047] The present invention will now be described in further detail with reference to the accompanying drawings:
[0048] See Figure 1 This invention provides a method for designing equilateral triangle formations for gravitational wave detection based on second-order CW equations, comprising the following steps:
[0049] Using the CW equations, a nominal configuration of an equilateral triangle formation for gravitational wave detection was designed, and the nominal relative positions and nominal relative velocities of the three spacecraft were obtained.
[0050] An optimization design model for the gravitational wave detection equilateral triangle formation based on the second-order CW equation is constructed. Using nominal relative position and nominal relative velocity as the reference, the correction amount of the gravitational wave detection equilateral triangle formation is obtained.
[0051] The relative positions and relative velocities of the three spacecraft are corrected based on the correction amount of the gravitational wave detection equilateral triangle formation, resulting in a new gravitational wave detection equilateral triangle formation.
[0052] The nominal relative position is determined by the vector r0 = [x0, y0, z0]. T The nominal relative velocity is expressed as a vector. In this representation, x0 represents the satellite's coordinates relative to the formation center along the orbital radial direction, y0 represents the satellite's coordinates relative to the formation center along the flight direction, and z0 represents the satellite's coordinates relative to the formation center along the orbital plane normal. This represents the first derivative of x0 with respect to time. This represents the first derivative of y0 with respect to time. Let z0 be the first derivative with respect to time.
[0053] Specifically, the nominal relative position and nominal relative velocity can be calculated based on the CW equation spatial circular formation formula:
[0054]
[0055] Where n represents the orbital angular velocity of the virtual reference center of the formation, r represents the formation size, α is the phase angle of the spacecraft, and the superscript / subscript i = 1, 2, 3 represents the numbers of the three spacecraft in the equilateral triangle formation.
[0056] The optimal design model for the equilateral triangle formation of gravitational wave detection based on the second-order CW equation is as follows:
[0057]
[0058]
[0059] Where, Δα i This represents the phase angle correction for spacecraft i, and its specific physical meaning is as follows: Figure 4 As shown; Δl ij (t) represents the difference between the arm length formed by spacecraft i and j in an equilateral triangle formation and the nominal arm length l0 of the formation. Its specific expression is:
[0060]
[0061] In equation (3), x, y, and z can be calculated from the approximate analytical solution of the second-order CW equation. The specific calculation method is as follows:
[0062]
[0063] In equation (4), ε = n 2 / R is a parameter without specific physical meaning, where α0~α8, β0~β7, and γ0~γ6 are related to x0, y0, z0, ... The relevant algebraic expression has the following specific form:
[0064]
[0065]
[0066]
[0067]
[0068]
[0069]
[0070]
[0071]
[0072]
[0073]
[0074]
[0075]
[0076]
[0077]
[0078]
[0079]
[0080]
[0081]
[0082]
[0083]
[0084]
[0085]
[0086]
[0087] t∈[t0,t f ] represents the optimization variable of the inner-layer optimization problem, represents the running time of the formation, t0 represents the start time, t f Indicates the termination time; Δα i For the outer optimization problem, δα represents the correction amount of the phase angle of each spacecraft in the formation relative to the nominal phase angle; δα represents the range of values for the outer optimization variable.
[0088] The nonlinear function f represents the nonlinear periodic correction formula based on energy matching:
[0089]
[0090] Where R represents the orbital radius of the virtual center of the formation.
[0091] The specific method for calculating the corrected relative positions and relative velocities of the three spacecraft based on the correction amount is as follows:
[0092]
[0093] The present invention will be further described below with reference to specific embodiments:
[0094] Example 1:
[0095] This invention considers Figure 2 The space gravitational wave detection mission scenario shown depicts three spacecraft forming an equilateral triangle with an arm length of 3 Mkm. The center of the triangle is located in Earth's orbit and is 20° ahead of Earth. The overall stability requirements for this formation in this mission scenario are shown in Table 1. The specific physical meanings of the indicators in Table 1 are as follows: Figure 3 As shown.
[0096] Table 1 Formation Stability Requirements
[0097] Task cycle arm length Rate of change of arm length breathing angle 4 to 10 years 3±0.035Mkm 0±10m / s 60±1°
[0098] The formation design method for the above scenario is given below, with specific steps as follows:
[0099] According to the requirements of the "Tai Chi" plan, the formation scale is taken as r = 1.7321 × 10⁻⁶. 9 Given the phase angles of the three spacecraft as α1 = 0°, α2 = 120°, and α3 = 240°, respectively, the orbital angular velocity of the virtual reference center of the formation is calculated as n = 1.9913 × 10⁻⁶ m. -7 Given rad / s, the nominal relative position and nominal relative velocity of the three spacecraft can be calculated as follows:
[0100]
[0101]
[0102]
[0103] Constructing an optimal design model for gravitational wave detection equilateral triangle formation based on the second-order CW equation:
[0104]
[0105]
[0106] For the above optimization problem, the inner optimization problem is solved using a global optimization algorithm, and the outer optimization problem is solved using a pattern search algorithm. The final optimization results are: Δα1=0, Δα2=-0.2257°, Δα3=0.0977°.
[0107] Based on the optimization results, the corrected relative positions and relative velocities of the three spacecraft are as follows:
[0108]
[0109]
[0110]
[0111] like Figure 5 and Figure 6 As shown, the evolution of the arm length and breathing angle of the formation, considering only the nonlinear periodic condition correction based on energy matching, is presented within a 10-year mission cycle. Figure 7 and Figure 8 As shown, the evolution of arm length and breathing angle of the formation designed based on the method of this invention over a 10-year mission cycle is presented. The comparison shows that the optimization effect on the maximum peak arm length of the formation designed based on the method of this invention is not significant over the 10-year mission cycle, but it still reduces the average maximum offset of the three arm lengths from 0.42% to 0.32%. However, regarding the breathing angle, although the breathing angle is not included in the optimization index in the formation optimization design model constructed in this invention, by selecting the spacecraft phase angle as the optimization variable, the optimization result still reduces the average maximum offset of the three breathing angles from 0.79% to 0.46%. In summary, the simulation results demonstrate the effectiveness of the formation design method proposed in this invention.
[0112] The above are merely preferred embodiments of the present invention and are not intended to limit the present invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.
Claims
1. A method for designing an equilateral triangle formation for gravitational wave detection based on the second-order CW equation, characterized in that, Includes the following steps: Using the CW equations, a nominal configuration of an equilateral triangle formation for gravitational wave detection was designed, and the nominal relative positions and nominal relative velocities of the three spacecraft were obtained. An optimization design model for the gravitational wave detection equilateral triangle formation based on the second-order CW equation is constructed. Using nominal relative position and nominal relative velocity as the reference, the correction amount of the gravitational wave detection equilateral triangle formation is obtained. The relative positions and relative velocities of the three spacecraft are corrected based on the correction amount of the gravitational wave detection equilateral triangle formation, resulting in a new gravitational wave detection equilateral triangle formation; The gravitational wave detection equilateral triangle formation optimization design model based on the second-order CW equation is as follows: in, Represents spacecraft in an equilateral triangle formation and spacecraft The arm length formed between them compared to the nominal arm length of the formation The difference, Let be the optimization variable for the inner optimization problem, representing the formation's running time. Indicates the start time. Indicates the end time. Let be the optimization variable for the outer-layer optimization problem, representing the correction amount of the phase angle of each spacecraft in the formation relative to the nominal phase angle. Indicates the range of values for the outer optimization variables; Spacecraft in the equilateral triangle formation and spacecraft The arm length formed between them compared to the nominal arm length of the formation The expression for the difference is: in, , , Calculated from the approximate analytical solution of the second-order CW equation: in, It is a parameter that does not have a specific physical meaning. , , They are respectively with , , , , , The relevant algebraic expressions.
2. The gravitational wave detection equilateral triangle formation design method based on the second-order CW equation as described in claim 1, characterized in that, The nominal relative position is The nominal relative speed is in, This represents the coordinates of the satellite relative to the center of the formation along the radial direction of the orbit. This represents the coordinates of the satellite relative to the center of the formation along the flight direction. This represents the coordinates of the satellites relative to the center of the formation along the normal direction of the orbital plane. express The first derivative with respect to time, express The first derivative with respect to time, express The first derivative with respect to time.
3. The gravitational wave detection equilateral triangle formation design method based on the second-order CW equation as described in claim 1, characterized in that, The nominal relative position and nominal relative velocity are calculated using the CW equation spatial circular formation formula: in, This represents the orbital angular velocity of the formation's virtual reference center. Indicates the formation scale. Phase angle of the spacecraft, superscript / subscript This indicates the numbers of the three spacecraft in the equilateral triangle formation.
4. The gravitational wave detection equilateral triangle formation design method based on the second-order CW equation as described in claim 1, characterized in that, The corrections for the gravitational wave detection equilateral triangle formation include corrections to the phase angle of each spacecraft in the formation relative to the nominal phase angle and corrections to the relative motion period conditions of each spacecraft in the formation.
5. The method for designing an equilateral triangle formation for gravitational wave detection based on the second-order CW equation as described in claim 1, characterized in that, The method for calculating the corrected relative positions and relative velocities of the three spacecraft is as follows: Among them, nonlinear functions This represents a nonlinear periodic correction formula based on energy matching.
6. The gravitational wave detection equilateral triangle formation design method based on the second-order CW equation as described in claim 5, characterized in that, The nonlinear periodic correction formula based on energy matching is as follows: Where R represents the orbital radius of the virtual center of the formation.