CN-116384102-B - Gravitational wave detection regular triangle formation design method based on second-order CW equation

CN116384102BCN 116384102 BCN116384102 BCN 116384102BCN-116384102-B

Abstract

The invention discloses a gravitational wave detection regular triangle formation design method based on a second-order CW equation, which comprises the following steps of utilizing the CW equation to design a gravitational wave detection regular triangle formation nominal configuration so as to obtain nominal relative positions and nominal relative speeds of three spacecrafts; and correcting the relative positions and the relative speeds of three spacecrafts based on the correction quantity of the gravitational wave detection regular triangle formation to obtain a new gravitational wave detection regular triangle formation. The correction of regular triangle formation designed based on CW equation is realized, and the divergence of formation arm length and respiratory angle under the two-body nonlinear attraction field is reduced.

Inventors

DANG CHAOHUI
Jiao Bohan
ZHANG YONGHE
WANG PENGCHENG
GUO MING
ZHOU HAO
LIU PEIDONG

Assignees

西北工业大学

Dates

Publication Date: 20260512
Application Date: 20230331

Claims (6)

1. The gravitational wave detection regular triangle formation design method based on the second-order CW equation is characterized by comprising the following steps: Designing a gravitational wave detection regular triangle formation nominal configuration by using a CW equation to obtain nominal relative positions and nominal relative speeds of three spacecrafts; Constructing an optimized design model of the gravitational wave detection regular triangle formation based on a second-order CW equation, and obtaining correction quantity of the gravitational wave detection regular triangle formation by taking a nominal relative position and a nominal relative speed as references; correcting the relative positions and the relative speeds of the three spacecrafts based on the correction amount of the gravitational wave detection regular triangle formation to obtain a new gravitational wave detection regular triangle formation; The gravitational wave detection regular triangle formation optimization design model based on the second-order CW equation is as follows: Wherein, the Representing spacecraft in regular triangle formation And spacecraft The arm length formed between the two is compared with the formation nominal arm length Is used for the difference in (a), Optimization variables for the inner layer optimization problem, representing the run time of the formation, The start time is indicated as such, The time of termination is indicated as such, An optimization variable, which is an outer layer optimization problem, representing a correction of the phase angle of each spacecraft in the formation relative to the nominal phase angle, Representing the value range of the outer layer optimization variable; Spacecraft in the regular triangle formation And spacecraft The arm length formed between the two is compared with the formation nominal arm length The expression of the difference of (c) is: Wherein, the 、、 Calculated from the approximate analytical solution of the second order CW equation: Wherein, the Is a parameter that does not have a specific physical meaning, 、、 Respectively is with 、、、、、 The algebraic expressions concerned.
2. The method of designing a gravitational wave detection equilateral triangle formation based on a second order CW equation according to claim 1, wherein the nominal relative position is The nominal relative speed is Wherein, the Representing the coordinates of the satellite along the orbital radial direction relative to the formation center, Representing the coordinates of the satellite relative to the formation center in the direction of flight, Representing the coordinates of the satellite normal to the orbital plane relative to the formation center, Representation of The first derivative with respect to time is, Representation of The first derivative with respect to time is, Representation of The first derivative with respect to time.
3. The method of claim 1, wherein the nominal relative position and nominal relative velocity are calculated by a CW equation space circular formation equation: Wherein, the Represents the angular velocity of the track forming the virtual reference center, The scale of the formation is represented, Superscript/subscript for phase angle of spacecraft The numbers of the three spacecraft in the regular triangle formation are represented.
4. The method of claim 1, wherein the correction of the gravitational wave detection triangle formation includes a correction of a phase angle of each spacecraft in the formation relative to a nominal phase angle and a correction of a relative motion cycle condition of each spacecraft in the formation.
5. The gravitational wave detection regular triangle formation design method based on the second-order CW equation according to claim 1, wherein the calculation method of the corrected relative position and relative speed of the three spacecraft is as follows: Wherein the nonlinear function A nonlinear periodic correction formula based on energy matching is represented.
6. The gravitational wave detection regular triangle formation design method based on the second-order CW equation according to claim 5, wherein the nonlinear periodic correction formula based on the energy matching is: Where R represents the track radius of the virtual center of the formation.

Description

Gravitational wave detection regular triangle formation design method based on second-order CW equation Technical Field The invention belongs to the technical field of aerospace, and relates to a gravitational wave detection regular triangle formation design method based on a second-order CW equation. Background The space gravitational wave detection task is to form a space regular triangle formation by utilizing three spacecrafts, and the gravitational wave signal is represented by measuring the arm length change between two adjacent spacecrafts according to the Michelson interference principle. In practical engineering, if the gravitational wave detection task of the principle is to be realized, extremely high precision requirements are required for three groups of Michelson interferometers formed among satellites, namely certain requirements are required for the overall stability of regular triangle formation. Therefore, how to design a regular triangle formation which can be stably maintained for a long time becomes an important problem of the space gravitational wave detection task. In order to solve the problem of long-time running stability of the formation configuration, nayak derives the relation between the formation arm length and the formation plane inclination angle based on a second-order CW equation, and finds that the fluctuation peak value of the formation arm length can be reduced by changing the formation plane inclination angle, so that the problem of configuration divergence of the regular triangle formation designed based on the CW equation under a two-body nonlinear gravitational field is solved to a certain extent. However, the method still has the following two defects that a CW equation with second-order precision is obtained by Nayak instead of the analysis solution of the second-order CW equation, so that the deep mechanism of the configuration divergence cannot be directly explained, the method of 2.Nayak is relatively complex in calculation and not intuitive, the physical mechanism of the better stable solution cannot be explained, and the optimization result of the configuration stability is still limited. Disclosure of Invention The invention aims to solve the problems in the prior art and provides a gravitational wave detection regular triangle formation design method based on a second-order CW equation. In order to achieve the purpose, the invention is realized by adopting the following technical scheme: a gravitational wave detection regular triangle formation design method based on a second-order CW equation comprises the following steps: Designing a gravitational wave detection regular triangle formation nominal configuration by using a CW equation to obtain nominal relative positions and nominal relative speeds of three spacecrafts; Constructing an optimized design model of the gravitational wave detection regular triangle formation based on a second-order CW equation, and obtaining correction quantity of the gravitational wave detection regular triangle formation by taking a nominal relative position and a nominal relative speed as references; and correcting the relative positions and the relative speeds of the three spacecrafts based on the correction amount of the gravitational wave detection regular triangle formation to obtain a new gravitational wave detection regular triangle formation. Further, the nominal relative position is r0＝[x0,y0,z0]T The nominal relative speed is Wherein x 0 represents the coordinates of the satellite along the track radial direction relative to the formation center, y 0 represents the coordinates of the satellite along the flight direction relative to the formation center, z 0 represents the coordinates of the satellite along the track surface normal direction relative to the formation center,Representing the first derivative of x 0 with respect to time,Representing the first derivative of y 0 with respect to time,Representing the first derivative of z 0 with respect to time. Further, the nominal relative position and nominal relative velocity are calculated by a CW equation space circular formation formula: wherein n represents the orbital angular velocity of the virtual reference center of the formation, r represents the formation scale, 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 regular triangle formation. Further, the gravitational wave detection regular triangle formation optimization design model based on the second-order CW equation is as follows: Wherein Δl ij (t) represents the arm length formed between spacecraft i and spacecraft j in the regular triangle formation, compared with the difference value of the nominal arm length l 0 of the formation, t e [ t 0,tf ] is an optimization variable of the inner layer optimization problem, t 0 represents the start time, t f represents the end time, Δα i is an optimization variable of the outer laye