A multi-pulse parallel construction method for gravitational wave detection formation considering timing optimization

Abstract

The present application relates to the field of aerospace technology, specifically to a kind of multi-pulse parallel construction method of gravitational wave detection formation considering timing optimization, each spacecraft of formation simultaneously departs from virtual center, each spacecraft is moved to the preset specified position by applying multiple pulses, so as to form a kind of spacecraft formation that can be used for space gravitational wave detection, by giving the steps of calculating the initial position and velocity of each spacecraft and applying initial control to reach the initial position and velocity, the state quantity of each spacecraft at each time in the formation construction process is calculated, so as to provide reference for the design of satellite formation task in actual engineering application, and the accuracy of the linearized relative motion dynamics model established by the multi-pulse parallel space gravitational wave detection formation is improved by differential correction, the error in the relative motion dynamics model is reduced, the initial state set in advance can be effectively reached, and the working efficiency of formation component is improved.

Description

Technical Field

[0001] This invention relates to the field of aerospace technology, specifically to a method for constructing multi-pulse parallel gravitational wave detection formations with consideration of timing optimization. Background Technology

[0002] Spacecraft formations are widely discussed in various research fields such as astronomy and physics, and research and application of related technologies are being carried out extensively. In recent years, with LIGO announcing the first detection of gravitational waves, research on gravitational waves by major research institutions around the world has become increasingly active. Space-based gravitational wave detection technology using spacecraft formations can compensate for the shortcomings of ground-based gravitational wave detection devices, detecting even lower-frequency gravitational waves. Therefore, research on gravitational wave detection formation technologies is becoming increasingly important. Currently, research on the construction methods of space gravitational wave detection formations is relatively lacking. The correct order in which spacecraft form a formation and how to construct a formation to save fuel are all urgent issues that need to be studied. Summary of the Invention

[0003] In order to overcome the defects of the prior art, the purpose of this invention is to provide a multi-pulse parallel construction method for gravitational wave detection formations that considers timing optimization, so as to solve the technical problem that the linearized relative motion dynamics model established by the multi-pulse parallel space gravitational wave detection formations in the prior art has errors, which prevents the formation components from reaching the preset initial state and thus causes the formation components to fail.

[0004] This invention is achieved through the following technical solution:

[0005] A method for constructing multi-pulse parallel gravitational wave detection arrays considering timing optimization includes the following steps:

[0006] Step 1: Input the parameters for the multi-pulse parallel space gravitational wave detection formation;

[0007] Step 2: Calculate the terminal state constraints of the multi-pulse parallel construction in the formation based on the spacecraft parameters;

[0008] Step 3: Based on the terminal state constraints and the space circular formation conditions, the target state of the multi-pulse is obtained, and a multi-pulse control mathematical model suitable for the initialization of the space gravitational wave detection formation is constructed.

[0009] Step 4: In the multi-pulse control mathematical model, an optimization algorithm is used to calculate the optimal application time of each pulse and the velocity increment under each pulse in the formation construction multi-pulse control.

[0010] Step 5: Calculate the state variables of each spacecraft at each moment during the formation building process based on the optimal application time of each pulse and each velocity increment in the multi-pulse control of formation building, and correct the linearization error through differential correction.

[0011] Preferably, in step 1, the space gravitational wave detection formation parameters include the formation's orbit and parameters determined according to mission requirements, as well as the spacecraft parameters within the formation. The formation's orbit and parameters include the semi-major axis, eccentricity, orbital inclination, right ascension of the ascending node, argument of perihelion, and true anomaly. The spacecraft parameters within the formation include the spacecraft's orbital radius. Pulse count .

[0012] Preferably, in step 2, the specific process of calculating the terminal state constraints of the multi-pulse parallel construction in the formation based on spacecraft parameters and the Clohessy-Wiltshire equations is as follows:

[0013] Step 21: Derive the initial conditions for the circular configuration of the spacecraft based on the Clohessy-Wiltshire equations according to the spacecraft parameters. The initial condition formula is as follows:

[0014] ;

[0015] in, d Indicates the orbital radius of the spacecraft. α The phase angle of the spacecraft is represented; n is the orbital angular velocity of the virtual center of the formation; nd is the reference value of the linear velocity of the spacecraft during its orbital flight.

[0016] Step 22: Calculate the terminal state constraints of the multi-pulse parallel construction in the formation based on the initial conditions of the spacecraft's circular configuration;

[0017]

[0018] in, t f This indicates the time when the parallel construction is completed.

[0019] Preferably, in step 3, the specific method for constructing a multi-pulse control mathematical model suitable for the initialization of a space gravitational wave detection formation is as follows:

[0020] Step 31: Using the spatial circular formation conditions under the Clohessy-Wiltshire equations in spatial relative motion, calculate the initial relative positions and velocities of all satellites, which is the target state of the multi-pulse.

[0021] Step 32: Based on the initial relative positions and velocities of all satellites, the control terms in the Clohessy-Wiltshire equations are approximated using pulses, resulting in the following multi-pulse control mathematical model:

[0022]

[0023] The multi-pulse control mathematical model is transformed into a multiplicative form as follows:

[0024] ;

[0025] in, Construct completion time for parallel formation t f The column vector of terminal target state variables of the spacecraft; For the spacecraft from the initial reference time t 0 to terminal completion time t f Position-velocity state transition matrix; For the spacecraft at the initial reference time t The initial relative state quantity column vector of 0; This represents the velocity increment of each pulse; For the spacecraft from the initial reference time t From 0 to any time t Position-velocity state transition matrix; N The total number of pulses applied during formation building; From arrive The pulse state transition matrix.

[0026] Furthermore, in step 32, the control term is expanded into a state-space model as follows:

[0027]

[0028] Among them, control items ,

[0029] in, This represents the components of the control quantity in each direction; n is the orbital angular velocity of the virtual center of the formation.

[0030] Preferably, in step 4, the specific process is as follows:

[0031] Step 41: Perform a transposition transformation on the multi-pulse control mathematical model to obtain the transposed multi-pulse control mathematical model, as follows:

[0032]

[0033] Step 42: Based on the target state of the multi-pulse, use an optimization algorithm to find the optimal pulse application time in the multi-pulse control mathematical model after the term transformation;

[0034] Step 43: Based on the target state of the multi-pulse combined with the optimal pulse application time, obtain the velocity increment under each pulse.

[0035] Furthermore, in step 43, the velocity increment expression for each pulse is as follows:

[0036]

[0037] in, ;

[0038] ;

[0039] ;

[0040] in, These respectively represent spacecraft one, two, and three in the gravitational wave detection formation. The velocity increment of the next pulse.

[0041] Preferably, in step 5, the state variables of each spacecraft at each moment during the formation building process are calculated based on the optimal application time of each pulse and each velocity increment in the multi-pulse control for formation building. The specific formulas are as follows:

[0042]

[0043] Where, x k0 y k0 z k0 Represents the three-dimensional relative position components of the k-th spacecraft at the initial moment (k=1,2,3); , , θ1, θ2, and θ3 represent the three-dimensional relative velocity components of spacecraft k at the initial moment (k=1,2,3); θ1, θ2, and θ3 represent the initial phase angles of spacecraft 1, 2, and 3, respectively.

[0044] Preferably, in step 5, the specific method for correcting the nonlinearization error through differential correction is as follows:

[0045] S1, given a set of initial velocity increments Through a dynamic nonlinear model Integrating, we obtain the actual pulse termination state:

[0046] ;

[0047] S2, calculate the error amount of the actual pulse termination state:

[0048] ;

[0049] S3, combining the error amount of the actual pulse terminal state, the correction value of the pulse velocity is determined in reverse through the state transition matrix:

[0050] ;

[0051] S4, based on the initial velocity increment and the pulse velocity correction value, the corrected pulse velocity is obtained as follows:

[0052] ;

[0053] S5, using the nonlinear model integration, and calculating the terminal state error, i.e., repeating S1-S2, if the corrected pulse velocity... This can make the error amount If the value is less than the given value, the iterative correction ends; otherwise, the correction performed in S3-S4 continues.

[0054] Compared with the prior art, the present invention has the following beneficial technical effects:

[0055] This invention provides a multi-pulse parallel construction method for gravitational wave detection formations that considers timing optimization. Each spacecraft in the formation starts simultaneously from a virtual center, and multiple pulses are applied to move each spacecraft to a preset, designated position, thus forming a spacecraft formation suitable for space gravitational wave detection. By providing steps for calculating the initial position and velocity of each spacecraft and applying initialization control to reach that initial position and velocity, the method calculates the state variables of each spacecraft at each moment during formation construction, providing a reference for the design of satellite formation missions in practical engineering applications. Furthermore, differential correction is used to improve the accuracy of the linearized relative motion dynamics model established for the multi-pulse parallel space gravitational wave detection formation, reducing errors in the relative motion dynamics model and effectively reaching the preset initial state, thereby improving the working efficiency of the formation components.

[0056] Furthermore, this invention can construct a space gravitational wave detection formation with a preset configuration. First, the orbital radius of the spacecraft in the desired gravitational wave detection formation is input. Then, an optimization algorithm is used to optimize and find the pulse application time. Subsequently, the velocity increments at each pulse application time in the sub-pulse initialization control are calculated. When the error is large and does not meet the requirements, differential correction is used to correct the velocity increments at each pulse application time.

[0057] Furthermore, the parallel initialization method of this invention involves three spacecraft simultaneously launching from the orbital center and performing orbital transfers via pulses to reach the final state. Determining the timing of pulse application is crucial; selecting an appropriate pulse application time contributes to the stability and accuracy of spacecraft formation initialization, while also conserving fuel. Attached Figure Description

[0058] Figure 1 This is a flowchart of the multi-pulse parallel construction method for space gravitational wave detection formations that considers timing optimization in this invention;

[0059] Figure 2 This is a schematic diagram of the orbit in which the formation is located in this invention;

[0060] Figure 3 This is a schematic diagram of the terminal state constraints in the parallel construction method of this invention;

[0061] Figure 4 This is a schematic diagram of the parallel construction method in this invention;

[0062] Figure 5 This is a schematic diagram showing the changes in the distance between the three spacecraft at different moments during the formation construction process in this invention;

[0063] Figure 6 This is a schematic diagram showing the changes in the distance between the three spacecraft and the formation center at different moments during the formation construction process in this invention;

[0064] Figure 7 This is a schematic diagram illustrating the iterative process of differential correction optimization in this invention. Detailed Implementation

[0065] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0066] It should be noted that the terms "first," "second," etc., in the specification, claims, and accompanying drawings of this invention are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate so that the embodiments of the invention described herein can be implemented in orders other than those illustrated or described herein. Furthermore, the terms "comprising" and "having," and any variations thereof, are intended to cover a non-exclusive inclusion; for example, a process, method, system, product, or apparatus that comprises a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products, or apparatus.

[0067] The present invention will now be described in further detail with reference to the accompanying drawings:

[0068] The purpose of this invention is to provide a multi-pulse parallel construction method for gravitational wave detection formations that considers timing optimization, in order to solve the technical problem that the linearized relative motion dynamics model established by the multi-pulse parallel space gravitational wave detection formations in the prior art has errors, which prevents the formation components from reaching the preset initial state and thus causes the formation components to fail.

[0069] Specifically, according to Figure 1 As shown, the multi-pulse parallel construction method for this space gravitational wave detection formation includes the following steps:

[0070] Step 1: Input the parameters for the multi-pulse parallel space gravitational wave detection formation;

[0071] Specifically, the parameters of the space gravitational wave detection formation include the formation's orbit and parameters determined according to mission requirements, as well as the parameters of the spacecraft within the formation. The formation's orbit and parameters include its semi-major axis, eccentricity, orbital inclination, right ascension of the ascending node, argument of perihelion, and true anomaly. The spacecraft parameters include the spacecraft's orbital radius. Pulse count .

[0072] Step 2: Calculate the terminal state constraints of the multi-pulse parallel construction in the formation based on the spacecraft parameters. The specific process is as follows:

[0073] Step 21: Derive the initial conditions for the circular configuration of the spacecraft based on the Clohessy-Wiltshire equations according to the spacecraft parameters. The initial condition formula is as follows:

[0074] ;

[0075] in, d Indicates the orbital radius of the spacecraft. α The phase angle of the spacecraft is represented; n is the orbital angular velocity of the virtual center of the formation; nd is the reference value of the linear velocity of the spacecraft during its orbital flight.

[0076] Step 22: Calculate the terminal state constraints of the multi-pulse parallel construction in the formation based on the initial conditions of the spacecraft's circular configuration;

[0077]

[0078] in, t f This indicates the moment when the parallel construction is completed.

[0079] Step 3: Based on the terminal state constraints and the space circular formation conditions, the target state of the multi-pulse is obtained, and a multi-pulse control mathematical model suitable for the initialization of the space gravitational wave detection formation is constructed.

[0080] Specifically, the method for constructing a multi-pulse control mathematical model suitable for the initialization of space gravitational wave detection formations is as follows:

[0081] Step 31: Using the spatial circular formation conditions under the Clohessy-Wiltshire equations in spatial relative motion, calculate the initial relative positions and velocities of all satellites, which is the target state of the multi-pulse.

[0082] Step 32: Based on the initial relative positions and velocities of all satellites, the control-related terms in the Clohessy-Wiltshire equations are approximated using pulses, resulting in the following multi-pulse control mathematical model:

[0083]

[0084] The multi-pulse control mathematical model is transformed into a multiplicative form as follows:

[0085] ;

[0086] in, Construct completion time for parallel formation t f The column vector of terminal target state variables of the spacecraft; For the spacecraft from the initial reference time t 0 to terminal completion time t f Position-velocity state transition matrix; For the spacecraft at the initial reference time t The initial relative state quantity column vector of 0; This represents the velocity increment of each pulse; For the spacecraft from the initial reference time t From 0 to any time t Position-velocity state transition matrix; N The total number of pulses applied during formation building; From arrive The pulse state transition matrix.

[0087] The control terms, expanded into a state-space model, are as follows:

[0088]

[0089] Among them, control items ,

[0090] in, This represents the components of the control term in each direction; n is the orbital angular velocity of the virtual center of the formation.

[0091] Step 4: In the multi-pulse control mathematical model, an optimization algorithm is used to calculate the optimal application time of each pulse and the velocity increment under each pulse in the formation construction multi-pulse control.

[0092] The specific process is as follows:

[0093] Step 41: Perform a transposition transformation on the multi-pulse control mathematical model to obtain the transposed multi-pulse control mathematical model, as follows:

[0094]

[0095] Step 42: Based on the target state of the multi-pulse, use an optimization algorithm to find the optimal pulse application time in the multi-pulse control mathematical model after the term transformation;

[0096] Step 43: Based on the target state of the multi-pulse combined with the optimal pulse application time, obtain the velocity increment under each pulse.

[0097] The velocity increment expressions for each pulse are as follows:

[0098]

[0099] in, ;

[0100] ;

[0101] ;

[0102] in, These respectively represent spacecraft one, two, and three in the gravitational wave detection formation. The velocity increment of the next pulse.

[0103] Step 5: Calculate the state variables of each spacecraft at each moment during the formation building process based on the optimal application time of each pulse and each velocity increment in the multi-pulse control of formation building, and correct the linearization error through differential correction.

[0104] Specifically, based on the optimal application time of each pulse and each velocity increment in the multi-pulse control for formation building, the state variables of each spacecraft at each moment during the formation building process are calculated, and the specific formulas are as follows:

[0105] .

[0106] Where, x k0 y k0 z k0 Represents the three-dimensional relative position components of the k-th spacecraft at the initial moment (k=1,2,3); , , θ1, θ2, and θ3 represent the three-dimensional relative velocity components of spacecraft k at the initial moment (k=1,2,3); θ1, θ2, and θ3 represent the initial phase angles of spacecraft 1, 2, and 3, respectively.

[0107] The specific method for correcting the mislinearization error through differential correction is as follows:

[0108] S1, given a set of initial velocity increments Through a dynamic nonlinear model Integrating, we obtain the actual pulse termination state:

[0109] ;

[0110] S2, calculate the error amount of the actual pulse termination state:

[0111] ;

[0112] S3, combining the error amount of the actual pulse terminal state, the correction value of the pulse velocity is determined in reverse through the state transition matrix:

[0113] ;

[0114] S4, based on the initial velocity increment and the pulse velocity correction value, the corrected pulse velocity is obtained as follows:

[0115] ;

[0116] S5, using the nonlinear model integration, and calculating the terminal state error, i.e., repeating S1-S2, if the corrected pulse velocity... This can make the error amount If the value is less than the given value, the iterative correction ends; otherwise, the correction performed in S3-S4 continues.

[0117] Example

[0118] This embodiment presents a multi-pulse parallel construction method for space gravitational wave detection formations using a parallel initialization approach. To demonstrate the versatility of this method, an equilateral triangle formation, the most common formation in space gravitational wave detection missions, is selected. Three spacecraft are positioned at the three vertices of the triangle, rotating around the flight center, as shown below. Figure 2 As shown, the formation formation is constructed by the simultaneous initialization of three spacecraft departing from the flyby center.

[0119] This embodiment places the formation's center position on the solar orbit, with an orbital radius of 149,579,870.7 km. Its orbital six elements are set as follows: semi-major axis... eccentricity track inclination Right ascension of ascending node Perihelion angle True close angle .

[0120] Specifically, it includes the following steps:

[0121] S1, based on mission requirements, determine the required formation size of the spacecraft in the formation. km, flight radius Pulse count .

[0122] S2, Derivation of the initial conditions for the circular configuration of the spacecraft based on the Clohessy-Wiltshire equations:

[0123] (1)

[0124] in The initial phase angle of the spacecraft on the fly-around circle, parameter This represents the orbital angular velocity of the primary star located in a circular orbit. The initial phase angle of three spacecraft in a triangular formation can be set as follows: , , The virtual flyby center of the spacecraft formation is located on the Sun's orbit, with an angular velocity of: .

[0125] The state variables of the three spacecraft are:

[0126] (2)

[0127] Based on the geometrical positional relationship of the three spacecraft in the formation constellation for gravitational wave detection, such as Figure 3 As shown, the terminal state constraints of each spacecraft after parallel construction are calculated:

[0128] (3)

[0129] S3 uses an optimization algorithm to calculate the optimal application time and velocity increment of each pulse in multi-pulse initialization control.

[0130] S4 further derives and calculates the state quantities of each spacecraft at each moment during the entire gravitational wave detection formation construction process, providing a basis for the implementation of actual engineering missions.

[0131] The determination of input parameters in S1 can be broken down into the following steps. First, based on mission requirements, the orbit of the formation and related parameters are determined. Then, the formation size, i.e., the orbital radius of the spacecraft within the formation, is determined. and the number of pulses applied .

[0132] The initial relative positions and velocities of all satellites in the formation are calculated according to equation (2). The calculated state variables containing relative positions and velocities are the target states of the multi-pulse required in subsequent steps.

[0133] according to Figure 4 As shown, based on the linearized Clohessy-Wiltshire equations for a circular orbit, the relationship between the velocity pulse increment and the initial and target states at each time step is derived. Considering the addition of control terms and expanding it into a state-space model, the equations are as follows:

[0134] (4)

[0135] Among them, control items , .

[0136] The analytical solution is:

[0137] (5)

[0138] in ,

[0139] .

[0140] Since multi-pulse control is used, the control-related terms in equation (5) are approximated by pulses to obtain:

[0141] (6)

[0142] in .

[0143] Order to take For the moment when parallel construction is completed, for the first One spacecraft, Let the state variables of the spacecraft represent the state variables of the spacecraft at the moment when the formation is completed. Then, it becomes a matrix multiplication form as follows:

[0144] (7)

[0145] In the formula, This indicates the time when each pulse is applied.

[0146] In S3, it is necessary to calculate the velocity increment at each pulse application time in the multi-pulse initialization control. This can be described in detail below.

[0147] The expressions for each velocity increment during the multi-pulse parallel initialization process of an equilateral triangular formation for space gravitational wave detection are as follows:

[0148] (8)

[0149] in,

[0150] (9)

[0151] (10)

[0152] (11)

[0153] In the formula They represent spacecraft number one, two, and three, respectively. The velocity increment of the next pulse. Based on the initial formation state, i.e., the pulse target state, calculated in equation (2), the optimal pulse application time in this equation is determined using an optimization algorithm. We searched for and identified its optimal performance indicator as fuel efficiency.

[0154] Fuel is extremely precious in space missions; therefore, this method constructs a performance index function:

[0155] (12)

[0156] definition The mathematical description of the optimization problem at this point is:

[0157] (13)

[0158] The goal is to minimize fuel consumption, achieving fuel optimization, with the timing of each pulse application being the optimization variable. Then, based on the determined target state after the pulse and the pulse application time, the velocity increment under each pulse can be calculated.

[0159] Step 4 further calculates the state variables of each spacecraft at each moment during the formation process and corrects them using differential correction. This step will be described in more detail below:

[0160] Based on the Clohessy-Wiltshire equation, the state variables at each moment during the formation construction process are calculated from the target state of the pulse, the pulse application time, and the velocity increment at each pulse application time calculated in S3.

[0161] Then, the linearization error in the above process is corrected by using the target shooting iterative algorithm. The steps of differential correction are as follows.

[0162] (1) Given a set of initial velocity increments Through a dynamic nonlinear model Integrating the data yields the actual terminal status:

[0163] (14)

[0164] (2) Calculate the error amount of the pulse termination state:

[0165] (15)

[0166] (3) Determine the correction value of the pulse velocity in reverse through the state transition matrix:

[0167] (16)

[0168] (4) The corrected pulse velocity is obtained as follows:

[0169] (17)

[0170] (5) Integrate again using the nonlinear model and calculate the terminal state error, i.e., repeat the aforementioned steps (1)-(2). If the corrected pulse velocity This can make the error amount If the error is less than the given value (determined according to the control accuracy requirements in actual problems), the iterative correction ends; if the error is greater than the given value, continue with steps (3)-(4) for another correction, and repeat the iteration. The given value is determined according to the requirements of the actual detection mission for the stability of the formation configuration.

[0171] For the initialization problem of this space gravitational wave detection formation, all three spacecraft in the formation satisfy equation (8), and the formation initialization is completed in parallel. The initial positions of the three spacecraft are the same. .

[0172] After optimization, the obtained pulse application time and velocity increment are as follows:

[0173] Table 1. Velocity increments at each time point

[0174]

[0175] After differential correction optimization, the changes in the distance between the three spacecraft (i.e., the formation arm length) at each moment during the formation formation process are as follows: Figure 5 As shown, the changes in the distances of the three spacecraft from the formation center at various moments during the formation formation process are as follows: Figure 6 As shown, the target optimization is as follows: Figure 7 As shown.

[0176] In summary, this invention provides a multi-pulse parallel construction method for gravitational wave detection formations that considers timing optimization. By applying subpulses, each spacecraft in the formation moves to a preset designated position, thereby forming a spacecraft formation suitable for space gravitational wave detection. By providing steps for calculating the initial position and velocity of each spacecraft and applying initialization control to reach the initial position and velocity, the state variables of each spacecraft at each moment during the formation construction process are calculated. This provides a reference for the design of satellite formation missions in practical engineering applications. Furthermore, by using differential correction, the accuracy of the linearized relative motion dynamics model established for the multi-pulse parallel space gravitational wave detection formation is improved, and the errors existing in the relative motion dynamics model are reduced. This method can effectively reach the preset initial state and improve the working efficiency of the formation components.

[0177] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the specific implementation of the present invention. Any modifications or equivalent substitutions that do not depart from the spirit and scope of the present invention should be covered within the scope of protection of the claims of the present invention.

Claims

1. A method for constructing a multi-pulse parallel of a gravitational wave detection formation considering timing optimization, characterized in that, Includes the following steps: Step 1: Input the parameters for the multi-pulse parallel space gravitational wave detection formation; Step 2: Calculate the terminal state constraints of the multi-pulse parallel construction in the formation based on the spacecraft parameters; Step 3: Based on the terminal state constraints and the space circular formation conditions, the target state of the multi-pulse is obtained, and a multi-pulse control mathematical model suitable for the initialization of the space gravitational wave detection formation is constructed. Step 4: In the multi-pulse control mathematical model, an optimization algorithm is used to calculate the optimal application time of each pulse and the velocity increment under each pulse in the formation construction multi-pulse control. Step 5: Calculate the state variables of each spacecraft at each moment during the formation building process based on the optimal application time of each pulse and each velocity increment in the multi-pulse control of formation building, and correct the linearization error through differential correction.

2. The method for constructing a multi-pulse parallel gravitational wave detection array considering timing optimization according to claim 1, characterized in that, In step 1, the parameters of the space gravitational wave detection formation include the formation's orbit and parameters determined according to mission requirements, as well as the parameters of the spacecraft within the formation. The formation's orbit and parameters include its semi-major axis, eccentricity, orbital inclination, right ascension of the ascending node, argument of perihelion, and true anomaly. The spacecraft parameters include the spacecraft's orbital radius. Pulse count .

3. The method for constructing a multi-pulse parallel gravitational wave detection array considering timing optimization according to claim 1, characterized in that, In step 2, the specific process of calculating the terminal state constraints of the multi-pulse parallel construction in the formation based on spacecraft parameters and the Clohessy-Wiltshire equations is as follows: Step 21: Derive the initial conditions for the circular configuration of the spacecraft based on the Clohessy-Wiltshire equations according to the spacecraft parameters. The initial condition formula is as follows: ； in, d Indicates the orbital radius of the spacecraft. α The phase angle of the spacecraft is represented; n is the orbital angular velocity of the virtual center of the formation; nd is the reference value of the linear velocity of the spacecraft during its orbital flight. Step 22: Calculate the terminal state constraints of the multi-pulse parallel construction in the formation based on the initial conditions of the spacecraft's circular configuration; in, t f This indicates the time when the parallel construction is completed.

4. The method for constructing a multi-pulse parallel gravitational wave detection array considering timing optimization according to claim 1, characterized in that, In step 3, the specific method for constructing a multi-pulse control mathematical model suitable for the initialization of a space gravitational wave detection formation is as follows: Step 31: Using the spatial circular formation conditions under the Clohessy-Wiltshire equations in spatial relative motion, calculate the initial relative positions and velocities of all satellites, which is the target state of the multi-pulse. Step 32: Based on the initial relative positions and velocities of all satellites, the control terms in the Clohessy-Wiltshire equations are approximated using pulses, resulting in the following multi-pulse control mathematical model: The multi-pulse control mathematical model is transformed into a multiplicative form as follows: ； in, Construct completion time for parallel formation t f The column vector of terminal target state variables of the spacecraft; For the spacecraft from the initial reference time t 0 to terminal completion time t f Position-velocity state transition matrix; For the spacecraft at the initial reference time t The initial relative state quantity column vector of 0; This represents the velocity increment of each pulse; For the spacecraft from the initial reference time t From 0 to any time t Position-velocity state transition matrix; N The total number of pulses applied during formation building; From arrive The pulse state transition matrix.

5. A method for constructing a multi-pulse parallel gravitational wave detection array considering timing optimization as described in claim 4, characterized in that, In step 32, the control term is expanded into a state-space model as follows: Among them, control items , in, This represents the components of the control quantity in each direction; n is the orbital angular velocity of the virtual center of the formation.

6. The method for constructing a multi-pulse parallel gravitational wave detection array considering timing optimization according to claim 5, characterized in that, Step 4 involves the following specific steps: Step 41: Perform a transposition transformation on the multi-pulse control mathematical model to obtain the transposed multi-pulse control mathematical model, as follows: Step 42: Based on the target state of the multi-pulse, use an optimization algorithm to find the optimal pulse application time in the multi-pulse control mathematical model after the term transformation; Step 43: Based on the target state of the multi-pulse combined with the optimal pulse application time, obtain the velocity increment under each pulse.

7. A method for constructing a multi-pulse parallel gravitational wave detection array considering timing optimization as described in claim 6, characterized in that, In step 43, the velocity increment expression for each pulse is as follows: in, ; ； ； in, These respectively represent spacecraft one, two, and three in the gravitational wave detection formation. The velocity increment of the next pulse.

8. A method for constructing a multi-pulse parallel gravitational wave detection array considering timing optimization as described in claim 1, characterized in that, In step 5, the state variables of each spacecraft at each moment during the formation building process are calculated based on the optimal application time of each pulse and each velocity increment in the multi-pulse control of formation building. The specific formulas are as follows: Where, x k0 y k0 z k0 Represents the three-dimensional relative position components of the k-th spacecraft at the initial moment (k=1,2,3); , , θ1, θ2, and θ3 represent the three-dimensional relative velocity components of spacecraft k at the initial moment (k=1,2,3); θ1, θ2, and θ3 represent the initial phase angles of spacecraft 1, 2, and 3, respectively.

9. A method for constructing a multi-pulse parallel gravitational wave detection array considering timing optimization according to claim 7, characterized in that, In step 5, the specific method for correcting the non-linearization error through differential correction is as follows: S1, given a set of initial velocity increments Through a dynamic nonlinear model Integrating, we obtain the actual pulse termination state: ； S2, calculate the error amount of the actual pulse termination state: ； S3, combining the error amount of the actual pulse terminal state, the correction value of the pulse velocity is determined in reverse through the state transition matrix: ； S4, based on the initial velocity increment and the pulse velocity correction value, the corrected pulse velocity is obtained as follows: ； S5, using the nonlinear model integration, and calculating the terminal state error, i.e., repeating S1-S2, if the corrected pulse velocity... This can make the error amount If the value is less than the given value, the iterative correction ends; otherwise, the correction performed in S3-S4 continues.

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