Translation point configuration initial value design method based on sequential optimization

Abstract

The invention discloses a translation point configuration initial value design method based on sequential optimization, and belongs to the technical field of aerospace. The implementation method comprises the steps of designing an orbit of a translation point detector, namely, based on a circular restrictive three-body model, obtaining a periodic or quasi-periodic orbit near a translation point through differential correction, and then obtaining an orbit near the translation point under a high-precision ephemeris by using a multiple targeting method. And designing the orbit of the earth phase offset detector, obtaining an optimized initial value according to the characteristics of the instantaneous earth phase offset orbit, and performing optimization by taking the minimum phase angle variation as an index and utilizing a differential evolution algorithm. And taking a configuration stability index as an optimization objective, introducing a relaxation factor and a relative arrival factor, establishing a multi-objective optimization problem, and converting the multi-objective optimization problem into a multi-objective arrival problem. And an expected reaching target is given, high-precision numerical optimization is carried out, a configuration initial value with relatively high stability is obtained, and related technical problems in the translation point configuration application field are solved.

Description

Technical Field

[0001] The invention relates to a method for designing initial values ​​of a translation point configuration based on sequential optimization, and belongs to the technical field of aerospace. Background Art

[0002] Interferometry is an important means of detecting gravitational waves, observing Earth-like planets, and conducting high-precision cosmic observations. The space gravitational wave detection mission proposed in recent years can be achieved by using the three-body system libration point deployment triangle configuration for interferometry. The libration point configuration consists of a detector located near the L2 point of the three-body system and two phase-biased orbit detectors with a certain lead and lag phase with the second object. This configuration has the advantages of long interferometric baseline length and wide detection frequency range. Configuration stability is an important constraint in the design of space interferometry configuration and has an important impact on the accuracy of interferometry. Configuration stability indicators include arm length change / ratio, arm length change rate, and breathing angle change. In order to achieve high-precision interferometry, the configuration composed of three detectors needs to remain stable for a long time. However, the deep space dynamic environment is complex and there are many disturbance factors, which brings great challenges to the stable configuration design of the libration point configuration. For the libration point configuration, it is necessary to combine the deep space multi-celestial body orbital dynamics, study the efficient optimization method of the libration point configuration, determine the configuration that meets the index constraints, and provide technical support for the implementation of future libration point configuration interferometry missions.

[0003] Therefore, under the above background, the present invention proposes a method for initial value design of translation point configuration based on sequential optimization, which realizes efficient initial value design and optimization of high-stability translation point configuration that is affected by complex perturbations for a long time. Summary of the invention

[0004] In order to solve the problem of divergence of interferometric measurement accuracy caused by unstable libration point configuration, the purpose of the present invention is to provide a method for designing initial values ​​of libration point configuration based on sequential optimization, which obtains initial values ​​of libration point configuration that meet stability requirements by successively designing the libration point orbit, the Earth phase offset orbit and the libration point configuration.

[0005] The objective of the present invention is achieved through the following technical solutions:

[0006] The present invention discloses a method for designing initial values ​​of a libration point configuration based on sequential optimization. The orbit of a libration point detector is designed, that is, based on a circular restricted three-body model, a periodic or quasi-periodic orbit near the libration point is obtained by differential correction, and then the orbit near the libration point under high-precision ephemeris is obtained by a multiple shooting method. The orbit of an earth phase bias detector is designed, and the optimized initial value is obtained by the instantaneous earth phase bias orbit characteristics. The optimization is performed using a differential evolution algorithm with the minimum phase angle change as an indicator. The configuration stability index is used as the optimization target, and a relaxation factor and a relative arrival factor are introduced to establish a multi-objective optimization problem, and the multi-objective optimization problem is converted into a multi-objective arrival problem. Given the expected arrival target, high-precision numerical optimization is performed to obtain a configuration initial value with strong stability, thereby solving technical problems related to the application field of the libration point configuration.

[0007] The present invention discloses a method for initial value design of a translation point configuration based on sequential optimization, comprising the following steps:

[0008] Step 1: Design the orbit of the libration point detector SC1. Take the first-order approximate analytical solution near the libration point under the circular restricted three-body model as the initial orbit value. Use differential correction to obtain the initial orbit value of the detector SC1 under the circular restricted three-body model. Then use numerical integration and multiple shooting method to obtain the orbit of the detector SC1 under high-precision ephemeris.

[0009] The first-order approximate analytical solution near the translation point in the circular restricted three-body model is:

[0010]

[0011] Where x, y and z are the coordinate components of spacecraft SC1 in the conjunction coordinate system, A x ,ω p and φ are the amplitude, frequency and initial phase angle of the in-plane motion, respectively. z ,ω v and ψ are the amplitude, frequency and initial phase angle of the out-of-plane motion, respectively; t is the dimensionless time; κ is the amplitude constant;

[0012] For different types of orbits, when t = 0, substituting into equation (1) we get

[0013]

[0014] Where x0 is the initial value of the orbit of the circular restricted three-body model detector SC1, x0, y0, z0, and are the position and velocity components of the initial value of the orbit x0 in the conjunction coordinate system;

[0015] Perform differential correction on x0, the formula is as follows

[0016]

[0017] where δx0 and δx f are the initial and final state deviations, δt f is the time deviation, is the terminal derivative, Φ ij Represents the i-th row and j-th column element of the state transfer matrix Φ.

[0018] Using equation (3) to perform differential correction on x0, we can obtain the initial value of the orbit of the detector SC1 under the circular restricted three-body model, which is denoted as right Numerical integration is performed, and then the multiple shooting method is used to obtain the orbit of the detector SC1 under high-precision ephemeris, completing the orbit design of the detector SC1.

[0019] Step 2: Design the orbits of the Earth phase offset detectors SC2 and SC3, that is, use the differential evolution algorithm to search for the initial values ​​of the orbits of the detectors SC2 and SC3 with the minimum phase angle change as the indicator, and obtain the orbits of the detectors SC2 and SC3 through numerical integration;

[0020] The initial orbital elements of detectors SC2 and SC3 are

[0021] E 20 =[a 20 ,e 20 ,i 20 ,Ω 20 ,ω 20 ,f 20 ] (4)

[0022] E 30 =[a 30 ,e 30 ,i 30 ,Ω 30 ,ω 30 ,f 30 ] (5)

[0023] Where E 20 and E 30 is the initial orbital element number of detectors SC2 and SC3, a 20 and a 30 are the semi-major axes of the orbits of detectors SC2 and SC3, e 20 and e 30 are the orbital eccentricities of the detectors SC2 and SC3, i 20 and i 30 are the orbital inclinations of detectors SC2 and SC3, Ω 20 and Ω 30 are the ascending nodes of the orbits of the detectors SC2 and SC3, ω20 and ω 30 are the orbital perihelion arguments of detectors SC2 and SC3, θ 20 and θ 30 are the true anomalies of the orbits of detectors SC2 and SC3 respectively.

[0024] For the initial moment, the orbital elements of the Earth are as follows

[0025] E e =[a e ,e e ,i e ,Ω e ,ω e ,f e ] (6)

[0026] Where E e is the initial orbital element number of the Earth, a e is the semi-major axis of the Earth's orbit, e e is the Earth's orbital eccentricity, i e is the Earth's orbital inclination, Ω e is the right ascension of the ascending node of the Earth's orbit, ω e is the argument of perihelion of the Earth’s orbit, f e The true anomaly of the Earth's orbit.

[0027] The initial phase angle between detector SC2 and the Earth is The initial phase angle between detector SC3 and the Earth is The orbital elements of detectors SC2 and SC3 are related to the orbital elements of the Earth as follows:

[0028]

[0029] Take the optimization variable as [a 20 ,e 20 ,i 20 ,Ω 20 ] and [a 30 ,e 30 ,i 30 ,Ω 30 ], the optimization index is

[0030]

[0031] in, and are the phase angles of detectors SC2 and SC3 in the mission cycle, and They are and The magnitude of the change.

[0032] The initial orbit values ​​E of detectors SC2 and SC3 are obtained by optimizing indicators (9) and (10) through differential evolution algorithm. 20 and E 30 , and then the orbits of detectors SC2 and SC3 are obtained through numerical integration, completing the design of the orbits of detectors SC2 and SC3.

[0033] Step 3: Based on the orbits of the probes SC1, SC2 and SC3 obtained in Steps 1 and 2, the multi-objective optimization problem is transformed into a multi-objective arrival problem with the minimum stability index of the libration point configuration as the optimization objective;

[0034] At the initial moment, the initial orbit value of the detector SC1 obtained in step 1 is

[0035]

[0036] in are the position components of the initial value of the orbit of detector SC1, are the velocity components of the initial value of the orbit of detector SC1,

[0037] At the initial moment, the initial orbit values ​​of detectors SC2 and SC3 obtained in step 2 are

[0038]

[0039] in are the position components of the initial value of the orbit of detector SC2, are the velocity components of the initial value of the orbit of detector SC2, are the position components of the initial value of the orbit of detector SC3, They are the velocity components of the initial value of the orbit of detector SC3 respectively.

[0040] The stability index of the libration point configuration is

[0041]

[0042] Where ΔL is the maximum arm length change ratio, Δθ is the maximum breathing angle change, is the maximum arm length change rate, Δl 12 (t) is the arm length l 12 (t) change ratio, Δl 13 (t) is the arm length l 13 (t), Δθ1(t) is the change ratio of the breathing angle θ1(t), The arm length l 12 The rate of change of (t), The arm length l 13 The rate of change of (t).

[0043] Using ΔL, Δθ and As the optimization target, a multi-objective optimization problem is established, namely

[0044]

[0045] It should be pointed out that the initial orbital value of the detector SC1 is particularly sensitive, so it is no longer optimized, but is involved in the calculation of the stability index of the libration point configuration.

[0046] The multi-objective optimization problem (15) is transformed into a multi-objective arrival problem, namely

[0047]

[0048] Among them, u is the decision variable, which is the initial value of the orbit of detectors SC2 and SC3 and γ is the relaxation factor, F i (u) is the i-th objective function (a total of k objective functions), w i is the relative arrival factor, F i * (u) is the expected value of the objective function, c(u) and ceq(u) are the nonlinear inequality constraints and equality constraints respectively, A and b are matrices related to linear inequality constraints, Aeq and beq are matrices related to linear equality constraints, lb and ub are the upper and lower bounds of the decision variable u respectively;

[0049] It should be pointed out that for the multi-target reaching problem, before solving it, a set of idealized expected targets F corresponding to the objective function is given first. i * (u), i = 1, ..., k, the relative arrival factor corresponding to each target is w i ,i=1,...,k, when F i * To ensure that the percentages of overflow or undershoot of the activity target are the same when all values ​​of (u) are non-zero, w i Set to abs[F i * (u)], and adjust w according to the overflow i .

[0050] According to equations (15) and (16), the multi-objective optimization problem is transformed into a multi-objective arrival problem.

[0051] Step 4: Based on the multi-objective arrival problem in step 3, the expected value of the stability index of the translation point configuration is set, and the final solution is obtained through iterative optimization. The final solution corresponds to the optimized initial value of the configuration, that is, the initial value design of the translation point configuration is realized based on sequential optimization.

[0052] Set the expected value of the maximum arm length change ratio ΔL to ΔL * , the expected value of the maximum breathing angle change Δθ is Δθ * , maximum arm length change rate The expected value of Substitute into formula (16) for high-precision numerical optimization;

[0053] In each iteration, the goal is to minimize the relaxation factor γ by adjusting the decision variable u and the relaxation factor γ so that the deviation of the objective function meets the conditions and calculate each objective function F i (u) and its corresponding expected target F i * (u) The deviation between

[0054] d i (u) = F i (u)-F i * (u) (17)

[0055] where d i (u) is the objective function F i (u) and the desired target F i * (u) difference.

[0056] Update the relaxation factor γ by the difference and weight of the objective function

[0057]

[0058] Gradient update decision variables, use optimization algorithm to update u, make γ as small as possible, for the decision variable u in step n n ,but

[0059] u n+1 =u n -α n ▽F(u n ) (19)

[0060] After each iteration, check whether the following constraints are satisfied

[0061] d i (u n+1 )≤γ·w i ,i=1,2,...,k (20)

[0062] If not satisfied, adjust the weight w i , and then continue to optimize, and obtain the final solution through iterative optimization. The final solution corresponds to the optimized initial value of the configuration, that is, the initial value design of the translation point configuration is realized based on sequential optimization.

[0063] The libration point configuration consists of three detectors, one of which is located near the L2 point of the three-body system, denoted as the libration point spacecraft SC1, and the other two are located in the Earth phase offset orbits that are ahead of and behind the predetermined phase of the Earth, denoted as the Earth phase offset spacecraft SC2 and SC3 respectively.

[0064] It also includes step five: the initial value of the iteratively optimized libration point configuration obtained in step four greatly enhances the stability of the configuration compared to that before optimization. Applying the optimized initial value of the libration point configuration to the high-precision interferometric measurement task, so that the detector enters the designed configuration orbit, can significantly improve the detection accuracy and efficiency of gravitational waves, terrestrial planets, cosmic observation objects, etc., and ensure the stability of the configuration during the mission cycle. In addition, by optimizing the initial value of the libration point configuration, it is ensured that the arm length change and the breathing angle change are within a controllable range, thereby maintaining the precise distance and angle between the detectors and ensuring the accuracy of data collection during the mission. At the same time, the optimized initial value enables the detector to maintain a stable configuration with lower energy consumption during the orbit maintenance phase, reducing the need for frequent orbit adjustment, reducing the possibility of configuration deviation in subsequent tasks, and ensuring the reliability of long-term measurement tasks. The above technical solution is used to solve technical problems related to the application field of the libration point configuration.

[0065] The technical problems related to the application field of the libration point configuration include configuration stability, data acquisition accuracy, and configuration orbit maintenance.

[0066] Beneficial effects:

[0067] 1. The present invention discloses a method for designing initial values ​​of the libration point configuration based on sequential optimization. For the orbit of the libration point detector, the differential correction method is used to obtain the periodic or quasi-periodic orbit near the libration point, and then the orbit under high-precision ephemeris is obtained by combining the multiple shooting method. The long-term stability of the libration point detector in the libration point configuration is significantly enhanced, and the problem of configuration instability under complex perturbation environments is effectively solved, ensuring that the configuration remains stable during the mission cycle, reducing the need for frequent orbit adjustment, and reducing the fuel consumption for orbit maintenance.

[0068] 2. The present invention discloses a method for designing initial values ​​of the configuration of the libration point based on sequential optimization. It adopts a decomposition optimization method, takes the minimization of phase angle change as the optimization index, uses a differential evolution algorithm to optimize the initial value of the Earth phase offset orbit, further optimizes the initial value of the configuration of the libration point, and improves the optimization efficiency and accuracy of the configuration. It can quickly and effectively search and optimize the initial value of the configuration, ensuring the efficiency and reliability of mission execution.

[0069] 3. The present invention discloses a method for designing initial values ​​of the configuration of a translation point based on sequential optimization, which takes the configuration stability index as the optimization target, introduces the relaxation factor and the relative arrival factor, establishes a multi-objective optimization problem, and transforms the multi-objective optimization problem into a multi-objective arrival problem, thereby realizing high-precision optimization of the initial values ​​of the configuration of the translation point. The optimized initial values ​​of the configuration can meet the strict requirements of complex tasks on configuration stability and accuracy, and are particularly suitable for tasks that require long-term configuration stability, ensuring the accuracy of data collection during the task.

[0070] 4. The present invention discloses a method for designing initial values ​​of the configuration of a translation point based on sequential optimization. In the established multi-objective optimization problem, the optimization index and constraint conditions can be adjusted according to different mission requirements, and then high-precision numerical optimization can be performed. The method has strong adaptability and can be widely used in various deep space missions, providing efficient and reliable initial value optimization solutions for different space interferometry missions, thereby improving mission adaptability and overall execution efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0071] Figure 1 It is a flow chart of a method for designing initial values ​​of a translation point configuration based on sequential optimization disclosed in the present invention;

[0072] Figure 2 A schematic diagram of a stability index of a translation point configuration of a solution embodiment of a translation point configuration initial value design method based on sequential optimization disclosed in the present invention;

[0073] Figure 3 The present invention discloses a method for initial value design of a translation point configuration based on sequential optimization to solve Example 1 for solving a stability index curve; wherein, Figure a is a curve showing a change in the proportion of arm length change during a mission, Figure b is a curve showing a change in the rate of arm length change during a mission, and Figure c is a curve showing a change in breathing angle change during a mission.

[0074] Figure 4 The present invention discloses a method for initial value design of a libration point configuration based on sequential optimization to solve the mission configuration of the libration point configuration within two years obtained in Example 1. DETAILED DESCRIPTION

[0075] In order to better illustrate the purpose and advantages of the present invention, the present invention is explained in detail below in conjunction with specific implementation examples. In the context of this embodiment, the present invention is applied to a high-precision interferometric measurement task based on the three-body system libration point configuration. Interferometric measurement is an important means of detecting gravitational waves, observing terrestrial planets, and conducting precise cosmic observations. In recent years, in space gravitational wave detection missions, it has been proposed to use an isosceles triangle configuration deployed with libration points to achieve interferometric measurement. This configuration consists of three detectors, one located near the three-body system libration point, and the other two detectors are located at positions that are ahead or behind the second body by a certain phase angle.

[0076] In order to ensure high-precision interferometric measurement, the configuration must remain stable for a long time, especially in the complex dynamic environment of deep space, where various perturbation factors may cause serious divergence of the configuration and affect the measurement accuracy. The present invention adopts a sequential optimization design method, firstly takes the minimum phase angle change as the optimization target, optimizes the initial value of the Earth phase offset orbit through a differential evolution algorithm, and then further takes the configuration stability index as the optimization target, performs numerical optimization of the multi-target arrival problem, and finally obtains a configuration initial value with extremely strong stability.

[0077] Specifically, after the initial values ​​are designed, through high-precision numerical simulation, the arm length change ratio, breathing angle change, and arm length change rate of the triangular configuration are maintained within the expected error range within the two-year mission cycle. The optimized initial values ​​of the configuration can effectively improve the configuration stability within the mission cycle and ensure the collection of high-precision interferometric measurement data. In addition, the optimized initial values ​​of the configuration can also reduce the possibility of configuration deviation in subsequent missions, ensuring the accuracy and reliability of long-term measurement missions.

[0078] Embodiment 1:

[0079] Taking the libration point configuration of the Sun-Earth three-body system as an example, the initial time is 2034 / 01 / 01 00:00:00 UTC, the Earth phase bias detector is about 8° ahead and behind the Earth, and the libration point detector is located near the L2 point of the Sun-Earth three-body system. x The Lyapunov orbit is about 3000km, the mission duration is 2 years, and the corresponding maximum arm length change ratio before initial value optimization reaches ΔL0=12.3280%, the maximum arm length change rate reaches Δθ0=1.0119, and the breathing angle change reaches The configuration is severely deformed and the stability is extremely poor, so high-precision interferometry cannot be guaranteed. The present invention provides a method for designing the initial value of the translation point configuration based on sequential optimization, such as Figure 1 As shown, the specific implementation steps are as follows:

[0080] Step 1: Design the orbit of the libration point detector SC1. Take the first-order approximate analytical solution near the libration point under the circular restricted three-body model as the initial orbit value. Use differential correction to obtain the initial orbit value of the detector SC1 under the circular restricted three-body model. Then use the multiple shooting method to obtain the orbit of the detector SC1 under high-precision ephemeris.

[0081] The first-order approximate analytical solution near the translation point in the circular restricted three-body model is:

[0082]

[0083] Where x, y and z are the coordinate components of spacecraft SC1 in the conjunction coordinate system, A x ,ωp and φ are the amplitude, frequency and initial phase angle of the in-plane motion, respectively. z , ψ and ω v are the amplitude, frequency and initial phase angle of the out-of-plane motion respectively; t is the dimensionless time; κ is the amplitude constant;

[0084] For the Sun-Earth circular restricted three-body model, the Lyapunov orbit z-direction amplitude A z =0, and take A x =3000km, t=0, ω p =2.0572, φ=0, κ=3.1875, substituting into formula (21) we get

[0085] x0=[1.010012955309541,0,0,0,1.315922837265305×10 -4 ,0] T (twenty two)

[0086] Perform differential correction on x0, the formula is as follows

[0087]

[0088] Through the differential correction of formula (23), the corrected value is

[0089]

[0090] By numerically integrating equation (24), we can get the amplitude A x =3000km Lyapunov orbit, and then use the multiple shooting method to obtain the orbit of the detector SC1 under high-precision ephemeris, completing the design of the orbit of the detector SC1.

[0091] Step 2: Design the orbits of the Earth phase offset detectors SC2 and SC3, that is, use the differential evolution algorithm to search for the initial values ​​of the orbits of the detectors SC2 and SC3 with the minimum phase angle change as the indicator, and obtain the orbits of the detectors SC2 and SC3 through numerical integration;

[0092] UTC 2034 / 01 / 01 00:00:00 is selected as the initial time, and the initial orbital elements of detectors SC2 and SC3 are

[0093] E 20 =[a 20 ,e 20 ,i 20 ,Ω 20 ,ω 20 ,f 20 ] (25)

[0094] E30 =[a 30 ,e 30 ,i 30 ,Ω 30 ,ω 30 ,f 30 ] (26)

[0095] Where E 20 and E 30 is the initial orbital element number of detectors SC2 and SC3, a 20 and a 30 are the semi-major axes of the orbits of detectors SC2 and SC3, e 20 and e 30 are the orbital eccentricities of the detectors SC2 and SC3, i 20 and i 30 are the orbital inclinations of detectors SC2 and SC3, Ω 20 and Ω 30 are the ascending nodes of the orbits of the detectors SC2 and SC3, ω 20 and ω 30 are the orbital perihelion arguments of detectors SC2 and SC3, θ 20 and θ 30 are the true anomalies of the orbits of detectors SC2 and SC3 respectively.

[0096] For the initial moment, the orbital elements of the Earth are as follows

[0097] E e =[a e ,e e ,i e ,Ω e ,ω e ,f e ] (27)

[0098] Where E e is the initial orbital element number of the Earth, a e is the semi-major axis of the Earth's orbit, e e is the Earth's orbital eccentricity, i e is the Earth's orbital inclination, Ω e is the right ascension of the ascending node of the Earth's orbit, ω e is the argument of perihelion of the Earth’s orbit, f e The true anomaly of the Earth's orbit.

[0099] The initial phase angle between detector SC2 and the Earth is The initial phase angle between detector SC3 and the Earth is The orbital elements of detectors SC2 and SC3 are related to the orbital elements of the Earth as follows:

[0100]

[0101] Pick According to equations (25) to (29), the initial values ​​of detectors SC2 and SC3 are shown in Table 1.

[0102] Table 1 Initial values ​​of detectors SC2 and SC3 (heliocentric ecliptic, 2034 / 01 / 01 00:00:00UTC)

[0103]

[0104] Take the optimization variable as [a 20 ,e 20 ,i 20 ,Ω 20 ] and [a 30 ,e 30 ,i 30 ,Ω 30 ], the optimization index is:

[0105]

[0106] in, and are the phase angles of detectors SC2 and SC3 in the mission cycle, and They are and The magnitude of the change.

[0107] By optimizing the indicators (30) and (31) through the differential evolution algorithm, the initial orbit values ​​E of the detectors SC2 and SC3 are obtained. 20 and E 30 , and then the orbits of detectors SC2 and SC3 are obtained through numerical integration, completing the design of the orbits of detectors SC2 and SC3.

[0108] Step 3: Based on the orbits of the probes SC1, SC2 and SC3 obtained in Steps 1 and 2, the multi-objective optimization problem is transformed into a multi-objective arrival problem with the minimum stability index of the libration point configuration as the optimization objective;

[0109] At the initial moment, the initial orbit value of the detector SC1 obtained in step 1 is

[0110]

[0111] in are the position components of the initial value of the orbit of detector SC1, are the velocity components of the initial value of the orbit of detector SC1,

[0112] At the initial moment, the initial orbit values ​​of detectors SC2 and SC3 obtained in step 2 are

[0113]

[0114] in are the position components of the initial value of the orbit of detector SC2, are the velocity components of the initial value of the orbit of detector SC2, are the position components of the initial value of the orbit of detector SC3, They are the velocity components of the initial value of the orbit of detector SC3 respectively.

[0115] The initial orbit values ​​of detectors SC1, SC2 and SC3 obtained through steps 1 and 2 are shown in Table 2.

[0116] Table 2 The initial orbit values ​​of detectors SC1, SC2 and SC3 obtained in step 1 and step 2 (heliocentric ecliptic system, 2034 / 01 / 01 00:00:00 UTC)

[0117]

[0118] The stability index of the libration point configuration is

[0119]

[0120] Where ΔL is the maximum arm length change ratio, Δθ is the maximum breathing angle change, is the maximum arm length change rate, Δl 12 (t) is the arm length l 12 (t) change ratio, Δl 13 (t) is the arm length l 13 (t), Δθ1(t) is the change ratio of the breathing angle θ1(t), The arm length l 12 The rate of change of (t), The arm length l 13 The rate of change of (t).

[0121] Using ΔL, Δθ and As the optimization target, a multi-objective optimization problem is established, namely

[0122]

[0123] It should be pointed out that the initial orbital value of the detector SC1 is particularly sensitive, so it is no longer optimized, but is involved in the calculation of the stability index of the libration point configuration.

[0124] The multi-objective optimization problem (15) is transformed into a multi-objective arrival problem, namely

[0125]

[0126] Among them, u is the decision variable, which is the initial value of the orbit of detectors SC2 and SC3 and γ is the relaxation factor, F i (u) is the i-th objective function (k objective functions), w i is the relative arrival factor, F i * (u) is the expected value of the objective function, c(u) and ceq(u) are the nonlinear inequality constraints and equality constraints respectively, A and b are matrices related to linear inequality constraints, Aeq and beq are matrices related to linear equality constraints, lb and ub are the upper and lower bounds of the decision variable u respectively;

[0127] It should be pointed out that for the multi-target reaching problem, before solving it, a set of idealized expected targets F corresponding to the objective function is given first. i * (u), i = 1, ..., k, the relative arrival factor corresponding to each target is w i ,i=1,...,k, when F i * To ensure that the percentages of overflow or undershoot of the activity target are the same when all values ​​of (u) are non-zero, w i Set to abs[F i * (u)], and adjust w according to the overflow i .

[0128] Therefore, it is established to transform the multi-objective optimization problem into a multi-objective arrival problem.

[0129] is the arm length change ratio corresponding to the waist of the translation point configuration, θ is the angle between the two waists, that is, the breathing angle of the configuration, Δθ is the change in the breathing angle, and is the arm length change rate.

[0130] Step 4: Based on the multi-objective reaching problem in step 3, set the expected value of the stability index of the libration point configuration, obtain the final solution through iterative optimization, and solve the initial value design problem of the libration point configuration;

[0131] Set the expected value of the maximum arm length change ratio ΔL to ΔL * , the expected value of the maximum breathing angle change Δθ is Δθ * , maximum arm length change rate The expected value of Substituting into formula (37) for high-precision numerical optimization, the desired configuration stability index is shown in Table 3.

[0132] Table 3 Expected values ​​of stability indicators

[0133]

[0134] In each iteration, the goal is to minimize the relaxation factor γ by adjusting the decision variable u and the relaxation factor γ so that the deviation of the objective function meets the conditions and calculate each objective function F i (u) and its corresponding expected target F i * (u) The deviation between

[0135] d i (u) = F i (u)-F i * (u) (38)

[0136] where d i (u) is the objective function F i (u) and the desired target F i * (u) difference.

[0137] Update the relaxation factor γ by the difference and weight of the objective function

[0138]

[0139] Gradient update decision variables, use optimization algorithm to update u, make γ as small as possible, for the decision variable u in step n n ,but

[0140] u n+1 =u n -α n ▽F(u n ) (40)

[0141] After each iteration, check whether the following constraints are satisfied

[0142] d i (u n+1 )≤γ·w i ,i=1,2,...,k (41)

[0143] If not satisfied, adjust the weight w i , and then continue to optimize. After iterative optimization, the final results are shown in Table 4.

[0144] Table 4 Optimization results of detectors SC2 and SC3 (heliocentric ecliptic system, 2034 / 01 / 01 00:00:00UTC)

[0145]

[0146] After the initial values ​​were designed, through high-precision numerical simulation, the arm length change ratio, breathing angle change and arm length change rate of the triangular configuration were maintained within the expected error range within the 2-year mission cycle. Figure 3 The stability index curve obtained by solving the initial value design method of the translation point configuration based on sequential optimization disclosed in the present invention is shown in Example 1, where the blue and red curves are different components of the stability index. The calculation results show that the maximum arm length change of the obtained configuration within 2 years is 2.8965%, the maximum breathing angle change is 0.045 degrees, and the maximum arm length change rate is 78.7084 m / s. Figure 4 The present invention discloses a method for designing the initial value of the configuration of the translation point based on sequential optimization, and solves the configuration of the translation point obtained in Example 1 within two years of the mission configuration. As can be seen from the figure, the optimized initial value of the configuration can effectively improve the stability of the configuration within the mission cycle and ensure the collection of high-precision interferometric measurement data. In addition, the optimized initial value of the configuration can also reduce the possibility of configuration deviation in subsequent missions, ensuring the accuracy and reliability of long-term measurement missions.

[0147] Table 5 Stability index optimization results

[0148]

[0149] The sequential optimization method in the present invention is not only applicable to the libration point configuration of the Sun-Earth system, but can also be widely used in space missions of other three-body systems, such as interferometric measurement missions in other planetary systems. By combining the design of configuration initial values ​​and orbit optimization, the present invention can provide stable configuration initial values ​​for different deep space exploration missions, ensuring that the configuration stability is significantly improved in a complex dynamic environment, thereby meeting the strict requirements for accuracy of long-term observation and measurement missions. In addition, the optimization method proposed in the present invention is highly flexible and can adjust the optimization indicators and constraints according to different mission requirements, such as optimizing key parameters such as the arm length change ratio and the breathing angle change rate, further improving the mission adaptability and optimization efficiency, and contributing to the realization of the three-body system space mission, and also providing higher reliability and accuracy guarantees for the implementation of interferometric measurement missions.

[0150] Finally, as the demand for interferometric measurement tasks continues to increase, the method for initial value design of the translation point configuration based on sequential optimization proposed in this invention can be applied in engineering practice as a standard method for related tasks, effectively solving the instability problem in configuration design, and helping to improve the accuracy and efficiency of interferometric measurement in the field of deep space exploration.

[0151] The above-mentioned specific description further describes the purpose, technical scheme and advantages of the invention in detail. It should be understood that the above description is only a specific implementation example of the present invention, which is used to explain the present invention and is not intended to limit the scope of protection of the present invention. Any modification, equivalent substitution, improvement, etc. made within the spirit and principle of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for initial value design of translation point configuration based on sequential optimization, characterized by: The following steps are included: Step 1: Design the orbit of the libration point detector SC1. Take the first-order approximate analytical solution near the libration point under the circular restricted three-body model as the initial orbit value. Use differential correction to obtain the initial orbit value of the detector SC1 under the circular restricted three-body model. Then use numerical integration and multiple shooting method to obtain the orbit of the detector SC1 under high-precision ephemeris. Step 2: Design the orbits of the Earth phase offset detectors SC2 and SC3, that is, use the differential evolution algorithm to search for the initial values ​​of the orbits of the detectors SC2 and SC3 with the minimum phase angle change as the indicator, and obtain the orbits of the detectors SC2 and SC3 through numerical integration; Step 3: Based on the orbits of the probes SC1, SC2 and SC3 obtained in Steps 1 and 2, the multi-objective optimization problem is transformed into a multi-objective arrival problem with the minimum stability index of the libration point configuration as the optimization objective; Step 4: Based on the multi-objective arrival problem in step 3, the expected value of the stability index of the translation point configuration is set, and the final solution is obtained through iterative optimization. The final solution corresponds to the optimized initial value of the configuration, that is, the initial value design of the translation point configuration is realized based on sequential optimization.

2. The method for initial value design of translation point configuration based on sequential optimization according to claim 1, characterized in that: The implementation method of step one is: The first-order approximate analytical solution near the translation point in the circular restricted three-body model is: Where x, y and z are the coordinate components of spacecraft SC1 in the conjunction coordinate system, A x ,ω p and φ are the amplitude, frequency and initial phase angle of the in-plane motion, respectively. z ,ω v and ψ are the amplitude, frequency and initial phase angle of the out-of-plane motion, respectively; t is the dimensionless time; κ is the amplitude constant; For different types of orbits, when t = 0, substituting into equation (1) we get Where x0 is the initial value of the orbit of the circular restricted three-body model detector SC1, x0, y0, z0, and are the position and velocity components of the initial value of the orbit x0 in the conjunction coordinate system; Perform differential correction on x0, the formula is as follows where δx0 and δx f are the initial and final state deviations, δt f is the time deviation, is the terminal derivative, Φ ij Represents the i-th row and j-th column element of the state transfer matrix Φ; Using equation (3) to perform differential correction on x0, we can obtain the initial value of the orbit of the detector SC1 under the circular restricted three-body model, which is denoted as right Numerical integration is performed, and then the multiple shooting method is used to obtain the orbit of the detector SC1 under high-precision ephemeris, completing the orbit design of the detector SC1.

3. The method for initial value design of translation point configuration based on sequential optimization according to claim 2, characterized in that: The implementation method of step 2 is: The initial orbital elements of detectors SC2 and SC3 are E 20 =[a 20 ,e 20 ,i 20 ,Oh 20 ,oh 20 ,f 20 ] (4) E 30 =[a 30 ,e 30 ,i 30 ,Oh 30 ,oh 30 ,f 30 ] (5) Where E 20 and E 30 is the initial orbital element number of detectors SC2 and SC3, a 20 and a 30 are the semi-major axes of the orbits of detectors SC2 and SC3, e 20 and e 30 are the orbital eccentricities of the detectors SC2 and SC3, i 20 and i 30 are the orbital inclinations of detectors SC2 and SC3, Ω 20 and Ω 30 are the ascending nodes of the orbits of the detectors SC2 and SC3, ω 20 and ω 30 are the orbital perihelion arguments of detectors SC2 and SC3, θ 20 and θ 30 are the true anomalies of the orbits of detectors SC2 and SC3 respectively; For the initial moment, the orbital elements of the Earth are as follows E e =[a e ,e e ,i e ,Oh e ,oh e ,f e ] (6) Where E e is the initial orbital element number of the Earth, a e is the semi-major axis of the Earth's orbit, e e is the Earth’s orbital eccentricity, i e is the Earth's orbital inclination, Ω e is the right ascension of the ascending node of the Earth's orbit, ω e is the Earth's orbit perihelion argument, f e The true anomaly of the Earth's orbit; The initial phase angle between detector SC2 and the Earth is The initial phase angle between detector SC3 and the Earth is The orbital elements of detectors SC2 and SC3 are related to the orbital elements of the Earth as follows: Take the optimization variable as [a 20 ,e 20 ,i 20 ,Ω 20 ] and [a 30 ,e 30 ,i 30 ,Ω 30 ], the optimization index is in, and are the phase angles of detectors SC2 and SC3 in the mission cycle, and They are and the magnitude of the change; By optimizing the indicators (9) and (10) through the differential evolution algorithm, the initial orbit values ​​E of the detectors SC2 and SC3 are obtained. 20 and E 30 , and then the orbits of detectors SC2 and SC3 are obtained through numerical integration, completing the design of the orbits of detectors SC2 and SC3.

4. The method for initial value design of translation point configuration based on sequential optimization according to claim 3, characterized in that: The implementation method of step three is: At the initial moment, the initial orbit value of the detector SC1 obtained in step 1 is in are the position components of the initial value of the orbit of detector SC1, are the velocity components of the initial value of the orbit of detector SC1, At the initial moment, the initial orbit values ​​of detectors SC2 and SC3 obtained in step 2 are in are the position components of the initial value of the orbit of detector SC2, are the velocity components of the initial value of the orbit of detector SC2, are the position components of the initial value of the orbit of detector SC3, They are the velocity components of the initial value of the orbit of detector SC3; The stability index of the libration point configuration is Where ΔL is the maximum arm length change ratio, Δθ is the maximum breathing angle change, is the maximum arm length change rate, Δl 12 (t) is the arm length l 12 (t) change ratio, Δl 13 (t) is the arm length l 13 (t), Δθ1(t) is the change ratio of the breathing angle θ1(t), l 12 (t) is the arm length l 12 The rate of change of (t), The arm length l 13 (t) rate of change; Using ΔL, Δθ and As the optimization target, a multi-objective optimization problem is established, namely The multi-objective optimization problem (15) is transformed into a multi-objective arrival problem, namely Among them, u is the decision variable, which is the initial value of the orbit of detectors SC2 and SC3 and γ is the relaxation factor, F i (u) is the i-th objective function, w i is the relative arrival factor, is the expected value of the objective function, c(u) and ceq(u) are the nonlinear inequality constraints and equality constraints respectively, A and b are matrices related to linear inequality constraints, Aeq and beq are matrices related to linear equality constraints, lb and ub are the upper and lower bounds of the decision variable u respectively; For the multi-target reaching problem, before solving it, a set of idealized expected targets corresponding to the objective function is given. The relative arrival factor corresponding to each target is w i ,i=1,...,k, when To ensure that the percentages of overflow or undershooting the active target are the same when all values ​​of i Set to And adjust w according to the overflow i ; According to equations (15) and (16), the multi-objective optimization problem is transformed into a multi-objective arrival problem.

5. The method for initial value design of translation point configuration based on sequential optimization according to claim 4, characterized in that: The implementation method of step 4 is: Set the expected value of the maximum arm length change ratio ΔL to ΔL * , the expected value of the maximum breathing angle change Δθ is Δθ * , maximum arm length change rate The expected value of Substitute into formula (16) for high-precision numerical optimization; In each iteration, the goal is to minimize the relaxation factor γ by adjusting the decision variable u and the relaxation factor γ so that the deviation of the objective function meets the conditions and calculate each objective function F i (u) and its corresponding expected goal The deviation between where d i (u) is the objective function F i (u) Difference from expected target; Update the relaxation factor γ by the difference and weight of the objective function Gradient update decision variables, use optimization algorithm to update u, make γ as small as possible, for the decision variable u in step n n ,but in n+1 =in n -α n ▽F(u n ) (19) After each iteration, check whether the following constraints are satisfied d i (u n+1 )≤γ·w i ,i=1,2,...,k (20) If not satisfied, adjust the weight w i , and then continue to optimize, and obtain the final solution through iterative optimization. The final solution corresponds to the optimized initial value of the configuration, that is, the initial value design of the translation point configuration is realized based on sequential optimization.

6. A method for initial value design of translation point configuration based on sequential optimization as claimed in claim 1, 2, 3, 4 or 5, characterized in that: The libration point configuration consists of three detectors, one of which is located near the L2 point of the three-body system, denoted as the libration point spacecraft SC1, and the other two are located in the Earth phase offset orbits that are ahead of and behind the predetermined phase of the Earth, denoted as the Earth phase offset spacecraft SC2 and SC3 respectively.

7. The method for initial value design of translation point configuration based on sequential optimization according to claim 6, characterized in that: It also includes step five, Apply the optimized initial value of the libration point configuration to high-precision interferometry tasks, so that the detector enters the designed configuration orbit, improves the detection accuracy and efficiency of gravitational waves, terrestrial planets, and cosmic observation objects, and ensures the configuration stability within the mission cycle; By optimizing the initial value of the translation point configuration, the arm length change and breathing angle change are ensured to be within the controllable range, the precise distance and angle between the detectors are maintained, and the data collection accuracy during the mission is ensured; The optimized initial values ​​enable the probe to maintain a stable configuration with lower energy consumption during the orbit maintenance phase, reduce the need for frequent orbit adjustments, reduce the possibility of configuration deviations in subsequent missions, and ensure the reliability of long-term measurement missions.

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