Method for rapidly solving shortest reconstruction time of cluster spacecraft

Abstract

The invention relates to a method for rapidly solving the shortest reconstruction time of a cluster spacecraft. The method comprises the following steps: constructing a time optimal control problem of a single spacecraft, converting the time optimal control problem of the single spacecraft into a double-layer optimization problem, and designing a time optimal cluster reconstruction problem based on the double-layer optimization problem; based on recursive evolution Latin hypercube design and polynomial adaptive sparse, improving a pre-constructed augmented radial basis model to obtain an improved augmented radial basis model; and solving a time optimal cluster reconstruction problem according to the improved augmented radial basis model to obtain predicted shortest reconstruction time. By adopting the method, the efficiency of quickly solving the shortest reconstruction time of the cluster spacecraft can be improved.

Description

Technical Field

[0001] The present application relates to the technical field of spacecrafts, and particularly to a method for quickly solving the shortest reconstruction time of a cluster of spacecrafts. Background Art

[0002] Cluster spacecrafts have significant advantages such as low cost, high efficiency, and strong robustness when performing space missions. Specifically, a cluster of spacecrafts consists of multiple micro and small spacecrafts. The structure and function of a single spacecraft are relatively simple, with low manufacturing and launch costs, and can be produced modularly and in batches. In addition, relying on the numerical advantage, the spacecrafts within the cluster can carry out serial and parallel pipeline operations through division of labor and cooperation, thus accelerating the mission process. Moreover, the spacecrafts within the system are structurally separated and information - shared, which can effectively reduce the risk of the overall mission failure caused by the failure of a certain spacecraft. Cluster formation reconstruction is the basis for cluster spacecrafts to perform space missions such as formation flight, collaborative observation, and collaborative control. By optimizing the flight trajectory, the mission capabilities can be enhanced, fuel consumption can be reduced, and the satellite lifespan can be extended. As a key indicator in the mission design of cluster spacecrafts, the shortest reconstruction time refers to the shortest flight time required for all spacecrafts within the cluster to simultaneously transfer from their respective initial states to the desired terminal states. In the mission planning under complex constraints, the quick determination of the shortest reconstruction time is related to the rationality of the index threshold setting and the feasibility of the mission plan. If the set mission time is less than the shortest reconstruction time of the cluster, there must be a situation where a certain spacecraft cannot complete the established transfer within the mission time. If there is a large margin between the set mission time and the shortest reconstruction time, the timeliness of the mission implementation will be significantly reduced. The optimal trajectory planning of the cluster time is the premise for quickly calculating the shortest reconstruction time. Compared with the trajectory planning of a single large spacecraft, the optimal trajectory planning of the cluster time involves many problems such as target orbit allocation, optimal trajectory planning, collision avoidance, and fuel consumption balance. As the number of spacecrafts within the cluster increases, the problem constraints and solution difficulties of the cluster trajectory planning increase significantly. Currently, a variety of trajectory planning methods for cluster reconstruction have been proposed successively.

[0003] Traditional cluster reconfiguration trajectory planning methods mainly include the pseudospectral method, artificial potential function method, convex optimization algorithm, and heuristic algorithm, etc., which have been widely used in the cluster reconfiguration problem. For example, some people use the Legendre pseudospectral method to transform the small-thrust satellite formation reconfiguration problem into a nonlinear programming problem and solve it to obtain the fuel-optimal reconfiguration trajectory. Some people, on the basis of considering terminal configuration constraints and collision avoidance constraints, transform the reconfiguration problem into an open-loop optimal control problem and use the Gaussian pseudospectral method to solve it. Some people propose a distributed planning method based on the high-precision adaptive pseudospectral method and apply it to large-scale cluster reconfiguration problems. The artificial potential field method constructs a gravitational field to guide cluster reconfiguration and uses a repulsive force field to achieve collision avoidance between spacecraft. Some people construct a virtual velocity controller based on the artificial potential function and introduce an auxiliary input weighted by an impact function to overcome the deficiency that the artificial potential field method is prone to falling into local minima. Some people use two potential functions to simulate collision avoidance constraints and formation configuration constraints respectively and propose a distributed control method that satisfies communication and collision avoidance constraints. Some people use the artificial potential field method based on velocity threshold for formation consensus control and prove the stability of the control. The biggest feature of the convex optimization algorithm is that the local minimum is equal to the global minimum. Some people propose using a rotating hyperplane constraint to replace the non-convex collision avoidance constraint for two-dimensional collision avoidance problems, and then some people extend it to three-dimensional space. Heuristic algorithms mainly include genetic, tabu search, and simulated annealing, etc. Some people use the solution of the genetic algorithm as the initial value of SQP, thus accelerating the iterative convergence speed of long-term cluster reconfiguration under the influence of J2 perturbation. Some people transform the formation reconfiguration problem into a convex optimization problem to quickly generate control laws and perform task allocation based on the genetic algorithm. Some people obtain the time-optimal control for the transfer of a fourth-order system between two arbitrary states through the optimal control synthesis method they proposed, and it can be applied to the optimal guidance of spacecraft autonomous approach maneuvers.

[0004] For the time-optimal cluster trajectory planning problem, the above traditional methods have their own advantages and disadvantages in terms of applicable scenarios, computational efficiency, optimality, etc. Specifically, direct methods represented by the pseudospectral method are simple to implement, do not require the derivation of complex optimal control conditions, and can handle complex constraints such as path constraints and terminal constraints, with strong applicability. However, its computational cost is large, high-dimensional problems are prone to falling into local optima, and the optimality of the solution depends on the discretization accuracy and initial value guess. Indirect methods can theoretically provide optimal solutions and satisfy necessary conditions, but their numerical stability is poor and often accompanied by complicated equation derivations, and their engineering practicability needs to be improved. For example, some people have solved the time-optimal formation problem around a slowly rotating asteroid well by indirect methods, but also pointed out in the conclusion that this method depends on second-order approximation and second-order gravitational fields, and inevitably produces inherent errors in actual mission scenarios. Convex optimization methods have high computational efficiency, can guarantee global optimality, and are suitable for real-time online planning. However, this method needs to sacrifice model accuracy for convexification and is difficult to directly handle strong nonlinear or non-convex constraints. Heuristic algorithms have strong global search capabilities, can avoid local optima, but the quality of the solutions is relatively random, and there is a lack of theoretical optimality guarantee. With the rapid development of artificial intelligence technology, more and more research has introduced the combination of surrogate models and traditional methods to solve trajectory planning problems. Some people calculate sub-optimal solutions of stochastic nonlinear optimal control problems under continuous-time chance constraints based on sequential convex programming of generalized polynomial chaos. This method enables the spacecraft system to perform motion planning under uncertain conditions. Some people address the multi-objective mission planning problem between planets and use a deep neural network (DNN) to predict the optimal transfer time and fuel consumption of low-thrust transfer trajectories. Some people consider the problems of optimal allocation, trajectory optimization, and collision avoidance, and study a fast planning method for the safe migration of cluster spacecraft along Halo orbits based on adaptive Gaussian process regression. The simulation results show that this method greatly improves the efficiency of the safe migration planning of the cluster. Some people propose a reentry guidance strategy based on long short-term memory networks, using the advantages of DNN in network mapping and real-time performance to generate bank angle commands in real time according to the motion state of the reentry vehicle, thus avoiding a large number of integral calculations. It should be noted that there are inevitably errors between the prediction results of the approximate model and the optimal control commands. If simply relying on the simple combination of traditional planning methods and approximate models, there is a risk of certain control failures. Even so, introducing an approximate model is still an effective way to improve the efficiency of the cluster trajectory planning problem. Therefore, the generalization ability of the approximate model adopted should be improved as much as possible to reduce prediction errors, and at the same time, effective strategies should be set to avoid the failure risks that may be caused by prediction errors. Summary of the Invention

[0005] Based on this, it is necessary to provide a fast solution method for the shortest reconstruction time of cluster spacecraft that can improve the solution efficiency of the shortest reconstruction time of the cluster in view of the above technical problems.

[0006] A fast solution method for the shortest reconstruction time of cluster spacecraft, the method comprising:

[0007] Construct a time-optimal control problem for a single spacecraft, transform the time-optimal control problem of the single spacecraft into a two-layer optimization problem, and design a time-optimal cluster reconstruction problem based on the two-layer optimization problem;

[0008] Improve the pre-constructed augmented radial basis model based on recursive evolutionary Latin hypercube design and polynomial adaptive sparsity to obtain an improved augmented radial basis model;

[0009] Solve the time-optimal cluster reconstruction problem according to the improved augmented radial basis model to obtain the predicted shortest reconstruction time.

[0010] For the above fast solution method for the shortest reconstruction time of cluster spacecraft, the present application proposes a two-layer optimization framework, constructs an optimization problem equivalent to the time-optimal control problem in the inner layer, derives the characteristics of the optimal control and the terminal error, and constructs the single-root optimization problem in the outer layer based on this, thereby transforming the original problem into an optimization problem and a root-finding problem that are easy to solve. At the same time, the fast construction of the improved augmented radial basis model is realized based on polynomial adaptive sparsity and recursive evolutionary experimental design. The recursive evolutionary LHD is used to dynamically generate an experimental design with good uniformity, and the polynomial terms in the augmented radial basis model are adaptively sparse based on cross-validation. The results of standard test cases verify the effectiveness of the proposed approximate model construction method. Finally, for the fast solution of the shortest reconstruction time of the cluster, a fast screening strategy based on the n-σ probability interval is proposed. Using the n-σ interval of the maximum predicted value relative error as a filter, the spacecraft with longer transfer times are screened out from the cluster, thereby reducing the number of calls to the precise optimization algorithm and greatly improving the calculation efficiency. Description of the Drawings

[0011] Figure 1 It is a schematic flow chart of a fast solution method for the shortest reconstruction time of cluster spacecraft in an embodiment;

[0012] Figure 2 It is a schematic diagram of coordinate system definition in an embodiment;

[0013] Figure 3 It is a schematic diagram of problem decoupling based on two-layer optimization in an embodiment;

[0014] Figure 4 It is a schematic diagram of recursive splitting of the sample space in another embodiment;

[0015] Figure 5 Schematic diagram of PIO operation in an embodiment

[0016] Figure 6 Example diagram of RELHD in an embodiment Figure 6 (a) Initial design schematic diagram Figure 6 (b) Schematic diagram after one round of expansion Figure 6 (c) Schematic diagram after two rounds of expansion

[0017] Figure 7 Box plot of RMSE results comparison in an embodiment Figure 7 (a) Schematic diagram of f1-10d Figure 7 (b) Schematic diagram of f1-20d Figure 7 (c) Schematic diagram of f2-10d Figure 7 (d) Schematic diagram of f2-20d Figure 7 (e) Schematic diagram of f3-10d Figure 7 (f) Schematic diagram of f3-20d

[0018] Figure 8 Schematic diagram of different solution strategies in an embodiment

[0019] Figure 9 Schematic diagram of the prediction interval of the shortest flight time in an embodiment

[0020] Figure 10 Schematic diagram of the model convergence process curve in an embodiment Figure 10 (a) Curve of the ARBF model convergence process Figure 10 (b) Curve of the DNN model convergence process

[0021] Figure 11 Schematic diagram of the ARBF test performance in an embodiment Figure 11 (a) Schematic diagram of the relative error distribution Figure 11 (b) Schematic diagram of the predicted value and the true value

[0022] Figure 12 Schematic diagram of the cluster reconstruction trajectory in an embodiment Figure 12 (a) Schematic diagram of Scenario 1 - two-dimensional perspective Figure 12 (b) Schematic diagram of Scenario 1 - three-dimensional perspective Figure 12 (c) Schematic diagram of Scenario 2 - two-dimensional perspective Figure 12 (d) Schematic diagram of Scenario 2 - three-dimensional perspective

[0023] Figure 13 Schematic diagram of the true value and the predicted value of the shortest reconstruction time in an embodiment Figure 13 (a) Schematic diagram of the true value and the predicted value of the shortest reconstruction time in Scenario 1Figure 13 (b) Schematic diagram of the true value and predicted value of the shortest reconstruction time in Scenario 2. Specific implementation manners

[0024] In order to make the objectives, technical solutions and advantages of this application clearer and more understandable, the following further details this application in combination with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain this application and are not used to limit this application.

[0025] In one embodiment, as Figure 1 shown, a method for quickly solving the shortest reconstruction time of a cluster spacecraft is provided, including the following steps:

[0026] Step 102, construct the time-optimal control problem of a single spacecraft, transform the time-optimal control problem of a single spacecraft into a two-layer optimization problem, and design the time-optimal cluster reconstruction problem based on the two-layer optimization problem.

[0027] The position vectors of the master spacecraft and the slave spacecraft in the Earth-centered inertial coordinate system O E -X E Y E Z E (Earth centered inertial, ECI) are respectively denoted as r c and r d , as Figure 2 shown. The origin of the ECI coordinate system is located at the Earth's center of mass O E , O E X E Y E plane coincides with the equatorial plane, the X E axis points to the mean equinox point, the Z E axis points to the North Pole, and the Y E axis is determined by the right-hand rule. Considering that the relative distances between the spacecrafts in the cluster are relatively close, the relative state is usually used to describe the motion of the spacecrafts. Select the master spacecraft as the reference point and construct an orbital coordinate system. The origin of the orbital coordinate system is fixedly connected to the center of mass of the master spacecraft, the x-axis coincides with the geocentric radius vector r c of the master spacecraft, the z-axis is perpendicular to the orbital plane of the master spacecraft and points to the direction of the angular momentum, and the y-axis is determined by the right-hand rule. When various perturbation forces and the orbital control force of the master spacecraft are not considered, let the relative position of the slave spacecraft in the orbital coordinate system be r = [x, y, z] T , and the relative velocity be Then the scalar form of the nonlinear relative motion equation is:

[0028]

[0029] In the formula: a x , a y , az They are the acceleration components of the thrust of the spacecraft in the directions of the three coordinate axes of the orbital coordinate system respectively. The instantaneous angular velocity ω and angular acceleration ε of the main spacecraft are respectively:

[0030]

[0031] where: e and f are the orbital eccentricity and instantaneous true anomaly of the main spacecraft respectively, and μ is the gravitational constant of the celestial body.

[0032] When the main spacecraft is located in a nearly circular orbit and the relative distance between the main and slave spacecraft is much smaller than the orbital radius of the main spacecraft, Equation (1) can be simplified to a linear differential equation with constant coefficients:

[0033]

[0034] where: is the average orbital angular velocity of the main spacecraft, and this equation is also known as the C-W equation.

[0035] When considering the active control of the slave spacecraft, Equation (3) can be transformed into:

[0036]

[0037] where, A and B are coefficient matrices, and u = [u x , u y , u z T are the relative state vector and control vector respectively.

[0038] The single spacecraft time-optimal trajectory planning problem can be described in the optimal control framework as: solving an optimal control function to minimize the flight time required for the spacecraft to transfer from the initial state to the desired terminal state, while satisfying the state equation constraint Equation (4), the boundary condition constraint Equation (6), and the control saturation constraint Equation (7):

[0039] x(t0) = x0, m(t0) = m0, x(t f ) = x f (6)

[0040] ||T||2 ≤ T max (7)

[0041] where: t0 and t f represent the initial and terminal times; x0 and x f are the initial relative state and the desired terminal relative state respectively; m0 is the spacecraft mass at the initial time. It is assumed that the magnitude of the engine thrust T is variable, and T max is the maximum thrust of the engine.

[0042] ​When continuous thrust is adopted, the change of the spacecraft mass follows:

[0043]

[0044] Where: I sp is the vacuum specific impulse of the engine, and g0 is the gravitational acceleration at sea level.

[0045] In summary, the time-optimal control problem of a single spacecraft can be denoted as:

[0046]

[0047] With the help of the Pontryagin maximum principle, it can be deduced that: except for individual discrete points, the optimal control thrust of Equation (9) is always the maximum thrust.

[0048] Taking the terminal boundary conditions in Equation (9) as the optimization objective, that is, taking the error between the actual and expected terminal states as the optimization objective, then the minimum terminal error problem for the specified flight time is:

[0049]

[0050] Where t f is the set flight time, let the shortest flight time be Then has the following characteristics:

[0051] Proposition (1): When the minimum value of the objective function is greater than 0.

[0052] When the minimum value of the objective function is equal to 0.

[0053] Proposition (2): When the minimum value of the objective function decreases with the increase of t f .

[0054] Combining the above propositions, it can be seen that the shortest flight time corresponds to the inflection point when J2 = 0. Since the gradient at the inflection point is zero and the optimal objective function value is always zero after this point, traditional root-finding algorithms such as Newton's method and bisection method are not applicable to this problem. Therefore, an index J3 as shown in Equation (11) is constructed:

[0055]

[0056] Then:

[0057] Proposition (3): When J3 > 0; when J3 = 0, when J3 < 0.

[0058] The index J3 has only a single root in the time domain, which is the shortest flight time. By constructing the index J3, the situation of iterative stagnation can be avoided, thereby reducing the search difficulty.

[0059] In summary, the time-optimal control problem in Equation (9) can be transformed into a two-layer optimization problem. Among them, the inner layer is the minimum terminal error problem with a given flight time, and the outer layer is the root-finding problem. The specific relationship is as Figure 3 shown.

[0060] Denote the shortest flight time required for the i-th spacecraft to transfer from the initial state to the desired state as Then the shortest reconstruction time of the cluster satisfies:

[0061]

[0062] In the formula: N s is the number of spacecraft in the cluster.

[0063] When designing the cluster mission, if the given mission time is less than the shortest reconstruction time of the cluster then there must be a spacecraft that cannot complete the relative state transfer within the mission time, resulting in mission failure. The time-optimal cluster reconstruction problem refers to solving a set of optimal controls such that the maximum value of the shortest flight times of each spacecraft is minimized, while satisfying state constraints, control saturation constraints, boundary conditions, etc., specifically as

[0064]

[0065] Step 104: Improve the pre-constructed augmented radial basis model based on recursive evolutionary Latin hypercube design and polynomial adaptive sparsity to obtain an improved augmented radial basis model.

[0066] The approximate model method can replace a complex model through a finite number of sample points, thereby further reducing the computational time consumption. Its main links include experimental design and model training, etc. To improve the generalization ability of the augmented radial basis model, based on recursive evolutionary experimental design, uniformly good training samples are dynamically provided for the model, and the training efficiency of the polynomial terms in the model is further improved through the polynomial adaptive sparsity algorithm. On the basis of transforming the single-spacecraft time-optimal control into a two-layer optimization problem, the dynamic and rapid construction of the improved augmented radial basis model is realized, and the trained model is used for the prediction of the shortest flight time of each spacecraft to further improve the solution efficiency of the shortest reconstruction time of the cluster, specifically including:

[0067] This application introduces the Recursive evolutionary Latin hypercube design (RELHD) method to conduct experimental design for generating samples. This method divides the sample set into multiple smaller similar subsets through a recursive splitting algorithm and updates and optimizes the existing design using a permutation information inheritance algorithm, enabling dynamic sampling while ensuring sample performance, including:

[0068] When the given number of design samples is N, N can be split into the sum of two adjacent or identical integers. Similarly, if the two resulting integers are still greater than 1, they can be recursively split into the sum of four smaller integers in the same way. Assuming that after k splits, 2 k subsets are obtained, and it can be proven that the maximum difference in the number of samples between subsets is 1, that is, the number of samples in each subset can be represented as p or p + 1. Taking the one-dimensional projection of 14 samples as an example, the recursive splitting of the sample space is as Figure 4 shown.

[0069] Any two columns P i ={x 1i ,x 2i ,...,x Ni} T and P j ={x 1j ,x 2j ,…,x Nj} T of the LHD design matrix are both integer permutations of X L ≤x≤X U , then any P j permutation can be obtained from P i through the transformation in Equation (14):

[0070] P j =T ij P i (14)

[0071] where T ij is a transformation matrix of size N×N. This matrix is an N-order diagonal matrix, uniquely determined by Equation (15):

[0072]

[0073] where diag(·) is the symbol for a diagonal matrix.

[0074] Through the above transformation, each column of the LHD can be associated with its i-th column. Setting i = 1, the entire design matrix can be obtained through a series of transformation matrices T i＝1 ={T 11 ,T 12 ,…,T1N} is associated with the first column of the design. Keeping the transformation matrix group T i＝1 unchanged, by performing column element exchanges on the first column of the design matrix, the update of the entire design can be achieved. This operation is named the Permutation inheritance operation (PIO). Taking a 5×4 design as an example, the PIO result is as Figure 5 shown. Setting p = 20, the expansion process of RELHD is as Figure 6 shown, where Figure 6 (a) is the initial design, Figure 6 (b) is the first round of expansion, Figure 6 (c) is the second round of expansion. It can be seen that this method can achieve the linear sequence expansion of samples while taking into account the spatial uniformity of the newly added samples and the existing samples.

[0075] Due to the different basic principles, basis function selections, and parameter training methods of different models, there are significant differences in the accuracy of the established approximation models. Therefore, in order to improve the generalization ability of the approximation model using different basis functions, a radial basis function (RBF) augmented by polynomial chaos expansions (PCE) is constructed as follows:

[0076]

[0077] where λ j and ω i are the coefficients of PCE and RBF respectively, and are the corresponding basis functions. Note that is the Gauss basis function as follows:

[0078]

[0079] In the formula: c i is the shape parameter of the i-th basis function.

[0080] The coefficients λ and ω are calculated by solving Equation (18):

[0081]

[0082] where and P are respectively and P ij the coefficient matrices composed of.

[0083] Due to the uneven spatial distribution of samples, selecting appropriate shape parameters based on sample density can effectively improve the accuracy of the meta-model. The local density of samples is used to characterize the sparsity of samples, which is specifically expressed as follows:

[0084]

[0085] where represents the influence of the sample on the local density. The influence volume of the training samples is normalized to obtain the reference kernel width of the shape parameter corresponding to each sample and dimension as:

[0086]

[0087] In the formula: V ij is the influence volume of the training samples.

[0088] The final kernel width is obtained by multiplying the scale factor λ j by as:

[0089]

[0090] Therefore, the anisotropic expression in Equation (17) can be constructed as:

[0091]

[0092] The final augmented radial basis function model improved by the anisotropic method is:

[0093]

[0094] Among them, the number of terms of PCE increases exponentially with the increase of the input variable dimension and the expansion order, resulting in the "curse of dimensionality". To improve the generalization ability of the polynomial terms in ARBF, this paper adaptively sparsifies the polynomial based on the cross-validation method. The best orthogonal polynomial basis is selected according to the cross-validation error, and the algorithm steps of the adaptive selection strategy are as follows.

[0095] Step 1. Determination of the initial basis and the initial order p0. Determine the function type of the initial basis according to the distribution type of the samples. Given a finite number of initial truncations, the leave-one-out (LOO) error corresponding to the truncation is obtained through the cross-validation method, and the truncation with the smallest LOO error is selected as the initial order p0.

[0096] Step 2. Calculate the PCE coefficients and the corresponding cross-validation error ε:

[0097]

[0098] Where: Var[y] is the response variance, and N is the total number of test samples. Generally, the smaller ε is, the better the prediction performance of the model.

[0099] Step 3. Compare ε with the preset value ε T . If ε < ε T , the iteration stops; otherwise, set p0 = p0 + 1 and return to Step 1.

[0100] Step 106. Solve the time-optimal cluster reconstruction problem according to the improved augmented radial basis model to obtain the predicted shortest reconstruction time.

[0101] After determining the improved augmented radial basis model, the process of solving the time-optimal cluster reconstruction problem according to the improved augmented radial basis model is the same as the prior art and will not be elaborated in this application.

[0102] In the above method for quickly solving the shortest reconstruction time of cluster spacecraft, in the above method for quickly solving the shortest reconstruction time of cluster spacecraft, in this application, by constructing the time-optimal control problem of a single spacecraft, the control variables and constraint conditions are clearly defined, and then it is transformed into a two-layer optimization problem, splitting the complex global problem into upper-layer macro decision-making and lower-layer micro execution. The upper layer can perform tasks such as target orbit allocation, and the lower layer makes trajectory planning for each spacecraft, making the solution more efficient and facilitating parallel computing to improve the overall operation speed. Finally, through recursive evolutionary Latin hypercube design and polynomial adaptive sparsity, the augmented radial basis model is effectively improved. Polynomial adaptive sparsity removes redundant terms by adaptively adjusting the model polynomials, simplifies the model structure, enhances the model generalization ability, avoids overfitting, reduces the complexity of the large-scale cluster reconstruction problem, and at the same time quickly screens the spacecraft that may be involved in the accurate calculation of the shortest reconstruction time according to the probability interval to further improve the calculation efficiency and reliability of the proposed method.

[0103] In one embodiment, the time-optimal control problem of a single spacecraft is constructed as:

[0104]

[0105] where, t f represents the terminal time, x0 and x f are the initial relative state and the desired terminal relative state respectively; m0 is the mass of the spacecraft at the initial time, T max is the maximum thrust of the engine, is the mass of the spacecraft, I sp is the vacuum specific impulse of the engine, g0 is the gravitational acceleration at sea level, A and B are coefficient matrices, and T represents the engine thrust.

[0106] In one embodiment, the inner layer of the two-layer optimization problem is the minimum terminal error problem for a given flight time, and the outer layer is the root-finding problem, including:

[0107] The two-layer optimization problem is

[0108]

[0109] where, t0 and t f represent the initial and terminal times, x0 and x f are the initial relative state and the desired terminal relative state respectively; m0 is the mass of the spacecraft at the initial time, T max is the maximum thrust of the engine, is the mass of the spacecraft, I sp is the vacuum specific impulse of the engine, g0 is the gravitational acceleration at sea level, A and B are coefficient matrices, and T represents the engine thrust.

[0110] In one embodiment, the time-optimal cluster reconfiguration problem designed based on the two-layer optimization problem is:

[0111]

[0112] where, is the shortest reconfiguration time of the cluster, is the shortest flight time required for the i-th spacecraft to transfer from the initial state to the desired state, A and B are coefficient matrices, t0 represents the initial time, x0 and x f are the initial relative state and the desired terminal relative state respectively; m0 is the mass of the spacecraft at the initial time, T max is the maximum thrust of the engine, is the mass of the spacecraft.

[0113] In one embodiment, the pre-constructed augmented radial basis model is improved based on the recursive evolutionary Latin hypercube design and polynomial adaptive sparsity to obtain the improved augmented radial basis model, including:

[0114] The design space is scaled to a d-dimensional unit hypercube based on the recursive evolutionary Latin hypercube method, and through RELHD sampling, the training set [x i , y i (i = 1, 2,..., n) is obtained by substituting it into the simulation model; the polynomial of the pre-constructed augmented radial basis model is adaptively sparsed, and the anisotropic shape parameters of the ARBF are estimated based on the local density and fast cross-validation method to construct an augmented radial basis approximation model. The cross-validation error threshold ε T is set to 0.01 to judge whether the model accuracy meets the requirements. If the accuracy is insufficient, the training set is expanded by RELHD until the model converges to obtain the improved augmented radial basis model.

[0115] In one of the embodiments, the pre - constructed augmented radial basis model is

[0116]

[0117] where λ j and ω i are the coefficients of PCE and RBF respectively, and are the corresponding basis functions, j and k are the basis function numbers of PCE and RBF respectively, m represents the order of PCE, and n represents the number of samples.

[0118] In one of the embodiments, the polynomial of the pre - constructed augmented radial basis model is adaptively sparsified, including:

[0119] Step 1: Determine the function type of the initial basis according to the distribution type of the samples. Given a finite number of primary truncations, obtain the LOO error corresponding to the truncation through cross - validation, and select the truncation with the minimum LOO error as the initial order p0;

[0120] Step 2: Calculate the PCE coefficients based on the polynomial form and truncation method determined by the initial order and calculate the corresponding cross - validation error ε;

[0121] Step 3: Compare ε with the preset value ε T , if ε < ε T , the iteration stops; otherwise, set p0 = p0 + 1 and return to Step 1.

[0122] In one of the embodiments, the corresponding cross - validation error is calculated as:

[0123]

[0124] where: Var[y] is the response variance, N is the total number of test samples, f(x) represents the prediction result of the current surrogate model, and f PCE (x) represents the prediction result of PCE.

[0125] In one of the embodiments, estimating the anisotropic shape parameter of ARBF based on local density and fast cross - validation method includes:

[0126] Estimating the anisotropic shape parameter of ARBF based on local density and fast cross - validation method as

[0127]

[0128] where c ij represents the kernel width, x ,j represents the current sample without considering local density, xi,j The current sample considering the local density is denoted as, and d represents the dimension of the design variable.

[0129] In one embodiment, the anisotropic shape parameters of the ARBF are estimated based on the local density and the fast cross-validation method, and an augmented radial basis approximation model is constructed, including:

[0130] The anisotropic shape parameters of the ARBF are estimated based on the local density and the fast cross-validation method, and the augmented radial basis approximation model is constructed as:

[0131]

[0132] where λ j and ω i are the coefficients of the PCE and RBF respectively, and are the corresponding basis functions, j and k are the serial numbers of the basis functions of the PCE and RBF respectively, m represents the order of the PCE, and n represents the number of samples.

[0133] In a specific embodiment, the detpep10exp, Griewangk, and Schwefel three standard test cases are selected to verify the effectiveness of the proposed method. The case information is shown in the table, and the dimensions of the test functions are 3, 5, and 10 respectively. The proposed method is compared with common methods such as RBF, Kriging model, PCE, and support vector regression (SVR). Among them, Kriging, PCE, and SVR can be constructed through the SURROGATES toolbox. The training samples are all generated by the RELHD method, and the number of samples is set to 10 times and 20 times the function dimension. The root mean square error (RMSE) shown in Equation (25) is used to evaluate the generalization ability of the model, and the number of test samples is set to 10 4 times the dimension, and the test is repeated 10 times under the same conditions.

[0134]

[0135] In the formula: x i is the i-th sample point, N is the number of test samples, y(x i ) is the true response of the sample x i , is the predicted output.

[0136] Table 1

[0137]

[0138] The test results of each algorithm are shown in Table 2 and Figure 7As shown, the specific average values of RMSE of each algorithm are given in Table 2, while Figure 7 the effects of each algorithm are more intuitively displayed and compared in the form of box plots. Among them, Figure 7 (a) is a schematic diagram of f1-10d, Figure 7 (b) is a schematic diagram of f1-20d, Figure 7 (c) is a schematic diagram of f2-10d, Figure 7 (d) is a schematic diagram of f2-20d, Figure 7 (e) is a schematic diagram of f3-10d, Figure 7 (f) is a schematic diagram of f3-20d; from Table 2 and Figure 7 it can be seen that the method proposed in this application has obvious advantages in terms of prediction accuracy and robustness, and is more competitive as the dimension and complexity of the test function increase.

[0139] Table 2

[0140]

[0141] To solve the problem of the shortest reconstruction time of the cluster in Equation (13), this application gives three different solution strategies as Figure 8 shown. Among them, the most direct idea is to traverse the shortest flight time of each spacecraft in the cluster and take it as the shortest reconstruction time of the cluster However, when the cluster scale is large, the computational efficiency of the traversal strategy will decrease sharply. Another relatively faster strategy is to predict the shortest flight time of all spacecraft based on an approximate model, and then use the root-finding algorithm to calculate the true shortest flight time of the spacecraft with the maximum predicted value. Due to the existence of certain prediction errors in the approximate model, there is a risk that the full approximation strategy cannot find the true value. This paper proposes a fast screening strategy based on the n-σ interval as a compromise between the above two strategies. First, use the trained ARBF to predict the shortest flight time of all spacecraft, then use the n-σ interval of the maximum prediction error as a filter to screen out the spacecraft with longer transfer times from the cluster, and finally call the root-finding algorithm to solve the exact values of the shortest flight times required by these spacecraft.

[0142] The schematic of the fast screening strategy is as Figure 9 shown, where the stars represent the predicted values of the shortest flight times of each spacecraft. In the figure, the predicted value of the shortest flight time of spacecraft No. 1 is the largest, which is represented by a hexagram. Only the interval of spacecraft No. 3 coincides with that of No. 1 among all spacecraft, so only the true values of the shortest flight times of spacecraft No. 3 and No. 1 need to be calculated and then the solution of the shortest reconstruction time can be completed.

[0143] Select a sequence composed of the initial and terminal relative states as the input, where the relative motion space range is [-5 km, 5 km], and the initial and terminal relative velocity ranges are [-2 m / s, 2 m / s]. Based on RELHD, dynamic sampling is performed within the input space. The number of augmented samples per round is set to 50, and the minimum transfer time of each spacecraft sample is solved through the optimization framework as the model output. Train the ARBF model according to the corresponding input-output results. The model convergence process is as shown in Figure 10 (a). It can be seen from the figure that the model meets the convergence condition when the number of samples reaches 400. To further ensure convergence, the number of samples is extended to 500 and the algorithm is terminated. The trained ARBF can directly predict the minimum transfer time of a single spacecraft based on the initial and terminal states.

[0144] To further verify the effectiveness of the proposed method, a DNN model is introduced for comparison. The above 500 samples are divided into a training set, a validation set, and a test set according to the ratio of 8:1:1, and the DNN is trained according to the parameters shown in Table 3.

[0145] Table 3Table 3DNN training parameter settings

[0146]

[0147] The DNN parameter update process is as follows:

[0148] 1) Initialize the network weights and bias parameters according to the dataset, and set the learning rate;

[0149] 2) Calculate the error of the network output using forward propagation;

[0150] 3) Calculate the gradient of the loss function using error backpropagation;

[0151] 4) Update the weights and biases according to the gradient calculated in step 3;

[0152] 5) Repeat steps 2-4 until a sufficient number of iterations are reached, or the loss function meets the preset conditions.

[0153] Since only 500 samples are provided, DNN is mostly difficult to converge during training. Therefore, random sampling with a sample number of 5×10^4 is performed in the input space, and the DNN is trained in the same way as before. The training processes of DNNs with different scales are as shown in Figure 10 (b), where the label "7×50" indicates that the number of hidden layers and neurons in the hidden layer of the DNN are 7 and 50 respectively. This neural network is denoted as DNN 7×50As can be seen from the figure, the RMSE of different networks during the training process decreases rapidly as the number of training steps increases and finally fluctuates in the order of 10 -2 magnitude.

[0154] The performance of the trained DNN and ARBF on the test set is shown in Table 4. The relative error between the actual value and the predicted value of the shortest flight time approximately follows a Gaussian distribution, and the mean of the relative error is close to zero. The mean and standard deviation of the relative error of ARBF on the test set are 0.14% and 2.33% respectively, slightly higher than those of each DNN model. The 3σ interval of the relative error of ARBF is [-6.85%, 7.13%], indicating that the probability that the relative error between the predicted value and the true value is within this interval is 99.74%. The performance of ARBF in the test set is as Figure 11 shown. As can be seen from Figure 11 (a), most of the prediction errors are within 5%. Figure 11 In (b), the predicted shortest flight time can better fit the actual shortest flight time, where the blue dots and the red line represent the predicted value and the reference line (y = x) respectively. To sum up, the proposed ARBF can converge within the order of magnitude of 10 2 for the shortest time prediction problem, and the required samples are significantly less than the 10 4 order of magnitude required by common DNN methods. The prediction accuracy of ARBF is slightly lower than that of the fully converged DNN, but the prediction accuracy is fully sufficient to meet the requirements of subsequent rapid screening. And because ARBF has a fast convergence speed and less sample consumption, it can be more flexibly applied to the rapid prediction task of the shortest reconstruction time.

[0155] Table 4

[0156]

[0157] To further verify the effectiveness of the method proposed in this application, two sets of problems for solving the cluster reconstruction time of 8 spacecraft are designed as simulation examples. In the two scenarios, the initial relative velocity of each spacecraft is zero, and the initial relative positions are shown in Equations (26) and (27) respectively. The terminal relative positions are equally phased and distributed on the natural elliptical orbiting path with b = 1500m, c = 0m, y c = 0m, ψ = π. Among them, b is the semi-minor axis of the ellipse, c is the maximum offset in the z-axis direction, y c is the offset of the ellipse center in the y-axis direction, and ψ represents the initial phase angle of the out-of-plane simple harmonic motion.

[0158] Scenario 1:

[0159]

[0160] Scenario 2:

[0161]

[0162] Table 5

[0163]

[0164] Select a sequence composed of the initial and terminal relative states as the input, where the relative motion space range is [-5 km, 5 km], and the initial and terminal relative velocity ranges are within [-2 m / s, 2 m / s]. Based on RELHD, dynamic sampling is performed within the input space, and the minimum transfer time of each spacecraft sample is solved through the Newton iteration method and the optimization framework as the model output. The ARBF model is trained according to the corresponding input-output results. The trained ARBF can directly predict the minimum transfer time of a single spacecraft based on the initial and terminal states. Finally, taking the 3σ interval of the maximum prediction error value as the filter, spacecraft with longer transfer times are screened out from the cluster based on the strategy. The root-finding algorithm is called to solve the exact value of the shortest flight time required by these spacecraft, and the maximum value among the obtained exact flight time values is the shortest reconstruction time of the cluster.

[0165] The transfer trajectories of the cluster spacecraft are as Figure 12 shown, Figure 12 (a) is a schematic diagram of Scenario 1 - two-dimensional perspective, Figure 12 (b) is a schematic diagram of Scenario 1 - three-dimensional perspective, Figure 12 (c) is a schematic diagram of Scenario 2 - two-dimensional perspective, Figure 12 (d) is a schematic diagram of Scenario 2 - three-dimensional perspective. In different scenarios, 8 spacecraft in the cluster can all generate transfer trajectories that meet the constraints. The calculation results of different strategies are shown in Table 5. Taking the results of the traversal strategy as the benchmark, in Scenario 1, the predicted results of the shortest reconstruction time of the first four strategies are consistent, while the results of the full approximation strategy are significantly different from other strategies. This is mainly because the full approximation strategy only gives the boundary conditions of the spacecraft with the maximum predicted value based on the approximate model, and there is a certain risk of failure. In Scenario 2, the shortest reconstruction times of the cluster obtained by all strategies are similar, indicating that the spacecraft with the longest flight time have been screened out.

[0166] In the two scenarios, the comparison of the true values and predicted values of the shortest transfer time of each spacecraft and the corresponding probability intervals is as Figure 13 shown, Figure 13 (a) Scenario 1 is a schematic diagram of the true value and predicted value of the shortest reconstruction time, Figure 13(b) Schematic diagram of the true and predicted values of the shortest reconstruction time for Scenario 2. Since there are differences in the number of spacecraft screened in different probability intervals, the corresponding calculation times are different. For example, in Scenario 1, the numbers of spacecraft screened by ARBF(3σ), ARBF(2σ), and ARBF(1σ) are 3, 2, and 1 respectively, so the corresponding calculation times are 29.43 s, 21.22 s, and 11.66 s respectively. In Scenario 2, the numbers of spacecraft screened by ARBF(3σ), ARBF(2σ), and ARBF(1σ) are 2, 1, and 1 respectively, and only Spacecraft 2 is screened in the latter two intervals. Therefore, the calculation times of ARBF(2σ) and ARBF(1σ) are the same, which is 11.88 s, and the calculation time of ARBF(3σ) is more, which is 20.14 s. In summary, the proposed n-σ fast screening strategy can not only ensure the reliability of the results, but also has a significantly better calculation efficiency than the traversal strategy.

[0167] It should be understood that although Figure 1 the steps in the flowchart of Figure 1 are shown in sequence according to the indication of the arrows, these steps are not necessarily executed in the order indicated by the arrows. Unless there is a clear indication in this article, there is no strict order restriction for the execution of these steps, and these steps can be executed in other orders. Moreover,

[0168] The technical features of the above embodiments can be combined arbitrarily. For the sake of concise description, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, it should be considered as the scope recorded in this specification.

[0169] The above-described embodiments only represent several implementation manners of the present application, and their descriptions are relatively specific and detailed, but they should not be construed as limiting the scope of the invention. It should be noted that for those of ordinary skill in the art, without departing from the concept of the present application, several deformations and improvements can still be made, and these all belong to the protection scope of the present application. Therefore, the protection scope of the present application should be subject to the appended claims.

Claims

1. A method for quickly solving the shortest reconstruction time of a cluster spacecraft, characterized in that The method includes: Construct the time-optimal control problem of a single spacecraft, transform the time-optimal control problem of a single spacecraft into a two-layer optimization problem, and design the time-optimal cluster reconfiguration problem based on the two-layer optimization problem; Improve the pre-constructed augmented radial basis model based on recursive evolutionary Latin hypercube design and polynomial adaptive sparsity to obtain an improved augmented radial basis model; Solve the time-optimal cluster reconfiguration problem according to the improved augmented radial basis model to obtain the predicted shortest reconfiguration time.

2. The method according to claim 1, wherein Construct the time-optimal control problem of a single spacecraft as: where, t f represents the terminal time, x0 and x f are the initial relative state and the desired terminal relative state respectively; m0 is the mass of the spacecraft at the initial time, T max is the maximum thrust of the engine, is the mass of the spacecraft, I sp is the vacuum specific impulse of the engine, g0 is the gravitational acceleration at sea level, A and B are coefficient matrices, and T represents the engine thrust.

3. The method according to claim 1, wherein The inner layer of the two-layer optimization problem is the minimum terminal error problem given the flight time, and the outer layer is the root-finding problem, including: The two-layer optimization problem is where, t0 and t f represent the initial and terminal times, x0 and x f are the initial relative state and the desired terminal relative state respectively; m0 is the mass of the spacecraft at the initial time, T max is the maximum thrust of the engine, is the mass of the spacecraft, I sp is the vacuum specific impulse of the engine, g0 is the gravitational acceleration at sea level, A and B are coefficient matrices, and T represents the engine thrust.

4. The method according to claim 1, wherein Design the time-optimal cluster reconfiguration problem based on the two-layer optimization problem as: Among them, is the shortest reconstruction time of the cluster, is the shortest flight time required for the i-th spacecraft to transfer from the initial state to the desired state. A and B are coefficient matrices, t0 represents the initial time, and x0 and x f are the initial relative state and the desired terminal relative state respectively; m0 is the mass of the spacecraft at the initial time, and T max is the maximum thrust of the engine, is the mass of the spacecraft.

5. The method according to claim 1, wherein Improve the pre-constructed augmented radial basis model based on recursive evolutionary Latin hypercube design and polynomial adaptive sparsity to obtain an improved augmented radial basis model, including: The design space is scaled to a d-dimensional unit hypercube based on the recursive evolutionary Latin hypercube method. Through RELHD sampling, it is substituted into the simulation model to obtain the training set [x i , y i (i = 1, 2,..., n); adaptively sparse the polynomial of the pre-constructed augmented radial basis model, estimate the anisotropic shape parameters of the ARBF based on the local density and fast cross-validation method, construct the augmented radial basis approximation model, and set the cross-validation error threshold ε T to 0.01, judge whether the model accuracy meets the requirements. If the accuracy is insufficient, expand the training set through RELHD until the model converges to obtain the improved augmented radial basis model.

6. The method according to claim 1, wherein The pre-constructed augmented radial basis model is where λ j and ω i are the coefficients of PCE and RBF respectively, and are the corresponding basis functions, j and k are the basis function numbers of PCE and RBF respectively, m represents the order of PCE, and n represents the number of samples.

7. The method according to claim 5, characterized in that, Perform adaptive sparsity on the polynomials of the pre-constructed augmented radial basis model, including: Step 1: Determine the function type of the initial basis according to the distribution type of the samples, given a finite number of primary truncations, obtain the LOO error corresponding to the truncation through cross-validation, and select the truncation with the smallest LOO error as the initial order p0; Step 2: Calculate the PCE coefficients and the corresponding cross-validation error ε based on the polynomial form and truncation method determined by the initial order; Step 3: Compare ε with a preset value ε T , if ε < ε T , the iteration stops; otherwise, set p0 = p0 + 1 and return to Step 1.

8. The method according to claim 6, wherein Calculate the corresponding cross-validation error as: Where: Var[y] is the response variance, N is the total number of experimental samples, f(x) represents the prediction result of the current surrogate model, and f PCE (x) represents the prediction result of PCE.

9. The method according to claim 1, characterized in that, Estimate the anisotropic shape parameters of the ARBF based on local density and fast cross-validation method, including: Estimate the anisotropic shape parameters of the ARBF based on local density and fast cross-validation method as Among them, c ij represents the core width, x ,j represents the current sample without considering the local density, x i,j represents the current sample considering the local density, and d represents the dimension of the design variable.

10. The method according to claim 1, characterized in that, Estimate the anisotropic shape parameters of the ARBF based on local density and fast cross-validation method, and construct an augmented radial basis approximation model, including: Estimate the anisotropic shape parameters of the ARBF based on local density and fast cross-validation method, and construct an augmented radial basis approximation model as: Among them, λ j and w k are the coefficients of PCE and RBF respectively, and are the corresponding basis functions, j and k are the basis function numbers of PCE and RBF respectively, m represents the order of PCE, and n represents the number of samples.

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