A fast estimation method of fuel consumption for low-thrust orbit transfer

Abstract

The application provides a small-thrust orbit transfer fuel consumption rapid estimation method, and belongs to the technical field of space navigation control. The method comprises the following steps: linearly transforming starting orbit elements and target orbit elements in small-thrust orbit transfer to GEO (Geostationary Earth Orbit) orbits adopting classical orbit elements; calculating the orbit inclination difference, the ascending node right ascension difference and the phase difference of the starting orbit and the target orbit based on the transformed starting orbit elements and the target orbit elements; combining the three parameters and the maximum spacecraft thrust acceleration to form orbit characteristics; normalizing each parameter in the orbit characteristics respectively; inputting the normalized orbit characteristics into a preset fuel consumption estimator model; outputting an estimated value of the normalized fuel consumption mass from the model; and performing inverse normalization on the estimated value to obtain an estimation result of the small-thrust orbit transfer fuel consumption. The application can quickly obtain the fuel consumption under time-optimal transfer, and has high accuracy and time efficiency.

Description

Technical Field

[0001] This invention belongs to the field of aerospace navigation and control technology, and specifically relates to a rapid estimation method for fuel consumption during low-thrust orbital transfer. Background Technology

[0002] Currently, the planning of GEO orbit "many-to-many" satellite in-orbit refueling missions primarily employs heuristic search algorithms, with the optimization objective of minimizing fuel consumption. There are two main orbit transfer thrust methods: traditional chemical pulse transfer and low-thrust transfer. However, when using the low-thrust orbit transfer strategy, the numerical optimization methods employed to calculate the entire low-thrust transfer orbit are extremely time-consuming, limiting the search space and speed of subsequent heuristic search algorithms. Therefore, rapid estimation of fuel consumption for low-thrust GEO orbit transfers is crucial for the subsequent rapid solution of mission planning.

[0003] Existing research often trains estimators based on a narrow initial and final orbit range. Once the propulsion parameters or the initial and final orbit range change, mission designers must regenerate the training dataset based on the new parameters and retrain the estimator, resulting in excessive time consumption. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings of existing technologies and propose a fast estimation method for fuel consumption in low-thrust orbital transfers. This invention can quickly obtain fuel consumption under time-optimal transfer conditions, with high accuracy and time efficiency. It can be applied to fitness calculations in heuristic search algorithms to increase population size, expand the search space, and accelerate the search speed.

[0005] This invention proposes a rapid estimation method for fuel consumption during low-thrust orbital transfer, comprising:

[0006] The initial orbital elements and the target orbital elements in the low-thrust orbit transfer are linearly transformed to GEO orbits using classical orbital elements;

[0007] Based on the transformed initial orbital elements and target orbital elements, the orbital inclination difference, ascending node right ascension difference, and phase difference between the initial orbit and the target orbit are calculated. The orbital inclination difference, ascending node right ascension difference, phase difference, and spacecraft maximum thrust acceleration are combined to form orbital characteristics.

[0008] Each parameter in the orbital features is normalized, and then the normalized orbital features are input into a preset fuel consumption estimator model. The model outputs a normalized estimate of fuel consumption mass.

[0009] The estimated value is inversely normalized to obtain the estimated fuel consumption for low-thrust orbital transfer.

[0010] In one specific embodiment of the present invention, the fuel consumption estimator model includes: 8 fully connected layers and 2 batch normalization layers; wherein, the first to third fully connected layers are connected sequentially, the third fully connected layer is followed by the first batch normalization layer, the first batch normalization layer is followed by the fourth fully connected layer, the fourth fully connected layer is followed by the second batch normalization layer, and the second batch normalization layer is followed by the fifth to eighth fully connected layers sequentially; each fully connected layer is followed by the ReLU activation function.

[0011] In one specific embodiment of the present invention, before inputting the normalized orbital features into a preset fuel consumption estimator model, the method further includes:

[0012] Train the fuel consumption estimator model;

[0013] Training the fuel consumption estimator model includes:

[0014] 1) Determine the value range of each parameter in the orbital features, and randomly and uniformly select values ​​from the value range of each parameter to construct multiple sets of orbital feature samples;

[0015] 2) Generate the target orbital roots for each orbital feature sample and convert them into the corresponding vernal equinox orbital roots;

[0016] 3) Based on the number of orbital elements at the vernal equinox for each orbital feature sample, calculate the corresponding fuel consumption as the label for the orbital feature sample, and combine each orbital feature sample and its corresponding label into a training set;

[0017] 4) Train the fuel consumption estimator model using the training set.

[0018] In one specific embodiment of the present invention, the orbital elements of the vernal equinox are denoted as x = [pfghk L]. T ,in:

[0019]

[0020] Where, [ae iωΩθ] T Let be the target orbital elements, a represent the semi-major axis of the orbit, e represent the orbital eccentricity, i represent the orbital inclination, w represent the argument of perigee, Ω represent the right ascension of the ascending node, and θ represent the true anomaly.

[0021] In one specific embodiment of the present invention, calculating the corresponding fuel consumption as a label for the orbital feature sample based on the vernal equinox orbital elements of each orbital feature sample includes:

[0022] 1) Construct the orbital dynamics equations for the spacecraft's motion:

[0023]

[0024] in, The time derivative of the orbital elements at the vernal equinox. Fuel consumption rate;

[0025] T max It is the maximum thrust amplitude with small thrust, u is the ratio of the actual thrust amplitude to the maximum thrust amplitude, 0≤u≤1; f=[f r ,f θ ,f h ] T It is a unit vector in the direction of thrust, g0 is the average gravitational acceleration at sea level, and I sp It is the specific impulse of the thruster;

[0026] The expressions for the perturbation influence matrix B(x) and the central gravitational influence matrix D(x) are as follows:

[0027]

[0028] Where W = 1 + fcosL + gsinL, S = 1 + h 2 +k 2 μ is the Earth's gravitational constant;

[0029] 2) Based on the results of step 1), construct the optimal fuel problem:

[0030]

[0031] Where t0 is the transfer start time, t f λ is the termination time of the transition; λ0 is a positive multiplier;

[0032] in, Δθ is the difference in longitude between the initial and final positions;

[0033] True longitude difference:

[0034]

[0035] Among them, the largest final time difference Normalized maximum thrust acceleration p1, p2, p3, p4, q1, and q2 are constants;

[0036] 3) Using the vernal equinox orbital elements of each orbital feature sample, solve the optimal fuel problem obtained in step 2), specifically including:

[0037] Establish the Hamiltonian function:

[0038]

[0039] H1 introduces a costate λ = [λ] with respect to the orbital elements at the vernal equinox. p ,λ f ,λ g ,λ h ,λ k ,λ L ] T And the costate λ of mass m , λ p ,λ f ,λ g ,λ h ,λ k ,λ L Let p, f, g, h, k, and L be the costates, respectively.

[0040] Based on the principle of minimization, by minimizing H1, the optimal thrust direction and magnitude are obtained as follows: Where ρ is the switching function.

[0041] Solving the target equation using MinPack-1 The solution to the fuel optimization problem is obtained, where the function costate multipliers [λ,λ] are used. m The sum of the numerical multipliers λ0 is denoted as the Lagrange multiplier vector, and the multipliers are normalized to satisfy the following conditions:

[0042] Using the method of randomized initial values ​​for costate variables, random shooting is performed to satisfy the equation Solving for;

[0043] Define seven random angle quantities α7∈[0,2π],[λ0,λ,λ m The relationship between the angle variable and the angle variable is as follows:

[0044]

[0045] Given the semi-major axis a and the maximum thrust acceleration a max Maximum thrust T max Gravitational acceleration g0, specific impulse I sp Normalization is performed separately, and the orbital elements at the vernal equinox of each orbital feature sample are used to solve the fuel optimization problem, yielding the changed dimensionless mass. Multiplying this dimensionless mass by the initial mass of the spacecraft yields the fuel consumption mass for each orbital feature sample; where T... max =ma max T max is the maximum thrust amplitude with low thrust, and m is the mass of the transfer vehicle.

[0046] Features and beneficial effects of the present invention:

[0047] This invention uses a neural network as a fuel consumption estimator to fit and map fuel consumption under low thrust transfer, which greatly reduces the calculation time for fuel consumption and has extremely high accuracy.

[0048] This invention prunes the number of features by utilizing symmetry and GEO orbital properties, reducing the theoretically required 14-dimensional vector as input to 4 input features. This reduces the number of samples needed for training, significantly lowering the difficulty of training the neural network.

[0049] The model trained by this invention exhibits a relative error of less than one percent in its predictions on the test set, demonstrating extremely high accuracy. Furthermore, the model can predict 1000 data points in parallel in just seconds, significantly improving computational efficiency compared to the homotopy method, which requires several seconds or even tens of seconds to calculate a single data point. Attached Figure Description

[0050] Figure 1 This is an overall flowchart of a method for rapidly estimating fuel consumption in low-thrust orbital transfer according to an embodiment of the present invention.

[0051] Figure 2 This is a logarithmic coordinate graph showing the changes in the loss function of the training set and test set during model training in a specific embodiment of the present invention.

[0052] Figure 3 This is a schematic diagram illustrating the accuracy of the training and test sets under a set relative error limit during the model training process in a specific embodiment of the present invention.

[0053] Figure 4 This is a graph showing the change in the average relative error between the test set and the training set during model training in a specific embodiment of the present invention. Detailed Implementation

[0054] This invention proposes a rapid estimation method for fuel consumption during low-thrust orbital transfer, which is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0055] This embodiment proposes a rapid estimation method for fuel consumption during low-thrust orbital transfer, including:

[0056] The initial orbital elements and the target orbital elements in the low-thrust orbit transfer are linearly transformed to GEO orbits using classical orbital elements;

[0057] Based on the transformed initial orbital elements and target orbital elements, the orbital inclination difference, ascending node right ascension difference, and phase difference between the initial orbit and the target orbit are calculated. The orbital inclination difference, ascending node right ascension difference, phase difference, and spacecraft maximum thrust acceleration are combined to form orbital characteristics.

[0058] Each parameter in the orbital features is normalized, and then the normalized orbital features are input into a preset fuel consumption estimator model. The model outputs a normalized estimate of fuel consumption mass.

[0059] The estimated value is inversely normalized to obtain the estimated fuel consumption for low-thrust orbital transfer.

[0060] In a specific embodiment of the present invention, the overall process of the rapid estimation method for fuel consumption in low-thrust orbital transfer is as follows: Figure 1 As shown, it includes the following steps:

[0061] 1) Based on the parameters of low-thrust orbit transfer, obtain multiple orbit feature samples to form an orbit feature dataset.

[0062] In this embodiment, the parameters for the low-thrust orbital transfer include Δi, ΔΩ, ΔL, and a. max Where Δi is the orbital inclination difference, ΔΩ is the ascending node longitude difference, ΔL represents the true longitude difference (phase difference), and a max Indicates the maximum thrust acceleration of the spacecraft. T max is the maximum thrust amplitude with low thrust, and m is the mass of the transfer vehicle.

[0063] Determine Δi, ΔΩ, ΔL, a respectively max The range of values ​​for is determined by the orbital transfer range.

[0064] A value is randomly and uniformly selected from the range of values ​​for each parameter to form an orbital feature sample, denoted as x=[ΔiΔΩΔL a max ] T All orbit feature samples constitute the orbit feature dataset.

[0065] In one specific embodiment of the present invention, the radius of the GEO orbit is concentrated around 42,164 kilometers and is approximately a circular orbit. In this embodiment, the semi-major axis is set to a constant value of 42,164,000 meters and the eccentricity is set to a constant value of 0.

[0066] Due to symmetry, the initial orbital elements of any two initial and final GEO orbits can be set to [4216400000000] through a linear transformation of rotation and translation. TIn this embodiment, the initial orbital inclination, right ascension of the ascending node, and true longitude L (due to the characteristics of geosynchronous orbit, the vernal equinox orbital element is used to describe the orbital position) of the transfer spacecraft are set to 0°. The GEO orbital inclination range of the target star is set to 0-5°, and the mean longitude L and right ascension of the ascending node are both set to 0-360°. Here, the mean longitude represents more the relative phase relationship between the initial and final orbits.

[0067] The initial mass (fully loaded) of the transfer vehicle is 6200 kg, and the empty mass is 1000 kg.

[0068] In summary, in this embodiment, Δi∈[0,5°], ΔΩ, ΔL∈[0,360°], and m∈[1000,6200]kg. A total of 74,000 orbital feature samples were generated in this embodiment, and each parameter in each sample is a random number that follows a uniform distribution within the corresponding value range.

[0069] 2) Based on the results of step 1), generate the target orbital elements [ae iωΩθ] corresponding to each orbital feature sample, combined with the initial orbital elements. T .

[0070] Where a represents the semi-major axis of the orbit, e represents the orbital eccentricity, i represents the orbital inclination, w represents the argument of perigee, Ω represents the right ascension of the ascending node, and θ represents the true anomaly.

[0071] In one specific embodiment of the present invention, e = 0, i = Δi, θ = 0, Ω = ΔΩ, ω = ΔL - ΔΩ.

[0072] Then, the target orbital elements corresponding to each orbital feature sample are converted into the vernal equinox orbital elements x = [p fg hk L] T The conversion expression is as follows:

[0073]

[0074] In this embodiment, a is 42164000, e, θ, and ω are 0, and i and Ω are randomly generated. During the conversion process, L is directly set to a random number between 0 and 2π; the initial number of orbital elements at the vernal equinox is [4216400000000]. T .

[0075] 3) Based on the results of step 2), calculate the corresponding fuel consumption using the six roots of the vernal equinox for each orbital feature sample, and use this as the label for that orbital feature sample. Then, combine each orbital feature sample and its corresponding label into a training set. The specific steps are as follows:

[0076] 3-1) Construct the orbital dynamics equations.

[0077] In this embodiment, the spacecraft motion is described by the following equations:

[0078]

[0079] in, The time derivative of the orbital elements at the vernal equinox. This refers to the fuel consumption rate.

[0080] T max It is the maximum thrust amplitude with small thrust, u (0≤u≤1) is the ratio of the actual thrust amplitude to the maximum thrust amplitude, f=[f r ,f θ ,f h ] T It is the unit vector of thrust direction (each component is a projection in the LVLH coordinate system, the X-axis is along the position vector direction, the Z-axis is along the orbital plane normal, and the Y-axis follows the right-hand rule, perpendicular to the X and Z axes, and points in the velocity direction), g0 = 9.80665 m / s 2 It is the average gravitational acceleration at sea level on Earth, I sp It is the specific impulse of the thruster.

[0081] The expressions for the perturbation influence matrix B(x) and the central gravitational influence matrix D(x) are as follows:

[0082]

[0083] Where W = 1 + fcosL + gsinL, S = 1 + h 2 +k 2 μ is the Earth's gravitational constant.

[0084] 3-2) Based on the results of step 3-1), construct the optimal fuel problem.

[0085] In this embodiment, we consider the optimal fuel transfer from a certain initial orbit to GEO, that is, to find the control inputs u(t) and f(t) at time t that minimize the following performance indices when a spacecraft with an initial mass of m0 transfers from its initial point x0 to the GEO orbit:

[0086]

[0087] Where t0 is the transfer start time, t f Let λ0 be the transition termination time. λ0 is an introduced positive multiplier, and ε is the homotopy parameter, ε∈[0,1]. When ε=0, the above equation corresponds to the fuel optimization problem, and when ε=1, it corresponds to the energy optimization problem.

[0088] The transfer termination time is estimated using this formula: Where Δθ is the difference in longitude between the initial and final positions.

[0089] True longitude difference:

[0090]

[0091] Among them, the largest final time difference Normalized maximum thrust acceleration p1, p2, p3, p4, q1, and q2 are constants. In a specific embodiment of the present invention, p1 = 0.04978, p2 = 7.48, p3 = 50.08, p4 = 6.73, q1 = 14.49, q2 = 15.94, and t f The unit is days.

[0092] The fuel optimization problem can then be expressed as follows:

[0093]

[0094] Among them, the orbital dynamics equations obtained in step 3-1) are the constraints for this optimal fuel problem.

[0095] 3-3) Using the orbital elements of the vernal equinox obtained in step 2), solve the optimal fuel problem obtained in step 3-2).

[0096] In this embodiment, a costate λ = [λ] is introduced regarding the orbital elements at the vernal equinox. p ,λ f ,λ g ,λ h ,λ k ,λ L ] T And the costate λ of mass m , λ p ,λ f ,λ g ,λ h ,λ k ,λ L Let p, f, g, h, k, and L be the costates.

[0097] Establish the Hamiltonian function:

[0098]

[0099] Based on the principle of minimization, by minimizing H1, the optimal thrust direction and magnitude are obtained as follows: Where ρ is the switching function.

[0100] Solving the target equation using MinPack-1 The solution to the fuel optimization problem can be obtained, where the function costate multipliers [λ,λ] are used. mThe sum of the numerical multipliers λ0 is denoted as the Lagrange multiplier vector, and the multipliers are normalized to satisfy the following conditions:

[0101] Initial value estimation of costate variables;

[0102] In this embodiment, a method of randomizing the initial values ​​of costate variables is used to randomly fire at targets to satisfy the equation. Solving for this problem. Define seven random angle quantities. α7∈[0,2π],[λ0,λ,λ m The relationship between the angle variable and the angle variable is as follows:

[0103]

[0104] In this embodiment, T max =ma max All variables include the semi-major axis a and the maximum thrust acceleration a. max Maximum thrust T max Gravitational acceleration g0, specific impulse I sp Normalization was performed on each sample, with a unit length of Earth's radius of 6,378,137.0 m and a unit time of 806.811 s. Using the vernal equinox orbital elements of each orbital feature sample obtained in step 2), the fuel optimization problem was solved to obtain the dimensionless mass after the change. Multiplying this by the initial mass of the spacecraft yields the fuel consumption mass, which is the label for the orbital feature sample corresponding to the six roots of the vernal equinox.

[0105] Each orbital feature sample and its corresponding label are combined to form a training sample, and all training samples constitute the training set.

[0106] 4) Train the fuel consumption estimator model using the training set obtained in step 3). The specific steps are as follows:

[0107] 4-1) Normalize each data point in each training sample obtained in step 3) to obtain the updated training set.

[0108] 4-2) Construct a fuel consumption estimator model.

[0109] In this embodiment, a fully connected neural network is used as the fuel consumption estimator model, comprising: 8 fully connected layers and 2 batch normalization layers, with a ReLU activation function applied after each fully connected layer. Specifically, the first to third fully connected layers are connected sequentially, followed by the first batch normalization layer, then the fourth fully connected layer, then the second batch normalization layer, and finally the fifth to eighth fully connected layers.

[0110] 4-3) Use the updated training set from step 4-1) to train the fuel consumption estimator model constructed in step 4-2).

[0111] In this embodiment, the PyTorch framework is used to train the neural network. The settings for each component and hyperparameter during training are as follows: learning rate is set to 1e-4; batch size is set to 16; the ratio of training set data size to test set data size is 4; the Adam optimizer is selected; and the root mean square error (MSE) function is used as the loss function. The training converges after 60 epochs, and the entire training process takes about ten minutes.

[0112] 5) Use the fuel consumption estimator model trained in step 4) to make predictions.

[0113] =In this embodiment, the fuel consumption prediction for low-thrust GEO orbital transfer is performed using a trained fuel consumption estimator model. The specific steps are as follows:

[0114] 5-1) The initial orbital elements [ae iωΩθ] for the transfer T Perform a linear transformation to convert the orbital to a classical orbital element [4216400000000]. T On the GEO orbit, the target orbital elements are then linearly transformed to obtain the transformed target orbital elements.

[0115] 5-2) Based on the results of step 5-1), calculate the difference in orbital inclination, difference in right ascension of the ascending node, and phase difference between the initial orbit and the target orbit, where Δi = i, ΔΩ = Ω, and ΔL = L. Then, calculate a based on the satellite parameters. max The orbital characteristic x = [ΔiΔΩΔL a] is obtained. max ] T .

[0116] 5-3) Normalize the orbital features obtained in step 5-2):

[0117]

[0118] where k∈{1,2,3,4}, x1=Δi, x2=ΔΩ, x3=ΔL, x4=a max ;x kmax For feature x k The maximum value corresponding to x in the orbital feature dataset obtained in step 1). kmin For feature x k The minimum value corresponding to the orbital feature dataset obtained in step 1).

[0119] 5-4) Input the track features normalized in step 5-3) into the fuel consumption estimator model trained in step 3). The model outputs a normalized estimate of the fuel consumption mass. By inversely normalizing the model output, the transferred fuel consumption mass can be obtained.

[0120] In one specific embodiment of the present invention, during the training process of the fuel consumption estimator model, the logarithmic coordinate graph of the changes in the loss function of the training set and the test set is shown below. Figure 2 As shown, the dark line corresponds to the training set and the light line corresponds to the test set. It can be seen that in this embodiment, the model converged after about 60 training rounds, and the loss function value dropped to 0.004.

[0121] The loss function value cannot intuitively display the accuracy and reliability of the model's predictions. Therefore, this embodiment of the invention statistically analyzes the relative error of the model's prediction results, records the average relative error, and considers a result where the relative error between the predicted value and the true value is within a set range as "correct." The accuracy changes during the training process are statistically analyzed, with the relative error limit set to 0.1%. The accuracy changes of the training set and test set during training are as follows: Figure 3 As shown, the change in average relative error is as follows: Figure 4 As shown, Figure 3 and Figure 4 In the diagram, dark lines correspond to the training set, and light lines correspond to the test set. From... Figure 3 and Figure 4 It can be seen that the accuracy improved rapidly while the relative error decreased rapidly. The accuracy was rapidly improved in the first two rounds of training, reaching 83%. After sixty rounds of training, the model had an average relative error of 0.067% on the test set. The proportion of samples with a relative error within 0.1% reached 85.87% of the total, and the maximum relative error was 0.99%, indicating that the model has high prediction accuracy and reliability.

[0122] In this embodiment, the trained model is used to predict the matrix composed of 1000 sets of test data, and the calculation time is only 1.0 to 4.0 seconds (with fluctuation).

Claims

1. A rapid estimation method for fuel consumption in low-thrust orbital transfer, characterized in that, include: The initial orbital elements and the target orbital elements in the low-thrust orbit transfer are linearly transformed to GEO orbits using classical orbital elements; Based on the transformed initial orbital elements and target orbital elements, the orbital inclination difference, ascending node right ascension difference, and phase difference between the initial orbit and the target orbit are calculated. The orbital inclination difference, ascending node right ascension difference, phase difference, and spacecraft maximum thrust acceleration are combined to form orbital characteristics. Each parameter in the orbital features is normalized, and then the normalized orbital features are input into a preset fuel consumption estimator model. The model outputs a normalized estimate of fuel consumption mass. The estimated value is inversely normalized to obtain the estimated fuel consumption for low-thrust orbital transfer.

2. The method according to claim 1, characterized in that, The fuel consumption estimator model includes: 8 fully connected layers and 2 batch normalization layers; wherein, the first to third fully connected layers are connected sequentially, the third fully connected layer is connected to the first batch normalization layer, the first batch normalization layer is connected to the fourth fully connected layer, the fourth fully connected layer is connected to the second batch normalization layer, and the second batch normalization layer is connected to the fifth to eighth fully connected layers sequentially; each fully connected layer is followed by the ReLU activation function.

3. The method according to claim 2, characterized in that, Before inputting the normalized orbital features into a preset fuel consumption estimator model, the method further includes: Train the fuel consumption estimator model; Training the fuel consumption estimator model includes: 1) Determine the value range of each parameter in the orbital features, and randomly and uniformly select values ​​from the value range of each parameter to construct multiple sets of orbital feature samples; 2) Generate the target orbital roots for each orbital feature sample and convert them into the corresponding vernal equinox orbital roots; 3) Based on the number of orbital elements at the vernal equinox for each orbital feature sample, calculate the corresponding fuel consumption as the label for the orbital feature sample, and combine each orbital feature sample and its corresponding label into a training set; 4) Train the fuel consumption estimator model using the training set.

4. The method according to claim 3, characterized in that, The orbital elements at the vernal equinox are denoted as x = [pfgh kL]. T ,in: Where, [ae iωΩθ] T Let be the target orbital elements, a represent the semi-major axis of the orbit, e represent the orbital eccentricity, i represent the orbital inclination, w represent the argument of perigee, Ω represent the right ascension of the ascending node, and θ represent the true anomaly.

5. The method according to claim 4, characterized in that, The calculation of the corresponding fuel consumption as a label for each orbital feature sample, based on the vernal equinox orbital elements of each orbital feature sample, includes: 1) Construct the orbital dynamics equations for the spacecraft's motion: in, The time derivative of the orbital elements at the vernal equinox. Fuel consumption rate; T max It is the maximum thrust amplitude with small thrust, u is the ratio of the actual thrust amplitude to the maximum thrust amplitude, 0≤u≤1; f=[f r ,f θ ,f h ] T It is a unit vector in the direction of thrust, g0 is the average gravitational acceleration at sea level, and I sp It is the specific impulse of the thruster; The expressions for the perturbation influence matrix B(x) and the central gravitational influence matrix D(x) are as follows: Where W = 1 + fcosL + gsinL, S = 1 + h 2 +k 2 μ is the Earth's gravitational constant; 2) Based on the results of step 1), construct the optimal fuel problem: Where t0 is the transfer start time, t f λ is the termination time of the transition; λ0 is a positive multiplier; in, Δθ is the difference in longitude between the initial and final positions; True longitude difference: Among them, the largest final time difference Normalized maximum thrust acceleration p1, p2, p3, p4, q1, and q2 are constants; 3) Using the vernal equinox orbital elements of each orbital feature sample, solve the optimal fuel problem obtained in step 2), specifically including: Establish the Hamiltonian function: H1 introduces a costate λ = [λ] with respect to the orbital elements at the vernal equinox. p ,λ f ,λ g ,λ h ,λ k ,λ L ] T And the costate λ of mass m , λ p ,λ f ,λ g ,λ h ,λ k ,λ L Let p, f, g, h, k, and L be the costates, respectively. Based on the principle of minimization, by minimizing H1, the optimal thrust direction and magnitude are obtained as follows: Where ρ is the switching function. Solving the target equation using MinPack-1 The solution to the fuel optimization problem is obtained, where the function costate multipliers [λ,λ] are used. m The sum of the numerical multipliers λ0 is denoted as the Lagrange multiplier vector, and the multipliers are normalized to satisfy the following conditions: Using the method of randomized initial values ​​for costate variables, random shooting is performed to satisfy the equation Solving for; Define seven random angle quantities α7∈[0,2π],[λ0,λ,λ m The relationship between the angle variable and the angle variable is as follows: Given the semi-major axis a and the maximum thrust acceleration a max Maximum thrust T max Gravitational acceleration g0, specific impulse I sp Normalization is performed separately, and the orbital elements at the vernal equinox of each orbital feature sample are used to solve the fuel optimization problem, yielding the changed dimensionless mass. Multiplying this dimensionless mass by the initial mass of the spacecraft yields the fuel consumption mass for each orbital feature sample; where T... max =ma max T max is the maximum thrust amplitude with low thrust, and m is the mass of the transfer vehicle.

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