A method for fast calculation of solid rocket carrying capacity in maneuvering condition

Abstract

The present application relates to a kind of solid rocket carrying capacity in the quick calculation method of motor condition, belong to aerospace engineering technical field, solve the problems such as long time-consuming and process complex that traditional rocket carrying calculation mode exists, including the following steps: step one, data preparation for training: the latitude of predetermined area, launch point height, orbit inclination and orbit height are gridded;Step two, neural network design and training: a multilayer perceptron neural network is used, and the network structure is as follows: input layer is 4 neurons;Hidden layer is 15 neurons, responsible for the nonlinear mapping of input variable;Output layer is 1 neuron;In actual launch process, the latitude of predetermined area, launch point height, orbit inclination and orbit height are input into the trained multilayer perceptron neural network, and the carrying capacity is obtained. The maximum carrying capacity is calculated quickly using the trained neural network to improve the efficiency and flexibility of launch task.

Description

Technical Field

[0001] This invention belongs to the field of aerospace engineering technology, specifically relating to a rapid calculation method for the carrying capacity of solid rockets under maneuvering conditions. Background Technology

[0002] In space launch missions, the payload capacity of solid rockets is a critical parameter that directly impacts the feasibility and effectiveness of the launch. Traditional methods for calculating rocket payload capacity often require lengthy trajectory optimization iterations, which are insufficient to meet the demands of rapid decision-making and response in real-time scenarios, such as maneuverable launch missions. While existing payload capacity calculation software can calculate payload capacity relatively accurately, the calculation process is complex and time-consuming, making it unsuitable for the high computational speed requirements of maneuverable launches. Summary of the Invention

[0003] The purpose of this invention is to overcome the shortcomings of the prior art and provide a rapid calculation method for the carrying capacity of solid rockets under maneuvering conditions, which solves the technical problems of complex process and long time consumption in traditional carrying capacity calculation methods.

[0004] To address the above problems, the technical solution of this invention is: a method for rapidly calculating the carrying capacity of a solid rocket under maneuvering conditions, comprising the following steps: Step 1: Training Data Preparation: Grid the latitude, launch point altitude, orbital inclination, and orbital altitude of the predetermined region; calculate the carrying capacity at each grid point, and organize the calculated data into a training database. The inputs are latitude, launch point altitude, orbital inclination, and orbital altitude, and the output is the carrying capacity. Step 2, Neural Network Design and Training: A multilayer perceptron neural network is used, with the following structure: The input layer has 4 neurons, corresponding to the four input variables: latitude, launch point altitude, orbital inclination, and orbital altitude; the hidden layer has 15 neurons, responsible for nonlinear mapping of the input variables; the output layer has 1 neuron, corresponding to the carrying capacity output; the multilayer perceptron neural network uses a fully connected approach, and the parameters of the multilayer perceptron neural network include weights and biases. From the input layer to the hidden layer: the weight matrix has a dimension of 4×15 and is initialized to Gaussian random numbers multiplied by 0.1; the bias has a dimension of 1×15 and is initialized to 0, used to adjust the starting point of the linear combination of the hidden layers; From hidden layer to output layer: The weight matrix has a dimension of 15×1 and is initialized to Gaussian random numbers multiplied by 0.1; the bias has a dimension of 1×1 and is initialized to 0, used to adjust the starting point of the linear combination of the output layer; The multilayer perceptron neural network is trained using the training database obtained in step one. After training is completed, the trained neural network model is saved. Step 3: Calculate the carrying capacity: During the actual launch, the latitude of the predetermined area, the launch point altitude, the orbital inclination angle, and the orbital altitude are input into the trained neural network model to obtain the carrying capacity.

[0005] Preferably, the hidden layers of the multilayer perceptron neural network use the sigmoid activation function to introduce a nonlinear transformation, enabling the multilayer perceptron neural network to fit complex input-output relationships. Its derivative is used for gradient calculation in backpropagation; the output layer has no activation function.

[0006] Preferably, in step two, the neural network is trained using gradient descent, with mean squared error as the loss function. The parameter gradient is calculated using the chain rule, and the weights and biases are updated in the direction of decreasing loss.

[0007] Preferably, in step two, the training data in the training database is divided into a training set and a validation set. The training set is used to train the multilayer perceptron neural network, and the validation set is used to monitor the training process and prevent overfitting. The neural network training process is as follows: Let the neural network prediction value be The actual value is y i The training set has n elements, and the mean squared error L is defined as follows: ; Let the input vector be x, the weight matrix from the input layer to the hidden layer be w1, the bias vector of the hidden layer be b1, the activation function be σ, the weight matrix from the hidden layer to the output layer be w2, the bias vector of the output layer be b2, α be the learning rate (set to 0.1), z1 be the output of a single neuron in the hidden layer, and z2 be the output of a single neuron in the output layer. The output of the hidden layer is T, which represents the matrix transpose. The hidden layer calculation is as follows: , ; Output layer computation: , ; Calculate the output layer gradient using the chain rule: , , ; Hidden layer gradient: , , , Θ represents element-wise multiplication. Activation function The derivative; Update the weights and biases using gradient descent: , , , .

[0008] Preferably, in step two, the number of iterations is set to 10,000.

[0009] Preferably, in step two, the ratio of the training set to the validation set is 7:3.

[0010] Preferably, in step two, the method for calculating the carrying capacity is as follows: the carrying capacity of the rocket is estimated using the Tsiolkovsky formula, and the calculation model for the final stage velocity of a multi-stage rocket is as follows: , In the formula: I spi λ is the specific impulse of the i-th stage engine; i ε is the ratio of the effective load of the i-th stage to the initial mass of that stage. i Let v be the ratio of the mass of the i-th stage structure to the initial mass of that stage, and v be the velocity of the final stage of the rocket. loss This indicates the loss of velocity caused by air and gravity.

[0011] Preferably, in step two, for rockets in the application stage, the rocket's structural parameters and guidance scheme are already fixed, and the rocket's final stage velocity v is determined by the orbital altitude, orbital inclination, launch point latitude, and launch point altitude. The analysis method for rocket carrying capacity includes the following steps: S1. Analysis of the impact of carrying capacity, orbital inclination, and launch point latitude; Let Φ be the latitude of the launch point and i be the target orbit inclination. Then the inclination angle to be adjusted is Δi = |i - Φ|. The fuel requirement for orbit inclination adjustment is determined by the change in velocity vector direction, and the required velocity increment Δv follows the formula... v N The current orbital velocity is given, and Δi represents the change in inclination angle; fuel consumption is directly related to Δv. I sp Let g be the engine's specific impulse, g be the acceleration due to gravity, and m0 be the acceleration due to gravity. The mass before and after launch; the greater the Δv requirement, the higher the fuel consumption. An exponential increase directly impacts carrying capacity; S2. Analyze the relationship between carrying capacity and track altitude; Rocket final stage velocity v requirement and Positive correlation, R e The radius represents the Earth's radius, and h is the altitude of the satellite's orbit above the Earth's surface.

[0012] Compared with the prior art, the beneficial effects of the present invention are as follows: This invention fully utilizes the advantages of multilayer perceptron neural networks (MLPNs) in their strong data fitting capabilities and fast computation speed to solve the problem of rapidly calculating the maximum payload capacity of rockets under mobile launch conditions. By combining neural networks with factors affecting the maximum payload capacity, a trained neural network is used to quickly calculate the maximum payload capacity, thereby improving the efficiency and flexibility of launch missions. Attached Figure Description

[0013] Figure 1 This is a comparison chart of the predicted and actual carrying capacity values ​​in the embodiment. Figure 2 This is a prediction error distribution diagram from the example; Figure 3 This is a bar chart showing the error distribution in the example. Detailed Implementation

[0014] The present invention will now be described in further detail with reference to the accompanying drawings and embodiments.

[0015] Example: Figures 1-3 As shown in the figure, this embodiment provides a rapid calculation method for the carrying capacity of a solid rocket under maneuvering conditions, including the following steps: I. Analyze the parameters affecting carrying capacity; This paper primarily analyzes the factors affecting the payload capacity of solid-propellant launch vehicles, using these factors as input parameters for a neural network. The payload capacity is estimated using the Tsiolkovsky formula; the final stage velocity calculation model for a multi-stage rocket is as follows: , In the formula: I spi λ is the specific impulse of the i-th stage engine; i ε is the ratio of the effective load of the i-th stage to the initial mass of that stage. i Let v be the ratio of the mass of the i-th stage structure to the initial mass of that stage, and v be the velocity of the rocket's final stage (i.e., the velocity at which the rocket payload enters orbit). A larger payload capacity is equivalent to a larger v. loss This indicates the loss of velocity caused by air and gravity, primarily originating from aerodynamic and gravitational losses. Different rocket trajectories will result in varying degrees of aerodynamic and gravitational losses.

[0016] For rockets in the application stage, their structural parameters and guidance schemes are already fixed. The payload capacity (v) is determined by the orbital altitude, orbital inclination, launch site latitude, and launch site altitude. V demand is positively correlated with orbital altitude; the higher the altitude, the lower the payload capacity. The rocket's payload capacity is closely related to the satellite's orbital inclination; the greater the deviation of the orbital inclination from the launch site latitude, the greater the V consumption. Different latitudes offer different usable Earth rotation angular velocities; launches near the equator maximize the use of Earth's rotation speed. The launch site's altitude primarily affects the rocket's payload capacity by increasing its initial gravitational potential energy and reducing its atmospheric flight range.

[0017] 1. Analyze the impact of carrying capacity, orbital inclination, and launch point latitude; At the same orbital altitude, the velocity is the same, but different orbital inclinations mean different lateral velocity components are needed, and the available Earth rotation velocity varies. If the target inclination and the launch site latitude are inconsistent, the rocket will decompose the initial Earth rotation velocity into the orbital plane, and the residual component must be offset by fuel.

[0018] When launching from a polar orbit, the Earth's rotational velocity is entirely decomposed into components perpendicular to the orbital plane, which must be completely offset, resulting in maximum fuel consumption. When the rocket launches eastward, the eastward velocity of the Earth's rotation is completely converted into orbital velocity, requiring no additional fuel.

[0019] Let Φ be the latitude of the launch point and i be the target orbit inclination. Then the inclination angle to be adjusted is Δi = |i - Φ|. The fuel requirement for orbit inclination adjustment is determined by the change in velocity vector direction, and the required velocity increment Δv follows the formula... v N The current orbital velocity is given, and Δi represents the change in inclination angle; fuel consumption is directly related to Δv. I sp Where is the engine's specific impulse (fuel efficiency), g is the acceleration due to gravity, and m0, Considering the mass before and after launch; it's easy to see that the greater the Δv requirement, the higher the fuel consumption. An exponential increase directly impacts carrying capacity; 2. Analyze the relationship between carrying capacity and track altitude; The higher the orbit, the greater the energy required to escape Earth's gravity. The higher the orbit, the greater the energy demand (v) and the lower the carrying capacity. (v demand and...) Positive correlation, R e The radius represents the Earth's radius, and h is the altitude of the satellite orbit above the Earth's surface. The higher the altitude, the lower the carrying capacity.

[0020] 3. Analyze the relationship between carrying capacity and launch site altitude; The altitude of the launch site affects the rocket's carrying capacity mainly through the following physical mechanisms: the higher the altitude, the lower the atmospheric density, the less air resistance the rocket experiences during ascent; the higher the altitude, the shorter the integral interval of the dense atmosphere, reducing total drag loss; the rocket engine nozzle design needs to match the external environmental pressure; the lower air pressure at high altitudes allows the first-stage engine nozzle expansion ratio to be closer to the vacuum optimization state, increasing specific impulse and carrying capacity; simultaneously, increased altitude leads to a greater distance from the Earth's center, increasing initial gravitational potential energy and further enhancing the rocket's carrying capacity.

[0021] 4. Parameter considerations in this embodiment; This embodiment uses latitude, launch point altitude, orbital inclination, and orbital altitude as input parameters for the neural network, precisely because these parameters have a significant impact on carrying capacity. By gridding these parameters, it can comprehensively cover situations under different parameter combinations, ensuring that the trained neural network can accurately capture the complex relationship between each parameter and carrying capacity, including the individual effects of each parameter and their interactions, thereby enabling rapid and accurate calculation of carrying capacity in practical applications.

[0022] II. Training Data Preparation; The latitude, launch site altitude, orbital inclination, and orbital altitude of the Chinese region are gridded, with a denser grid being preferable. For example, the latitude of the Chinese region can be divided at certain intervals; the launch site altitude, orbital inclination, and orbital altitude are also gridded in a similar manner. During gridding, both computational accuracy and data volume must be considered, and the grid should be refined as much as possible while ensuring sufficient accuracy.

[0023] The carrying capacity at each grid point is calculated using proven carrying capacity calculation software, and this data will be used as training data for the subsequent neural network. Existing carrying capacity calculation software should be selected from those that have been validated and have high accuracy to ensure the reliability of training data.

[0024] III. Training and Application of Neural Networks; A suitable neural network can be selected, and trained using the previously obtained training data (inputs are latitude, launch point altitude, orbital inclination, and orbital altitude; output is payload capacity). The trained neural network is then saved. In actual launches, the payload capacity can be obtained simply by inputting the latitude, launch point altitude, orbital inclination, and orbital altitude into the neural network, avoiding a lengthy trajectory optimization iteration process and making it suitable for real-time applications. The design of a neural network needs to consider the structure of the input layer, hidden layers, and output layer. The input layer has 4 nodes, corresponding to latitude, launch point altitude, orbital inclination, and orbital altitude, respectively; the output layer has 1 node, corresponding to the carrying capacity. The number of hidden layers and nodes can be adjusted according to actual conditions to improve the fitting and generalization abilities of the neural network. During training, it is necessary to select an appropriate loss function and optimization algorithm, and continuously adjust the parameters of the neural network to make the output of the neural network as close as possible to the actual output of the training data. After training, the neural network is validated and tested to ensure that it can accurately calculate carrying capacity under different conditions.

[0025] Taking solid-propellant launch vehicles as an example, the specific implementation of the above method is introduced.

[0026] (1) Training data preparation; The latitude range of China was determined to be [18°N~53°N], and the grid was formed at 5° intervals, generating a total of 8 discrete launch point latitudes. The launch point altitude ranges from 0 to 3 km, and is gridded at 1 km intervals, generating a total of 4 discrete points.

[0027] The orbital inclination ranges from [launch point latitude to (180° - launch point latitude)], and is gridded at 5° intervals. The discrete size of this variable is related to the launch point latitude: when the launch point latitude is 18° and 53°, the number of discrete points generated is 29 and 13 respectively, with an average of 21 discrete points.

[0028] The orbital altitude ranges from 300km to 700km, and the points are gridded at 10km intervals, generating a total of 41 discrete points.

[0029] In summary, after discretizing the input variables, approximately 27,552 (8×4×21×41) grid points are generated.

[0030] Using existing high-precision carrying capacity calculation software, the carrying capacity corresponding to each of the above grid points is calculated. The software takes about 5 minutes to calculate each grid point, and the total calculation time is 2296 hours, or about 96 days.

[0031] These data are compiled into a training database, with latitude, launch point altitude, orbital inclination, and orbital altitude as inputs, and carrying capacity as output.

[0032] (2) Neural network design and training; Here, a multilayer perceptron neural network is used. The structure of the multilayer perceptron neural network is as follows: the input layer has 4 neurons, corresponding to the 4 input variables of latitude, launch point altitude, orbital inclination angle and orbital altitude; the hidden layer has 15 neurons, which is the only hidden layer in the multilayer perceptron neural network and is responsible for nonlinear mapping of the input variables; the output layer has 1 neuron, corresponding to the 1 output of carrying capacity.

[0033] Multilayer perceptron neural networks use a fully connected approach, meaning that every neuron in each layer is connected to all neurons in the next layer. Parameters include weights and biases. ① Input layer → Hidden layer: The weight matrix has a dimension of 4×15 (number of input features × number of hidden layer neurons), and is initialized to a Gaussian random number multiplied by 0.1 (to avoid the activation function from saturating due to an excessively large initial value); the bias dimension has a dimension of 1×15, and is initialized to 0, which is used to adjust the starting point of the linear combination of the hidden layer.

[0034] ②Hidden Layer → Output Layer: The weight matrix has a dimension of 15×1 (number of hidden layer neurons × number of output layer neurons), and is initialized in the same way as ①. The bias dimension is 1×1, initialized to 0, and adjusts the starting point of the linear combination of the output layer.

[0035] ③ Activation function: The hidden layer uses the sigmoid activation function, which introduces a non-linear transformation, enabling the multilayer perceptron neural network to fit complex input-output relationships. Its derivative is used for gradient calculation in backpropagation. The output layer has no activation function (linear output), which is a typical design for regression tasks, ensuring that the output can be any real number (rather than being restricted to a specific range by the activation function, such as sigmoid [0,1]).

[0036] The training of the multilayer perceptron neural network employs gradient descent, with the core being an iterative process of "forward propagation to calculate predicted values ​​→ backpropagation to update parameters." Mean squared error is used as the loss function, and the parameter gradient is calculated using the chain rule. Weights and biases are updated along the direction of decreasing loss. The number of iterations is set to 10,000 to ensure model convergence. The learning rate is set to 0.1 to control the parameter update step size; too large a rate may cause oscillations, while too small a rate slows convergence.

[0037] The training data in the training database is divided into a training set and a validation set in a 7:3 ratio. The multilayer perceptron neural network is trained using the training set, and the training process is monitored using the validation set to prevent overfitting. After training is complete, the trained neural network model is saved.

[0038] Specifically, the neural network training process is as follows: Let the neural network prediction value be The actual value is y iThe training set has n elements, and the mean squared error L is defined as follows: ; Let the input vector be x, the weight matrix from the input layer to the hidden layer be w1, the bias vector of the hidden layer be b1, the activation function be σ, the weight matrix from the hidden layer to the output layer be w2, the bias vector of the output layer be b2, α be the learning rate (set to 0.1), z1 be the output of a single neuron in the hidden layer, and z2 be the output of a single neuron in the output layer. The output of the hidden layer is T, which represents the matrix transpose. The hidden layer calculation is as follows: , ; Output layer computation: , ; Calculate the output layer gradient using the chain rule: , , ; Hidden layer gradient: , , , Θ represents element-wise multiplication. Activation function The derivative; Update the weights and biases using gradient descent: , , , .

[0039] Figures 1-3 This analysis examines the load capacity prediction results of a multilayer perceptron neural network. Figure 1 , Figure 2 This indicates that the actual carrying capacity is compared with the predicted value of the trained model, and the error value is small, showing a good match between the predicted and actual values. Figure 3 The error value is represented by a statistical distribution. After training, the average prediction error of the carrying capacity is 1.39 kg, and the probability of the prediction error being less than 1 kg is over 90%.

[0040] In the actual launch process, the latitude of the predetermined area, the launch point altitude, the orbital inclination angle and the orbital altitude are input into the trained neural network model, and the carrying capacity result can be obtained in 0.984 seconds, which is much less than the time (5 minutes) generated by the carrying capacity calculation software.

[0041] This embodiment designs a simple and effective shallow neural network to fit moderate regression tasks. The linearity of the output layer ensures the flexibility of the regression, and the gradient descent combined with mean squared error loss is suitable for continuous value prediction.

[0042] This method uses a pre-trained dataset to predict rocket payload capacity. By comparing the software calculation time with the neural network response time, it demonstrates extremely high efficiency. By using a multilayer perceptron neural network to replace ballistic calculation software, it can quickly and accurately calculate the payload capacity of launch vehicles under different launch point latitudes, altitudes, orbital inclinations, and orbital altitudes. It reduces the calculation time for the payload capacity of a single trajectory from 5 minutes to less than 1 second, while keeping the payload capacity error within 1%. This transforms a highly real-time calculation task into routine calculation work, achieving rapid and accurate calculation of payload capacity and strongly supporting launch missions under maneuvering conditions.

Claims

1. A rapid calculation method for the payload capacity of a solid rocket under maneuvering conditions, characterized in that, Includes the following steps: Step 1: Training data preparation: Grid the latitude, launch point altitude, orbital inclination, and orbital altitude of the predetermined region; Calculate the carrying capacity at each grid point, and organize the calculated data into a training database. The inputs are latitude, launch point altitude, orbital inclination, and orbital altitude, and the output is the carrying capacity. Step 2, Neural Network Design and Training: A multilayer perceptron neural network is used, with the following structure: The input layer has 4 neurons, corresponding to the four input variables: latitude, launch point altitude, orbital inclination, and orbital altitude; the hidden layer has 15 neurons, responsible for nonlinear mapping of the input variables; the output layer has 1 neuron, corresponding to the carrying capacity output; the multilayer perceptron neural network uses a fully connected approach, and the parameters of the multilayer perceptron neural network include weights and biases. From the input layer to the hidden layer: the weight matrix has a dimension of 4×15 and is initialized to Gaussian random numbers multiplied by 0.1; the bias has a dimension of 1×15 and is initialized to 0, used to adjust the starting point of the linear combination of the hidden layers; From hidden layer to output layer: The weight matrix has a dimension of 15×1 and is initialized to Gaussian random numbers multiplied by 0.1; the bias has a dimension of 1×1 and is initialized to 0, used to adjust the starting point of the linear combination of the output layer; The multilayer perceptron neural network is trained using the training database obtained in step one. After training is completed, the trained neural network model is saved. Step 3: Calculate the carrying capacity: During the actual launch, the latitude of the predetermined area, the launch point altitude, the orbital inclination angle, and the orbital altitude are input into the trained neural network model to obtain the carrying capacity.

2. The method for rapidly calculating the payload capacity of a solid rocket under maneuvering conditions according to claim 1, characterized in that, The hidden layers of a multilayer perceptron neural network use the sigmoid activation function to introduce a nonlinear transformation, enabling the multilayer perceptron neural network to fit complex input-output relationships. Its derivative is used for gradient calculation during backpropagation. The output layer has no activation function.

3. The method for rapidly calculating the payload capacity of a solid rocket under maneuvering conditions according to claim 1, characterized in that, In step two, the neural network is trained using gradient descent, with mean squared error as the loss function. The parameter gradient is calculated using the chain rule, and the weights and biases are updated in the direction of decreasing loss.

4. The method for rapidly calculating the payload capacity of a solid rocket under maneuvering conditions according to claim 3, characterized in that, In step two, the training data in the training database is divided into a training set and a validation set. The training set is used to train the multilayer perceptron neural network, and the validation set is used to monitor the training process and prevent overfitting. The neural network training process is as follows: Let the neural network prediction value be The actual value is The number of training sets is Define mean square error : ; Let the input vector be The weight matrix from the input layer to the hidden layer is: The bias vector of the hidden layer is The activation function is The weight matrix from the hidden layer to the output layer is: The bias vector of the output layer is , The learning rate is set to 0.

1. For the output of a single neuron in the hidden layer, For the output of a single neuron in the output layer, For output of the hidden layer, Represents matrix transpose and hidden layer calculation: 、 ； Output layer computation: , ; Calculate the output layer gradient using the chain rule: , , ; Hidden layer gradient: , , , , This represents element-wise multiplication. Activation function The derivative; Update the weights and biases using gradient descent: , , , .

5. The method for rapidly calculating the payload capacity of a solid rocket under maneuvering conditions according to claim 3, characterized in that, In step two, the number of iterations is set to 10,000.

6. The method for rapidly calculating the payload capacity of a solid rocket under maneuvering conditions according to claim 4, characterized in that, In step two, the ratio of the training set to the validation set is 7:

3.

7. The method for rapidly calculating the payload capacity of a solid rocket under maneuvering conditions according to claim 1, characterized in that, In step one, the method for calculating the carrying capacity is as follows: The carrying capacity of the rocket is estimated using the Tsiolkovsky formula. The calculation model for the final stage velocity of a multi-stage rocket is as follows: In the formula: For the first Specific impulse of a stage engine; For the first The ratio of the stage's effective payload to its initial mass. For the first The ratio of the mass of the stage structure to the initial mass of the stage. The velocity of the rocket's final stage. This indicates the loss of velocity caused by air and gravity.

8. The method for rapidly calculating the payload capacity of a solid rocket under maneuvering conditions according to claim 1, characterized in that, In step one, for rockets in the application stage, the rocket's structural parameters and guidance scheme are already fixed, and the rocket's final stage velocity... The launch capacity is determined by the orbital altitude, orbital inclination, launch site latitude, and launch site altitude; the analysis method for rocket payload capacity includes the following steps: S1. Analysis of the impact of carrying capacity, orbital inclination, and launch point latitude; set up The latitude of the launch point. If the target orbital inclination is [value], then the required inclination adjustment is [value]. The fuel requirement for adjusting the orbital inclination is determined by the change in the velocity vector direction, which requires a velocity increment. Follow the formula , The current orbital velocity, For the change in tilt angle; fuel consumption and Directly related , For the engine's specific impulse. It is the acceleration due to gravity. , Mass before and after launch; The greater the demand, the higher the fuel consumption. An exponential increase directly impacts carrying capacity; S2. Analyze the relationship between carrying capacity and track altitude; rocket final stage speed demand and Positive correlation Represents the Earth's radius. This refers to the altitude of the satellite's orbit above the Earth's surface.

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