Geofire transfer orbit real-time two-dimensional calculation method based on variable step size method

Abstract

The invention discloses a real-time two-dimensional calculation method for a ground fire transfer orbit based on a variable step size method, and the method mainly comprises the following steps: 1, setting flight time, and obtaining gravitational constants of the earth, the Mars and the sun; 2, initializing the system; 3, judging a time stage; 4, screening time steps; 5, calculating the motion trail of the spacecraft based on the Newton's law of motion and the universal gravitation law; 6, solving the position and speed of the spacecraft through a numerical integration method; 7, updating the mass change of the spacecraft in real time; and 8, drawing a spacecraft orbit and outputting related data. The method can be used for simulating the orbit behavior of the spacecraft between the earth and the Mars, especially under the condition that a nuclear power propulsion system is considered. According to the method, the motion trail of the spacecraft in the complex gravitational field can be accurately calculated based on classical mechanics and numerical simulation technologies, and theoretical support is provided for design and optimization of aerospace tasks.

Description

Technical Field

[0001] This invention relates to the field of spacecraft orbit simulation technology, specifically to a real-time two-dimensional calculation method for Earth-Mars transfer orbit based on the variable step size method, particularly for simulating the orbital dynamics of a spacecraft between Earth and Mars when considering a nuclear electric propulsion (NEP) system. Background Technology

[0002] Numerical integration is one of the core techniques for solving differential equations in dynamical systems, and it is widely used in orbital dynamics, molecular dynamics, computational fluid dynamics, and other fields. In orbital dynamics simulations, spacecraft trajectory calculations typically rely on numerical integration methods to solve the equations of motion under the influence of gravity, thrust, and other external forces. Common numerical integration methods include the Euler method, the Runge-Kutta method, and symplectic integration methods. However, fixed-step methods may have limitations in efficiency and accuracy in complex orbital environments, especially when spacecraft trajectories involve strong gravitational gradient changes, long-distance deep space missions, or unstable perturbations. In these cases, traditional fixed-step methods may struggle to balance computational accuracy and efficiency.

[0003] Variable-step numerical integration is a technique that adaptively adjusts the time step based on the system state to improve computational efficiency while maintaining accuracy. Its core idea is to use a smaller step size when state variables change drastically to reduce errors, and to increase the step size when state changes are gradual to reduce computational costs. Variable-step methods are widely used in orbital dynamics simulations because they can automatically adjust the step size, ensuring stable accuracy and computational efficiency in trajectory calculations at different stages.

[0004] Existing orbital dynamics simulations include the variable-step-size Runge-Kutta method and the variable-step-size symplectic integral method. The variable-step-size Runge-Kutta method employs an embedded error estimation mechanism, possessing multi-perturbation adaptability, and is mainly applied to short-term orbit prediction missions. However, this method has significant drawbacks: error accumulation easily occurs during long-term integration, accuracy fluctuates significantly under strong perturbation environments, and the computational cost is high due to the need for additional embedded solution calculations per step. The variable-step-size symplectic integral method adjusts the step size through time transformation, maintaining the symplectic property of the Hamiltonian system, and is suitable for long-term deep-space exploration orbit integration. However, it suffers from a compatibility issue between step size adjustment and symplectic properties, and has poor adaptability to non-conservative forces such as electric propulsion. Both techniques share common drawbacks: insufficient adaptability to multi-scale orbital scenarios, and error tolerance settings rely on engineering experience, lacking adaptive optimization mechanisms. Summary of the Invention

[0005] To accurately simulate the orbital dynamics of spacecraft in complex gravitational fields, especially considering nuclear electric propulsion (NEP) systems, this invention proposes a real-time two-dimensional calculation method for Earth-Mars transfer orbits based on the variable step-size method. This method accurately calculates the spacecraft's trajectory through numerical integration and Newton's laws of motion, and updates the spacecraft's mass changes in real time, providing theoretical support for the design and optimization of space missions.

[0006] To achieve the above objectives, the present invention adopts the following technical solution: A real-time two-dimensional calculation method for Earth-Mars transfer trajectory based on the variable step size method is proposed, with the following steps: Step 1: First, set the total flight time of the spacecraft and divide the total flight time into propulsion phase and unpowered gliding phase; and obtain the gravitational constants of Earth, Mars and Sun. The gravitational constants of Earth, Mars and Sun are adjusted according to the research purpose. The calculation formula is shown in formula (1). Formula (1) Represents the gravitational constant, N·m 2 ·kg -2 The mass of the sun is expressed in kg. The mass of the Earth is expressed in kg. The mass of Mars is expressed in kg. The vector representing the Earth's position relative to the Sun, in meters (m); The first component, m, represents the vector array representing the Earth's position relative to the Sun. The second component of the spacecraft's velocity vector relative to Earth, m s-1; Step 2: Initialize the nuclear propulsion system by setting the initial mass, initial position, initial velocity, and initial orbital parameters of the spacecraft, and setting the parameters of the nuclear propulsion system, including thrust and fuel consumption rate; Step 3: Determine the current time period; Step 4: Perform time step filtering to obtain the time step for this calculation. The calculation formula is shown in formula (2): Formula (2) In the formula: The vector representing the spacecraft's real-time position relative to the sun, in m; Represents the spacecraft's position vector The transpose vector, m; This represents the spacecraft's real-time velocity vector relative to the Sun at time t, in m·s. -1 ; Represents the spacecraft velocity vector The transpose vector, m·s -1 ; The velocity vector of the spacecraft relative to the Earth, m s-1; The vector representing the position of Mars relative to the Sun is m; The velocity vector m of the spacecraft relative to Mars. s-1; The ratio of the spacecraft's position modulus to its velocity modulus relative to the Sun, expressed in s; The ratio of the position modulus to the velocity modulus of a spacecraft relative to Earth, expressed in s; The ratio of the spacecraft's position modulus to its velocity modulus relative to Mars, expressed in s; This represents the time step size (in seconds) for variable-step integration. Step 5: Based on Newton's laws of motion and the law of universal gravitation, establish a spacecraft orbital dynamics model that includes the gravitational constants of Earth, Mars, and the Sun, as shown in formula (3). If it is the gliding phase, the acceleration vector of the electric thruster on the spacecraft is not included when calculating the total acceleration vector in formula (3). ; Formula (3) In the formula: The vector representing the position of Mars relative to the Sun is m; The unit vector representing the first component of the spacecraft's velocity direction, in m·s. -1 ; The unit vector representing the second component of the spacecraft's velocity direction, in m·s. -1 ; The thrust of an electric thruster is expressed in N. Indicates the number of electric thrusters; This indicates the real-time mass of the spacecraft, expressed in kg. The vector representing the acceleration of the spacecraft by the sun, in m·s -2 ; This represents the Earth's acceleration vector over the spacecraft, in m·s. -2 ; This represents the acceleration vector of Mars on the spacecraft, in m·s. -2 ; The acceleration vector of the electric thruster on the spacecraft is expressed in m·s. -2 ; Represents the total acceleration vector, m·s -2 ; Step 6: Use the variable step size method to perform time integration on the spacecraft's motion equations, update the spacecraft's position and velocity in real time, and record the spacecraft's position and velocity data, as shown in formulas (4) and (5); Formula (4) Formula (5) The vector representing the spacecraft's position relative to the sun at time t is m; express The position vector of the spacecraft relative to the sun at any given moment, in meters; This represents the spacecraft's real-time velocity vector relative to the Sun at time t, in m·s. -1 ; express The spacecraft's velocity vector relative to the sun at any given moment, in m·s -1 ; This represents the time step size (in seconds) for variable-step integration. The vector of total acceleration at time t, in m·s -2 ; express The total acceleration vector at time t, m·s -2 ; Step 7: Consider the mass change of the nuclear power propulsion system and update the mass of the spacecraft in real time, as shown in formula (6); Formula (6) express Real-time mass of spacecraft, kg; express Real-time mass of spacecraft, kg; The mass flow rate of a single nuclear electric thruster is expressed in kg·s. -1 ; Step 8: Based on the recorded spacecraft position, velocity data and updated mass data, plot the spacecraft's orbital trajectory and output data on the changes in the spacecraft's position, velocity and mass over time.

[0007] The equation of motion in step 6, whose total acceleration vector is obtained by superimposing the acceleration vectors of the Sun on the spacecraft, the Earth on the spacecraft, the Mars on the spacecraft, and the electric thruster on the spacecraft, fully reproduces the real mechanical environment of the spacecraft under the combined gravitational influence of multiple sources during the Earth-Mars transfer process, and avoids the orbital calculation deviation caused by the assumption of a single gravitational source.

[0008] In step 6, the position and velocity of the spacecraft are obtained by time integration of the spacecraft's motion equation using the variable step size method. The variable step size method can adaptively adjust the time step based on the ratio of position to velocity magnitude in formula (2). This not only avoids the accumulation of integration errors caused by excessive step size, but also has the advantages of significantly reducing the amount of integration operations and improving real-time calculation efficiency while ensuring that the calculation accuracy meets the requirements.

[0009] In step 7, the change in the spacecraft's mass is determined by the fuel consumption rate of the nuclear power propulsion system. This can accurately reproduce the impact of fuel consumption on the spacecraft's physical characteristics and avoid orbital deviations caused by the fixed mass assumption. It is especially suitable for scenarios with a high proportion of fuel consumption in long-range Earth-Mars transfer missions.

[0010] Compared with the prior art, the present invention has the following advantages: 1. This method adopts a variable step size time integration strategy to realize real-time two-dimensional calculation of the Earth-Mars transfer orbit, which can update the mass of the spacecraft in real time and ensure the stability and accuracy of numerical calculation. 2. Since the gravitational field parameters of Earth, Mars and the Sun can be adjusted according to the research objectives in step 1, and the initial conditions of the spacecraft can also be adjusted according to the mission requirements, this method can handle complex orbital dynamics problems under multibody gravitational fields. 3. Since step 8 can plot the spacecraft's orbital trajectory and output relevant data, this method can intuitively display the spacecraft's motion state, providing intuitive theoretical support for the design and optimization of space missions.

[0011] In summary, this invention enables the simulation of more complex and extreme spacecraft orbital dynamics using a more economical and accurate method, providing strong theoretical support for the design and optimization of space missions. Attached Figure Description

[0012] Figure 1 This is a flowchart of a real-time two-dimensional calculation method for Earth-Mars transfer trajectory based on the variable step size method of the present invention.

[0013] Figure 2 This is an example of step 1 of the present invention—a model of a ground-fire transfer device.

[0014] Figure 3 This is a calculation example of step 7 of the present invention—a schematic diagram of the output results of a two-dimensional calculation model of a nuclear power-driven Earth-Mars transfer orbit. Detailed Implementation

[0015] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0016] like Figure 1 As shown, this invention is a real-time two-dimensional calculation method for Earth-Mars transfer orbit based on the variable step size method. The main steps are as follows: Step 1: First, set the total flight time, dividing it into propulsion and unpowered gliding phases. Obtain the gravitational constants of Earth, Mars, and the Sun. These constants can be adjusted according to the research objective; in this embodiment, they are set as shown in formula (1). Formula (1) Represents the gravitational constant, N·m 2 ·kg -2 The mass of the sun is expressed in kg. The mass of the Earth is expressed in kg. The mass of Mars is expressed in kg. The vector representing the Earth's position relative to the Sun, in meters (m); The first component, m, represents the vector array representing the Earth's position relative to the Sun. The second component of the spacecraft's velocity vector relative to Earth, m s-1; Step 2: Initialize the nuclear power propulsion system by setting the initial mass, initial position, initial velocity, and initial orbital parameters of the spacecraft, and setting the parameters of the nuclear power propulsion system, including thrust and fuel consumption rate. Step 3: Determine the current time period; Step 4: Perform time step filtering to obtain the time step for this calculation. The calculation formula is shown in formula (2): Formula (2) In the formula: The vector representing the spacecraft's real-time position relative to the sun, in m; Represents the spacecraft's position vector The transpose vector, m; This represents the spacecraft's real-time velocity vector relative to the Sun at time t, in m·s. -1 ; Represents the spacecraft velocity vector The transpose vector, m·s -1 ; The velocity vector of the spacecraft relative to the Earth, m s-1; The vector representing the position of Mars relative to the Sun is m; The velocity vector m of the spacecraft relative to Mars. s-1; The ratio of the spacecraft's position modulus to its velocity modulus relative to the Sun, expressed in s; The ratio of the position modulus to the velocity modulus of a spacecraft relative to Earth, expressed in s; The ratio of the spacecraft's position modulus to its velocity modulus relative to Mars, expressed in s; This represents the time step size (in seconds) for variable-step integration. Step 5: Based on Newton's laws of motion and the law of universal gravitation, establish a spacecraft orbital dynamics model that includes the gravitational field constants of Earth, Mars, and the Sun, as shown in formula (3). If it is the gliding phase, the total acceleration vector calculated in formula (3) does not include the gravitational field constants of Earth, Mars, and the Sun. The acceleration vector of the electric thruster on the spacecraft, and the physical model of the spacecraft as follows: Figure 2 As shown, the spacecraft mainly consists of modules such as the reactor core, electric propulsion system, hydrogen storage tank, and reentry capsule; Formula (3) In the formula: The vector representing the spacecraft's real-time position relative to the sun, in m; Represents the spacecraft's position vector The transpose vector, m; The vector representing the position of Mars relative to the Sun is m; This represents the spacecraft's real-time velocity vector relative to the Sun at time t, in m·s. -1 ; The transpose of the spacecraft's velocity vector v, m·s -1 ; The unit vector representing the first component of the spacecraft's velocity direction, in m·s. -1 ; The unit vector representing the second component of the spacecraft's velocity direction, in m·s. -1 ; This represents the time step size (in seconds) for variable-step integration. The thrust of an electric thruster is expressed in N. Indicates the number of electric thrusters; This indicates the real-time mass of the spacecraft, expressed in kg. The vector representing the acceleration of the spacecraft by the sun, in m·s -2 ; This represents the Earth's acceleration vector over the spacecraft, in m·s. -2 ; This represents the acceleration vector of Mars on the spacecraft, in m·s. -2 ; The acceleration vector of the electric thruster on the spacecraft is expressed in m·s. -2 ; Represents the total acceleration vector, m·s -2 ; Step 6: Use the variable step size method to perform time integration on the spacecraft's motion equations, update the spacecraft's position and velocity in real time, and record the spacecraft's position and velocity data, as shown in formulas (4) and (5); Formula (4) Formula (5) The vector representing the spacecraft's position relative to the sun at time t is m; express The position vector of the spacecraft relative to the sun at any given moment, in meters; This represents the spacecraft's real-time velocity vector relative to the Sun at time t, in m·s. -1; express The spacecraft's velocity vector relative to the sun at any given moment, in m·s -1 ; This represents the time step size (in seconds) for variable-step integration. The vector of total acceleration at time t, in m·s -2 ; express The total acceleration vector at time t, m·s -2 ; Step 7: Consider the mass change of the nuclear power propulsion system and update the mass of the spacecraft in real time, as shown in formula (6); Formula (6) express Real-time mass of spacecraft, kg; express Real-time mass of spacecraft, kg; This represents the mass flow rate of a single NEP thruster, in kg·s. -1 ; Step 8: Based on the recorded position, velocity data, and updated mass data, plot the spacecraft's orbital trajectory and output data on the changes in the spacecraft's position, velocity, and mass over time, such as... Figure 3 The results shown can output mission time, time step, and real-time trajectory of the spacecraft.

[0017] In summary, steps 1 to 3 initialize the gravitational parameters of the celestial body and the physical parameters of the spacecraft; steps 4 to 6 calculate the real-time position and velocity of the spacecraft relative to the Sun; step 7 updates the spacecraft mass in real time; and step 8 plots the spacecraft's trajectory in real time. Through these steps, the trajectory of the spacecraft during the Earth-Mars transfer was simulated, and the real-time position, mass, and arrival time of the spacecraft under electric propulsion were obtained. The mission time of the Earth-Mars transfer under different initial parameters was also calculated, thus providing strong theoretical support for the design and optimization of space missions.

[0018] This invention, based on the variable step-size method, proposes a novel approach for simulating spacecraft orbital dynamics. This method can accurately calculate the trajectory of a spacecraft in the gravitational fields of Earth, Mars, and the Sun, particularly when considering nuclear electric propulsion (NEP) systems. Because the thrust patterns of NEP systems vary significantly compared to traditional chemical propulsion, the thrust duration is longer, and thrust variations may be influenced by control strategies, traditional fixed-step-size methods struggle to balance accuracy and computational efficiency. By introducing a variable step-size strategy, this invention allows for improved accuracy during the spacecraft's acceleration phase and improved computational efficiency during the cruise phase, thereby enhancing the overall performance of orbital calculations while maintaining accuracy.

Claims

1. A real-time two-dimensional calculation method for Earth-Mars transfer trajectory based on the variable step size method, characterized in that: The steps are as follows: Step 1: First, set the total flight time of the spacecraft and divide the total flight time into propulsion phase and unpowered gliding phase; and obtain the gravitational constants of Earth, Mars and Sun. The gravitational constants of Earth, Mars and Sun are adjusted according to the research purpose. The calculation formula is shown in formula (1). Official (1) Represents the gravitational constant, N·m 2 ·kg -2 ; The mass of the sun is expressed in kg. The mass of the Earth is expressed in kg. The mass of Mars is expressed in kg. The vector representing the Earth's position relative to the Sun, in meters (m); The first component, m, represents the vector array representing the Earth's position relative to the Sun. The second component of the spacecraft's velocity vector relative to Earth, m s-1; Step 2: Initialize the nuclear propulsion system by setting the initial mass, initial position, initial velocity, and initial orbital parameters of the spacecraft, and setting the parameters of the nuclear propulsion system, including thrust and fuel consumption rate; Step 3: Determine the current time period; Step 4: Perform time step filtering to obtain the time step for this calculation. The calculation formula is shown in formula (2): Official (2) In the formula: The velocity vector of the spacecraft relative to the Earth, m s-1; The vector representing the position of Mars relative to the Sun is m; The velocity vector m of the spacecraft relative to Mars. s-1; The vector representing the spacecraft's real-time position relative to the sun, in m; Represents the spacecraft's position vector The transpose vector, m; This represents the spacecraft's real-time velocity vector relative to the Sun at time t, in m·s. -1 ; Represents the spacecraft velocity vector The transpose vector, m·s -1 The ratio of the spacecraft's position modulus to its velocity modulus relative to the Sun, expressed in s; The ratio of the position modulus to the velocity modulus of a spacecraft relative to Earth, expressed in s; The ratio of the spacecraft's position modulus to its velocity modulus relative to Mars, s, This represents the time step size (in seconds) for variable-step integration. Step 5: Based on Newton's laws of motion and the law of universal gravitation, establish a spacecraft orbital dynamics model that includes the gravitational constants of Earth, Mars, and the Sun, as shown in formula (3). If it is the gliding phase, the acceleration vector of the electric thruster on the spacecraft is not included when calculating the total acceleration vector in formula (3). ; Official (3) In the formula: The vector representing the position of Mars relative to the Sun is m; The unit vector representing the first component of the spacecraft's velocity direction, in m·s. -1 ; The unit vector representing the second component of the spacecraft's velocity direction, in m·s. -1 ; The thrust of an electric thruster is expressed in N. Indicates the number of electric thrusters; This indicates the real-time mass of the spacecraft, expressed in kg. The vector representing the acceleration of the spacecraft by the sun, in m·s -2 ; This represents the Earth's acceleration vector over the spacecraft, in m·s. -2 ; This represents the acceleration vector of Mars on the spacecraft, in m·s. -2 ; The acceleration vector of the electric thruster on the spacecraft is expressed in m·s. -2 ; Represents the total acceleration vector, m·s -2 ; Step 6: Use the variable step size method to perform time integration on the spacecraft's motion equations, update the spacecraft's position and velocity in real time, and record the spacecraft's position and velocity data, as shown in formulas (4) and (5); Official (4) Official (5) The vector representing the spacecraft's position relative to the sun at time t is m; express The position vector of the spacecraft relative to the sun at any given moment, in meters; This represents the spacecraft's real-time velocity vector relative to the Sun at time t, in m·s. -1 ; express The spacecraft's velocity vector relative to the sun at any given moment, in m·s -1 ; This represents the time step size (in seconds) for variable-step integration. The vector of total acceleration at time t, in m·s -2 ; express The total acceleration vector at time t, m·s -2 ; Step 7: Consider the mass change of the nuclear power propulsion system and update the mass of the spacecraft in real time, as shown in formula (6); Official (6) express Real-time mass of the spacecraft, in kg; express Real-time mass of the spacecraft, in kg; The mass flow rate of a single nuclear electric thruster is expressed in kg·s. -1 ; Step 8: Based on the recorded spacecraft position, velocity data and updated mass data, plot the spacecraft's orbital trajectory and output data on the changes in the spacecraft's position, velocity and mass over time.

2. The real-time two-dimensional calculation method for Earth-Mars transfer trajectory based on the variable step size method according to claim 1, characterized in that: The equation of motion in step 6 is obtained by superimposing the acceleration vector of the spacecraft on the sun, the acceleration vector of the spacecraft on the earth, the acceleration vector of the spacecraft on Mars, and the acceleration vector of the spacecraft on the electric thruster.

3. The real-time two-dimensional calculation method for Earth-Mars transfer trajectory based on the variable step size method according to claim 1, characterized in that: In step 6, the spacecraft's position and velocity are obtained by time integration of the spacecraft's equations of motion using the variable step size method.

4. The real-time two-dimensional calculation method for Earth-Mars transfer trajectory based on the variable step size method according to claim 1, characterized in that: In step 7, the change in the spacecraft's mass is determined by the fuel consumption rate of the nuclear electric propulsion system.

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