An orbit optimization method for a lunar transfer mission

Abstract

The present application relates to the technical field of earth-moon transfer, and specifically discloses an orbit optimization method for an earth-moon transfer mission, comprising: S1, simulating an initial orbit of a spacecraft in an earth-moon space through a relative motion trajectory of a sun around a moon-earth mass center in a double-circle restricted four-body model; S2, obtaining orbit initial values through searching of an initial orbit according to search parameters; S3, screening the orbit initial values through preset constraint conditions to obtain an optimized trajectory; S4, performing multi-segment optimization on the optimized trajectory through a multiple shooting method to obtain a target function of a nonlinear programming; and S5, solving the target function to obtain a target orbit. The present application can significantly improve the quality of trajectory design and the success probability of a mission, and provides a more flexible, efficient and reliable orbit design means for future lunar exploration and deep space exploration missions.

Description

Technical Field

[0001] This invention relates to the field of Earth-Moon transfer technology, and more specifically to an orbit optimization method for an Earth-Moon transfer mission. Background Technology

[0002] In Earth-Moon transfer missions, the traditional approach typically employs the Hohmann transfer orbit. While simple to design and reliable to implement, the Hohmann transfer orbit usually requires a higher velocity increment, which translates to a significant increase in fuel consumption. This limits the widespread use of fuel-sensitive small spacecraft (such as CubeSats) in missions. Furthermore, the Hohmann transfer has significant limitations on launch windows and lower flexibility, resulting in substantial constraints on mission planning.

[0003] To address these issues, various low-energy transfer methods have been proposed in recent years, including WeakStability Boundary (WSB), low-thrust transfer (such as electric propulsion), and planetary or lunar gravitational assist. These methods significantly reduce fuel requirements and improve the fuel efficiency of trajectory design by utilizing the characteristics of complex dynamic systems (such as gravitational boost, unstable manifolds, and solar perturbations). However, these methods are typically accompanied by long transfer times, increasing the complexity of mission planning and placing higher demands on trajectory design and optimization. Existing trajectory design methods often face convergence problems when faced with more constraints. This is especially true when specific mission requirements are involved (such as avoiding spacecraft entry into the Earth's shadow or ensuring mission completion within a specific mission time window), making it even more difficult to design effective trajectories. This is because trajectory design in the Earth-Moon space not only needs to satisfy physical constraints in nonlinear dynamic systems but also requires trade-offs between multiple objectives, such as fuel efficiency, mission time, mission safety, and trajectory reliability. These constraints increase the complexity of trajectory design, causing optimization algorithms to often search for suitable solutions in a wide parameter space, easily getting trapped in local optima or convergence problems. In addition, there are a large number of unstable regions in the Earth-Moon space. These regions are extremely sensitive to the initial conditions of the trajectory, which means that even a slight deviation in the initial state of the spacecraft may lead to the failure to reach the target orbit.

[0004] With the continuous evolution of mission requirements, exploring more efficient, flexible, and robust trajectory design methods has become an important direction in current Earth-Moon transfer research. By combining novel dynamic modeling tools with more advanced numerical optimization algorithms, such as using the restricted four-body problem model (considering the gravitational interactions of the Sun, Earth, and Moon) for more accurate modeling, scientists can better describe the trajectory evolution of spacecraft in Earth-Moon space. Simultaneously, intelligent trajectory design methods improve the globality and convergence of solutions, avoiding getting trapped in local optima. These new methods not only significantly reduce fuel consumption but also effectively maintain trajectory stability and mission feasibility under complex constraints. Therefore, future Earth-Moon trajectory design will not only focus on fuel economy but also find the optimal balance between mission flexibility, trajectory robustness, and planning feasibility, thereby contributing to more Earth-Moon exploration missions and human deep space exploration.

[0005] In recent years, the design of Earth-Moon transfer missions has typically used solutions from the three-body model as initial trajectory conditions. However, in the initial condition-sensitive regions of Earth-Moon space, solutions generated by the three-body model often fail to achieve optimal trajectory convergence, increasing the complexity of trajectory design. This sensitivity manifests in the fact that, due to the combined gravitational influence of the Earth and Moon in the Earth-Moon system, the orbital state is highly sensitive to subtle changes in initial conditions, especially near Lagrange points or other unstable regions, where the trajectory is significantly affected by nonlinear perturbations. This phenomenon not only complicates trajectory design but also places higher demands on the stability and convergence of optimization algorithms, making it difficult to obtain accurate optimization results. Therefore, traditional three-body model methods face many limitations when dealing with complex constraints (such as avoiding the Earth's shadow region, strict time windows, specific orbital shapes, etc.).

[0006] Furthermore, current trajectory design research remains insufficient in terms of the complexity of the dynamic environment, particularly lacking a systematic and comprehensive study of the gravitational perturbations and interactions between the Sun, Earth, and Moon. While the Sun's gravity is relatively weak in Earth-Moon transfer missions, its cumulative effect over long-term trajectories cannot be ignored, especially when considering long-duration, low-energy transfers. Solar gravitational perturbations can affect the spacecraft's trajectory evolution and orbital stability. Therefore, neglecting the Sun's influence may result in trajectory designs that fail to meet mission requirements in real-world environments, which would be extremely detrimental to the mission's successful implementation. Summary of the Invention

[0007] To address the aforementioned problems, the purpose of this invention is to provide an orbit optimization method for Earth-Moon transfer missions, which can significantly improve the quality of trajectory design and the probability of mission success, providing a more flexible, efficient, and reliable orbit design method for future lunar and deep space exploration missions.

[0008] This invention provides a method for orbit optimization of a lunar transfer mission, comprising:

[0009] Step S1: Simulate the initial orbit of the spacecraft in Earth-Moon space by using the relative motion trajectory of the Sun around the Earth-Moon mass center in the double-circle restricted four-body model;

[0010] Step S2: Search for the initial orbit according to the search parameters to obtain the initial orbit value; the search parameters include: the angle between the initial velocity and the line connecting the Earth and the Moon, the scaling factor of the spacecraft's initial velocity relative to the local circular velocity, and the initial phase of the Sun relative to the Earth-Moon line;

[0011] Step S3: Filter the initial trajectory values ​​using preset constraints to obtain an optimized trajectory;

[0012] Step S4: The optimized trajectory is optimized in multiple segments using the multiple target method to obtain the objective function of the nonlinear programming.

[0013] Step S5: Solve the objective function to obtain the target orbit.

[0014] In one possible implementation, step S1 includes:

[0015] The initial orbit of a spacecraft can be represented by the following formula:

[0016]

[0017] In the formula, r is the spacecraft's position, v is the spacecraft's velocity, and Q is the spacecraft's state. Let r be the acceleration of the spacecraft, t be the time, g(r,t) be the gravitational acceleration at position r and time t, and h(v) be the acceleration component related to velocity v.

[0018] In one possible implementation, step S1 further includes:

[0019] The spacecraft's state Q can be represented by the following formula:

[0020] Q = [r, v] T

[0021] In the formula, r is the position of the spacecraft, v is the velocity of the spacecraft, and T is the transpose.

[0022] In one possible implementation, step S1 further includes:

[0023] g(r,t) can be expressed by the following formula:

[0024]

[0025] h(v) can be expressed by the following formula:

[0026]

[0027] In the formula, Ω⁴ is the angular velocity, Ω⁴ is the partial derivative, x is the position coordinate, y is the position coordinate. For the spacecraft's velocity component in the x-direction, Let be the velocity component of the spacecraft in the y-direction.

[0028] In one possible implementation, step S1 further includes:

[0029] According to the following formula

[0030]

[0031] In the formula, μ s For the mass of the Sun, a s θ is the distance from the Earth-Moon barycenter to the Sun. s It is the solar phase angle, ω s Ω3 is the relative angular velocity of the Sun, x is the position coordinate, y is the position coordinate, t is the time, Ω3 is the angular velocity, μ is the ratio of the total mass of the Earth-Moon system to the mass of the Sun, m1 is the mass of the Earth, and m2 is the mass of the Moon.

[0032] In one possible implementation, step S2 includes:

[0033] The search parameters are discretized within a preset value range to obtain a multidimensional parameter grid;

[0034] The initial orbital values ​​are obtained by searching on the parameter grid.

[0035] In one possible implementation, step S2 further includes:

[0036] The initial orbital parameters of a spacecraft in the Earth-Moon space are expressed by the following formula:

[0037]

[0038] In the formula, α is the angle between the initial velocity and the line connecting the Earth and the Moon, β is the scaling factor of the spacecraft's initial velocity relative to its local circular velocity, and θ is the angle between the initial velocity and the line connecting the Earth and the Moon. s,0 The initial phase of the Sun relative to the Earth-Moon line, x i and y i The initial position coordinates of the spacecraft in Earth-Moon space. and Let r be the initial velocity components of the spacecraft in the x and y directions. i Let be the distance from the Earth's center to the spacecraft, μ be the ratio of the total mass of the Earth-Moon system to the mass of the Sun, and v be the distance from the Earth's center to the spacecraft. iThe initial velocity of the spacecraft.

[0039] In one possible implementation, step S4 includes:

[0040] The objective function J(q) can be expressed by the following formula:

[0041] J(q)=Δv1+Δv N

[0042] q={Q j ,t1,t N},j=1,2,...,N

[0043]

[0044] t1=t i ,t N =t f

[0045] In the formula, Δv1 and Δv N Let q be the change in velocity, and Q be the variable. j Let j be the state, t1, and t2. N These represent the time to enter a lower orbit and the time to orbit, respectively, t i For the time to enter a lower orbit than the transfer orbit, t f Let j be the orbital insertion time, j be the number of segments in the optimized trajectory, and N be the total number of segments in the optimized trajectory. For time t j The spacecraft's position coordinates at time t j The velocity coordinates of the spacecraft at any given time.

[0046] In one possible implementation, step S4 further includes:

[0047] The boundary constraints can be expressed using the formulas above:

[0048]

[0049] In the formula, x1 and y1 are the position coordinates of the spacecraft at the initial moment, and x... N and y N For the spacecraft's position coordinates at the final moment, and The spacecraft's velocity coordinates at the initial moment, and The spacecraft's velocity coordinates at the final moment, μ is the ratio of the total mass of the Earth-Moon system to the mass of the Sun, r1 is the spacecraft's distance from the Earth-Moon's center of mass at the initial moment, and r N This represents the distance the spacecraft will reach from the Earth-Moon center of mass at the final moment.

[0050] The following formula represents the constraint on task accuracy:

[0051]

[0052] In the formula, R E For Earth's radius, R m The radius of the moon;

[0053] The constraints on task time are expressed by the following formula:

[0054] τ=t1-t N <0

[0055] In the formula, τ represents the time difference, t1 represents the start time of the task, and t N Indicates the time point when the task ends.

[0056] In one possible implementation, step S5 includes:

[0057] The target trajectory is obtained by solving multiple nonlinear programming functions in parallel using the solver in MATLAB.

[0058] The orbit optimization method for Earth-Moon transfer missions provided by this invention not only overcomes the limitations of traditional three-body models, such as sensitivity to initial conditions and difficulty in convergence of trajectory optimization, but also incorporates the combined gravitational effects of the Sun, Earth, and Moon, making the trajectory design closer to the real mission environment. This can significantly improve the quality of trajectory design and the probability of mission success, providing a more flexible, efficient, and reliable orbit design method for future lunar and deep space exploration missions. Attached Figure Description

[0059] Figure 1 A schematic flowchart of the trajectory optimization method provided in an embodiment of the present invention. Detailed Implementation

[0060] The embodiments of the present invention will be further described in detail below with reference to the accompanying drawings and examples. The following detailed description of the embodiments and the accompanying drawings are used to illustrate the principles of the present invention by way of example, but should not be used to limit the scope of the present invention. That is, the present invention is not limited to the described preferred embodiments, and the scope of the present invention is defined by the claims.

[0061] In the description of this invention, it should be noted that, unless otherwise stated, "a plurality of" means two or more; the terms "first," "second," etc., are used for descriptive purposes only and should not be construed as indicating or implying relative importance; those skilled in the art can understand the specific meaning of the above terms in this invention as appropriate.

[0062] This invention proposes a trajectory optimization method for Earth-Moon transfer missions. This method comprehensively considers the gravitational effects of the Earth, Moon, and Sun, describing them under the same model, thus making the trajectory design more consistent with the actual dynamic environment. Compared with traditional three-body problem models, the BCR4BP model, by introducing the Sun as a third primary celestial body, not only improves the accuracy of trajectory modeling but also better describes the trajectory evolution of the spacecraft under the influence of multi-body gravity in Earth-Moon space.

[0063] Under this model, this invention first generates initial trajectory solutions through grid search within a two-circle constrained four-body model, providing a relatively stable initial value for trajectory optimization. Compared to the initial values ​​of a three-body model, these initial solutions better adapt to various constraints in complex dynamic systems, reducing trajectory deviations in unstable regions and decreasing dependence on the optimization process. Secondly, by optimizing the generated initial trajectory using nonlinear programming methods, this invention can effectively find the optimal solution under multi-objective conditions (e.g., minimizing fuel consumption, reducing Δv, controlling transfer time, etc.), enhancing the flexibility and robustness of trajectory design.

[0064] This orbit optimization design method based on the four-body model has significant application value in practical mission planning. It not only solves the problem of trajectory convergence difficulty in initial value-sensitive regions, but also makes the designed trajectory more stable and reliable in lunar space by incorporating the influence of solar gravity into the trajectory optimization process. Furthermore, the optimized trajectory significantly reduces propulsion fuel consumption while meeting mission constraints, substantially improving mission economy and execution efficiency. This has great practical significance and application potential for future lunar exploration missions, especially small spacecraft missions with strict limitations on propulsion systems and fuel budgets (such as CubeSats).

[0065] Figure 1 A schematic flowchart of the trajectory optimization method provided in the embodiments of the present invention is shown below. Figure 1 As shown, this invention provides an orbit optimization method for a lunar transfer mission, comprising:

[0066] Step S1: Simulate the initial orbit of the spacecraft in Earth-Moon space by using the relative motion trajectory of the Sun around the Earth-Moon mass center in the double-circle restricted four-body model;

[0067] In one possible approach, deep space exploration has long been a key national research area, and multibody models have been a subject of study for many scholars. The two-circle restricted four-body problem is a special case of the four-body problem, where two celestial bodies are in circular Keplerian motion around their center of mass, and their center of mass is in circular Keplerian motion with another celestial body. The mass of the fourth body is negligible compared to the masses of the others. This problem is called the two-circle restricted four-body problem. The two-circle restricted four-body model describes the dynamic properties of the system more accurately than the circular restricted three-body model, and orbits designed based on the four-body model are closer to reality. The equations of motion for the two-circle restricted four-body model can be derived from Newton's law of universal gravitation, allowing for the study of the influence of the fourth body's characteristic parameters on the position and frequency of the translational point, providing a framework for future deep space exploration. The two-circle restricted four-body model incorporates the gravitational influence of the Earth, Moon, and Sun. Under this model, the relative motion of the Sun around the Earth-Moon center of mass is simplified to a circular orbit. In this way, the dynamic behavior of spacecraft in the Earth-Moon space can be described more accurately.

[0068] The initial orbit of a spacecraft can be represented by the following formula:

[0069]

[0070] In the formula, r is the spacecraft's position, v is the spacecraft's velocity, and Q is the spacecraft's state. Let r be the acceleration of the spacecraft, t be the time, g(r,t) be the gravitational acceleration at position r and time t, and h(v) be the acceleration component related to velocity v.

[0071] The spacecraft's state Q can be represented by the following formula:

[0072] Q = [r, v] T

[0073] In the formula, r is the position of the spacecraft, v is the velocity of the spacecraft, and T is the transpose.

[0074] g(r,t) can be expressed by the following formula:

[0075]

[0076] h(v) can be expressed by the following formula:

[0077]

[0078] In the formula, The boundary of region Ω4 is represented by Ω4, where x and y are coordinates. and For the velocity component.

[0079] According to the following formula

[0080]

[0081] In the formula, μ s For the mass of the Sun, a s θ is the distance from the Earth-Moon barycenter to the Sun. s It is the solar phase angle, ω s Ω3 is the relative angular velocity of the Sun, x is the position coordinate, y is the position coordinate, t is the time, Ω3 is the angular velocity, μ is the ratio of the total mass of the Earth-Moon system to the mass of the Sun, m1 is the mass of the Earth, and m2 is the mass of the Moon.

[0082] Assume the initial track is a circular track with a height of h. i =167km, final lunar orbit altitude h f =100km, calculate the time t for the double-pulse transfer to enter the Earth-Moon transfer orbit. i The corresponding pulse size is Δv i The orbital insertion time is t f The corresponding pulse size is Δv f Then we have:

[0083]

[0084] Where, r i and r f These represent the distances from the center of the Earth and the Moon to the spacecraft, respectively.

[0085] The transition time is Δt = Δt i +Δt f The transfer velocity increment is Δv = Δv i +Δv f

[0086] Step S2: Search for the initial orbit based on the search parameters to obtain the initial orbit value;

[0087] The search parameters include: the angle between the initial velocity and the line connecting the Earth and the Moon, the scaling factor of the spacecraft's initial velocity relative to the local circular velocity, and the initial phase of the Sun relative to the Earth-Moon line;

[0088] In one possible implementation, the search parameters are discretized within a preset range to obtain a multidimensional parameter grid; the initial orbital value is obtained by searching on the parameter grid.

[0089] Define the search parameters as α, β, θ s,0 Where α is the angle between the initial velocity and the line connecting the Earth and the Moon, and β is the scaling factor of the spacecraft's initial velocity relative to its local circular velocity. θ s,0This represents the initial phase of the Sun relative to the Earth-Moon line.

[0090] Table 1 shows the parameter range.

[0091] Table 1

[0092]

[0093] The above parameters are all discretized within a reasonable range of values, forming a multidimensional parameter grid. By searching on this grid, better initial conditions can be provided for subsequent trajectory optimization.

[0094] In one possible implementation, the initial orbit of the spacecraft in Earth-Moon space is represented by the following formula:

[0095]

[0096] In the formula, α is the angle between the initial velocity and the line connecting the Earth and the Moon, β is the scaling factor of the spacecraft's initial velocity relative to its local circular velocity, and θ is the angle between the initial velocity and the line connecting the Earth and the Moon. s,0 The initial phase of the Sun relative to the Earth-Moon line, x i and y i The initial position coordinates of the spacecraft in Earth-Moon space. and Let r be the initial velocity components of the spacecraft in the x and y directions. i Let be the distance from the Earth's center to the spacecraft, μ be the ratio of the total mass of the Earth-Moon system to the mass of the Sun, and v be the distance from the Earth's center to the spacecraft. i The initial velocity of the spacecraft.

[0097] For each set of initial values, the spacecraft's trajectory is calculated through numerical integration. During the integration process, the spacecraft's state is continuously updated to evaluate whether the initial values ​​can generate a valid trajectory. During integration, a large number of invalid solutions exist; without adding constraints, computational resources would be wasted.

[0098] Step S3: Filter the initial trajectory values ​​using preset constraints to obtain the optimized trajectory;

[0099] In one possible implementation, the preset constraints for the initial orbit values ​​are expressed by the following formula:

[0100] Δv≤3200m / s,t≤200day

[0101] or

[0102] Δv≤3140m / s,t≤200day

[0103] In the formula, Δv represents the change in velocity and t represents time.

[0104] Step S4: The optimized trajectory is optimized in multiple segments using the multiple target shooting method to obtain the objective function of the nonlinear programming.

[0105] In one possible implementation, the continuous trajectory obtained from the initial guess is discretized, and the trajectory is optimized in multiple segments using a multi-shot method, thus transforming the optimal control problem into a nonlinear programming (NLP) problem.

[0106] Divide the initial trajectory into N-1 segments, evenly distributed across N nodes. Define the variables for the nonlinear programming problem:

[0107] q={Q j ,t1,t N},j=1,2,...,N

[0108] The state at time j:

[0109]

[0110] The objective function J(q) can be expressed by the following formula:

[0111]

[0112] In the formula, Δv1 and Δv N Let q be the change in velocity, and Q be the variable. j Let j be the state, t1, and t2. N These represent the time to enter a lower orbit and the time to orbit, respectively, t i For the time to enter a lower orbit than the transfer orbit, t f Let j be the orbital insertion time, j be the number of segments in the optimized trajectory, and N be the total number of segments in the optimized trajectory. For time t j The spacecraft's position coordinates at time t j The velocity coordinates of the spacecraft at any given time.

[0113] In one possible implementation, step S4 further includes:

[0114] The boundary constraints can be expressed using the formulas above:

[0115]

[0116] In the formula, x1 and y1 are the position coordinates of the spacecraft at the initial moment, and x... N and y N For the spacecraft's position coordinates at the final moment, and The spacecraft's velocity coordinates at the initial moment, and The spacecraft's velocity coordinates at the final moment, μ is the ratio of the total mass of the Earth-Moon system to the mass of the Sun, r1 is the spacecraft's distance from the Earth-Moon's center of mass at the initial moment, and r N This represents the distance the spacecraft will reach from the Earth-Moon center of mass at the final moment.

[0117] The following formula represents the constraint on task accuracy:

[0118]

[0119] In the formula, R E For Earth's radius, R m The radius of the moon;

[0120] The constraints on task time are expressed by the following formula:

[0121] τ=t1-t N <0

[0122] In the formula, τ represents the time difference, t1 represents the start time of the task, and t N Indicates the time point when the task ends.

[0123] Step S5: Solve the objective function to obtain the target orbit.

[0124] In one possible implementation, multiple nonlinear programming functions are solved in parallel using the solver in MATLAB to obtain the target trajectory.

[0125] Specifically, the constraint solver "fmincon" in MATLAB is used to solve the nonlinear programming problem. This solver uses a sequential quadratic programming (SQP) algorithm, which is suitable for optimization problems with complex constraints.

[0126] To ensure computational accuracy, all tolerances were set to 10⁻¹⁰ to meet the accuracy requirements of the trajectory under a high-fidelity model. The trajectory was solved using the 7th and 8th order Runge-Kutta methods.

[0127] By utilizing MATLAB's parallel computing capabilities, different initial trajectory solutions can be processed in parallel using the "parfor" function, thereby accelerating the solution process.

[0128] In one possible implementation approach, low-thrust transfer and weak stability boundary methods are more suitable for missions requiring significant reduction in fuel consumption and adaptation to complex dynamic environments; while for short-duration missions, dual-pulse Hohmann transfer and lunar leverage methods may be more appropriate. In practical mission planning, due to the complexity and diversity of missions, a single trajectory design method is often insufficient to meet all mission requirements.

[0129] Therefore, hybrid strategies are gradually becoming an important trajectory design approach in Earth-Moon transfer missions. Hybrid strategies combine multiple trajectory design methods to fully leverage the advantages of different approaches, thereby further improving mission flexibility and fuel efficiency. For example, low-thrust transfer can be combined with lunar gravitational pull: initially using low thrust to achieve slow trajectory changes to save fuel, while accelerating closer to the Moon by leveraging lunar gravity, thus reducing further fuel requirements.

[0130] Similarly, weak stability boundaries and double-pulse Hohmann shifts can be combined: first, the spacecraft enters the weak gravitational region near the Moon at low energy using the WSB method, and then a single double pulse propels the spacecraft rapidly into the target lunar orbit. This hybrid strategy not only saves fuel but also shortens transfer time, making orbit design more flexible and efficient.

[0131] This invention proposes a trajectory optimization method for Earth-Moon transfer missions. By comprehensively modeling the gravitational interactions between the Earth, Moon, and Sun, it significantly improves the accuracy of trajectory design and its adaptability in practical applications. Under the double-circle constrained four-body model, the gravitational influences of the Sun, Earth, and Moon are integrated into a unified dynamic framework, enabling trajectory design to not only reduce fuel consumption but also significantly expand the launch window, providing more flexible mission planning capabilities.

[0132] This invention first establishes a mathematical description of the gravitational forces of the Earth, Moon, and Sun through dynamic modeling, and then uses this description to find initial orbits suitable for various mission requirements. These initial orbits exhibit good robustness and can adapt to the complex Earth-Moon space environment. After the initial orbits are determined, high-precision ephemeris data is used to further optimize the orbit design, ensuring that the orbits remain stable even under higher-precision models, thus improving the feasibility and reliability of the orbits. By selecting different initial orbit values, the method can be customized according to specific mission requirements, ensuring the flexibility of orbit design, such as optimization based on various constraints like mission time and fuel budget.

[0133] In the trajectory optimization process, this invention transforms the trajectory optimization problem into a nonlinear programming (NLP) problem, thereby improving computational stability and convergence. By using the NLP framework, trajectory optimization is no longer limited to traditional sequential optimization, but can find the optimal solution in the entire solution space, thus achieving comprehensive optimization of fuel efficiency, trajectory accuracy, and mission time. Compared with traditional multi-objective optimization, this approach is more stable and can guarantee the globality of optimization in multibody dynamics systems.

[0134] To further improve optimization efficiency, this invention employs parallel computing technology to handle complex numerical computation tasks. Since orbit optimization in the Earth-Moon space involves a large amount of numerical integration and function evaluation, the computational cost is often very high. Parallel computing allows for the simultaneous calculation and optimization of multiple trajectory solutions, significantly reducing computation time and accelerating the solution process. This enables computation to be completed within a reasonable timeframe, even with high-precision ephemeris data, saving valuable time resources for mission planning and design.

[0135] This invention significantly improves the efficiency and flexibility of Earth-Moon transfer trajectory design by effectively combining dynamic modeling, trajectory optimization, and parallel computing. This method not only significantly reduces fuel requirements and expands the mission launch window, but also exhibits good computational stability and robustness, making it particularly suitable for future deep space exploration missions and small spacecraft missions with tight fuel budgets. By fully utilizing the gravitational interactions of the Sun, Earth, and Moon, this method provides a novel solution for optimizing trajectory design, showing broad application prospects in future deep space and lunar exploration missions.

[0136] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A method for orbit optimization in a lunar transfer mission, characterized in that, include: Step S1: Simulate the initial orbit of the spacecraft in Earth-Moon space by using the relative motion trajectory of the Sun around the Earth-Moon mass center in the double-circle restricted four-body model; Step S2: Search for the initial orbit according to the search parameters to obtain the initial orbit value; the search parameters include: the angle between the initial velocity and the line connecting the Earth and the Moon, the scaling factor of the spacecraft's initial velocity relative to the local circular velocity, and the initial phase of the Sun relative to the Earth-Moon line; Step S3: Filter the initial trajectory values ​​using preset constraints to obtain an optimized trajectory; Step S4: The optimized trajectory is optimized in multiple segments using the multiple target method to obtain the objective function of the nonlinear programming. Step S5: Solve the objective function to obtain the target orbit.

2. The trajectory optimization method according to claim 1, characterized in that, Step S1 includes: The initial orbit of a spacecraft can be represented by the following formula: In the formula, r is the spacecraft's position, v is the spacecraft's velocity, and Q is the spacecraft's state. Let r be the acceleration of the spacecraft, t be the time, g(r,t) be the gravitational acceleration at position r and time t, and h(v) be the acceleration component related to velocity v.

3. The trajectory optimization method according to claim 2, characterized in that, Step S1 further includes: The spacecraft's state Q can be represented by the following formula: Q=[r,v] T In the formula, r is the position of the spacecraft, v is the velocity of the spacecraft, and T is the transpose.

4. The trajectory optimization method according to claim 2, characterized in that, Step S1 further includes: g(r,t) can be expressed by the following formula: h(v) can be expressed by the following formula: In the formula, Ω⁴ is the angular velocity, Ω⁴ is the partial derivative, x is the position coordinate, y is the position coordinate. For the spacecraft's velocity component in the x-direction, Let be the velocity component of the spacecraft in the y-direction.

5. The trajectory optimization method according to claim 4, characterized in that, Step S1 further includes: According to the following formula In the formula, μ s For the mass of the Sun, a s θ is the distance from the Earth-Moon barycenter to the Sun. s It is the solar phase angle, ω s Ω3 is the relative angular velocity of the Sun, x is the position coordinate, y is the position coordinate, t is the time, Ω3 is the angular velocity, μ is the ratio of the total mass of the Earth-Moon system to the mass of the Sun, m1 is the mass of the Earth, and m2 is the mass of the Moon.

6. The trajectory optimization method according to claim 1, characterized in that, Step S2 includes: The search parameters are discretized within a preset value range to obtain a multidimensional parameter grid; The initial orbital values ​​are obtained by searching on the parameter grid.

7. The trajectory optimization method according to claim 6, characterized in that, Step S2 further includes: The initial orbital parameters of a spacecraft in the Earth-Moon space are expressed by the following formula: In the formula, α is the angle between the initial velocity and the line connecting the Earth and the Moon, β is the scaling factor of the spacecraft's initial velocity relative to its local circular velocity, and θ is the angle between the initial velocity and the line connecting the Earth and the Moon. s,0 The initial phase of the Sun relative to the Earth-Moon line, x i and y i The initial position coordinates of the spacecraft in Earth-Moon space. and Let r be the initial velocity components of the spacecraft in the x and y directions. i Let be the distance from the Earth's center to the spacecraft, μ be the ratio of the total mass of the Earth-Moon system to the mass of the Sun, and v be the distance from the Earth's center to the spacecraft. i The initial velocity of the spacecraft.

8. The trajectory optimization method according to claim 1, characterized in that, Step S4 includes: The objective function J(q) can be expressed by the following formula: J(q)=Δv1+Δv N q={Q j ,t1,t N },j=1,2,...,N t1=t i ,t N =t f In the formula, Δv1 and Δv N Let q be the change in velocity, and Q be the variable. j Let j be the state, t1, and t2. N These represent the time to enter a lower orbit and the time to orbit, respectively, t i For the time to enter a lower orbit than the transfer orbit, t f Let j be the orbital insertion time, j be the number of segments in the optimized trajectory, and N be the total number of segments in the optimized trajectory. For time t j The spacecraft's position coordinates at time t j The velocity coordinates of the spacecraft at any given time.

9. The trajectory optimization method according to claim 1, characterized in that, Step S4 further includes: The boundary constraints can be expressed using the formulas above: In the formula, x1 and y1 are the position coordinates of the spacecraft at the initial moment, and x... N and y N For the spacecraft's position coordinates at the final moment, and The spacecraft's velocity coordinates at the initial moment, and The spacecraft's velocity coordinates at the final moment, μ is the ratio of the total mass of the Earth-Moon system to the mass of the Sun, r1 is the spacecraft's distance from the Earth-Moon's center of mass at the initial moment, and r N This represents the distance the spacecraft will reach from the Earth-Moon center of mass at the final moment. The following formula represents the constraint on task accuracy: In the formula, R E For Earth's radius, R m The radius of the moon; The constraints on task time are expressed by the following formula: τ=t1-t N <0 In the formula, τ represents the time difference, t1 represents the start time of the task, and t N Indicates the time point when the task ends.

10. The trajectory optimization method according to claim 1, characterized in that, Step S5 includes: The target trajectory is obtained by solving multiple nonlinear programming functions in parallel using the solver in MATLAB.

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