Method for solving Kepler equation close to critical eccentricity
By combining the series expansion method and the Newton iteration method, the computational efficiency and accuracy problems of the Kepler equations near the critical eccentricity are solved, achieving high-precision and fast convergence, which is suitable for large eccentricity orbit design and deep space exploration missions.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-12-26
- Publication Date
- 2026-04-07
AI Technical Summary
Existing technologies suffer from low computational efficiency or non-convergence when solving Kepler equations near the critical eccentricity, especially in the design of large eccentricity orbits, where it is difficult to balance computational convergence, speed, and accuracy.
A hybrid strategy combining series expansion and Newton's iteration method is adopted. The Lagrange series expansion is used to provide high-quality initial values for iteration, and Newton's iteration method is used for accurate solution. The tolerance error and the maximum number of iterations are set to ensure convergence.
It achieves high accuracy and fast convergence under near-critical eccentricity conditions, broadens the reliable solution range of Kepler equations, and improves the computational efficiency and accuracy of orbit design and deep space exploration missions.
Smart Images
Figure CN121808176A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of aerospace dynamics technology, specifically relating to a method for solving the Kepler equation near critical eccentricity. Background Technology
[0002] Kepler's equations are transcendental equations describing the relationship between the mean anomaly and the off-anomaly angle when a spacecraft is orbiting an elliptical or hyperbolic orbit. They are fundamental problems in celestial mechanics and space dynamics. These equations connect the easily calculable time variable—the mean anomaly—with the geometric variable—the off-anomaly angle—which indirectly reflects the spacecraft's true spatial position. Once the off-anomaly angle is solved, the spacecraft's precise position in its orbit can be further determined.
[0003] Because Kepler's equations are transcendental equations, exact analytical solutions cannot be obtained through finite algebraic operations; their solutions must rely on numerical methods. Currently, commonly used numerical methods mainly fall into two categories: numerical iteration methods and series expansion methods. Numerical iteration methods, such as Newton's iteration method, have the advantage of fast convergence speed, but when the orbital eccentricity approaches the critical value of 1, they become extremely sensitive to the initial values of the iterations. Inappropriate initial value selection can easily lead to iteration divergence and computational failure. Series expansion methods, such as Lagrange series expansions, can provide effective analytical approximations for small eccentricities, but as the eccentricity increases (e.g., exceeding 0.7), the convergence speed of the series slows down drastically, becoming computationally inefficient or even impractical near the critical eccentricity.
[0004] With the continuous development of deep space exploration missions, spacecraft orbit design often involves elliptical orbits with large eccentricities or hyperbolic orbits approaching parabolic eccentricities, which are often very close to the critical value of 1. In this context, existing conventional methods face severe challenges, struggling to balance computational convergence, speed, and accuracy. Therefore, developing a robust, efficient, and high-precision Kepler equation solving algorithm suitable for near-critical eccentricity conditions is of urgent and significant practical importance for improving the reliability of orbit determination, mission analysis, and spacecraft navigation. Summary of the Invention
[0005] To address the aforementioned technical problems, this invention provides a method for solving Kepler equations with near-critical eccentricity. Based on a combination of series expansion and numerical iteration, this method can achieve high-precision solutions for the near-point angles of Kepler equations for ellipses or hyperbolas with eccentricity close to 1.
[0006] To achieve the above objectives, the present invention adopts the following technical solution:
[0007] A method for solving the Kepler equation near the critical eccentricity, the method comprising:
[0008] Step 1: Obtain the orbital eccentricity parameters and the horizontal approach angle parameters;
[0009] Step 2: Based on the eccentricity and the mean aperimeter angle, calculate the initial value of the aperimeter angle using the Lagrange series expansion method;
[0010] Step 3: Set the tolerance error condition for iterative convergence and the maximum number of iterations;
[0011] Step 4: Based on Newton's iteration method, construct the iterative equation corresponding to the eccentricity;
[0012] Step 5: Using the initial value of the iteration, perform iterative calculations using the iterative equation until the tolerance error condition is met and the maximum number of iterations is reached, and output the final near-point angle.
[0013] Furthermore, in step 2, when the eccentricity e is less than 1, the initial iteration value is based on the ellipse's plan anterior angle. The calculation results of the 5th order Lagrange series expansion are as follows:
[0014] ,
[0015] This represents the initial value of the iteration of the near-point angle of the ellipse.
[0016] Furthermore, in step 2, when the eccentricity e is greater than 1, the initial iteration value is based on the hyperbolic planar anterior angle. The calculation results of the 5th order Lagrange series expansion are as follows:
[0017] ,
[0018] This represents the initial value of the hyperbolic approach angle iteration.
[0019] Furthermore, in step 3, the tolerance error condition is that the absolute value of the difference between the near point angles obtained from two adjacent iterations is less than a set threshold.
[0020] Furthermore, in step 4, when the eccentricity e is less than 1, the iterative equation is the application form of Newton's iteration method to Kepler's equations for elliptical orbits:
[0021] ,
[0022] This represents the angle of the ellipse's closest point at the k-th iteration. This represents the angle at the plane anterior point of the ellipse. This represents the iteration error of the ellipse's near-point angle.
[0023] Furthermore, in step 4, when the eccentricity e is greater than 1, the iterative equation is the application form of Newton's iteration method on Kepler's equations for hyperbolic orbits:
[0024] ,
[0025] This represents the hyperbolic approach angle at the k-th iteration. Indicates the hyperbolic plane anterior angle. This represents the iteration error of the hyperbolic near-point angle.
[0026] Furthermore, in step 3, the maximum number of iterations is set to 20.
[0027] Furthermore, the method also includes preprocessing the input mean aperimeter angle before step 1. When the mean aperimeter angle exceeds the range of [0, 2π) or the corresponding hyperbolic range, it is reduced to the standard principal value range by modulo operation.
[0028] In a second aspect, the present invention provides an electronic device, comprising: one or more processors; a memory for storing one or more programs; wherein, when the one or more programs are executed by the one or more processors, the one or more processors implement the aforementioned method for solving the Kepler equation with near-critical eccentricity.
[0029] Thirdly, the present invention provides a computer-readable storage medium having executable instructions stored thereon, which, when executed by a processor, enable the processor to implement the aforementioned method for solving the Kepler equation with near-critical eccentricity.
[0030] The beneficial effects of this invention are as follows:
[0031] The problem of convergence in the critical region is effectively solved by adopting a hybrid strategy that combines the series expansion method with the Newton iteration method. The Lagrange series expansion provides high-quality initial values for the Newton iteration that are close to the true solution. This fundamentally overcomes the defect of the traditional Newton iteration method, which is prone to divergence due to improper initial value selection when the eccentricity is close to 1. At the same time, it also makes up for the deficiency of the pure series expansion method in that region, which is slow or non-convergent, and ensures the robust convergence of the algorithm in the critical region.
[0032] This invention achieves a balance between computational efficiency and accuracy: it combines the analytical advantages of series expansion in initial value estimation with the rapid convergence of Newton's iteration method in subsequent approximation. Examples show that for elliptical orbits with eccentricities as high as 0.995 and hyperbolic orbits with eccentricities as high as 1.005, high computational accuracy can be achieved with only a few iterations, and the convergence speed is much faster than the traditional fixed-point iteration method, effectively improving computational efficiency.
[0033] This method expands the reliable solution range of the Kepler equations: It is specifically optimized for the special case of "near-critical eccentricity" where existing technologies are not effective, significantly expanding the applicable boundary of high-precision numerical solutions to the Kepler equations. It provides a more reliable and powerful technical tool for engineering practices such as the design of Earth spacecraft with large eccentric orbits and the orbit calculation of deep space exploration missions. Attached Figure Description
[0034] Figure 1 This is a flowchart of a method for solving the Kepler equation near the critical eccentricity according to the present invention;
[0035] Figure 2 Iterative optimization curve for solving the Kepler equation for elliptical orbits approaching critical eccentricity;
[0036] Figure 3 To find the optimal curve for the Kepler equation when the hyperbolic orbit approaches the critical eccentricity, an iterative solution is needed. Detailed Implementation
[0037] The present invention will be further described below with reference to the accompanying drawings and embodiments.
[0038] like Figure 1 As shown, the present invention provides a method for solving the Kepler equations near the critical eccentricity, applicable to spacecraft orbit design, orbit determination, and orbit transfer control, comprising the following steps:
[0039] Step 1: Obtain the orbital eccentricity parameters Parameters of the near point .
[0040] Before this, the input mean aperimeter angle is preprocessed. When the mean aperimeter angle exceeds the range of [0, 2π) or the corresponding hyperbolic range, it is reduced to the standard principal value range by modulo operation.
[0041] Step 2: Based on the eccentricity e and the level anterior angle M (without loss of generality, the level anterior angle of the ellipse is used...), Indicates that the hyperbolic plane anterior angle is expressed as... (represented), the initial values for the iterative calculation of the near-point angle are obtained using the Lagrange series expansion method, as follows:
[0042] Taking into account both the convergent range and computational cost, the series expansion method is used to design the initial values for the iteration.
[0043] When eccentricity If the value is less than 1, it indicates an elliptical orbit; the initial value for iteration is taken from the angle of the closest point on the ellipse. The following is a 5th-order Lagrange series expansion:
[0044] ,
[0045] When eccentricity If the value is greater than 1, it indicates a hyperbolic orbit; the initial value for the iteration is taken from the hyperbolic near-point angle. The following is a 5th-order Lagrange series expansion:
[0046] .
[0047] Step 3: Set the tolerance error condition for iterative convergence and the maximum number of iterations;
[0048] To meet the accuracy requirements of the calculation and to avoid the iteration from falling into an infinite loop, we can take it as the absolute value of the iteration error when the point angle is too close. or If the error exceeds the set tolerance threshold and the number of iterations does not exceed the set maximum number of iterations, continue iterating; otherwise, it indicates that the iteration has converged.
[0049] Step 4: Based on Newton's iteration method, construct the iterative equation corresponding to the eccentricity e, as follows:
[0050] When eccentricity If the value is less than 1, it indicates an elliptical orbit, with the ellipse deviating from the nearest point angle. The iterative equation is:
[0051] ,
[0052] This represents the angle of the ellipse's closest point at the k-th iteration.
[0053] When eccentricity If the value is greater than 1, it indicates a hyperbolic orbit, with the hyperbolic approach angle being greater than 1. The iterative equation is:
[0054] ,
[0055] It represents the hyperbolic approach angle at the k-th iteration.
[0056] Step 5: Using the initial value of the iteration, perform iterative calculations using the iterative equation until the tolerance error condition is met and the maximum number of iterations is reached, and output the final near-point angle.
[0057] Example 1:
[0058] This embodiment provides a method for solving the Kepler equation for an elliptical orbit approaching the critical eccentricity, as detailed below.
[0059] In this embodiment, the eccentricity of the elliptical orbit is =0.995, angle of approach =0.8°. According to the method of the present invention, Newton's iteration method is used for solving: initial value of iteration. The result, calculated using the 5th-order Lagrange series expansion, is as follows:
[0060] ,
[0061] The iterative formula is:
[0062] ,
[0063] The criterion for exiting the iteration loop is taken as Or the number of iterations exceeds 20. The iteration result is:
[0064] .
[0065] The result has an error of no more than 0.05°, with 7 iterations. For the iteration convergence process, please refer to [link / reference needed]. Figure 2 Curve 1 in the diagram.
[0066] Compared with the method of this invention, the asymptotic angle is directly calculated using the above-mentioned 5th order Lagrange series expansion. Compared with the higher accuracy obtained by the above iteration The error between them is approximately 19.1°; when using the fixed-point iteration method, the initial value of the iteration is taken as the angle closest to the plane using the conventional method. Equal, that is After 20 iterations, the result still did not converge, and the last obtained near-angle result was... With higher precision The error is approximately 8.8°. Refer to the iterative process. Figure 2 Curve 2 in the figure. It can be seen that the method provided by this invention has fast convergence speed and high convergence accuracy.
[0067] Example 2:
[0068] This embodiment provides a method for solving the Kepler equation for a hyperbolic orbit approaching the critical eccentricity, as detailed below.
[0069] In this embodiment, the hyperbolic orbital eccentricity is =1.005, angle of approach =2.5°. According to the method of the present invention, Newton's iteration method is used for solving: initial value of iteration. The result, calculated using the 5th-order Lagrange series expansion, is as follows:
[0070] ,
[0071] The iterative formula is:
[0072] ,
[0073] The criterion for exiting the iteration loop is taken as Or the number of iterations exceeds 20. The iteration result is:
[0074] .
[0075] The result has an error of no more than 0.02°, with 7 iterations. For the iteration convergence process, please refer to [link / reference needed]. Figure 3 Curve 1 in the diagram.
[0076] Compared with the method of this invention, the asymptotic angle is directly calculated using the above-mentioned 5th order Lagrange series expansion. Compared with the higher accuracy obtained by the above iteration The error between them is approximately 25.3°; when using the fixed-point iteration method, the initial value of the iteration is taken as the angle closest to the plane using the conventional method. Equal, that is After 20 iterations, the result still did not converge, and the last obtained near-angle result was... With higher precision The error is approximately 2.7°. Refer to the iterative process. Figure 3 Curve 2 in the figure. It can be seen that the method provided by this invention has fast convergence speed and high convergence accuracy.
[0077] In a second aspect, the present invention provides an electronic device, comprising: one or more processors; a memory for storing one or more programs; wherein, when the one or more programs are executed by the one or more processors, the one or more processors implement the aforementioned method for solving the Kepler equation with near-critical eccentricity.
[0078] Thirdly, the present invention provides a computer-readable storage medium having executable instructions stored thereon, which, when executed by a processor, enable the processor to implement the aforementioned method for solving the Kepler equation with near-critical eccentricity.
[0079] The specific embodiments described above further illustrate the purpose, technical solution, and beneficial effects of the present invention. It should be understood that the above descriptions are merely specific embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.
Claims
1. A method for solving the Kepler equation near the critical eccentricity, characterized in that, The method includes: Step 1: Obtain the orbital eccentricity parameters and the horizontal approach angle parameters; Step 2: Based on the eccentricity and the mean aperimeter angle, calculate the initial value of the aperimeter angle using the Lagrange series expansion method; Step 3: Set the tolerance error condition for iterative convergence and the maximum number of iterations; Step 4: Based on Newton's iteration method, construct the iterative equation corresponding to the eccentricity; Step 5: Using the initial value of the iteration, perform iterative calculations using the iterative equation until the tolerance error condition is met and the maximum number of iterations is reached, and output the final near-point angle.
2. The method for solving the Kepler equation near the critical eccentricity according to claim 1, characterized in that, In step 2, when the eccentricity e is less than 1, the initial iteration value is based on the ellipse's plan anterior angle. The calculation results of the 5th order Lagrange series expansion are as follows: ， This represents the initial value of the iteration of the near-point angle of the ellipse.
3. The method for solving the Kepler equation near the critical eccentricity according to claim 1, characterized in that, In step 2, when the eccentricity e is greater than 1, the initial iteration value is based on the hyperbolic planar anterior angle. The calculation results of the 5th order Lagrange series expansion are as follows: ， This represents the initial value of the hyperbolic approach angle iteration.
4. The method for solving the Kepler equation near the critical eccentricity according to claim 1, characterized in that, In step 3, the tolerance error condition is that the absolute value of the difference between the near point angles obtained from two adjacent iterations is less than a set threshold.
5. The method for solving the Kepler equation near the critical eccentricity according to claim 1, characterized in that, In step 4, when the eccentricity e is less than 1, the iterative equation is the application form of Newton's iteration method to Kepler's equations for elliptical orbits: ， This represents the angle of the ellipse's closest point at the k-th iteration. This represents the angle at the plane anterior point of the ellipse. This represents the iteration error of the ellipse's near-point angle.
6. The method for solving the Kepler equation near the critical eccentricity according to claim 1, characterized in that, In step 4, when the eccentricity e is greater than 1, the iterative equation is the application form of Newton's iteration method on Kepler's equations for hyperbolic orbits: ， This represents the hyperbolic approach angle at the k-th iteration. Indicates the hyperbolic plane anterior angle. This represents the iteration error of the hyperbolic near-point angle.
7. The method for solving the Kepler equation near the critical eccentricity according to claim 1, characterized in that, In step 3, the maximum number of iterations is set to 20.
8. The method for solving the Kepler equation near the critical eccentricity according to claim 1, characterized in that, The method further includes preprocessing the input mean aperimeter angle before step 1. When the mean aperimeter angle exceeds the range of [0, 2π) or the corresponding hyperbolic range, it is reduced to the standard principal value range by modulo operation.
9. An electronic device, characterized in that, include: One or more processors; Memory, used to store one or more programs; When one or more programs are executed by the one or more processors, the one or more processors implement the method for solving the Kepler equation with near-critical eccentricity as described in any one of claims 1-8.
10. A computer-readable storage medium, characterized in that, It stores executable instructions that, when executed by a processor, enable the processor to implement a method for solving the Kepler equation with near-critical eccentricity as described in any one of claims 1-8.