A longitude-height-based orbit extrapolation method

Abstract

The application discloses a longitude-altitude coordinate system-based orbit extrapolation method, and solves the problem of low numerical propagation efficiency, and belongs to the field of orbit extrapolation; the method comprises the following steps: constructing a substar point longitude model based on orbit elements; obtaining a true anomaly model by solving; obtaining an optimized true anomaly according to a latitude amplitude iteration parameter; obtaining a first-order derivative and a second-order derivative of the true anomaly based on the optimized true anomaly, and combining the optimized true anomaly to form a derivative parameter; introducing the derivative parameter to obtain a first-order derivative and a second-order derivative of the substar point longitude; constructing an orbit altitude model from the difference between the spacecraft geocentric distance and the earth radius; introducing the derivative parameter to obtain a first-order derivative and a second-order derivative of the orbit altitude, and combining the first-order derivative and the second-order derivative of the substar point longitude to construct a state space equation; integrating the state space equation by using a numerical integrator, extrapolating a four-dimensional space state vector, and obtaining a longitude-altitude propagation model; and the method improves the numerical propagation efficiency.

Description

Technical Field

[0001] This invention belongs to the field of aerospace dynamics and orbit determination technology, and relates to an orbit extrapolation method based on a longitude-altitude coordinate system. Background Technology

[0002] Orbit extrapolation is fundamental for mission planning and on-orbit control, requiring accurate and rapid prediction of spacecraft state information at any given time. However, for spacecraft orbits with arbitrary inclinations, most existing methods rely on state variables. To conduct high-precision orbit propagation; among which, This represents the position in the geocentric inertial rectangular coordinate system. Let T be the velocity in the geocentric inertial Cartesian coordinate system, and T be the transpose. While this high-dimensional description is theoretically complete, it faces significant challenges in engineering practice.

[0003] Firstly, the computational efficiency is not particularly outstanding, especially when calculating orbits with large eccentricities. In order to capture large changes in state, extremely small step sizes must be used, which directly leads to a decrease in computational efficiency. For example, Zhang Haocheng. Spacecraft Orbital Dynamics Modeling and Control Based on Quaternions [D]. Harbin Institute of Technology, 2014.

[0004] Secondly, the state variables cannot intuitively reflect the geometric information of the orbit. Engineers must use complex mathematical transformations to convert Cartesian coordinates into the longitude and orbital altitude of the sub-satellite point directly required at the mission level.

[0005] Current technology lacks a way to directly define state vectors in four-dimensional space. The existing framework for comprehensive dynamic modeling limits the ability to perform control design and rapid analysis directly within an intuitive parameter space. Therefore, reducing the orbital dynamics description from six dimensions to an intuitive four dimensions and establishing its complete system has become a key breakthrough for improving the effectiveness of space missions. Summary of the Invention

[0006] To address the challenge of improving the efficiency of space missions, this invention proposes an orbit extrapolation method based on a longitude-altitude coordinate system. By establishing a precise dynamic relationship between the longitude of the sub-satellite point, the orbital altitude, and their rate of change, a closed state-space equation is constructed, which improves the efficiency of orbit propagation, reduces computation time, and enables high-precision numerical propagation in the state space of interest to the mission.

[0007] The objective of this invention is specifically achieved through the following technical solutions:

[0008] This invention discloses a method for orbit extrapolation based on a longitude-height coordinate system, the method comprising:

[0009] Step 1: Based on orbital elements and considering Earth's rotation, construct a nadir longitude model; solve the nadir longitude model to obtain the true anomaly model; iterate the parameters in the true anomaly model according to the quadrant of the latitude argument to obtain the converged optimal true anomaly.

[0010] Step 2: Based on the preferred true anterior angle, obtain the first derivative of the true anterior angle. Differentiate the first derivative of the true anterior angle to obtain the second derivative of the true anterior angle. The differentiation parameters are composed of the preferred true anterior angle, the first derivative of the true anterior angle, and the second derivative of the true anterior angle.

[0011] Step 3: Introduce the differentiation parameter to differentiate the nadir longitude model, and obtain the first and second derivatives of the nadir longitude;

[0012] Step 4: Construct an orbital altitude model based on the difference between the spacecraft's distance from the Earth's center and the Earth's radius; introduce differentiation parameters to differentiate the orbital altitude model, and obtain the first and second derivatives of the orbital altitude.

[0013] Step 5: Construct the state-space equation of the four-dimensional state vector using the first and second derivatives of the longitude of the sub-satellite point and the first and second derivatives of the orbital altitude; integrate the state-space equation using a numerical integrator to extrapolate the four-dimensional state vector required for the mission, thus obtaining the longitude-altitude propagation model.

[0014] In step one, the longitude model of the sub-satellite point is as follows:

[0015] ;

[0016] In the formula, The longitude of the sub-satellite point. Right ascension of the ascending node, It is an integer. The value is determined based on the latitude argument. Quadrant determination: when When, k=1; when or When k=0; The perigee argument, For true near point angle, Pi For the track inclination angle, The Greenwich Mean Sidereal Time at the moment the spacecraft passes through the ascending node. The angular velocity of Earth's rotation. It is a time variable.

[0017] In step one, the true near angle model is as follows:

[0018] .

[0019] In step one, the method for iterating the parameters in the true anomaly model according to the quadrant of the latitude argument to obtain the converged optimal true anomaly angle is as follows:

[0020] When S11, k=0, calculate the true anterior angle in the true anterior angle model. initial value ;

[0021] S12, will Argument of perigee The summation yields the iterative latitude argument. ;

[0022] S13, according to Quadrant adjustment k: when When, k=1; when or When k=0;

[0023] S14, Substitute the adjusted k into the true anterior angle model to obtain the iterative value of the true anterior angle. ;

[0024] S15, the iterative value of the true nearest angle. Replace the initial value of the true anterior angle in S12. Repeat steps S12 to S14 until convergence, and obtain the converged preferred true anterior angle. .

[0025] In step two, the first derivative of the true anterior angle is:

[0026] ;

[0027] The second derivative of the true anterior angle is:

[0028] ;

[0029] In the formula, The first derivative of the true anterior angle. The second derivative of the true anterior angle. The gravitational constant of celestial bodies, It is the semi-nominal diameter. For eccentricity, For true near point angle, This is the argument of perigee.

[0030] In step three, the first derivative of the longitude of the sub-satellite point is:

[0031] ;

[0032] The second derivative of the longitude of the sub-satellite point is:

[0033] ;

[0034] In the formula, The first derivative of the longitude of the point beneath the star. It is the second derivative of the longitude of the point below the star; The first derivative of the right ascension of the ascending node. The first derivative of the argument of latitude. It is the first derivative of the argument of perigee.

[0035] In step four, the orbital height model is as follows:

[0036] ;

[0037] In the formula, For the orbital height, The distance from the Earth's center to the spacecraft. The radius is the Earth's radius.

[0038] In step four, the first derivative of the orbital altitude for:

[0039] ;

[0040] Second derivative of orbital altitude for:

[0041] .

[0042] In step five, the state-space equation is:

[0043] ;

[0044] In the formula, It is a four-dimensional space state vector.

[0045] In step five, the method of integrating the state-space equations using a numerical integrator to extrapolate the four-dimensional state vector required for the task is as follows:

[0046] S21. Starting from the four-dimensional spatial state vector at the initial moment, a numerical integrator is used to integrate the state space equation. Within the time step of the integration process, the first and second derivatives of the longitude and altitude of the sub-satellite point are calculated inversely.

[0047] In step S22, the first and second derivatives of the longitude and altitude of the sub-satellite point are returned to the numerical integrator to update the four-dimensional space state vector. The updated four-dimensional space state vector is then pushed to the next time step. Steps S21 to S22 are repeated until the termination time when the extrapolation stops, thus obtaining the four-dimensional space state vector required by the task.

[0048] The beneficial effects of this invention are:

[0049] 1. State Space Innovation: A state space equation with a four-dimensional state vector as its core was constructed, which greatly improves the intuitiveness and efficiency of task analysis.

[0050] 2. Universality of Inclination Angle: By integrating the state-space equations using a numerical integrator, the orbital parameters of the four-dimensional spatial state vector corresponding to the time required by the task are extrapolated. This organically combines the accurate calculation of the true anomaly angle with the numerical integration of the four-dimensional spatial state vector, solving the problem of calculating the longitude derivative under any non-zero inclination angle. This allows the direct dynamics model to be extended from the equatorial orbit to any orbit, ensuring the practicality and accuracy of the method.

[0051] 3. Closure of derivatives: The first and second derivatives of orbital altitude and sub-satellite longitude were derived, and their derivative relationship with the true anomaly angle was disclosed, forming a closed state-space equation system of differential equations.

[0052] 4. Numerical stability: An iterative calculation method for the true anomaly angle, which includes quadrant determination of latitude argument, is proposed to ensure the numerical stability and accuracy of the core variable calculation.

[0053] 5. A new paradigm for direct orbital dynamics modeling and propagation in the "longitude-altitude" state space was proposed, changing the traditional approach that relies on inertial frames or orbital elements.

[0054] 6. The constructed longitude-altitude propagation model significantly reduces the complexity of orbit calculations in spacecraft mission planning and analysis, providing core technical support for rapid and intuitive orbit decision support. Attached Figure Description

[0055] The present invention will now be described in further detail with reference to the accompanying drawings and embodiments.

[0056] Figure 1 This is a schematic diagram comparing the propagation results of the longitude-height propagation model and the two-body propagation model provided in the embodiments of the present invention.

[0057] Figure 2 This is a schematic diagram comparing the propagation time of the longitude-altitude propagation model and the two-body propagation model provided in the embodiments of the present invention. Detailed Implementation

[0058] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0059] This invention provides a method for orbit extrapolation based on a longitude-height coordinate system, the method comprising:

[0060] Step 1: Based on orbital elements and considering Earth's rotation, construct a nadir longitude model; solve the nadir longitude model to obtain the true anomaly model; iterate the parameters in the true anomaly model according to the quadrant of the latitude argument to obtain the converged optimal true anomaly.

[0061] Step 2: Based on the preferred true anterior angle, obtain the first derivative of the true anterior angle. Differentiate the first derivative of the true anterior angle to obtain the second derivative of the true anterior angle. The differentiation parameters are composed of the preferred true anterior angle, the first derivative of the true anterior angle, and the second derivative of the true anterior angle.

[0062] Step 3: Introduce the differentiation parameter to differentiate the nadir longitude model, and obtain the first and second derivatives of the nadir longitude;

[0063] Step 4: Construct an orbital altitude model based on the difference between the spacecraft's distance from the Earth's center and the Earth's radius; introduce differentiation parameters to differentiate the orbital altitude model, and obtain the first and second derivatives of the orbital altitude.

[0064] Step 5: Construct the state-space equation of the four-dimensional state vector using the first and second derivatives of the longitude of the sub-satellite point and the first and second derivatives of the orbital altitude; integrate the state-space equation using a numerical integrator to extrapolate the four-dimensional state vector required for the mission, thus obtaining the longitude-altitude propagation model.

[0065] In step one, the longitude model of the sub-satellite point is as follows:

[0066] ;

[0067] In the formula, The longitude of the sub-satellite point. Right ascension of the ascending node, It is an integer. The value is determined based on the latitude argument. Quadrant determination: when When, k=1; when or When k=0; The perigee argument, For true near point angle, Pi For the track inclination angle, The Greenwich Mean Sidereal Time at the moment the spacecraft passes through the ascending node. The angular velocity of Earth's rotation. It is a time variable.

[0068] In step one, the true near angle model is as follows:

[0069] The derivation process is as follows:

[0070] because: ;

[0071] ;

[0072] Let the first parameter ;

[0073] The formula then simplifies to: ;

[0074] so: ;

[0075] ;

[0076] ;

[0077] so ;

[0078] In summary, the complete solution for the true anterior angle f is:

[0079] .

[0080] In step one, the method for iterating the parameters in the true anomaly model according to the quadrant of the latitude argument to obtain the converged optimal true anomaly angle is as follows:

[0081] When S11, k=0, calculate the true anterior angle in the true anterior angle model. initial value ;

[0082] S12, will Argument of perigee The summation yields the iterative latitude argument. ;

[0083] S13, according to Quadrant adjustment k: when When, k=1; when or When k=0;

[0084] S14, Substitute the adjusted k into the true anterior angle model to obtain the iterative value of the true anterior angle. ;

[0085] S15, the iterative value of the true nearest angle. Replace the initial value of the true anterior angle in S12. Repeat steps S12 to S14 until convergence, and obtain the converged preferred true anterior angle. .

[0086] In step two, the first derivative of the true anterior angle is:

[0087] ;

[0088] The second derivative of the true anterior angle is:

[0089] ;

[0090] In the formula, The first derivative of the true anterior angle. The second derivative of the true anterior angle. The gravitational constant of celestial bodies, It is the semi-nominal diameter. For eccentricity, For true near point angle, Let be the argument of the perigee. A point above the parameter represents the first derivative, and a point above the parameter represents the second derivative.

[0091] The derivation of the first derivative of the true anterior angle is as follows:

[0092] ;

[0093] The derivation of the second derivative of the true anterior angle is as follows:

[0094] ;

[0095] We can obtain: .

[0096] In step three, the first derivative of the longitude of the sub-satellite point is:

[0097] ;

[0098] The derivation process is as follows:

[0099] ;

[0100] We can obtain: ;

[0101] Since the orbital inclination angle i is a constant, let From the above formula, we can obtain:

[0102] ;

[0103] Then the middle term The differentiation process is as follows:

[0104] ;

[0105] in The above equation can be simplified to:

[0106] ;

[0107] Will , Substitution ;

[0108] Among them, the denominator ;

[0109] so ;

[0110] Summarized as follows: .

[0111] The second derivative of the longitude of the sub-satellite point is:

[0112] ;

[0113] The derivation process is as follows:

[0114] ;

[0115] so ;

[0116] Let the second parameter B, the third parameter C, and the fourth parameter D be respectively:

[0117] Then the fifth parameter inside the parentheses ;

[0118] ;

[0119] B is a constant, so ;

[0120] in, ;

[0121] Substitution ;

[0122] so ;

[0123] When perturbation is not considered, we can obtain: .

[0124] In the formula, The first derivative of the longitude of the point beneath the star. It is the second derivative of the longitude of the point below the star; The first derivative of the right ascension of the ascending node. The first derivative of the argument of latitude. It is the first derivative of the argument of perigee.

[0125] In step four, the orbital height model is as follows:

[0126] ;

[0127] In the formula, For the orbital height, The distance from the Earth's center to the spacecraft. The radius is the Earth's radius.

[0128] In step four, the first derivative of the orbital altitude for:

[0129] ;

[0130] The derivation process is as follows:

[0131] ;

[0132] ;

[0133] .

[0134] Second derivative of orbital altitude for:

[0135] ;

[0136] The derivation process is as follows:

[0137] ;

[0138] Let the first sub-parameter ,but ;

[0139] in ,calculate :

[0140] ;

[0141] therefore, .

[0142] In step five, the state-space equation is:

[0143] ;

[0144] In the formula, It is a four-dimensional space state vector.

[0145] In step five, the method of integrating the state-space equations using a numerical integrator to extrapolate the four-dimensional state vector required for the task is as follows:

[0146] S21. Starting from the four-dimensional spatial state vector at the initial moment, a numerical integrator is used to integrate the state space equation. Within the time step of the integration process, the first and second derivatives of the longitude and altitude of the sub-satellite point are calculated inversely.

[0147] In step S22, the first and second derivatives of the longitude and altitude of the sub-satellite point are returned to the numerical integrator to update the four-dimensional space state vector. The updated four-dimensional space state vector is then pushed to the next time step. Steps S21 to S22 are repeated until the termination time when the extrapolation stops, thus obtaining the four-dimensional space state vector required by the task.

[0148] Verification experiment:

[0149] To verify the effectiveness of the technical solution of the present invention, a verification experiment is provided for illustration:

[0150] Scenario: Calculate the changes in nadir longitude and orbital altitude of an inclined geosynchronous orbit satellite over the next 48 hours. The initial orbital elements are shown in Table 1.

[0151] Table 1

[0152]

[0153] Experimental procedure:

[0154] Initialization: Based on the aforementioned orbital elements, the initial value of the true anomaly angle is calculated using the technical solution disclosed in this invention. Initial value of longitude of the sub-satellite point and initial orbital altitude And calculate the first derivative of the initial value of the longitude of the sub-satellite point. The first derivative of the initial orbital altitude Thus, the initial moment is complete. Initialization of the four-dimensional state vector.

[0155] Numerical integrator setup: Fourth-order Runge-Kutta method is used for numerical integration.

[0156] The time step Δt = 60 seconds.

[0157] Total propagation steps N = 2880 steps (48 hours).

[0158] relative tolerance absolute tolerance .

[0159] Loop propagation: Numerical integrator from To begin, the state-space equations are invoked.

[0160] At each intermediate state point, the state space equations are used to inversely calculate the first and second derivatives of the longitude and altitude of the nadir point. The numerical integrator uses the first and second derivatives of the longitude and altitude of the nadir point to update the four-dimensional space state vector until the four-dimensional space state vector is pushed to the termination time and the extrapolation stops, thus obtaining the four-dimensional space state vector required by the task.

[0161] The dynamic propagation results of the longitude-height propagation model are compared with the two-body propagation results of the two-body propagation model, such as... Figure 1 As shown, it can be seen that the dynamic propagation results obtained by using the longitude-height propagation model proposed in this invention are highly consistent with the two-body propagation results of the two-body model, proving the accuracy and reliability of this method.

[0162] To evaluate computational efficiency, the computation times of the two methods were statistically compared, such as... Figure 2 As shown, through Figure 2 It can be seen that the computation time of the longitude-altitude propagation model is significantly lower than that of the two-body model, achieving the best balance between accuracy and efficiency. This demonstrates that the proposed method effectively improves computational efficiency while ensuring the accuracy of orbit propagation, thus solving the problem of high computational overhead in traditional methods.

[0163] This verification experiment successfully demonstrated the effectiveness and superiority of the method of this invention in inclined geosynchronous orbit. Compared with traditional methods, this invention directly establishes state-space equations in the longitude-altitude state space of interest to the mission. Through a precise numerical integration process, it fully executes the core calculation steps of the patent at each time step, avoiding frequent coordinate system transformations while maintaining considerable accuracy. This method is particularly suitable for applications requiring rapid orbit analysis and real-time mission planning, providing a more intuitive and efficient analysis tool for space missions.

[0164] 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 extrapolation based on a longitude-height coordinate system, characterized in that, The method includes: Step 1: Based on orbital elements and considering Earth's rotation, construct a nadir longitude model; solve the nadir longitude model to obtain the true anomaly model; iterate the parameters in the true anomaly model according to the quadrant of the latitude argument to obtain the converged optimal true anomaly. Step 2: Based on the preferred true anterior angle, obtain the first derivative of the true anterior angle. Differentiate the first derivative of the true anterior angle to obtain the second derivative of the true anterior angle. The differentiation parameters are composed of the preferred true anterior angle, the first derivative of the true anterior angle, and the second derivative of the true anterior angle. Step 3: Introduce the differentiation parameter to differentiate the nadir longitude model, and obtain the first and second derivatives of the nadir longitude; Step 4: Construct an orbital altitude model based on the difference between the spacecraft's distance from the Earth's center and the Earth's radius; introduce differentiation parameters to differentiate the orbital altitude model, and obtain the first and second derivatives of the orbital altitude. Step 5: Construct the state space equation of the four-dimensional spatial state vector using the first and second derivatives of the longitude of the sub-satellite point and the first and second derivatives of the orbital altitude; integrate the state space equation using a numerical integrator to extrapolate the four-dimensional spatial state vector required for the mission, thus obtaining the longitude-altitude propagation model. In step one, the longitude model of the sub-satellite point is as follows: ； In the formula, The longitude of the sub-satellite point. Right ascension of the ascending node, It is an integer. The value is determined based on the latitude argument. Quadrant determination: when When, k=1; when or When k=0; The perigee argument, For true near point angle, Pi For the track inclination angle, The Greenwich Mean Sidereal Time at the moment the spacecraft passes through the ascending node. The angular velocity of Earth's rotation. It is a time variable; The true near angle model is: ； The method for iterating through the parameters in the true anomaly model based on the quadrant of the latitude argument to obtain the converged optimal true anomaly angle is as follows: When S11, k=0, calculate the true anterior angle in the true anterior angle model. initial value ; S12, will Argument of perigee The summation yields the iterative latitude argument. ; S13, according to Quadrant adjustment k: when When, k=1; when or When k=0; S14, Substitute the adjusted k into the true anterior angle model to obtain the iterative value of the true anterior angle. ; S15, the iterative value of the true nearest angle. Replace the initial value of the true anterior angle in S12. Repeat steps S12 to S14 until convergence, and obtain the converged preferred true anterior angle. ; In step two, the first derivative of the true anterior angle is: ； The second derivative of the true anterior angle is: ； In the formula, The first derivative of the true anterior angle. The second derivative of the true anterior angle. The gravitational constant of celestial bodies, It is the semi-nominal diameter. For eccentricity, For true near point angle, Let be the argument of the perigee. A point above the parameter represents the first derivative, and two points above the parameter represent the second derivative.

2. The method as described in claim 1, characterized in that, In step three, the first derivative of the longitude of the sub-satellite point is: ； The second derivative of the longitude of the sub-satellite point is: ； In the formula, The first derivative of the longitude of the point beneath the star. It is the second derivative of the longitude of the point below the star; The first derivative of the right ascension of the ascending node. The first derivative of the argument of latitude. It is the first derivative of the argument of perigee.

3. The method as described in claim 2, characterized in that, In step four, the orbital height model is as follows: ； In the formula, For the orbital height, The distance from the Earth's center to the spacecraft. The radius is the Earth's radius.

4. The method as described in claim 3, characterized in that, In step four, the first derivative of the orbital altitude for: ； Second derivative of orbital altitude for: 。 5. The method as described in claim 4, characterized in that, In step five, the state-space equation is: ； In the formula, It is a four-dimensional space state vector.

6. The method as described in claim 5, characterized in that, In step five, the method of integrating the state-space equations using a numerical integrator to extrapolate the four-dimensional state vector required for the task is as follows: S21. Starting from the four-dimensional spatial state vector at the initial moment, a numerical integrator is used to integrate the state space equation. Within the time step of the integration process, the first and second derivatives of the longitude and altitude of the sub-satellite point are calculated inversely. In step S22, the first and second derivatives of the longitude and altitude of the sub-satellite point are returned to the numerical integrator to update the four-dimensional space state vector. The updated four-dimensional space state vector is then pushed to the next time step. Steps S21 to S22 are repeated until the termination time when the extrapolation stops, thus obtaining the four-dimensional space state vector required by the task.

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