低轨卫星非奇异平均根数估计算法研究

doi:10.12305/j.issn.1001-506X.2025.10.25

摘要/Abstract

摘要：

针对低轨（low Earth orbit, LEO）巨型星座这类近圆轨道卫星自主构型保持的需求，研究一种基于非奇异轨道根数和扩展卡尔曼滤波（extended Kalman filter，EKF）的自主实时平均轨道根数估计算法。推导非球形摄动带谐项、田谐项和大气阻力摄动作用下的平均根数动力学模型，并以此建立滤波模型，设计基于EKF的自主实时平均根数估计算法。仿真结果表明，该算法可以长期应用于平均根数的在轨估计，精度相比数值迭代法和快速傅里叶变换方法可以提升一个量级，其中半长轴估计误差可达米级，而且全部轨道根数的估计误差均具有稳定性。最后将算法应用于卫星真实测量数据，验证了算法在实际工程中的可行性以及对不同轨道高度的适用性。

关键词: 低轨巨型星座, 平均根数估计, 扩展卡尔曼滤波, 非奇异轨道根数, 轨道摄动

Abstract:

In order to meet the requirements of autonomous configuration maintenance of near circular orbit satellites in the low Earth orbit （LEO） mega constellations, an real-time algorithm for estimating the mean orbital elements autonomously based on nonsingular orbital elements and extended Kalman filter （EKF） is studied. The dynamic model of mean elements perturbed by the non-spherical perturbation zonal harmonic, tesseral harmonic and atmospheric drag is derived. Using the dynamic model as the filtering model, an autonomous real-time mean elements estimation algorithm based on EKF is designed. Simulation results prove that the algorithm can be applied to the long-term onboard estimation of mean elements. The accuracy can be improved by one order of magnitude compared with the traditional numerical iterative method and the fast Fourier transform method, in which the semimajor axis estimation error can reach the meter level. Estimation errors of all elements are stability. Finally, the algorithm is applied to the real observation data of a satellite, which proves the feasibility in practical engineering and applicability to different orbit altitudes.

Key words: low Earth orbit (LEO) mega constellations, mean elements estimation, extended Kalman filter (EKF), nonsingular orbital elements, orbit perturbation

中图分类号:

V 412.4

王志豪, 仲惟超, 张皓. 低轨卫星非奇异平均根数估计算法研究[J]. 系统工程与电子技术, 2025, 47(10): 3411-3425.

Zhihao WANG, Weichao ZHONG, Hao ZHANG. Nonsingular mean elements estimation algorithm for LEO satellites[J]. Systems Engineering and Electronics, 2025, 47(10): 3411-3425.

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瞬时轨道根数	数值
$ a $/$ {\text{m}} $	7266924.2915
$ e $	0.0012248334
$ i $/$ \left(^{\circ}\right) $	99.1178843000
$ \varOmega $/$ \left(^{\circ}\right) $	96.7987637000
$ \omega $/$ \left(^{\circ}\right) $	136.0618208000
$ M $/$ \left(^{\circ}\right) $	56.4723910000
$ \xi $	−0.000582400688
$ \eta $	0.001096622612
$ \lambda $/$ \left(^{\circ}\right) $	192.5342118000

参数	数值
$ {J_2} $	$ 1\;082.626\;683\;55 \times {10^{ - 6}} $
$ {J_3} $	$ - 2.532\;656\;485\;33 \times {10^{ - 6}} $
$ {C_{22}} $	$ 1.574\;460\;374\;56 \times {10^{ - 6}} $
$ {S_{22}} $	$ - 9.038\;038\;066\;39 \times {10^{ - 7}} $
$ {C_{31}} $	$ 2.192\;638\;529\;17 \times {10^{ - 6}} $
$ {S_{31}} $	$ 2.684\;248\;902\;97 \times {10^{ - 7}} $
$ {C_{32}} $	$ 3.089\;892\;068\;81 \times {10^{ - 7}} $
$ {S_{32}} $	$ - 2.114\;376\;124\;37 \times {10^{ - 7}} $
$ {C_{33}} $	$ 1.005\;487\;780\;64 \times {10^{ - 7}} $
$ {S_{33}} $	$ 1.972\;225\;590\;06 \times {10^{ - 7}} $
地球平均半径$ {{{r_e}} \mathord{\left/ {\vphantom {{{r_e}} {{\text{km}}}}} \right. } {{\text{km}}}} $	6378.137
引力常数$ \mu /({\text{m}}^{\text{3}} \cdot {\text{s}}^{{-2}} )$	$ 3.986\;004\;36 \times {10^{14}} $
标高$ {H \mathord{\left/ {\vphantom {H {{\text{km}}}}} \right. } {{\text{km}}}} $	83.7
大气密度$ \rho /\left(\text{kg} \cdot {\text{m}}^{\text{-3}} \right)$	$ 9.549 \times {10^{ - 15}} $
阻力系数$ {C_{\text{d}}} $	2.2
卫星质量m/kg	100
迎风面积$ {S \mathord{\left/ {\vphantom {S {{{\text{m}}^{\text{2}}}}}} \right. } {{{\text{m}}^{\text{2}}}}} $	4.1233
地球角速度$ {\omega }_{e}/\left(\mathrm{rad} \cdot {{\mathrm{s}}}^{-1}\right) $	$ 7.292\;115\;855\;3 \times {10^{ - 5}} $

轨道根数过程噪声方差	数值
$ \delta _{{Q_a}}^2/{{\text{m}}^{\text{2}}} $	$ {\text{1}} \times {\text{1}}{{\text{0}}^{ - 4}} $
$ \delta _{{Q_i}}^2/{\text{ra}}{{\text{d}}^{\text{2}}} $	$ 1 \times {10^{ - 19}} $
$ \delta _{{Q_\varOmega }}^2 /{\text{ra}}{{\text{d}}^{\text{2}}}$	$ 1 \times {10^{ - 20}} $
$ \delta _{{Q_\xi }}^2 $	$ 1 \times {10^{ - 19}} $
$ \delta _{{Q_\eta }}^2 $	$ 1 \times {10^{ - 18}} $
$ \delta _{{Q_\lambda }}^2/{\text{ra}}{{\text{d}}^{\text{2}}} $	$ 1 \times {10^{ - 16}} $

轨道根数测量噪声方差	数值
$ \delta _{{R_a}}^2 /{{\text{m}}^{\text{2}}}$	$ 538.9 $
$ \delta _{{R_i}}^2/{\text{ra}}{{\text{d}}^{\text{2}}} $	$ 6.111 \times {10^{ - 12}} $
$ \delta _{{R_\varOmega }}^2/{\text{ra}}{{\text{d}}^{\text{2}}} $	$ 8.452 \times {10^{ - 13}}$
$ \delta _{{R_\xi }}^2 $	$ {\text{8}}{\text{.349\;36}} \times {10^{ - 12}} $
$ \delta _{{R_\eta }}^2 $	$ {\text{1}}{\text{.957\;05}} \times {10^{ - 12}} $
$ \delta _{{R_\lambda }}^2/{\text{ra}}{{\text{d}}^{\text{2}}} $	$ 5.910\;74 \times {10^{ - 6}} $

误差	EKF	数值迭代法	FFT
$ {{{\varepsilon _a}} \mathord{\left/ {\vphantom {{{\varepsilon _a}} {\text{m}}}} \right. } {\text{m}}} $	0.9656	32.0303	16.9100
$ {{{\varepsilon _i}} \mathord{\left/ {\vphantom {{{\varepsilon _i}} {{\text{rad}}}}} \right. } {{\text{rad}}}} $	$ 1.228\;9 \times {10^{ - 6}} $	$ 5.353\;9 \times {10^{ - 6}} $	$ 3.015\;4 \times {10^{ - 6}} $
$ {{{\varepsilon _\varOmega }} \mathord{\left/ {\vphantom {{{\varepsilon _\varOmega }} {{\text{rad}}}}} \right. } {{\text{rad}}}} $	$ 4.953\;2 \times {10^{ - 7}} $	$ 5.063\;9 \times {10^{ - 6}} $	$ 4.244\;3 \times {10^{ - 6}} $
$ {\varepsilon _\xi } $	$ 3.036\;8 \times {10^{ - 7}} $	$ 3.402\;8 \times {10^{ - 6}} $	$ 2.551\;8 \times {10^{ - 6}} $
$ {\varepsilon _\eta } $	$ 5.923\;2 \times {10^{ - 7}} $	$ 3.182\;2 \times {10^{ - 6}} $	$ 2.386\;7 \times {10^{ - 6}} $