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

Abstract

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

CLC Number:

V 412.4

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

Figures/Tables 14

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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}} $

Nonsingular mean elements estimation algorithm for LEO satellites

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采样周期	单步运行时间
10	0.0035610
60	0.0043358
600	0.0051561

标准差	EKF	数值迭代法	FFT
$ {{{\sigma _a}} \mathord{\left/ {\vphantom {{{\sigma _a}} {\text{m}}}} \right. } {\text{m}}} $	1.0124	27.3539	10.0154
$ {{{\sigma _i}} \mathord{\left/ {\vphantom {{{\sigma _i}} {{\text{rad}}}}} \right. } {{\text{rad}}}} $	$ 1.882\;2 \times {10^{ - 7}} $	$ 2.535\;3 \times {10^{ - 6}} $	$ 4.258\;3 \times {10^{ - 7}} $
$ {{{\sigma _\varOmega }} \mathord{\left/ {\vphantom {{{\sigma _\varOmega }} {{\text{rad}}}}} \right. } {{\text{rad}}}} $	$ 4.258\;3 \times {10^{ - 7}} $	$ 8.281\;3 \times {10^{ - 6}} $	$ 5.354\;1 \times {10^{ - 6}} $
$ {\sigma _\xi } $	$ 1.715\;4 \times {10^{ - 7}} $	$ 6.749\;1 \times {10^{ - 6}} $	$ 3.305\;6 \times {10^{ - 6}} $
$ {\sigma _\eta } $	$ 1.016\;0 \times {10^{ - 7}} $	$ 6.574\;8 \times {10^{ - 6}} $	$ 3.216\;7 \times {10^{ - 6}} $

位置测量误差/m	速度测量误差/（m/s）	平均半长轴估计误差$ {\varepsilon _a} $/m
5	0.005	0.2178
5	0.02	0.9656
5	0.05	5.0621
5	0.1	9.2874
0.1	0.005	0.1045
0.1	0.02	0.6087
1	0.02	1.1080
10	0.02	1.3256

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