CC BY
28Прикладная математика и механика
ISBN 669.713.7
A HIGH PRECISION DYNAMIC ORBIT DETERMINATION METHOD BASED ON GPS PSEUDORANGE FOR LOW EARTH ORBIT SATELLITES
Xinlong Wang1, Qing Zhang1, Hengnian Li 2
1School of Astronautics, Beihang University Beijing, 100191, China. E-mail: xlwon@163.com 2State Key Lab of Astronautical Dynamics Xi'an Satellite Control Centere, Xi'an, 710043, China
To improve the precision of orbit determination for Low Earth Orbit (LEO) satellites, a dynamic orbit determination (DOD) approach based on pseudorange measurements is proposed. A high precision force model is established to get precise orbital estimates (position and velocity). It includes nonspherical perturbation, third-body attraction, drag, solar radiation pressure, tides, relativistic effects etc. The Global Position System (GPS) observation data is corrected to obtain precise position with the principle of Single Point Positioning (SPP). And the corrections contains clock offset, ephemeris error, ionospheric path delay, antenna phase center offset, aberration and Sagnac effect. Then the difference of the position from force model and SPP is chosen as the observed variable, and Least-Squares (LSQ) is used to achieve Precise Orbit Determination (POD) for LEO satellites. At last, the actual information of CHAllenging Minisatellite Payload (CHAMP) is adopted to simulate, and the results demonstrate that this approach can fulfill POD with dm precision (3-dimensional).
Keywords: LEO satellites; force model; GPS; SPP; POD; DOD.
In recent years, LEO satellites get well-development and play an important role in the field of meteorological observation, environmental protection and geodetic survey. As POD is essential to the space mission, the precision is definitely key to the LEO satellites and depend on the orbit measure technology and orbit determination approach [1-2]. There are several common orbit measure technologies, namely Satellite Laser Ranging (SLR), Doppler Orbitography and Radipositioning Integrated by Satellite (DORIS), Precise Range And Range rate Equipment (PR ARE) and GPS. Compared to others, GPS has the advantages of all-weather, low-cost, light-equipment, high-accuracy and day-night observation, and has become the main orbit measure technology [3].
The orbit determination approach with GPS is mainly divided into three groups: kinematic, reduced-dynamic and dynamic [4]. The kinematic approach uses only GPS measurements to determine a time series of positions of the satellite; therefore, it is a simple and efficient method. However, the orbit quality is strongly dependent on the geometry and continuity of the GPS signals [5]. The reduced-dynamic approach introduces kinematic components to the dynamic force models in the form of the process noise parameters that is not real. This means that the dynamic parameters are estimated first; then the state vector is re-estimated using the Kalman filter along with the stochastic process noise, which usually can't be chosen correctly, but is crucial to the precision of this approach [6]. For the dynamic POD, all forces acting on the satellite are computed and numerically integrated to estimate the initial state vector and other unknown dynamic parameters [7]. Thus, accurate force modeling is a critical issue for a successful POD in the dynamic approach. Due to the increasing errors of the force model, the precision of LEO satellites is not high. But the GPS data can estimate empirical parameters, which provide an optimum compensation of unmodelled forces. In contrast
to other approaches, this dynamic POD, as the main approach of GPS orbit determination, not only smooth the orbit by dynamic model, but also reduce the influences of model errors and various stochastic errors.
A method of dynamic orbit determination based on pseudorange is proposed, which fully takes account of various kinds of perturbations and adopts precise dynamic models. Besides, with the consecutive GPS observations, the empirical forces are used to absorb the variability of unmodeled forces and make the force model really precise. The difference of the position from force model and SPP is chosen as the observed variable in this approach, which not only reduces the order of the state vector and increases the computation, but also improves the precision and reaches POD for LEO satellites.
Force Model. Except for the gravitational attraction, the motion of LEO satellites is affected by various kinds of perturbations, including nonspherical perturbation, third-body attraction, atmospheric drag, solar radiation pressure and some empirical force terms.
The motion equation of LEO satellites is generally expressed in the Earth Centered Inertial (ECI) frame:
r = -GMJL7 + f(t,y,j, q^..., qn) (1)
r
where 7 and 7 are the position and velocity of the satellite, respectively, G is the gravitational constant, and Me denotes the Earth's mass, r = |l^l is the radial distance from the center of the Earth, t represents the time, at which the acceleration is calculated, f is the total perturbing forces, and q1, —, qn are the dynamic parameters.
Dynamic Orbit Determination Based on
Pseudorange. The main process of DOD based on pseudorange are: after initialization, the perturbing forces as well as the Keplerian accelerations are computed at each epoch and then numerically integrated to get the position and velocity of the satellite in the next epoch.
Решетневскуе чтения. 2013
The difference of position vectors from the integration and SPP is the observed variable, and the initial state vector and dynamic parameters can be obtained by batch LSQ. When the precision meet the requirement, the orbit is determined, or the orbit should be reinitialized with the updated parameters.
Conclusions. A dynamic orbit determination method based on pseudorange is proposed. Various perturbations that influence the precision of orbit determination are analyzed, and their models are established. The high-precision position is obtained with SPP, and the errors of observation are analyzed and corrected. The difference of the position from numerical integrator and SPP is the observed variable in this DOD method. The LSQ is exploited to get the estimates of initial position and velocity, as well as other dynamic parameters. And then, with the high-precision initial state vector (position and velocity) and force model, the orbit of the LEO satellite can be determined precisely. With the actual data of CHAMP from GeoForschungsZentrum Potsdam, Germany (GFZ), the force model and this DOD method are tested. Simulation results demonstrate that the precision of the force model is high enough to be used in the integrator, and this DOD method not only restrain the divergence due to model errors, but also decrease the influence of random error. As a result, the precision of the orbit determination is obviously improved.
References
1. Tapley B. D., et al. Precision Orbit Determination for TOPEX/POSEIDON[J]. Journal of Geophysical Research, 1994, 99(12): 24383-24404.
2. Melbourne W. G., Davis E. S., Yunck T. P., Tapley B. D. The GPS Fight Experiment on TOPEX/POSEIDON // Geophysical Research Letters, 1994, 2171-2174.
3. Perosanz F., Marty J. C., Balmino G. Dynamic Orbit Determination and Gravity Field Model Improvement from GPS,DORIS and Laser Measurements on TOPEX/POSEIDON Satellite // Journal of Geodesy,1997, 71:160-170.
4. Bisnath S. B., Langley R. B. Precise Orbit Determination of Low Earth Orbiters with GPS Point Positioning // IN: Proceedings of the Institute of Navigation National Technical Meeting. California, 2001, 725-733.
5. Byun S. H. Satellite Orbit Determination Using Triple-Differenced GPS Carrier Phase in Pure Kinematic Mode // Journal of Geodesy, 2003, 76: 569-585.
6. Bertiger W., Bar-Sever Y., et al. GPS Precise Tracking of TOPEX/POSEIDON: Result and Implication // Journal of Geophysical Research, 1994, 99(12): 2444924464.
7. Tapley B. D., et al. Precision Orbit Determination for TOPEX/POSEIDON // Journal of Geophysical Research, 1994, 99(12): 24383-24404.
© Xinlong Wang, Qing Zhang, Hengnian Li, 2013
УДК 518.61
РАСЧЕТ ХАРАКТЕРИСТИК СИСТЕМЫ ПЕРЕДАЧИ ОПТИЧЕСКОГО ИЗОБРАЖЕНИЯ
В МАЛОУГЛОВОМ ПРИБЛИЖЕНИИ
О. Б. Браславская, И. Ю. Гендрина
Томский государственный университет Россия, 634050, г. Томск, просп. Ленина, 36. Е-mail: olechka90@inbox.ru
Реализуется алгоритм построения функции размытия точки и оптической передаточной функции с использованием малоуглового приближения (МУП) для различных оптических и геометрических условий. На основе этих объектов строятся изображения в пространственной и частотной области.
Ключевые слова: уравнение переноса излучения, малоугловое приближение.
THE CALCULATION OF CHARACTERITICS OF THE PASSING SYSTEM OF OPTICAL IMAGE IN THE SMALL-ANGLE APPROXIMATION
O. B. Braslavskaya, I. Y. Gendrina
Tomsk State University 36, Lenina prosp., Tomsk, 634050, Russia. Е-mail: olechka90@inbox.ru
An algorithm of constructing the point spread function and optical transfer function using small-angle approximation for various optical and geometrical conditions is implemented. On the basis of these objects the images in direct and spatial frequency domain.
Keywords: the radiation transport equation, the small angle approximation.
Фундаментом теории переноса изображения в рассеивающих средах являются два раздела современной науки: теория линейных систем и теория переноса излучения.
Для описания процесса распространения света в среде необходимо знание таких оптических характеристик, как показатели рассеяния и поглощения, а также индикатрисы рассеяния, которые определя-
