ORBIT DETERMINATION SYSTEM FOR LOW EARTH ORBIT
SATELLITES
Yossi Elisha a, Maxim Hankin b, Haim Shyldkrot a
a Israel Aerospace Industries, MBT space, Israel
b Israel Aerospace Industries, MLM, Israel
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Abstract
如何做腊八蒜The IAI/MBT Preci Orbit Determination system for Low Earth Orbit satellites is prented. The system is bad on GPS pesudorange and carrier pha measurements and implements the Reduced Dynamics method. The GPS measurements model, the dynamic model, and the least squares orbit determination are discusd. Results are shown for data from the CHAMP satellite and for simulated data from the ROKAR GPS receiver. In both cas the one sigma 3D position and velocity accuracy is about 0.2 m and 0.5 mm/c respectively.
Introduction
Satellite tracking and orbit determination are esntial elements of most satellite missions. Knowledge of the spacecraft position at any time is a requirement for communication, mission planning and for geolocation purpos.
Several Orbit Determination systems for Low Earth Orbit satellites have been implemented in IAI/MBT. Both, on-board and ground station systems are ud. The on-board system is bad on the ROKAR GPS receiver. The ground gment orbit determination is bad mostly on GPS samples, with a backup mode, which us the tracking antenna measurements in ca GPS signals are not available.
The GPS code or GRAPHIC (code + pha) measurements form the basis for veral types of integrated solutions for satellite orbit determination, using either dynamic or kinematic models (or a combination of the two). The existence of independent undetermined parameters requires an overall solution using batch post-processing or real-time processing techniques. The batch solutions u least-squares methods, while the real-time solutions apply Kalman filters.
The dynamic approach is to u force and satellite models in order to compute the satellite’s acceleration. The satellite’s position as a function of time is then computed by numerical integration.
This result is compared with the orbit predicted by the GPS measurements. In the batch least-squares solution, the independent force parameters are chon so as to minimize the differences between the predicted trajectory and the actual measurements. In Yunck’s “Kinematic Orbit Determination” [1], a Kalman filter is ud to apply geometric corrections to the dynamic trajectory as a result of the GPS measurements. In Yunck’s “Reduced Dynamic Orbit Determination” [1], the corrections are both geometric and dynamic in nature.
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Montenbruck has propod a “Kinematic” solution [2], which us no dynamic model at all (as oppod to Yunck’s kinematic solution). It merely computes a least squares solution for all the locations and the bias relative to the GPS predictions. High accuracy solutions are obtained, however, kinematic methods are vulnerable and nsitive to bad measurements or bad geometry.
Montenbruck et al. have propod a “Reduced Dynamic” solution [3], which involves estimation of empirical accelerations on top of a preci deterministic force model. It is compod of the dynamic models and the purely kinematic solution and combines the best of both worlds. Not only the accuracy of GPS measurements may be fully exploited but it also has a high robustness offered by dynamical orbit determination techniques.
In this paper we describe the Preci Orbit Determination system for Low Earth Orbit Satellites that has been implemented in IAI/MBT for future missions. The concept is bad on the Reduced Dynamics approach. In the following, we discuss the GPS measurements model, the dynamic model, and the least squares orbit determination. We show results for data from the CHAMP satellite and for simulated data from the ROKAR GPS receiver.
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The real-time navigation position and velocity accuracy provided by the ROKAR GPS receiver is about 5-10 meters and 2-3 cm/c respectively. Since this accuracy may be inadequate for geolocation implementations and for on-board orbit prediction, a real time orbital filter is implemented in the receiver. The orbital filter implements an Extended Kalman Filter (EKF) algorithm, which generates the refined estimates on the basis of the 3D fixed PVT solution supplied by the GPS receiver and the orbit dynamic equations. The orbital filter reduces the velocity error to 1 cm/c and in the abnce of sufficient visibility conditions (e.g. when the antenna is obscured) the GPS receiver us the orbital filter to generate the extrapolation estimate of the orbit for aided navigation (i.e. faster reacquisition).
The ROKAR GPS receiver provides L1 C/A code and carrier pha tracking on 12 channels with accuracies of 0.8 meters and 1 mm respectively. The code measurement is compod of the true distance between a GPS satellite and the receiver antenna, clock offts of both, the receiver and the GPS satellites, the ionospheric path delay, and the receiver noi of 0.8 meters. The carrier-pha measurements have much lower noi of 1 mm, but contain an unknown bias, which must be estimated as part of the orbit determination process. This bias is different for each obrved GPS Satellite but constant between epochs during uninterrupted carrier-pha tracking. The measurements are downlinked and procesd offline by Least Squares Fit (LSF) for preci orbit determination.
GPS data and GRAPHIC method
The main error sources of space navigation bad on a single frequency GPS receiver are: ionospheric range delay, inaccurate GPS ephemeris and clock data, and receiver noi.
The ionospheric range delay is the most significant error source even when a ionospheric model is ud in the post processing of GPS data. Therefore an alternative approach must be implemented to reduce this error source. As a result of the ionospheric layer characteristic, its effect on the carrier-ph
a measurements and on the code measurements is equal in magnitude but opposite in sign. By using the arithmetic mean of code and carrier measurements, the ionospheric path delays can be fully eliminated. This measurement called GRAPHIC (Group And Pha Ionospheric Calibration) [1], exhibits a noi level of about half the pudorange code noi, i.e., 0.4 m for the ROKAR GPS receiver.
Inaccuracy in the GPS satellites orbit is the cond error source that has to be handled. For post processing GPS navigation implementations, the GPS satellites ephemerides are known with high precision and are ud for preci orbit determination. The Center for Orbit Determination in Europe (CODE) provides special GPS ephemeris products [4]. The GPS orbit data is available in the standard SP3 format on a 15 min grid with typical position error of 5 cm. By using 9th order Lagrange interpolation, intermediate positions of similar accuracy can be calculated.
The Center for Orbit Determination in Europe (CODE) also provides high rate (30 c) GPS satellites clock drifts in a standard CLK file [4]. By using such a high rate clock product the range modeling error is less then 1 cm.
Updated differential code bias for the GPS satellites (DCB file) and the relative position of the GPS satellites antenna from the center of mass (Antex file) are also ud. Dynamical model
The additional u of orbit knowledge from the equations of motion may substantially improve the orbit determination accuracy. By using the fundamentals of Newtonian mechanics, given an initial position and velocity vector, the satellite’s orbit can be computed at arbitrary times by performing a double integration of the satellite accelerations over time. This computation is called orbit propagation and is compod of two main procedures that work concutively.
The first one, the acceleration model, calculates the satellite instantaneous acceleration as a function of time, position, and velocity. The acceleration model has to describe the forces that act on the satellite faithfully becau it affects significantly the prediction accuracy. The force model has to include the main forces that act on LEO satellites: earth gravity, atmospheric drag, and luni-solar gravity. Smaller forces like dynamic solid tide, solar radiation pressure, and albedo can be considered, however, their contribution is negligible and in many cas the inaccuracy in the main forces models is bigger than all the small forces together. The acceleration model in this work includes EGM96 of degree and order 70 as the Earth gravity model [5], CIRA72 as the atmospheric density model [6], Moon and Sun gravity with low precision Solar and
Lunar coordinates [7]. The dependence of the cross ction area on satellite attitude is taken into account in the drag model.
The other procedure is the numerical integration of the instantaneous acceleration for the solution of the equation of motion. The differential equation can be handled by common integration methods but cond sum methods are the most suitable for orbit calculation. Therefore, Gauss-Jackson [8, 9] of order 5 with step size of 30 conds is ud in this work. There is no need to u a more accurate method becau the inaccuracies, which result from the acceleration model, are much bigger than tho from the numerical integrator.
Orbit propagation of 10 hours for a LEO satellite at altitude of 400 Km above earth produces an error of 0.5 – 1 Km mainly due to inaccuracies in the force model. In order to reduce this error and to get a more reliable orbit, some parameters that characterize the forces at work might be determined as part of the estimation process. For example, a drag coefficient, C D, and a solar radiation pressure coefficient, C R, act as adjustable scaling factors in many orbit determination systems [1, 7]. However, even when the two empirical constants are estimated properly and the measurements have a very small error, there are errors of 2 - 5 meters in the estimated orbit.
In order to account for the deficiencies in the dynamic model, more degrees of freedom have to be enabled. Piecewi constant synthetic accelerations in the radial, tangential, and normal directions are estimated as part of the orbit determination process to compensate for faults in the acceleration
model [3].
The interval size for the synthetic accelerations has to be chon wily: on one hand short intervals increas the number of degrees of freedom and overweigh the GPS measurements, on the other hand, long intervals may not compensate sufficiently for the lacks in the acceleration model. Intervals of 5 – 15 minutes duration have been found to be suitable, and were implemented in the MBT ground orbit determination system. Least squares orbit determination
The notion of least squares estimation in the context of orbit determination is to find a t of model parameters for which the square of the difference between the modeled obrvations and the actual measurements becomes minimal. Using the parameters one can derive the position and velocity of the spacecraft at any instant within the time interval of the measurements and for some time into the future.
The unknown variables that are estimated during the orbit determination process are: Initial satellite position and velocity
南京周边旅游景点推荐The amplitude of the empirical accelerations
Receiver clock offts at each measurement epoch
Carrier-pha bias for each arc of continuous tracking of a single GPS satellite The practical solution of the least squares orbit determination problem is complicated by the fact that the obrvation model is a highly non-linear function of the unknown variables, which makes it difficult or impossible to locate the minimum of the loss function without additional information. Therefore, calculating a priori values for the unknown variables and estimating only small corrections to the initial values, simplifies the least squares problem considerably. As this is a nonlinear problem, we
reformulate it as one of computing a linear correction to the initial guess. Strictly speaking, this is still not a linear problem, but if the nominal trajectory is sufficiently clo to the true trajectory, it will be in the “linear regime”, where a linear correction is adequate. Yet, some iterations are required to cope with the non-linearity of the orbit determination problem.
Estimation of the drag coefficient, C D,in addition to the empirical accelerations is problematic due to the coupling between them. The estimated value of C D is highly dependent on its weight relative to the weight of the empirical accelerations in the estimator. Without strict calibration, the obtained C D value is unreliable and meaningless. In order to avoid this problem, first, the orbit is estimated using synthetic accelerations with a constant C D coefficient. Later on, the drag coefficient is estimated from the accurate orbit without estimating synthetic accelerations. When this approach
is ud, an accurate trajectory of 0.2 meters and 0.5 mm/c, and a preci drag coefficient are obtained becau the empirical accelerations and the atmospheric drag are now decoupled.黄安战役
Results
The validation of the ground orbit determination was performed in two stages. The first stage in the process was bad on data from the CHAMP satellite [10]. The orbit of the CHAMP satellite is known to a very high precision due to a high quality dual frequency GPS Receiver located on the satellite and advanced orbit determination techniques. However, only single frequency data was ud in the IAI/MBT orbit determination. In the cond stage of tests data was generated in a simulation tup which included a Spirent hardware in the loop GPS simulator that transmitted RF signals to the ROKAR GPS receiver.
In the CHAMP bad validation tests 48 orbital data arcs of GPS measurements in 2004 have been procesd and analyzed. Each orbital data arc contains 10 hours of data. Inputs of the Preci Orbit Determination system were: Rinex file (L1 frequency GPS measurements every 30 conds), preci GPS ephemeredes, and high rate (30 c) GPS satellite clock data. Input data was obtained from CODE, The Center for Orbit Determination in Europe.
wetThe Champ orbit was estimated from the GPS data in three cas:
Optimal - using all the available measurements
Realistic - GPS antenna is considered to be obscured due to the satellite crui law and only 75% of the available GPS measurements are ud
Near real-time - doesn’t incorporate clocks data (corresponds to 3 hours delay instead of 17 hours in the optimal ca)
In the optimal ca, a typical position accuracy of less than 20 cm 1-sigma has been achieved. Due to the robustness of the algorithm, the results of the realistic ca were only slightly wor - 25 cm 1-sigma. In the near real time ca a better then 40 cm, 1-sigma accuracy was achieved. In all cas the velocity accuracy was less than 0.5 mm/c.
