With the everincreasing number of satellites in Low Earth Orbit (LEO) for scientific missions, the precise determination of the position and velocity of the satellite is a necessity. GPS (Global Positioning System) based reduceddynamic orbit determination (RPOD) method is commonly used in the post processing with high precision. This paper presents a sequential RPOD strategy for LEO satellite in the framework of Extended Kalman Filter (EKF). Precise Point Positioning (PPP) technique is used to process the GPS observations, with carrier phase ambiguity resolution using Integer Phase Clocks (IPCs) products. A set of GRACE (Gravity Recovery And Climate Experiment) mission data is used to test and validate the RPOD performance. Results indicate that orbit determination accuracy could be improved by 15% in terms of 3D RMS error in comparison with traditional RPOD method with float ambiguity solutions.
Precise orbit information of satellites is the basis for many space missions, such as the earth gravity recovery, atmosphere sounding, and ocean surveillance. From the beginning of 1980s, Global Positioning System (GPS), which can provide continuous, allday, and highprecision observations, has been widely used in Positioning, Navigation, and Timing (PNT) applications. The concept of spaceborne GPS receivers tracking GPS signals for orbit determination has been proposed since then. During the past 30 years, a large number of satellites have been launched into Low Earth Orbit (LEO) with GPS receivers. GPS based Precise Orbit Determination (POD) has been considered as the fundamental and primary technique for spacecraft missions.
Several GPS based orbit determination strategies have been proposed in the literatures and applied in the real space missions as well. Generally, they are categorised into three groups: kinematic method (KPOD) [
The rest of the paper is organised as follows: Section
The GPS code phase and carrier phase observation equations are written as follows [
According to (
Combining (
In the IF observation equation, the carrier phase ambiguity,
Another two types of observable combinations are commonly used for GPS data quality screening as shown below [
The MW combination can eliminate the effects due to the ionosphere, the geometry and the satellite, and receiver clocks, which is given by [
GF observable is formed via the difference between the dualfrequency observations, which is also independent of geometry, receiver clocks, and satellite clocks [
In addition to ionospheric path delay, other errors in GPS observation are necessary to be corrected, which are including relativistic biases in clocks, phase windup effects, antenna phase centre offsets, and multipath errors. Note that in the IF observation (
For the spacecraft in nearearth orbit, a NewtonKepler system is traditionally used to describe the orbit for the twobody case. Real orbit modeling, however, should take into account additional gravitational and nongravitational perturbations. In general the accelerations acting on the satellite consist of terms for the geopotential gradient, the 3rdbody gravitational attraction of the sun and moon, the solar radiation pressure, and atmosphere drag on the spacecraft, if no active orbital manoeuvre is performed. The exact formulations for each term can be obtained from classical books, for example [
Generally there are two main coordinate systems involved in the GPSbased LEO satellite orbit determination: the earth centred inertial (ECI) system and the earth centred earth fixed (ECEF) system. The former one provides a formulation of the satellite propagation and coordinates of other spatial bodies, for example, the sun and the moon. The solar radiation pressure acting on the spacecraft is also formulated in this frame. On the other hand, the ECEF frame provides modeling of the earth gravity field and atmospheric drag affecting the spacecraft. Aside from that, GPS orbits and its signals are given in this frame. With aim of implementing the sequential filtering algorithm for spacecraft orbit determination, the coordinate systems have to be consistent. In this sense, all the acceleration components are calculated or transformed into the ECEF system in this paper. Additionally, two types of “fictitious” forces are necessary to be added into this rotating reference frame, that is, the centrifugal and Coriolis accelerations, which are written as [
The motion of the satellite along with the time
Note that
Assume that uncertainty of the orbital model could be absorbed into empirical accelerations, which are always modeled as a firstorder, stationary, GaussMarkov process [
This research uses EKF to determine the LEO satellite positions and velocities epoch by epoch [
Single point positioning algorithm is used for determining the positions and velocities of LEO satellite and receiver clock offsets at the first epoch. Carrier phase ambiguities are initialised by carrier phase measurement minus pseudorange measurement. As for the orbit parameters, namely, solar radiation pressure and atmospheric drag coefficients, their initial values are given empirically. For instance,
In the time update step of EKF, both the states and their covariance matrix should be predicted:
To use RPOD approach, state transition matrix
Traditional GPS data quality control methods by means of MW and GF observation combinations (see (
Within the measurement update step of the EKF, the predicted states and covariance matrix are required to be updated using new GPS observations:
Carrier phase integer ambiguity resolution was applied in RealTime Kinematic (RTK) positioning at early stage, since differencing technology could eliminate and mitigate most of the observation errors and biases so that the integer property of the ambiguity could be preserved. In the traditional PPP algorithm, however, it is hard to achieve doubledifferenced positioning accuracy.
Here the GPS IF observations for LEO satellite orbit determination in (
Since 2006, researchers started fixing integer ambiguity by correcting initial phase biases (namely, HDBs) in zerodifference PPP simulations. Wang and Gao found that phase biases of singledifference between receivers are unstable, while that of SingleDifference Between GPS Satellites (SDBS) are very stable in several days. Hence, they concluded that PPP ambiguities could be fixed as integers with GPS satellite phase biases well corrected [
Single Difference Between GPS Satellites (SDBS) [
DoubleDifferenced Ambiguities (DDA) [
Decoupled Satellite Clocks (DSCs) [
Integer Phase Clocks (IPCs) [
In summary, with receiving extra information from network solutions, standalone receivers are able to implement integer ambiguity resolution in PPP algorithm. Since 2009, IGS CNESCLS (Centre National d’Etudes Spatiales, Collecte Localisation Satellites) started to release GPS satellite IPCs products together with Widelane Satellite Biases (WSBs) [
The basic concept of this method is to separate pseudorange clocks and carrier phase clocks; the float widelane ambiguities
The IF observations given in (
After fixing the narrowlane ambiguity, the IF ambiguity is recalculated together with the integer widelane ambiguity. This precise information can be considered as new pseudoobservation and used for positions update in a new run of Kalman filter. The flow diagram of the IPCs based PPPAR procedure is shown in Figure
IPCs based PPPAR procedure.
Firstly, the IPCs based PPPAR method is tested with IGS station observations. Data collected from four IGS stations (AUCK, SHAO, TLSE, and WTZZ) on February 20, 2012, are used for both static and kinematic PPP testing. Table
IGS station solutions comparison.
PPPAR strategy  IGS station  StaticPPP  KinematicPPP  




Fixing Ratio 



Fixing Ratio  
Rounding  AUCK  0.019  0.024  0.038  97.3%  0.023  0.029  0.059  95.5% 
No  0.020  0.024  0.039  0.0%  0.039  0.030  0.059  0.0%  


Rounding  SHAO  0.028  0.025  0.085  94.5%  0.036  0.029  0.120  88.9% 
No  0.028  0.025  0.086  0.0%  0.040  0.032  0.121  0.0%  


Rounding  TLSE  0.009  0.019  0.028  97.4%  0.021  0.037  0.059  93.3% 
No  0.013  0.020  0.032  0.0%  0.022  0.036  0.061  0.0%  


Rounding  WTZR  0.010  0.025  0.020  97.3%  0.018  0.033  0.047  93.6% 
No  0.015  0.026  0.020  0.0%  0.025  0.035  0.049  0.0% 
Figure
WTZR positioning errors comparison using PPPAR and float PPP strategies in the kinematic mode.
Spaceborne GPS data from the GRACE mission is used for testing RPOD performance with ambiguity resolution, which has twin LEO satellites flying in a group for space missions of earth gravity field modeling and climate experiments [
GPS satellite clocks with consistent satellite orbits are provided weekly by the CNESCLS IGS analysis centre [
Orbit models used in this paper.
Satellite model  Cannonball model (constant surface and mass) [ 


Earth gravity field  GGM03S 75 × 75 [ 


Earth tides 



Atmospheric density  HarrisPriester [ 


Solar radiation pressure  Assuming the surface normal is always aligned with the sun 


Planetary ephemerides  Low precision model [ 


Coordinates transformations  IERS1996/IAU1980 transformations [ 
RPOD EKF parameters settings.
Initial values  A priori variance  Correlation time  Process noise  Observation noise  


100 

30 

500 

200 

80 

100 

0.1 

100 

600 

0.8 

100 

0.2  —  — 

400  —  — 

1.3 

0.4  —  — 

500  —  — 

2.3  —  —  —  —  —  —  —  — 
RPOD solutions in RIC coordinates with and without phase ambiguity fixing using 10 hours of GRACEB satellite data are shown in Figure
GRACEB data RPOD errors comparison (10 hours on February 20, 2012).
Figure
GRACEB data RPOD carrier phase ambiguity resolution analysis (10 hours on February 20, 2012).
The empirical accelerations estimated in RPOD are plotted in RIC coordinate system in Figure
GRACEB satellite empirical accelerations in RIC coordinate system (10 hours on February 20, 2012).
The IF observation residuals are plotted in Figure
GRACEB IF observations postfit residuals (10 hours on February 20, 2012).
This paper presents PPP ambiguity resolution in reduceddynamic orbit determination using GPS observations. Several centimetres of accuracy has been achieved. Carrier phase ambiguity resolution could significantly improve orbit determination performance by 15% in terms of 3D RMS in the GRACEB satellite scenario. In the RPOD algorithm, the empirical accelerations are in accordance with the orbital model precision. Hence, it is important to determine the value of empirical accelerations. On the other hand, the estimated empirical accelerations can help detect mismodeling part in the orbital models. Since GNSS realtime PPPAR method has attracted much more attention these days, it is potential to apply this technique into realtime precise orbit determination in the future work.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This study is financially supported by the National Natural Science Foundation of China (11172235) and the Doctoral Fund of Ministry of Education of China (20106102110003).