ORBIT DETERMINATION MODELING ANALYSIS BY GPS

Ahmed Adel Ahmed AbdElhamed The purpose of this paper was to analyze the modeling of a GPS satellite orbit, The Global Positioning System (GPS) is a satellite navigation system for determining position, velocity and time with high accuracy, using signals of the GPS constellation (RINEX files), with the aim of analyzing the performance of the orbit. One pursues to verify how differences of modeling can affect the final accuracy of orbit, To do that, the following effects were considered in the orbit propagation process: earth oblatness and Sun-Moon attraction.a RINEX(Receiver Independent EXchange) file with navigation data at the 8th of August 2015 was used as an input for an algorithm for determining the GPS satellite position and velocity , the algorithm was implemented using VB.net code with the ability of determining the position and velocity of any GPS satellite within the chosen orbit epoch of the used RINEX navigation file , in order to yield conclusive results about the position and velocity accuracy, a NGA(National Geospatial-Intelligence Agency) data file for precise GPS satellites positions and velocities at the chosen orbit epoch was used for the algorithm output verification. Numerical application will be given.


Broadcast Orbit:-
The satellite positions estimated in the Kalman filter process are next represented in the form of Keplerian elements with additional perturbation parameters. The parameters refer to a given reference epoch, t 0e for the ephemeris and t 0c for the clock, and they are based on a four hours curve fit (ICD, 1993).hence, the representation of the satellite trajectory is achieved through a sequence of different disturbed Keplerian orbits.
The first group of parameters is used to correct satellite time. The second group determines a Keplerian ellipse at the reference epoch, and the third group contains nine perturbation parameters. where Δn is the secular drift in dω/dt due to the second zonal harmonic (C 20 ); also it absorbs effects of the Sun's and Moon's gravitation and solar Radiation pressure over the interval of fit, Ω is the Secular drift in right ascension of the node due to the second zonal Harmonic; includes also effects of polar motion, i is the Rate of change of inclination, and Cus,Cuc short periodic effects of C20; also include higher order effects and Cis,Cic short periodic effects of lunar gravitation (during the closest approach Crs,Crc of the space vehicle to the Moon); also absorb further perturbations. Fig. 1 explains the Keplerian and the perturbation parameters. Note that the element Ω 0 in the GPS message is not measured from the vernal equinox, ϒ, but from the zero meridians, X T . In essence, Ω 0 is not a right ascension angle but a longitude. In recent literature the parameter is therefore designated as longitude of ascending node (LAN).

Computation of Satellite Time and Satellite Coordinates:-
The GPS system time is a continuous time scale, and is defined by the weighted mean of the atomic clocks in the monitor stations and the satellites.Using The satellite coordinates X k , Y k , Z k are computed for a given epoch, t, with respect to the Earth-fixed geocentric reference frame X T ,Y T , Z T . The time, t k , elapsed since the reference epoch, t 0e , is t k = t − t 0e . ` Then, we can use the following Computational Sequence to compute the GPS satellite position and velocity at given epoch 3. t k = t -t oe 4. n = n 0 +Δn 5. M k = M 0 + nt k 6. M k = n 7. M k = E k -esinE k 8. Solve for E k by iteration using NEWTON-RAPHSON method. = Ae sinE k + 26.
= r k cos u k 28.
= r k sin u k 29.
= cos u k -r k sin u k 30.
= − 33. x k = cos Ω k -cos i k sin Ω k 34. y k = sin Ω k + y cos i k cos Ω k 35. z k = sin i k 36. V x = cos Ω k -cos i k sin Ω k + sin i k sin Ω k -y k 37. V y = sin Ω k + cos i k cos Ω k -sin i k cos Ω k + x k 38. V z = sin i k + cos i k (Benjamin W. Remondi, 2004) The previous algorithm for determination of GPS satellite position and velocity from the navigation data broadcasted by the GPS satellites through the navigation message (using RINEX file: 7odm2200.15n as an input) was implemented using Microsoft Visual Studio.NET and the results compared with Ephemeris data by NGA (using nga18566.eph file) were listed below.

Conclusion
The differences between the algorithm implementation results and NGA precise ephemeris were: In

Propagation of satellite State vector:-
To accurately predict position and velocity of a spacecraft through certain interval we have to take into account the perturbing forces acting on satellite during its motion around earth.

Perturbations due to Earth Gravitational Field:-
The Earth is not a perfect sphere with homogeneous mass distribution, and cannot be considered as a material point. Such irregularities disturb the orbit of an artificial satellite and the keplerian elements that describe the orbit do not stay constant. The perturbing function can be given by: Now apply the Lagrange VOP equations with a disturbing function, the secular change due to earth oblatness is Perturbations due to Third-Body Potential:-These perturbations are due to Sun and Moon attraction force and they can be meaningful if the satellite is far from the Earth. As the orbital variations are of the same type, whatever is the Sun or the Moon the attractive body, they should be studied without distinguishing the third body. The luni-solar gravitational attraction mainly acts on Ω and ω, what causes precession of the orbit and the orbital plane. The secular change due to Third-Body Potential is: Calculation of R N from equation.
Calculation of GAST, GMST, GMST 0 and α from equations. The calculated state vector from the first algorithm was used as input data for propagation using the previous algorithm and the results were compared with reports generated by Satellite Tool Kit (AGI STK9) and the results depicted in the two figures below.

Conclusions:-
This paper describes a satellite orbit determination concept, based on the broadcasted GPS navigation message, As can be seen in the results of the first algorithm, the achieved position accuracy lies in the order of 1 m for the position components and 0.0005 m/s for the velocity components.
The outlined GPS based satellite point positioning has been applied successfully in the next algorithm for motion propagation of the state vector and the compared results with STK were nearly identical as shown in the graphs.