Initial Orbit Determination (Least Squares Method)

Initial orbit determination applying the Least Squares method

1) In test_LSQ_GEOS3.m, we define global variables: const (astronomical and mathematical constants), CS (unnormalized gravity field coefficients), AuxParam (struct that represents model parameters and time for orbit propagator), eopdata (Earth orientation parameters), PC (planetary coefficients for JPL ephemerides computation).

2) Download Earth orientation parameters from the following link (if it is out of date).

https://celestrak.org/SpaceData/EOP-All.txt

3) Earth orientation parameters are read from EOP-All.txt.

4) Observations are read from GEOS3.txt, and biases from table 4-4 of the book “Fundamentals of Astrodynamics and Applications, 4th Edition, 2013,” are subtracted. Then, observation noises are set from table 4-4. After that, the station’s ecef position is calculated using Position.m, and two sets of observations (azimuth, elevation, and range) are selected. Afterward, the LTC matrix (transformation from Greenwich meridian system to local tangent coordinates) is computed by LTCMatrix.m, and the initial guess of the satellite state vector (TOD) at the epoch (first observation’s time) is found by Elements.m and State.m.

5) The Least Squares’ initial parameters and parameters for the orbit propagator (AuxParam.Mjd_UTC = Mjd_UTC; AuxParam.n = 20; AuxParam.m = 20; AuxParam.sun = 0; AuxParam.moon = 0; AuxParam.planets = 0;) are set.

6) The epoch’s state vector is propagated to the times of all observations in an iterative procedure and corrected at each stage. Then, the epoch’s state vector is converted from the TOD to the J2000 coordinate system and compared with the true state vector. Finally, the epoch’s state vector is converted to the TEME coordinate system and sv2tle.m is used to produce the TLE.

Please note that numerical integration is done in TOD coordinate system. So, only Earth rotation is considered in Accel.m and VarEqn.m.

References:

O. Montenbruck, E. Gill, "Satellite Orbits: Models, Methods, and Applications," Springer Verlag, Heidelberg, 2005.

D. Vallado, "Fundamentals of Astrodynamics and Applications," 4th Edition, 2013.

https://celestrak.org/software/vallado/matlab.zip

https://ssd.jpl.nasa.gov/planets/eph_export.html