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Step-size influence on the time of processing for 4th order Runge-Kutta numerical integration to obtain GLONASS satellite coordinates

Aeronautics and Aerospace Open Access Journal
Loram Siqueira, João Francisco Galera Monico 


The modernization of GNSS (Global Navigation Satellite System) with the inclusion of new constellations and improvements in the already available systems has been assessed with great interest over the last decade. The combination of different satellite constellations expands the possible applications, so the term multi-GNSS appeared to designate the use of these multiple navigation systems. On that topic, one important element is the quality of the broadcast messages made available by each system to determine the satellites’ coordinates that will be used to calculate the user’s position and how much this information can be trusted. When merging systems for multi-GNSS, the Russian constellation, GLONASS, is still an outsider when the combination of systems is discussed. After regaining its full constellation in 2012 some studies have been conducted on efforts to increase the knowledge about its functionality. This paper discusses the step size needed for the RungeKutta 4th order, indicated to be used on the ICD documentation, interpolation that gets the coordinates of the satellite, and impacts on computational processing time, which has not been previously addressed in the literature. Three days of broadcast ephemeris were processed and compared with IGS final products to determine the residual mean squared error for the local components. It was tested steps of 0.1, 1, 2, 10,30, 45, 60, and 120 seconds, and the step of 1 second was determined as the best one. With the analyzed ideal step-size, as a result of this paper, a Python 3.8 library was made available for users to include on their own projects.


GLONASS, Runge-Kutta, broadcast message