## Question

Dear Professor

I am Tom Weng, a master student whose major in astronautics and geodesy at Taiwan. I would like to ask for your assistant to answer or give me comments based your rich knowledge on orbital mechanics.

My research now is focus on the GPS and geodesic relative topics. Currently, I am looking for a SP3-Rinex Nav conversion program. This type of program would be very useful for people who have software that only uses RINEX NAV files. I also think it is a very interesting problem from a satellite geodesy point of view. (Definitely, I only focus on a set of 15 quasi-keplerian elements, (i.e., A,e,I,...Crc,Crs,...) except for clock and others information in Rinex Nav.) I spend a lot to search it and it's difficult to look for an adequate program for that conversion, so now I am working on it.

Basically, I think it is a conversion or curve fiting procedures between SP3 (ECEF) and NAV file (15 quasi-keplerian elements). Firstly, I would take the partial derivative of these 3 long equations for x,y,z with respect to each of the 15 NAV elements (Keplerian elements, and correction terms, etc...). Obviously these equations will be very long, and difficult to differentiate and also I am not sure if I get a right differentiate. And maybe some of these 16 or so NAV elements will not be separable (?). So, I try another simplified approaches.

(I am not sure, if I can go to this approache?)
As you know, the ECF position and velocity can be converted to six keplerian elements directly (which are mentioned in many books). So that, I may estimate the 15 quasi-keplerian elements from a time series of 6 keplerians by least squares or others method.(?) I use the models which refer to ICD-GPS-200 document and page 114 (or at the bottom of the page - page 96) in reference 1.

PS. reference1 : GPS Theory and Practice, 4th ed. -- by Hoffman-Wellenhof, Lichtenegger, & Colllins (Springer 1997 ISBN 3-211-82839-7)
reference2: Satellite Orbits, Models, Methods and Applications -- by Oliver Montenbruck Eberhard Gill

But my algorithm with the simplified model (a least square method to fit a data set in 4-6 hr duration) does not work well. The comparisions between mine and IGS rapid orbit is still 800 meters in errors. My target is reducing the error to 10 meter after conversion. Actually, a obvious error was contributed from radis direction "r". That induces, A, e, Crs, Crc to be not good solution in the solution of my optimization.
There are not many text books describe how to inverse the ECF state to quasi-keplerians. I really need some comment and suggestions. Could you give me some comment and introduce or guide some references (books, papers, webs) with this topic to me?
Thanks for your time, and
look forward to here from you soon.

Best Regards,
Tom Weng, From Taipei, Taiwan

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TomWeng - 03 Apr 2007

## Answer

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At this time there is no application in the GPSTk that converts SP3 to RINEX nav files.

First, as for your approach, I have two questions for you.

- Does your curve fit also include the clock model? Clock errors introduce errors on the order of 100's of meters.
- Have you considered using the broadcast ephemeris? The actual performance of the broadcast ephemeris, while lower than that of the precise ephemeris, is much better than the performance specified in the ICD-GPS-200 (now superceded by IS-GPS-200)

The solution I had for this problem is simple. First create one ephemeris per SP3 epoch. This transforms the problem from an estimation (least squares) problem to a deterministic one.
Then, for each pair of epochs, you have the ECEF position of a satellite. If transform the ECEF to ECI parameters, this becomes a Lambert targeting problem. There are many references on this topic. I have a solution to Lambert's problem I spotted in a text by Escobal in his text "Methods of Orbit Determination" that I may code up and check into the GPSTk now that I know someone may use it other than me.

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BenHarris - 04 Apr 2007

but the 15 quasi-keplerian elements should be kept for an hour in RINEX; if looking for solution in adjacent epoches, these 15 quasi-keplerian elements will keep changing by every 15 seconds

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LeiYang - 25 Sep 2007

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