Linear equations for calculating position for the GPS

[guillefix]

I am doing my Extended essay about the maths involved in calculating your position using the GPS. I am reading a very complete book, but the maths are sometimes too hard or not enough deeply explained for my level and I am struggling to follow some parts. Here I post one page of the book where they derive a linear equation (14.8) which I can't finish to see where they get it from. I now understand that the weighting matrix entries are direction cosines but I don't really get how they derived that equation. Thank you.

 

[guillefix]

No one can help me..?

 

[jbunniii]

I work in GPS and am quite familiar with the linearized pseudorange equations, but the notation in your book is so horrible that I am having trouble making sense of it. (I recommend getting the Kaplan book instead!)

Try reading this brief PDF file for a more readable derivation:

http://scholar.lib.vt.edu/theses/available/etd-112516142975720/unrestricted/appa.pdf

 

[guillefix]

Thank you for that, I have finally understood the process of linearizaiton. The book I'm reading also includes a covariance matrix in its calculation, but I think that it's not very relevant because it simplifies to exactly the same normal equation than in yours. Anyway, I am now struggling to figure out how to calculate the initial position estimate. In both my book and your document it doesn't explain how to calculate it. How is it done?

BTW is the book you refer to this one https://www.amazon.com/dp/1580538940/?tag=pfamazon01-20 ?

 

[jbunniii]

Yes, that is the book.

For an initial position estimate, it depends on what is known when the receiver starts. If a previous position is known, then you can use that. If the receiver is in a cell phone, then the phone can give the receiver an approximate position based on the cell tower's location. If nothing at all is known, choose some nominal position, e.g. north pole.