Linear equations for calculating position for the GPS

In summary, the conversation discusses the difficulties in understanding the mathematics involved in calculating position using GPS. The person is struggling to follow some parts of a book and is looking for help. They receive a recommendation for a more readable derivation and have come to understand the linearization process. They are now trying to calculate the initial position estimate and the book recommended is "GPS Satellite Surveying" by Alfred Leick. Options for obtaining an initial position estimate are also discussed.
  • #1
guillefix
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0
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.
 

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  • #2
No one can help me..?
 
  • #3
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
 
  • #4
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 ?
 
  • #5
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.
 

FAQ: Linear equations for calculating position for the GPS

How does the GPS use linear equations to calculate position?

The GPS (Global Positioning System) uses a network of satellites orbiting the Earth to determine the location of a receiver on the ground. Each satellite broadcasts a signal containing its own position and the current time. By measuring the time it takes for the signal to reach the receiver, the distance between the satellite and the receiver can be calculated using the speed of light. Using multiple satellites, the GPS receiver can then use linear equations to determine its own position based on the known positions of the satellites.

What is the basic formula for calculating position using linear equations for the GPS?

The basic formula for calculating position using linear equations for the GPS is: Position = Speed x Time. This formula takes into account the speed of light and the time it takes for the signal to travel from the satellite to the receiver. By using this formula with multiple satellites, the receiver can determine its own position with a high degree of accuracy.

How many satellites are needed for the GPS to accurately calculate position?

The GPS requires a minimum of four satellites to accurately calculate position. This is because each satellite can only provide two pieces of information (its own position and the current time), and three pieces of information are needed to determine a unique position. The fourth satellite is needed for error correction and to improve the accuracy of the calculation.

Can linear equations also be used for altitude calculation in the GPS?

Yes, linear equations can also be used for altitude calculation in the GPS. This is done by measuring the distance between the receiver and the satellites, and then using trigonometric functions to calculate the altitude based on the known positions of the satellites. However, altitude calculations are generally less accurate than horizontal position calculations due to factors such as the curvature of the Earth and atmospheric interference.

Are there any limitations to using linear equations for calculating position in the GPS?

While linear equations are generally very accurate for calculating position in the GPS, there are some limitations to consider. The accuracy of the calculation can be affected by factors such as atmospheric interference, signal blockage by buildings or trees, and errors in the time signal from the satellites. Additionally, the GPS may not work well in areas with limited satellite visibility, such as deep canyons or dense forests. However, advancements in technology and the use of differential GPS techniques have helped to mitigate these limitations.

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