Project 1 - GPS, CONDITIONING, AND NONLINEAR LEAST SQUARES

Overview:

The Global Positioning System (GPS) is a satellite-based navigation system made up of a network of 24 satellites. In order to triangulate the position of the receiver, we require at least four satellites. With the four satellites are able to determine the position of the receiver and the error in the receiver's clock.

Team Members: Daniel West, Lado Tonia

Problem 3: Daniel/Lado
Problem 4: Lado
Problem 5: Lado
Problem 6: Daniel/Lado
Website: Daniel

Problem 3 | Problem 4 | Problem 5 | Problem 6 | All

Problem 4:

In this problem we were asked to change the inputs by modifying the time values by +/- 10^-8.

The maximum position error found was approximately 40 meters with an error magnification factor of 13.3.

Code - Function
Code - Polar/Init
Output

Problem 5:

This problem required us to tightly group the satellites within 5% of each other. With a group of tightly bunched satellites we expected to have a much higher error magnification factor.
We used a random number generator to give randomized data to establish the four satellites.

As expected we received a much higher maximum forward error of 726 meters and a maximum Error Magnification Factor of 242. Code - Function
Code - Polar/Init
Output

Problem 6:

In this problem we were expected to add on additional satellites to our equations. Normally this would require a combined least squares/multivariate Newton's method. However, Matlab, when given more equations than variables, already expects this and does the least squares method for us.

To accomplish this we modified our original f and dF matrix to be constructed within a for loop allowing us to dynamically establish the number of satellites used.

For our data we added 3 more satellites for a total of 7 unbunched satellites. We received a position error of 2.5 meters and an error magnification factor of .835

Code - Gauss Function
Code - Polar/Init
Output

Results:

Problem Max EMF Max Position Error

4 13.3 40 meters

5 242 726 meters

6 .835 2.5 meters

It is clear that plenty of unbunched satellites will give you a much more accurate location compared to bunched or fewer satellites.

Final Code:
Code - Multivariate Newtons Method Function
Code - Gauss Function
Code - Polar/Init
Full Output - See Individual Problems Individual Output/