George Mason University

Zakaria Tarik Zerhouni

Step 4.
We generalized our Multivariate Newton's method procedure from Part 1 into a function capable of estimating the position of an observer on the north pole based on the signal from any set of 4 satellites.

Unlike previous examples, which specified satellite positions as a set of (x, y, z) coordinates, our function for this part took as its arguments spherical coordinates \( \phi\) (ranged 0 to \(\pi\)) and
\(\theta\) (ranged 0 to \(2\pi\)). All satellites were assumed to be 20km above the earth's surface (\(\rho\)=26570 km). As shown below, we had MATLAB generate a 3d plot to roughly show the positioning of the satellites.

In order to determine the sensitivity of the measured coordinates to errors in the timing of the input data, the algorithm applied tiny changes (\(\pm d\) for some small \(d\), e.g. 10 nanoseconds) to the observed time delays and measured the largest coordinate error in the computed result.

All possible combinations of deviations (\(-d\) error all four time delays, \(+d\) error for all four time delays, no error, any combination thereof), a total of \(3^4=81\) possibilities, were tried in order to determine the maximum output error, and the unit-less error magnification factor (EMF) was calculated as follows:

\( \text{emf} = \frac{|| \Delta x, \Delta y, \Delta z ||_{\infty}}{c || \Delta t_1 \Delta t_2 \Delta t_3 \Delta t_4 ||_{\infty}} = \frac{|| \Delta x, \Delta y, \Delta z ||_{\infty}}{cd} \)

Example:
>> part4(10^-8, [0, 0.4, 0.8, 1.2], [0, pi/2, pi, 3*pi/2])
maxerr =
0.0132
ans =
4.3946

Thus, for the given coordinates, we have a maximum error magnification factor of 4.3946, meaning that a timing error of 10ns (2.998 meters backwards error, the distance that the GPS signal travels in that time) translates to a forward error of 13.2 meters.

Thus our estimate for the condition number of the problem at these coordinates is 4.3946.

Figure (4.1)

Click on the image to view the associated code.

Previous: Step 2

Project 2 Home

Next: Step 5