Testing of the Conditioning of the GPS

The fourth and fifth part of the project involved testing the conditioning system of the GPS problem. The three satellite positions were put into spherical coordinates such that \(A_i,B_i,C_i\) were now \begin{align*} A_i = \rho cos(\phi_i) cos(\theta_i)\\ B_i = \rho cos(\phi_i) sin(\theta_i)\\ C_i = \rho sin(\phi_i) \end{align*} where \(\rho\) = 26570 km, and \(\phi_i\) was four different values between 0 \(\leq \phi_i \leq \pi/2\) and \(\theta_i\) was four different values between 0 \(\leq \theta_i \leq 2\pi \). The four values used for \(\theta\) and \(\phi\) can be seen in the table below:

Using this, we were able to define an error magnification factor for this situation using the formula provided in the book for Reality Check 4. It was stated that the atomic clocks aboard the satellites are correct up to 10 nanoseconds, so \(\Delta t_i = 10^{-8}\).
This allows the EMF to be defined as \begin{align*} emf = \frac{\|\Delta x, \Delta y, \Delta z\|_\inf}{c\|\Delta t_1,...,\Delta t_m\|_\inf} \end{align*}

From this, the \( t_i\) were changed between \(+10^{-8}\) and \(-10^{-8}\), and the error magnification factor was calculated. The codes used are H.m, and DH.m. The results can be seen in the table below.

Based on the table, it was seen that the max position error was 0.0150798597733228 kilometers, and the condition number was 5.03009978100344.

The fifth part of the project was similar to the fourth, except the satellites were spaced within 5% of each other. This meant that all the \(\phi_i\) and \(\theta_i\) were all spaced within 5% of each other. The codes for this part of the project can be seen here I.m and DI.m. The error magnification factor was calculated for each of them and the results can be seen in the table below.

The maximum position error was 2.86965713370289, with a condition number of 957.214585332525. It was found that the EMF decreased as the satellites moved further apart from each other. Thus it was better if the satellites are unbunched.