GPS in Flat Land

Peter Hand FRS 144 Project 2

GPS in Flat Land

The goal of this project was to get us to think like a GPS receiver and to calculate our position, altitude, and the time given the times that three satellites sent signals to our receivers. In this project, we assumed that all of the satellites are in perfectly circular orbit, and are evenly spaced in that orbit. In reality, there are 6 satellites in 4 different planes, and the orbits are elliptical with respect to the earth. We were also provided with an almanac of information relating to the satellites. Not all of the columns that were given in out almanac would be in a real almanac. The ones that would are satellite number, and square root. A real almanac would also include PRN number, health of satellite, angles and reference times, clock correction times, etc. The ones that are missing because they are not applicable in flatland are inclination angle, argument of perigee, ascension, and corrected mean motion. All of the other ones that are missing from our almanac (time correction, eccentricity) are missing because of extra simplifications. The GPS system had a Y2K problem before Y2K because it counts in weeks, up to 20 years. Once the counter hit week 1024, or 20 years, the system restarted at 0, causing what we call a Y2K problem. This problem will occur again 20 years from the last problem date.

Getting through the calculations, after going over in class the process by which this problem can be solved, was not as big a task as I originally thought it to be. Using formulas we discussed in class, I was able to come up with the answer fairly easily. The big mistake I made in my last attempt was that I was not remembering that I could convert the period of each satellite into degrees to figure out the longitude. Without that, it was impossible to solve the problem! So now knowing to do this, I was able to figure out the longitudes of all 3 satellites very easily. After doing this, I could figure out the longitude of the receiver, knowing that it was closest to satellite 3, and closer to satellite 2 than to satellite 4. To get the exact position, the longitude and altitude were approximated until each was as precise as it could be. Knowing the position of the receiver, I was then able to figure out the distances from each satellite to the receiver, and then figure out the delay of transmission from each satellite, and thus was able to figure out the time at which all of the satellite signals reached the receiver ( I was able to figure out the current time of day. And that was that, and it was so much less painful and time consuming than my last effort.

As for the last question, taking into account the fact that the earth rotates, for a better calculation, you would have to account for the speed of rotation, and for the speed of the satellite. Doing this would allow you to get a better reading of where you are, as it accounts for the changing position of the satellites with respect to the rotating of the earth.

This paper represents my own work in accordance with University regulations.

Peter J Hand