Jonathan Lew12/1/2018

GPS in FlatLand Project

Pre-lab Questions:

We've assumed that all the satellites are in the same perfectly circular orbit and they are equally spaced around that orbit. What about the real GPS constellation?

  • In the real GPS constellation there are 4 satellites in each of the 6 orbital planes plus spares. Because the earth is oblate, when the satellites cross over the middle of the earth they are pulled to the middle, but when they are over the sides, they are pulled to the slightly to the sides. The real orbit is not circular.

The flat land almanac contains four columns. Which of these columns would be in a real almanac? What other columns are in the real almanac? Some of the column missing from the flat land almanac are missing because they're not applicable to flat land and some are missing because of the extra simplifications. Can you tell which are in each category? (This might be hard!)

  • The column “square root a” would be included in a real almanac.
  • Some terms omitted because of simplification: clock correction, eccentricity, corrected mean motion, argument of perigee, mean anomaly
  • Some columns omitted because of flat land: inclination angle, Rate right ascension

Did you know the GPS system had a Y2K (actually a W1024) problem before Y2K? When will it happen again?

  • The GPS format only allows for weeks up to 1024, so when week 1024 reached in August 1999 (1024 weeks from January 1980), the week reverted back to 0 rather than 1024. This will happen again in March 2018.

What can you learn from one signal?

  • The position (latitude) of the satellite can be calculated. However the position of where you are, and altitude can only be calculated after more than two signals are received, and the time is estimated.

Project

In order to calculate the receiver’s position (longitude and altitude) and time, a spreadsheet was created (shown below):

A / B / C / D / E / F / G / H
Satellite / Angle / longitude - satellite / longitude - you / delay time / altitude / time of almanac / receiver time
2
3
4

Angle was the angle at the center of the earth, between your position (longitude – you) and the satellite (longitude – satellite). Since we knew the starting angle of the satellite, its position at a given time could be calculated by the formula:

“=t*360/43082.0452694999+α” [where tis the time of almanac and α is the starting longitude of the satellite]

The value for longitude-youwas changed by trial and error, so both longitude-satellite and longitude-you could be used to calculate the Angle with this formula:

“=ABS(D2-C2)”[1]

The delay time was dependent on the angle and the altitude. Now that there is an angle value and since the altitude value was changed by trial and error. Delay time could be calculated with the given formula:

“=SQRT((26561819.3619*COS(RADIANS(B2))-6371000-F2)^2+(26561819.3619*SIN(RADIANS(B2)))^2)/299792458”

Finally the time of almanac is equivalent to the delay time + receiver time. Receiver time was a manipulated trial and error variable so this could be input as a formula too.

“=H2-E2”

Therefore we are now set to change longitude-you, altitude and receiver time in order to get the right time of almanac for all three satellites (given). While this seemed arbitrary at first there were some guiding principles.

We knew that the time of almanac was smallest for satellite four, this meant that there was a larger delay time (since receiver time was all the same). A larger delay time is caused by a greater distance, therefore the longitude must be away from satellite 4. Using the same kind of analysis for the other two satellites the longitude had to be somewhere between satellites 2 and 3. Changing the longitude of yourself was the main tool to get the ratio’s of time of almanac correct. If a time of almanac value was too big, then the longitude had to be moved further from the satellite, if the time of almanac value was too small it had to move closer.

Receiver time could be used to balance the whole set of satellites if all three values of time of almanac were too small or two big.

However at about 5 decimal places accuracy I noticed that the values of the third satellite were too small, while the other two values were too big. If you moved the longitude of yourself closer to satellite three this would only make satellite 4 bigger too, and it was already too big. Therefore I tweaked the altitude value. Altitude added the most to the almanac time for satellite three, then satellite two and then satellite four.

After many trials the correct values for time of almanac were calculated. Attached along with this file is the spreadsheet used and the following results:

Satellite / Angle / longitude - satellite / longitude - you / delay time / altitude / time of almanac / receiver time
2 / 32.68518 / 253.1916068 / 285.87679 / 0.071639274 / 152 / 22222.1505827258 / 22222.222222
3 / 12.31485 / 298.1916373 / 285.87679 / 0.067989075 / 152 / 22222.1542329248 / 22222.222222
4 / 57.31475 / 343.1915438 / 285.87679 / 0.079171148 / 152 / 22222.1430508522 / 22222.222222

Last question: we've ignored the fact that the Earth rotates. How would you handle that in a more realistic calculation?

You would have to include a column labeled rotation which would be the speed that the earth rotates at times the delay time. This rotation would affect the angle (added or subtracted) which in turn affects the other values.

[1] This formula could be used because we knew that the position of you given the three satellites was going to be along the horizon from longitude ≈180°-360°. Had it been straddling the 0° line along the horizon 90°-275° then the angle would require a more complex formula.