GPS Tracking - Anniversary Dinner

A ECE6390 Production By:
     "Can you hear me now?"
Actors in Alphabetical Order:
     Derek Campbell, Rod Drews, Chris Lee, Lisa Moyer, Jason Uher

Problem:

It is your anniversary, and you are meeting your spouse at a local Atlanta restaurant. Your spouse calls you to tell you the name of the restaurant, but the cell phone cuts out before you can receive the information. Return calls go to the voice-mail. Luckily, you notice that your phone received the pseudo-ranges, calculated from 4 GPS signals, of your spouse's location, via the cellular company's protocol. Based on the pseudo-ranges, you attempt to determine your spouse's location, and get yourself out of trouble.

Technical Data:

The received pseudo-ranges of your spouse's location are tabularized below. The pseudo-range values are relative to the first satellite position.

Satellite 1 Satellite 2 Satellite 3 Satellite 4
0.000000000 s -0.001587009 s -0.001373255 s -0.001092284 s

In the calculations, the values for these constants are used:

Radius of Earth = 6380.0 km
Altitude of GPS satellite = 20,200.0 km
Speed of light = 299,860.0 km

The equation used to calculate the location is:

(xi - ux)2+(yi - uy)2+(zi - uz)2 = (PRi - c τ)2 where i = satellite number

Technique:

  • Convert latitude and longitudes of each GPS satellite from spherical coordinates to Cartesian coordinates
  • Enter the Cartesian coordinate values into the coordinate equation above
  • Solve the equations to determine the Cartesian coordinates for the true location
  • Convert the resulting Cartesian coordinates back into spherical coordinates
  • Determine appropriate solution from results (look for one with valid radius)
  • Enter the latitude and longitude coordinates into Google Maps
  • Determine the restaurant from the map

Following the above technique, MATLAB 7 R14 was used to calculate the Cartersian coordinates of the latitudes and longitudes of the GPS satellites.

  Satellite 1 Satellite 2 Satellite 3 Satellite 4
Latitude (degrees) 43.2345 35.5221 29.0510 40.7777
Longitude (degrees) -62.3256 -85.2234 -90.9760 -95.0234
x (meters) 8.9940e6 1.8007e6 -0.3958e6 -1.7624e6
y (meters) -1.7150e7 -2.1550e7 -2.3233e7 -2.0050e7
z (meters) 1.8207e7 1.5455e7 1.2907e7 1.7360e7

These values were put into the equation form, and using Maple 9.02, the equations were solved.

  t ux uy uz
Solution 1 0.1271710729 -991640.1367 9954469.328 -6689184.662
Solution 2 -0.6896623642e-1 519396.8847 -5278149.767 3546822.260

The resultant Cartesian coordinates were converted back into spherical coordinates with MATLAB.

  Latitude (degrees) Longitude (degrees) Radius (meters)
Solution 1 -33.7694 95.6889 1.2034e7
Solution 2 33.7727 -84.3799 6.3803e6

The difference in the radius of the first solution states that the cellular handset is 6.5641e6 meters above the surface of the Earth. The difference in the radius of the second solution is 328.0661 meters. Although the radius for the second solution may not be exact, it is far more realistic than the first solution. The final value has the latitude and longitude coordinates: (33.7727, -84.3799).

Results:

The technique above gives the coordinates: (33.7727, -84.3799). Google Maps puts (33.7727, -84.3799) at the corner of Myrtle St. and Ponce de Leon Ave.

When driving to this location, you discover one of the most fabulous restaurants in the entire South. It is the one and only Mary Mac's Tea Room.

Directions:
Mary Mac's Tea Room
224 Ponce de Leon Ave.
Atlanta, GA 30308

Remarks:

The technique above appears to work quite well. However the equations did produce two different solutions. This requires some analysis to determine if the radius is appropriate. In this example, the comparison of the altitudes gave a clear result. One altitude was nowhere near the surface, whereas the other was under 1 km. We believe future examples will be as clear as this one. Using the coordinates from the second solution, the result put us at the front door of Mary Mac's Tea Room. This gives us 100% confidence in the technique and solution. Other factors not considered include Ephemeris error, rain, and the standard error already present in GPS. These factors would create a more complex computation of the the result. However for the problem stated, the technique provided an extremely accurate answer.

And in case you forgot our previous wonderful report.... sarsat.html