Just how do you find your position on earth using a Global Navigation Satellite System (GNSS) such as GPS or Galileo? Consider a traveler who has just picked up his rental car in Amsterdam. Turning on the GNSS receiver, the receiver itself has to figure out its position using only received signals from overhead satellites.

GNSS systems such as GPS and Galileo help us answer the question: Where am I? Understanding Space Fig. 1-14.

To solve for its position in three dimensions, we know that there are three unknowns (e.g. Latitude, Longitude, Altitude) so the receiver will need at least three independent equations to solve the problem.  This leads to the need for a constellation of satellites that provides 4 or more contacts with the receiver at all times.  (We’ll need the fourth independent equation because of the inaccuracy of our receiver’s time clock.) As part of the initial contact information, the receiver will receive the locations of the entire GNSS constellation.  Now the receiver is ready to find position.  Assume, for the moment, that the receiver could find its exact distance, d1, to any one satellite.  Since it also knows the satellite’s location, the receiver can now place itself on a sphere of radius d1 from that central point.  Now, if the receiver can solve again for distance from a second satellite, d2, the receiver will also be somewhere on a second sphere of radius d2 from the second satellite’s location.  The intersection of these two spheres will result in a circle on which the receiver resides.  Contacting a third satellite, and repeating the procedure, the receiver will now have three spheres on which it resides, with the intersection of the spheres being two different points in space. Fortunately at this point, one of those two locations will not be in Earth. And assuming you ARE on Earth, your receiver can reject that solution.  Voila.  The receiver has found its position using a technique known as trilateration.

Of course, its not quite that simple.  To get the solution using only 3 satellites, we assumed the receiver can find its exact distance to the satellites and we haven’t said how that is done.  This distance is estimated by finding the time it takes for the signal to leave the satellite and travel to the receiver.  Since the radio signal travels at the speed of light (in a vacuum), if we know the time of travel and multiply by the speed of light, the receiver will have a precise estimate of its distance to the satellite as long as it knows the exact time the signal arrives. Unfortunately, clocks we carry around in GNSS receivers aren’t as accurate as the atomic clocks onboard the satellites. So to compensate for this timing error in our receiver, we take in the distance from a fourth satellite (and a fifth, sixth, seventh, etc. satellite if they’re available). Together these can give us a precise fix to with a few meters, answering the question: Where am I?

This article was provided by Teaching Science & Technology Inc. (TSTI) delivers introductory and advanced courses in space systems engineering.