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The Theory Behind GPS May 4, 2015

Eschborn, adinegara.com. How does the receiver calculate its position?We will start by assuming that the clocks of the receiver and all of the satellites are perfectly synchronized. The receiver calculates its position through triangulation. The basic principle of triangulation methods is to determine where a person (object) is located by using some knowledge relating the position of the person (object) with respect to reference objects whose positions are known. In the case of the receiver of the GPS, it calculates its distance to the satellites, whose positions are known.

The receiver measures the time t1 it takes for the signal emitted from satellite P1 to reach it. Given that the signal travels at the speed of light c, the receiver can calculate its distance from the satellite as r1=ct1. The set of points situated at a distancer1from the satellite P1 forms a sphere S1centered at P1with radius r1. So we know that the receiver is on S1. Consider these points as defined in a Cartesian coordinate system. Let (x, y, z) be the unknown position of the receiver and let (a1,b1,c1) be the known position of the satellite P1. Then (x, y, z) must satisfy the equation describing points on the sphere S1.

This piece of information is insufficient to determine the precise position of the receiver. The receiver therefore records the signal of a second satellite P2, recording the time t2 that the signal took to arrive and calculating the distance r2 = ct2 to the satellite. As before, it must be that the receiver lies on the sphereS2of radius r2 centered at (a2,b2,c2). This narrows down our search, since the intersection of two overlapping spheres is a circle. Thus, we have now narrowed down the position of the receiver to a circle C1,2 on which the receiver must lie. However, we again do not know precisely where the receiver is on this circle.

In order for the receiver to calculate its final position, it needs to capture and process the signal received from a third satellite P3. Once again, the receiver measures the time t3 for the signal to arrive and calculates its distance r3=ct3 from it. As before, it follows that the receiver lies somewhere on the sphere S3 of radius r3 centered at (a3,b3,c3).

The receiver is therefore at the intersection of the circle C1,2 and the sphere S3. Since a sphere and a circle intersect at two points, it may seem that we are not yet sure of the position of the receiver. Fortunately, this is not the case. In fact, the satellites have been positioned such that one of the two solutions will be completely unrealistic, being quite faraway from the surface of the Earth. Thus, by finding the two solutions of the system of equations, and subsequently eliminating the spurious solution, the receiver may calculate its precise position.

Hélène Antaya, Christiane Rousseau, Chris Hamilton, Isabelle Ascah-Coallier, and Yvan Saint-Aubin. (2008) Mathematics and Technology. New York: Springer.

Categories Mathematics