Satellite Positioning
A Global Navigation Satellite System (GNSS) allows a mobile receiver to determine its exact position anywhere in the world. Currently, there are three GNSS among which only the first one is fully operational: the United States' Global Positioning System (GPS), the Russian Federation's Global'naya Navigatsionnaya Sputnikovaya Sistema (GLONASS), and the European Union's Galileo. All GNSS use trilateration to locate a mobile receiver through calculations involving information from a number of satellites.
Satellite positioning of vehicles relies on the knowledge of the exact position of satellites and the distance of the vehicle to those satellites.
Let 
denote the distance of vehicle 
to satellite 
and let 
denote the satellite's position in space.
The position 
of vehicle can be calculated with the help of the following equation.
This equation has three variables and three satellites are sufficient to determine the position of the vehicle. As illustrated in the following figure, two spheres intersect in a circle. The intersection of three spheres results in two distinct positions. Only one of them is near the Earth's surface whereas the other is far in space and can be discarded.
In an ideal scenario each satellite sends a signal which includes the exact time 
of transmission.
The signal travels with the speed of light 
(which is approximately 300 000 km/s) towards the receiver where it arrives at the time 
.
The distance between satellite and receiver is
In real-life, however, the clocks of satellites and receivers in the vehicles are not always running synchronously.
Due to the high speed of light an error of 1 
results in a difference of 300 metres.
The clocks of the satellites are very precise and are regularly synchronised by ground control.
The clocks of the receivers, however, are usually not as precise and are not synchronised with the satellite clocks.
Fortunately, the precise time of the internal clocks of the receiver is not required.
Instead, the time is calculated using a fourth satellite signal.
Let 
denote the locally determined time of satellite and
let
denote the time bias.
Analogously, let 
denote the locally determined time of the receiver and
let
denote the time bias.
The pseudo range
based on the locally determined time can be used to determine the position by
As the clocks of all satellites are synchronised regularly each satellite has approximately the same time bias 
.
If 
is substituted by 
the above equation has four variables and four satellites are sufficient to determine the position of the vehicle.
