Global Positioning System - Navigation Equations

Navigation Equations

The receiver uses messages received from satellites to determine the satellite positions and time sent. The x, y, and z components of satellite position and the time sent are designated as where the subscript i denotes the satellite and has the value 1, 2, ..., n, where When the time of message reception indicated by the on-board clock is, the true reception time is where is receiver's clock bias (i.e., clock delay). The message's transit time is . Assuming the message traveled at the speed of light, the distance traveled is . Knowing the distance from receiver to satellite and the satellite's position implies that the receiver is on the surface of a sphere centered at the satellite's position and with the radius being the distance traveled. Thus the receiver is at or near the intersection of the surfaces of the spheres if it receives signals from more than one satellite. In the ideal case of no errors, the receiver is at the intersection of the surfaces of the spheres.

The clock error or bias, b, is the amount that the receiver's clock is off. The receiver has four unknowns, the three components of GPS receiver position and the clock bias . The equations of the sphere surfaces are given by:

or in terms of pseudoranges, as

.

These equations can be solved by algebraic or numerical methods.