GPS Positioning Algorithm, its Errors and Solution

Undergraduate #243
Discipline: Mathematics and Statistics
Subcategory: Mathematics and Statistics

Mohammed Abdi - Benedict College
Co-Author(s): Hohite Fetene and Naima Naheed

The Global Positioning System is a space-centered satellite which consists of 24 basic satellites carrying atomic clocks navigation system that are responsible for delivering location and time information prudently, anywhere on Earth. The satellite’s mission is to transmit synchronized signals from predetermined position in space to the receiver; the receiver uses the information transmitted by the satellite to calculate the accurate coordinates of itself. If three satellites are available, then three spheres are known whose intersection consists of two points. One is the location of the receiver and the other is far away from the surface of the earth which can be ignored. As a result, the problem is to solve the three sphere equations. One major problem is that the receiver clock is not perfectly in sync with the satellite clock. The only way to fix this error is by adding one more satellite to solve the inaccurate timing which brings a prodigious difference by several kilometers on the positioning. We define d to be the difference between the coordinated time on the (four) satellite clocks and the earth-constrained receiver clock. Two further problems arise when GPS is deployed. One is the conditioning of the system of equations and another is the transmission speed of the signals, which is not precisely the speed of light (c). Because the signals may encounter blockage by different hindrances on earth before reaching the receiver, this is referred to as multipath interference. To be more accurate, we increased the number of satellites from four to eight in our calculation. Our goal was to solve the least squares system of eight equations in four unknown variables (x, y, z, d) using Gauss-Newton iteration method. We used two types of satellites such as tightly and loosely bunched. Results indicated that system becomes ill-conditioned when satellites are bunched closely in the sky. Future research involves working with two or more receivers to compute difference of position instead of absolute position. Errors that are shared by the receivers will be cancelled when we form the differences.

Funder Acknowledgement(s): HBCU UP

Faculty Advisor: Naima Naheed,