The Global Positioning System (GPS)

The Global Positioning System (GPS)
An Introduction

Computer Science Department

bernstdh@jmu.edu

The Satellites

The System:
- 24 satellites in 6 evenly-spaced orbits
- Each satellite circles the Earth every 11 hours 58 minutes
- The orbital radius of these satellites is 26,560km
- Each satellite weights about 900Kgs and is about 6m across (including the solar panels)
History:
- The first satellite was launched in 1978
  
  Source: USAF
- The complete constellation (of 24 satellites) became available in 1994
Maintenance:
- Each satellite has a life of about 7-10 years

The Signal

The Transmitter:
- About 50 watts
The Civilian Frequency (L1):
- 1575.42 MHz (which is in the UHF band)
The Data:
- An ID
- A date and time
- Almanac information (the position of all of the satellites)

Signal Errors

Atmospheric (Propogation) Delay:
- The signal slows as it passes through the atmosphere
Multipath Errors:
- The signal may be reflected (e.g. off tall buildings)
Orbital Errors:
- Inaccuricies in the reported locations

Coordinate System

Triangulation

Notation:
- $T^{j}$ is the "actual" time satellite j transmits
- $T$ is the "actual" time that the signal is received
Some Fundamentals:
- Distance: $c \cdot \Delta T^{j}$ (where $c$ is the speed of light)
- Each distance can be used to locate the receiver on a sphere.

Triangulation (cont.)

The Spheres Around Three Satellites

Triangulation (cont.)

The System to Solve:
- $\sqrt{ (x_{1}^{i} - x_{1})^{2} + (x_{2}^{i} - x_{2})^{2} + (x_{3}^{i} - x_{3})^{2} } = c \cdot \Delta T^{i}$
- $\sqrt{ (x_{1}^{j} - x_{1})^{2} + (x_{2}^{j} - x_{2})^{2} + (x_{3}^{j} - x_{3})^{2} } = c \cdot \Delta T^{j}$
- $\sqrt{ (x_{1}^{k} - x_{1})^{2} + (x_{2}^{k} - x_{2})^{2} + (x_{3}^{k} - x_{3})^{2} } = c \cdot \Delta T^{k}$

Triangulation (cont.)

An Observation:
- The clock on the GPS receiver differs from the correct time by $\delta$ . That is: $t = T + \delta$ .
The New System to Solve:
- $\sqrt{ (x_{1}^{i} - x_{1})^{2} + (x_{2}^{i} - x_{2})^{2} + (x_{3}^{i} - x_{3})^{2} } = c \cdot \Delta t^{i} - c \cdot \delta$
- $\sqrt{ (x_{1}^{j} - x_{1})^{2} + (x_{2}^{j} - x_{2})^{2} + (x_{3}^{j} - x_{3})^{2} } = c \cdot \Delta t^{j} - c \cdot \delta$
- $\sqrt{ (x_{1}^{k} - x_{1})^{2} + (x_{2}^{k} - x_{2})^{2} + (x_{3}^{k} - x_{3})^{2} } = c \cdot \Delta t^{k} - c \cdot \delta$
- $\sqrt{ (x_{1}^{m} - x_{1})^{2} + (x_{2}^{m} - x_{2})^{2} + (x_{3}^{m} - x_{3})^{2} } = c \cdot \Delta t^{m} - c \cdot \delta$

Longitude, Latitude and Altitude

Longitude and Latitude:
- The system above is in Cartesian coordinates
- We often must convert to spherical (or, actually, ellipsoidal) coordinates
Altitude:

There's Always More to Learn