The Global Positioning System (NAVSTAR)

The Global Positioning System (NAVSTAR)
An Introduction

Prof. David Bernstein
James Madison University

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:
- \(\sqrt{ (x_{1}^{i} - x_{1})^{2} + (x_{2}^{i} - x_{2})^{2} + (x_{3}^{i} - x_{3})^{2} } = c \cdot \Delta t^{i} - c \cdot \delta\)
- \(\sqrt{ (x_{1}^{j} - x_{1})^{2} + (x_{2}^{j} - x_{2})^{2} + (x_{3}^{j} - x_{3})^{2} } = c \cdot \Delta t^{j} - c \cdot \delta\)
- \(\sqrt{ (x_{1}^{k} - x_{1})^{2} + (x_{2}^{k} - x_{2})^{2} + (x_{3}^{k} - x_{3})^{2} } = c \cdot \Delta t^{k} - c \cdot \delta\)
- \(\sqrt{ (x_{1}^{m} - x_{1})^{2} + (x_{2}^{m} - x_{2})^{2} + (x_{3}^{m} - x_{3})^{2} } = c \cdot \Delta t^{m} - c \cdot \delta\)

Triangulation (cont.)

Solving Systems of Nonlinear Equations:
- Many algorithms exist
A Related Optimization Problem:
- Letting \(\rho^{i}(x) = \sqrt{ (x_{1}^{i} - x_{1})^{2} + (x_{2}^{i} - x_{2})^{2} + (x_{3}^{i} - x_{3})^{2}}\)
- Min \( (\rho^{i}(x) - c \cdot \Delta t^{i} + c \cdot \delta)^{2}+ (\rho^{j}(x) - c \cdot \Delta t^{j} + c \cdot \delta)^{2}+ (\rho^{k}(x) - c \cdot \Delta t^{k} + c \cdot \delta)^{2} \)

Longitude, Latitude and Altitude

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