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The Global Positioning System (NAVSTAR)
An Introduction


Prof. David Bernstein
James Madison University

Computer Science Department
bernstdh@jmu.edu

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The Satellites
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  • 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
      gps-block1-satellite
      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
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  • 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
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  • 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
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  • Geocentric (i.e., origin at the center of the Earth)
  • Earth-Centered-Earth-Fixed (Cartesian)
gps-range
Triangulation
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  • 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.)
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The Spheres Around Three Satellites

gps-triangulation
Triangulation (cont.)
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  • 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.)
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  • 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.)
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  • 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
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  • Longitude and Latitude:
    • The system above is in Cartesian coordinates
    • We must convert to spherical (or, actually, ellipsoidal) coordinates
  • Altitude:
    • gps-altitude
There's Always More to Learn
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