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Dead Reckoning
An Introduction


Prof. David Bernstein
James Madison University

Computer Science Department
bernstdh@jmu.edu

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Getting Started
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  • Definition:
    • The process of determining the location of a point from the location of another point, a speed, a duration of time, and a heading
  • Notation:
    • \(a \in \mathbb{R}^2\) denotes the original point on the plane
    • \(\alpha\) denotes the heading (by convention, \(0^\circ\) is to the right, \(90^\circ\) is up, etc.)
    • \(s\) is the speed
    • \(t\) is the time duration
Performing the Calculations
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  • A Visualization:
    • images/deadreckoning01.png
  • Some Trigonometry:
    • images/deadreckoning02.png
    • images/deadreckoning03.png
Performing the Calculations (cont.)
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  • The Solution:
    • images/deadreckoning04.png
  • Words of Caution:
    • These illustrations are for headings in the interval \([0^\circ, 90^\circ]\), you should think carefully about what happens in the other quadrants
    • You need to be careful about units
An Example
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  • The Initial Point:
    • \(0, 0\)
  • The Measured Data:
    • \(\alpha = 30^\circ\)
    • \(s = 30\text{mph}\)
    • \(t = 15\text{min} = 0.25\text{hr}\)
An Example (cont.)
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  • Distance Traveled:
    • \(s \cdot t = 30\text{mph} \cdot 0.25\text{hr} = 7.5\text{mi}\)
  • The Trigonometry:
    • \(\cos(\alpha) = \cos(30^\circ) = 0.866\)
    • \(\sin(\alpha) = \sin(30^\circ) = 0.500\)
  • The New Location:
    • \((0.0 + 0.866 \cdot 7.5, 0.0 + 0.500 \cdot 7.5) = (6.495, 3.750)\)
Two Important Sources of Error
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  • Discretizing Time:
    • Sensor measurements and calculations are not performed continuously, they are performed at discrete points in time
    • We assume (incorrectly) that the speed and heading are constant during each "time step"
  • Sensor Errors:
    • All sensors are inaccurate to some extent
Error Propagation
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  • The Idea:
    • Errors "accumulate"
  • In Dead Reckoning:
    • Errors are introduced at each "time step" and they accumulate over time
There's Always More to Learn
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