Dead Reckoning

Dead Reckoning
An Introduction

Prof. David Bernstein
James Madison University

Computer Science Department

bernstdh@jmu.edu

Getting Started

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

Performing the Calculations (cont.)

The Solution:
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

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.)

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

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

The Idea:
- Errors "accumulate"
In Dead Reckoning:
- Errors are introduced at each "time step" and they accumulate over time