9.5: A Practical Example - Dead Reckoning

The measurement of heading and longitudinal speed gives rise to one of the oldest methods of navigation: dead reckoning. Quite simply, if the estimated longitudinal speed over ground is \(U\), and the estimated heading is \(\phi\), ignoring the lateral velocity leads to the evolution of Cartesian coordinates:

\begin{align} \dot{x} \, &= \, U \cos \phi \\[4pt] \dot{y} \, &= \, U \sin \phi. \end{align}

Needless to say, currents and vehicle sideslip will cause this to be in error. Nonetheless, some of the most remarkable feats of navigation in history have depended on dead reckoning of this type.

Suppose that the heading is estimated from an angular rate gyro. We use

\begin{align} \dot{\phi} \, &= \, r \\[4pt] \dot{x} \, &= \, U \cos \phi \\[4pt] \dot{y} \, &= \, U \sin \phi, \end{align}

where \(r\) is the measured angular rate. As you might expect, long-term errors in this rule will be worse than for the previous, because integration of the rate gyro signal is subject to drift.

Suppose that we have, in addition to a sensor for \(U\) and \(r\), a sensor for the cross-body velocity \(V\). Our dead-reckoning problem is

\begin{align} \dot{\phi} \, &= \, r \\[4pt] \dot{x} \, &= \, U \cos \phi - V \sin \phi \\[4pt] \dot{y} \, &= \, U \sin \phi + V \sin \phi. \end{align}