User:Zeterraseca/sandbox

A Simple Example of a Schuler Oscillation
Schuler oscillations occur in an inertial navigation system (INS) due to errors in the estimate of the Earth's gravitational field as one moves across the Earth's curved surface. Here we give a simple example of how a Schuler oscillation can arise assuming a idealized spherical Earth and an INS that can perfectly measure gravity, but which has some uncertainty in its position estimate.

Assume the Earth is a perfectly uniform solid sphere. Define a cartesian coordinate system with origin at the Earth's center and with the positive z-axis extending from the origin up through the north pole. Let the positive x-axis extend from the origin out through the equator at the prime meridian.

Consider an inertial navigation system (INS) that is motionless and resting exactly on the north pole. The INS experiences a gravitational field $$\overrightarrow{g}=-g_0 \widehat{z} $$, which is indistinguishable from an acceleration, and is measured by the INS as $$\overrightarrow{a}_{INS} = g_0 \widehat{z} $$. In order for the INS to avoid thinking that it is accelerating in the $$\widehat{z} $$ direction, it must correct for the gravitational field by adding it to the measured acceleration to get a corrected acceleration $$\overrightarrow{a}_{cor} = \overrightarrow{a}_{INS} + \overrightarrow{g} = \overrightarrow{0} $$.

The INS infers the gravitational field vector based on its assumed position. Although the INS is motionless at the north pole, assume that it has an error in its estimated position. In particular, assume that the INS thinks that it displaced by a small angle $$\theta $$ along the prime meridian. It therefore thinks that the gravity vector is given by $$\overrightarrow{g}_{est} = -\widehat{x} g_0 \sin\theta - \widehat{z} g_0 \cos\theta $$. Using a small angle approximation, $$\overrightarrow{g}_{est} $$ can be approximated as $$\overrightarrow{g}_{est} \approx -\widehat{x} $$