Editing Gyrocompass (section)

== Mathematical model ==
We consider a gyrocompass as a gyroscope which is free to rotate about one of its symmetry axes, also the whole rotating gyroscope is free to rotate on the horizontal plane about the local vertical. Therefore there are two independent local rotations. In addition to these rotations we consider the rotation of the Earth about its north-south (NS) axis, and we model the planet as a perfect sphere. We neglect friction and also the rotation of the Earth about the Sun.

In this case a non-rotating observer located at the center of the Earth can be approximated as being an inertial frame. We establish cartesian coordinates <math>(X_{1},Y_{1},Z_{1})</math> for such an observer (whom we name as 1-O), and the barycenter of the gyroscope is located at a distance <math>R</math> from the center of the Earth.

=== First time-dependent rotation ===
Consider another (non-inertial) observer (the 2-O) located at the center of the Earth but rotating about the NS-axis by <math>\Omega.</math> We establish coordinates attached to this observer as
<math display="block">\begin{pmatrix}
X_{2}\\
Y_{2}\\
Z_{2}
\end{pmatrix} = \begin{pmatrix}
\cos\Omega t & \sin\Omega t & 0\\
-\sin\Omega t & \cos\Omega t & 0\\
0 & 0 & 1
\end{pmatrix}\begin{pmatrix}
X_{1}\\
Y_{1}\\
Z_{1}
\end{pmatrix}</math>
so that the unit <math>\hat{X}_{1}</math> versor <math>(X_{1}=1,Y_{1}=0,Z_{1}=0)^{T}</math> is mapped to the point <math>(X_{2} = \cos\Omega t, Y_{2}=-\sin\Omega t, Z_{2}=0)^{T}</math>. For the 2-O neither the Earth nor the barycenter of the gyroscope is moving. The rotation of 2-O relative to 1-O is performed with angular velocity <math>\vec{\Omega}=(0,0,\Omega)^{T}</math>. We suppose that the <math>X_{2}</math> axis denotes points with zero longitude (the prime, or Greenwich, meridian).

=== Second and third fixed rotations ===
We now rotate about the <math display="inline"> Z_{2}</math> axis, so that the <math display="inline"> X_{3}</math>-axis has the longitude of the barycenter. In this case we have
<math display="block">\begin{pmatrix}
X_{3}\\
Y_{3}\\
Z_{3}
\end{pmatrix}=\begin{pmatrix}
\cos\Phi & \sin\Phi & 0\\
-\sin\Phi & \cos\Phi & 0\\
0 & 0 & 1
\end{pmatrix} \begin{pmatrix}
X_{2}\\
Y_{2}\\
Z_{2}
\end{pmatrix}.</math>

With the next rotation (about the axis <math display="inline"> Y_{3}</math> of an angle <math display="inline"> \delta</math>, the co-latitude) we bring the <math display="inline"> Z_{3}</math> axis along the local zenith (<math display="inline"> Z_{4}</math>-axis) of the barycenter. This can be achieved by the following orthogonal matrix (with unit determinant)
<math display="block">\begin{pmatrix}
X_{4}\\
Y_{4}\\
Z_{4}
\end{pmatrix}=\begin{pmatrix}
\cos\delta & 0 & -\sin\delta\\
0 & 1 & 0\\
\sin\delta & 0 & \cos\delta
\end{pmatrix} \begin{pmatrix}
X_{3}\\
Y_{3}\\
Z_{3}
\end{pmatrix},</math>

so that the <math display="inline">\hat{Z}_{3}</math> versor <math display="inline"> (X_{3}=0,Y_{3}=0,Z_{3}=1)^{T}</math> is mapped to the point <math display="inline"> (X_{4}=-\sin\delta,Y_{4}=0,Z_{4}=\cos\delta)^{T}.</math>

=== Constant translation ===
We now choose another coordinate basis whose origin is located at the barycenter of the gyroscope. This can be performed by the following translation along the zenith axis
<math display="block">\begin{pmatrix}
X_{5}\\
Y_{5}\\
Z_{5}
\end{pmatrix}=\begin{pmatrix}
X_{4}\\
Y_{4}\\
Z_{4}
\end{pmatrix}- \begin{pmatrix}
0\\
0\\
R
\end{pmatrix},</math>

so that the origin of the new system, <math>(X_{5}=0,Y_{5}=0,Z_{5}=0)^{T}</math> is located at the point <math>(X_{4}=0,Y_{4}=0,Z_{4}=R)^{T},</math> and <math>R</math> is the radius of the Earth. Now the <math>X_{5}</math>-axis points towards the south direction.

=== Fourth time-dependent rotation ===
Now we rotate about the zenith <math>Z_{5}</math>-axis so that the new coordinate system is attached to the structure of the gyroscope, so that for an observer at rest in this coordinate system, the gyrocompass is only rotating about its own axis of symmetry. In this case we find
<math display="block">\begin{pmatrix}
X_{6}\\
Y_{6}\\
Z_{6}
\end{pmatrix}=\begin{pmatrix}
\cos\alpha & \sin\alpha & 0\\
-\sin\alpha & \cos\alpha & 0\\
0 & 0 & 1
\end{pmatrix}\begin{pmatrix}
X_{5}\\
Y_{5}\\
Z_{5}
\end{pmatrix}.</math>

The axis of symmetry of the gyrocompass is now along the <math>X_{6}</math>-axis.

=== Last time-dependent rotation ===
The last rotation is a rotation on the axis of symmetry of the gyroscope as in
<math display="block">\begin{pmatrix}
X_{7}\\
Y_{7}\\
Z_{7}
\end{pmatrix}=\begin{pmatrix}
1 & 0 & 0\\
0 & \cos\psi & \sin\psi\\
0 & -\sin\psi & \cos\psi
\end{pmatrix}\begin{pmatrix}
X_{6}\\
Y_{6}\\
Z_{6}
\end{pmatrix}.</math>