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Equipment orientation

The equipment is positioned using its two extremities and is oriented thanks to the three well-known Euler angles. These angles define three consecutive rotations which allow passing from the (X,Y,Z) global coordinates system to the local coordinates system (x,y,z). The center of rotation is placed at mid- height on the central axis.

  1. The first rotation is around the Z axis. The corresponding angle value is specified using the PHI parameter. The global frame (X,Y,Z) becomes (X1,Y1,Z1 = Z).

  2. The second rotation is around the X1 axis. The corresponding angle value is specified using the TETA parameter. The (X1,Y1,Z1) frame becomes (X2 = X1, Y2, Z2).

  3. The third rotation is around the Z2 axis. The corresponding angle value is specified using the PSI parameter. The (X2, Y2, Z2) frame finally becomes (x, y, z = Z2).

The transformation finally writes :

\[ \begin{pmatrix} x \\ y \\ z \end{pmatrix} = [P]\begin{pmatrix} X \\ Y \\ Z \end{pmatrix} \]

with

\[ [P] = \begin{bmatrix} \cos(\Psi) \cos(\phi) - \cos(\theta) \sin(\Psi) \sin(\phi) & \cos(\Psi) \sin(\phi) + \cos(\theta) \sin(\Psi) \cos(\phi) & \sin(\Psi) \sin(\theta) \\ - \sin(\Psi) \cos(\phi) - \cos(\theta) \cos(\Psi) \sin(\phi) & - \sin(\Psi) \sin(\phi) + \cos(\theta) \cos(\Psi) \cos(\phi) & \cos(\Psi) \sin(\theta) \\ \sin(\theta) \sin(\phi) & - \sin(\theta) \cos(\phi) & \cos(\theta) \end{bmatrix} \]

Note

The most common case is the particular case where \(\theta\) = \(\Psi\) = 0. The angle \(\phi\) then simply determines the equipment's heading - positive anti- clockwise when seen from above.