Dynamics of Aircraft Flight

9Quaternion Methods

9.1. Goal

The problem posed in this chapter is that of simplifying the calculation of the change of axes by not making use of Euler angles, thereby avoiding the calculation of trigonometric lines and at the same time excluding any angular indeterminacy (a problem which arises from using Euler angles).

A method for changing axes that uses only rational functions can save calculation time (in the case of ground resources) or lead to a reduction in the size of the calculator (in the case of on-board equipment).

Euler angles are widely used in mechanics and astronomy. They are useful when explicit knowledge of a mobile around its center of gravity is required. In addition, they correspond to degrees of freedom materialized physically.

Olinde-Rodrigues’s formulas make it possible to simplify and reduce the number of axis change calculations when explicit knowledge of angles Ψ, θ and Φ is not required. They use the direction of the instantaneous axis of rotation (two parameters) and the angle of rotation V around this axis as three independent parameters.

9.2. Reminder of the axis change formulas using Euler angles

The most general change of axes corresponds to a certain displacement. It consists of a translation (movement of the center of gravity) and of a rotation around an axis passing through the center of gravity (movement around the center of gravity, i.e. around the instantaneous axis of rotation).

If we are only interested in the rotation around the center ...