Use of Quaternions in the Representation of Angular Coordinates [Utilisation des Quaternions Pour la Représentation des Coordonnées Angulaires]

The classical representation of the angular coordinates of an aerodyne by means of Euler angles can be advantageously replaced, in the mathematical models of flight mechanics, by the use of the four homogeneous components of the quaternion of unit modulus associated with the space rotation which brings the frame bound to the moving body onto the fixed frame.

A comparison of the analytical expression of' the three differential equations of angular motion with the matrix of transformation of coordinates shows this method to be particularly effective in the numerical integration of motion on a data processing machine because of:

(1) A greater speed in the performance of calculations through the elimination of circular functions in favor of a simple arithmetical formulation;

(2) An increased accuracy of the numerical integration process, which remains of the same order, whatever the position of the moving body and of the instantaneous rotation.

The results are transposed into classical representation systems by means of simple formulas.

Moreover, this homogeneous representation is completely devoid of the singularity due to the polar indetermination of Euler angles, which permits calculations of angular motions of any amplitude with the above-mentioned effectiveness (study of spinning and rocking motions, etc.)

Finally, the remarkable properties of the quaternions associated in successive rotations find interesting applications in the study of the relative motion of two moving bodies (separations of composite systems).