A new angle on the Euler angles

Abstract

We present a generalization of the Euler angles to axes beyond the twelve conventional sets. The generalized Euler axes must satisfy the constraint that the first and the third are orthogonal to the second; but the angle between the first and third is arbitrary, rather than being restricted to the values 0 and pi/2, as in the conventional sets. This is the broadest generalization of the Euler angles that provides a representation of an arbitrary rotation matrix. The kinematics of the generalized Euler angles and their relation to the attitude matrix are presented. As a side benefit, the equations for the generalized Euler angles are universal in that they incorporate the equations for the twelve conventional sets of Euler angles in a natural way.