A vectorial method to derive the equivalent rotation of two successive finite rotations

Abstract

Originally demonstrated by Euler and later derived by Rodriguez, every two finite rotations in space can be represented by a single equivalent rotation. This paper presents a new geometric method to derive both the coordinates of the vector direction and the angle of the equivalent rotation of a rigid body that undergoes two successive finite rotations. The method is based on calculating the paths of two singular points which only undergo a single rotation during the two successive rotations. The first point is located on the first rotation axis and the second is a point whose first rotation relocates it to the axis of the second rotation. The equivalent axis of rotation is perpendicular to the trajectories of both singular points and the angle of rotation is found by dividing the travelled distance of one of the points by the distance to the equivalent axis of rotation. The expressions that we obtain are identical to the ones in the literature.