Unit Quaternion from Rotation Matrix

Often one must find the quaternion corresponding to a given rotation matrix. Of themanymethods that have been proposed for performing this computation [3–9], Shepperd’s algorithm [5], which is singularity free and requires only one square root, has been the most widely applied. In this Note, we review Shepperd’s algorithm and present a variant that always produces a normalized quaternion even if numerical errors cause the matrix A to be only approximately orthogonal. This modification of Shepperd’s algorithm also provides a very efficient method for computing an exactly orthogonal matrix that is close to an approximately orthogonal matrix.