Quaternions and Rotations in 3-Space: The Algebra and its Geometric Interpretation

Think of a quaternion Q as a vector augmented by a real number to make a four element entity. It has a real part Qcre and a vector part Qcve: If Qcre is zero, Q represents an ordinary vector; if Qcve is zero, it represents an ordinary real number. In any case, the ratio between the real part and the magnitude of the vector part jQcvej plays an important role in rotations, and is conveniently represented by the parameter = tan (jQcvej=Qcre): A unit magnitude quaternion U has a Pythagorean sum of 1 over its four elements, and its product with any vector Sv gives another vector having the same magnitude as Sv but rotated in space. If Sv ? Ucve; the rotation is by an angle about the vector Ucve (or simply about U). If Sv is arbitrary, however, certain cross-terms of the product spoil this convenient relationship. Even in this general case however, these cross-terms cancel in the triple product Rv = USvU ; where U 1 1=U . The rotations of the two successive products are in the same direction, so Rv represents a rotation of Sv about Ucve by an angle 2 ; which depends only on U: Thus, the operation USvU 1 performs a rotation of Sv which is entirely characterized by the unit quaternion U: The rotation occurs about an axis parallel to U by an amount 2 tan (jUcvej=Ucre): Quaternion notation conveniently handles composition of any number of successive rotations into one equivalent rotation: U = U1U2 Un where each unit quaternion Ui represents one of the succession of rotations. Other operations useful in inertial navigation problems are also presented.