3.5 Quaternion

One may view the quaternion as an alternative form to a rotation vector in which it is associated with the sine and the cosine of the rotation angle rather than the actual rotation angle associated with the rotation vector.

A quaternion, as its name imply, is a four-scalar set. It is represented by a 4-dimensional vector

or equivalently by a scalar and 3-dimensional vector Q = { q ₀, q} with q = [ q ₁ , q ₂ , q ₃]. Both forms are commonly used, as the former expression fits well for matrix implementations, while the later is useful for formula derivation and shorthand notations. A brief introduction to quaternion algebra is presented in Appendix B. We recall that the identity quaternion is

The product of the two quaternions P = { p ₀, p} and Q = { q ₀, q} is also a quaternion given by

The norm of the quaternion is

The inverse of a quaternion Q, denoted it by Q ^#, have the property

From Eq. (3.9) one can verify that the inverse of a quaternion Q is given by

A normalized quaternion is one whose norm equals 1, that is if Q = { q ₀, q} then

and its inverse is given by

The class of normalized quaternions plays a central role in matrix transformations as seen in the following.

Lemma 3.1

Suppose...