quaternions - understanding a formula

Question

Quaternions are new for me. I am trying to understand the following formula:

What are:

• $\large{q^x}$ ? I don't think it is a power.
• $\large{q^t}$ ? just a transposition of the quaternion $q$?

Do the subscripts next to the $q's$, represent entire rows or columns of the quaterion in question?

This should normally give me a $3 \times 3$ matrix $R$, if I understood it correctly.

source: http://www.dept.aoe.vt.edu/~cdhall/courses/aoe4140/attde.pdf

page 4-14

Answer 1

Addressing your questions in order:

• $\mathbf q^\times$ is the $3\times3$ matrix that represents the operation of taking the cross product with $\mathbf q$, i.e. $(\mathbf q^\times)\mathbf x=\mathbf q\times\mathbf x$.
• No, $\mathbf q$ is not a quaternion. The quaternion is being denoted by $\bar{\mathbf q}$, and $\mathbf q$ is the vector of its components $1$ to $3$. The symbol $^\top$ represents transposition.
• The subscripts refer to the four components of the quaternion, which are here being indexed with $1$ to $4$.