Explaining Quaternion Rotation Matrix

With a quaternion \({\bf q} = q_{0} + q_{1}i + q_{2}j + q_{3}k\), the rotation matrix is:

\[ \begin{bmatrix}
    q_{0}^{2}+q_{1}^{2}-q_{2}^{2}-q_{3}^{2} & 2q_{1}q_{2}-2q_{0}q_{3} & 2q_{1}q_{3}+2q_{0}q_{2} \\
    2q_{1}q_{2}+2q_{0}q_{3} & q_{0}^{2}+q_{2}^{2}-q_{1}^{2}-q_{3}^{2} & 2q_{2}q_{3}-2q_{0}q_{1}\\
    2q_{1}q_{3}-2q_{0}q_{2} & 2q_{2}q_{3}+2q_{0}q_{1} &  q_{0}^{2}+q_{3}^{2}-q_{1}^{2}-q_{2}^{2}  \\
\end{bmatrix}
\]

A quaternion is determined by two factors: normalized rotation axis \((a_{x},a_{y},a_{z})\) and rotation angle \(\theta\). \({\bf q} = cos(\theta/2) + (a_{x}i + a_{y}j + a_{z}k)sin(\theta/2)\) and \({\bf q^{-1}} = cos(\theta/2) - (a_{x}i + a_{y}j + a_{z}k)sin(\theta/2)\). This means that \(||{\bf q}|| = 1\).

The rotation matrix in term of \((a_{x},a_{y},a_{z})\) and \(\theta\) is:
\[ \begin{bmatrix}
1+(a_{x}^{2}-1)(1-cos(\theta)) & -a_{z}sin(\theta)+a_{x}a_{y}(1-cos(\theta)) & a_{y}sin(\theta)+a_{x}a_{z}(1-cos(\theta))\\
a_{z}sin(\theta)+a_{x}a_{y}(1-cos(\theta)) & 1+(a_{y}^2-1)(1-cos(\theta)) & -a_{x}sin(\theta)+a_{y}a_{z}(1-cos(\theta))\\
-a_{y}sin(\theta)+a_{x}a_{z}(1-cos(\theta)) & a_{x}sin(\theta)+a_{y}a_{z}(1-cos(\theta)) & 1+(a_{z}^{2}-1)(1-cos(\theta))
\end{bmatrix}
\]

From the expression of quaternion, we have
\[cos(\theta) = cos^{2}(\theta/2)-sin^{2}(\theta/2)
=q_{0}^2-(q_{1}^2+q_{2}^2+q_{3}^2)\]
\[1-cos(\theta) = 1-cos^{2}(\theta/2)+sin^{2}(\theta/2)
=2sin^{2}(\theta/2)\]
\[(a_{x},a_{y},a_{z})=(q_{1},q_{2},q_{3})/sin(\theta/2)\]

Based on these equations, we can derive the quaternion rotation matrix. For example,
\[1+(a_{x}^{2}-1)(1-cos(\theta)) = cos(\theta)+a_{x}^{2}(1-cos(\theta))\\
=q_{0}^2-(q_{1}^2+q_{2}^2+q_{3}^2)+2q_{1}^{2}\\
= q_{0}^2+q_{1}^2-q_{2}^2-q_{3}^2\]
\[-a_{z}sin(\theta)+a_{x}a_{y}(1-cos(\theta)) = \frac{-q_{3}}{sin(\theta/2)}sin(\theta)+\frac{q_{1}q_{2}}{sin^{2}(\theta/2)}2sin^{2}(\theta/2)\\
=2q_{1}q_{2}-2q_{0}q_{3}\]