Unit Quaternion

Define $q\in {\mathbb{R}}^4$ according to

$$q=\left[\begin{array}{c}{q}_1\\ q_2\\ q_3\\ q_r\end{array}\right]=\left[\begin{array}{c}\varphi \sin \frac{\theta }{2}\\ \cos \frac{\theta }{2}\end{array}\right]$$

where $\varphi$ is the rotational axis (unit vector) and $\theta \in [0,\pi ]$ is the angle.

Properties:

• $||q||=1$
• $q_r=\frac{1}{2}\sqrt{1+r_{11}+r_{22}+r_{33}}$, where the $r$‘s are the diagonal entries of the $3\times 3$ rotation matrix $R$ corresponding to $q$

Note that we can also write

$$\left[\begin{array}{c}{q}_1\\ q_2\\ q_3\end{array}\right]=\frac{1}{4q_r}\left[\begin{array}{c}{r}_{32}-r_{23}\\ r_{13}-r_{31}\\ r_{21}-r_{12}\end{array}\right]$$

for $q_r\ne 0$.

Note that we can write

$$R=\left[\begin{array}{ccc}{q}_r^2+q_1^2-q_2^2-q_3^2& 2(q_1{q}_2-q_r{q}_3)& 2(q_r{q}_2+q_1{q}_3)\\ 2(q_r{q}_3+q_1{q}_2)& q_r^2-q_1^2+q_2^2-q_3^2& 2(q_2{q}_3-q_r{q}_1)\\ 2(q_1{q}_3-q_r{q}_2)& 2(q_r{q}_1+q_2{q}_3)& q_r^2-q_1^2-q_2^2+q_3^2\end{array}\right]$$

ROS uses a unit quaternion which is denoted by a 4-tuple $[x,y,z,w]$.

Example