Finding rotation quaternion

Question

I am trying to use a quaternions for robotics and there is one thing I don't understand about it. Most likely because I don't understand how to define position with quaternions and how to define rotation with quaternions if there is any difference..
Please watch my "understanding steps" and correct if I am wrong somewhere.

Lets assume I we have 2 vehicle positions described by 2 rotation quaternions: $$ q_1 = w_1 + x_1i + y_1j +z_1k = \cos(\pi/4) + \sin(\pi/4)i $$ This quaternion is normalized and represents rotation over the $x$ axis for $\pi/2$ angle as I understand it. $$ q_2 = w_2 + x_2i + y_2j + z_2k = \cos(\pi/4) + \sin(\pi/4)k $$ And this one represents rotation for the same angle $\pi/2$ over the $y$ axis.

$q_1*q_2 = q_3$ which would be the same rotation as if we made $q_1$ first and $q_2$ second. $$q_3 = \frac{1}{2} + \frac{i}{2} +\frac{j}{2} +\frac{k}{2}$$

QUESTION 1: Given $q_2$ and $q_3$ how can I find $q_1$?

QUESTION 2: How to find a rotation angle over some other vector, given rotation quaternion? For example I want to find on what angle does $q_3$ turned over $2i+j-k$ quaternion.

Answer 1

Quaternions

Mathematical definition

$\mathcal{H} := \{ h_{1} + h_{2}i + h_{3}j + h_{4}k : h_{1},h_{2},h_{3},h_{4} \in \Re$ and $ i^{2} = j^{2} = k^{2} = ijk = -1 \}$

A pure quaternion can be defined as:

$\mathcal{H}_{p} := \{ p \in \mathcal{H} : Re(p)=0\}$

Pure quaternions can be associated to vectors.

Translation

Since pure quaternions are directly related to vectors, you can add two pure quaternions to form a third one. Observe the figure, were we represent a pure quaternion and the real axis is omitted:

Rotation

A general rotation of $\phi$ around an arbitrary unit norm rotation axis n is givem by:

QUESTION 1: Given $q_{2}$ and $q_{3}$ how can I find $q_{1}$?

If $q_{3} = q_{1}q_{2}$ and they are rotation quaternions:

$q_{3} = q_{2}q_{1}$

$q_{3}q_{2}^{*} = q_{1}q_{2}q_{2}^{*}$

Since $q_{2}^{*}q_{2} = 1 = q_{2}q_{2}^{*}$

$q_{1} = q_{3}q_{2}^{*}$

were the * states for the conjugate of a quaternion.

QUESTION 2: How to find a rotation angle over some other vector, given rotation quaternion? For example I want to find on what angle does $q_{3}$ turned over 2i+j−k quaternion.

If your vector is represented by $n$ and you know the $r$, your rotation angle will be the double of the angle you see.

Example:

$n = i + 0j + 0k$

$r$ = $q_{1} = cos(\frac{\pi}{4}) + sin(\frac{\pi}{4})i$

from the formula presented in the section of rotation:

$\frac{\phi}{2} = \frac{\pi}{4}$

$\phi = \frac{\pi}{2}$

Hope that was useful.

Best regards

Answer 2

Unit quaternions are used to define rotations, and has nothing to do with position. It defines all points in $SO(3)$, the 3D rotation group, and does not suffer from singularity, as in the case of Euler angles.

The quaternion ${\bf q}_1=\cos(\pi/4)+\sin(\pi/4)\bf i$ represents rotation about (not over) the $\bf i$ vector, which in general represents the $x$-axis, by an angle of $\pi/2$. This can be easily understood by the equivalent axis-angle representation. Any quaternion can be written as ${\bf q}=\cos(\phi/2) + \sin(\phi/2){\bf k}$, where ${\bf k}$ is a unit vector representing the axis of rotation and $\phi$ is the angle in radians. Following the logic, the quaternion ${\bf q}_2=\cos(\pi/4)+\sin(\pi/4)\bf k$ represents rotation about the $z$-axis and not $y$-axis.

The equivalent rotation of first performing ${\bf q}_1$ and then ${\bf q}_2$ is given by ${\bf q}_3={\bf q}_2{\bf q}_1$. Note the order of the multiplication. Also, the quaternion multiplication is not commutative. The quaternion multiplication is performed using the following rules:

1. ${\bf ij}={\bf k}$, ${\bf jk}={\bf i}$, ${\bf ki}={\bf j}$
2. ${\bf ji}={\bf -k}$, ${\bf jk}={\bf -i}$, ${\bf ki}={\bf -j}$
3. ${\bf i}^2={-1}$, ${\bf j}^2={-1}$, ${\bf k}^2={-1}$.

Following these rules, one can calculate ${\bf q}_3={\bf q}_2{\bf q}_1=\frac{1}{2}+\frac{1}{2}{\bf i}+\frac{1}{2}{\bf j}+\frac{1}{2}{\bf k}$.

In order to find ${\bf q}_1$ if ${\bf q}_2$ and ${\bf q}_3$ are given, one can write the equation as

${\bf q}_1={\bf q}^{-1}_2{\bf q}_3$. The inverse of a quaternion ${\bf q}^{-1}_2$ is given by its conjugate.

Hence, ${\bf q}^{'}_1=(\cos(\pi/4)-\sin(\pi/4){\bf k})*(\frac{1}{2}+\frac{1}{2}{\bf i}+\frac{1}{2}{\bf j}+\frac{1}{2}{\bf k})$, $*$ represents quaternion multiplication. This gives ${\bf q}^{'}_1={\bf q}_1$.

Finding the equivalent axis-angle representation:

The norm of the vector component gives $\sin(\phi/2)$ and the scalar component gives $\cos(\phi/2)$. Hence, the angle $\phi=2\operatorname{ArcTan}(\sin(\phi/2),\cos(\phi/2))$. Note the use of ArcTan and the double argument. This gives a unique answer, as opposed to taking ArcCos or ArcSin. The axis vector can then be found by dividing the vector component of quaternion by $\sin(\phi/2)$. For example, the quaternion ${\bf q}_3$ can be stated as rotation about the unit vector ${\bf v}=\frac{1}{\sqrt{3}}{\bf i}+\frac{1}{\sqrt{3}}{\bf j}+\frac{1}{\sqrt{3}}{\bf k}$ by an angle of $\frac{2\pi}{3}$.

Further references:

Quatenions and spatial rotations (Wikipedia)

Quaternions and Rotations in 3-Space: How it Works

Representing Attitude: Euler Angles, Unit Quaternions, and Rotation Vectors