Realization of transformations

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Quaternion notation for general transformations

Up to now transformations have been defined by homogeneous matrices combining a rotation matrix $\mathbf{R}$ and a translation vector $\vec{\mathbf{p}}$ . Now a new notation is introduced to represent a transformation using two quaternions $e$ and $p$ :

\mathbf{T} = \left[\begin{array}{cccc} & & & \\ & \mathbf{R} & & \vec{\mathbf{p}} \\ & & & \\ 0 & 0 & 0 & 1\end{array}\right] \quad \equiv \quad \left\{e,p\right\}

The quaternion $e$ is equivalent to $\mathbf{R}$ and describes the rotation while $p$ is defined as $0 \oplus \vec{\mathbf{p}}$ and so equivalent to the translation.

e \equiv \mathbf{R} \qquad \quad p = 0 \oplus \vec{\mathbf{p}}

Applying such a transformation to a quaternion $q$ is done by first rotating $q$ with $e$ corresponding to the rotation equation and then adding $p$ :

q' = e\ q\ e^*+ p

Assume a pure translation is to be described using the above notation. Thus the rotation quaternion $e_t$ of a pure translation describes a rotation with an angle $\phi=0^\circ$ which is actually no rotation at all. According to Rotations using quaternions $e$ is defined as:

e_t(\vec{\mathbf{u}},0^\circ)=\cos{0^\circ} + \sin{0^ \circ}\vec{\mathbf{u}} = 1+0\cdot\vec{\mathbf{u}} = 1

The rotation axis $\vec{\mathbf{u}}$ can be any arbitrary unit vector as the sine of $0^\circ$ is zero. So $e_t$ of a pure translation is always just $1$ .

The translation quaternion $p_t$ of a pure translation has the scalar part $0$ and the translation vector $\vec{\mathbf{p}}$ as vector part:

p_t=0\oplus\left[\begin{array}{c}p_x\\p_y\\p_z\end{array}\right] = p_xi+p_yj+p_zk

So a pure translation by vector $[p_x\ p_y\ p_z]^T$ is defined as:

\left\{e_t,p_t\right\}=\left\{1,\ p_xi+p_yj+p_zk\right\}

Example: Translation

A translation by $2$ on the $y$ -axis for example would be

\mathbf{Trans}(y,2) \equiv \left\{1,\ 2j\right\}

For a pure rotation there is no translation. Thus the translation quaternion $p_r$ of a pure rotation is a zero quaternion:

p_r=0 \oplus \left[\begin{array}{c}0\\0\\0\end{array}\right] = 0

The rotation quaternion $e_r$ of a pure rotation around an axis defined by a unit vector $\vec{\mathbf{u}}$ by the angle $\phi$ is:

e_r(\vec{\mathbf{u}},\phi)=\cos{\frac{\phi}{2}} + \sin{\frac{\phi}{2}}\vec{\mathbf{u}}

So a pure rotation is defined as:

\left\{e_r,p_r\right\}=\left\{\cos{\frac{\phi}{2}} + \sin{\frac{\phi}{2}}\vec{\mathbf{u}},\ 0\right\}

Example: Rotation

A rotation by $60^\circ$ around the $z$ -axis for example would be

\mathbf{Rot}(z,60^\circ) \equiv \left\{\cos{30^\circ} + \sin{30^\circ}\left[\begin{array}{c}0\\0\\1\end{array}\right],\ 0\right\} = \left\{\frac{\sqrt{3}}{2} + \frac{1}{2}k,\ 0\right\}

Combination of transformations

It is known that a combination of transformations is defined as:

\begin{align} \mathbf{T}_c = \mathbf{T}_2\mathbf{T}_1 = \left[\begin{array}{cccc} & & & \\ & \mathbf{R}_2\mathbf{R}_1 & & \vec{\mathbf{p}}_2+\mathbf{R}_2\vec{\mathbf{p}}_1 \\ & & & \\ 0 & 0 & 0 & 1\end{array}\right] = \left[\begin{array}{cccc} & & & \\ & \mathbf{R}_c & & \vec{\mathbf{p}}_c \\ & & & \\ 0 & 0 & 0 & 1\end{array}\right] \quad \equiv \quad \left\{e_c,p_c\right\}=\left\{e_2,p_2\right\}\left\{e_1,p_1\right\} \end{align}

But how can the two quaternions $e_c$ and $p_c$ of the quaternion notation be calculated based on the quaternions of the individual transformations? The first transformation leads to

q' = e_1\ q\ e_1^*+ p_1

Now the second transformation is applied on $q'$ . The resulting equation can be solved using the distributive law for quaternions to determine $e_c$ and $p_c$ :

\begin{align} q'' &= e_2\ q'\ e_2^*+ p_2 \\ &= e_2\Big(e_1qe_1^*+ p_1\Big)e_2^*+ p_2 \\ &= \Big(e_2e_1qe_1^*+ e_2p_1\Big)e_2^*+ p_2 \\ &= \underbrace{e_2e_1}_{e_c}q\underbrace{e_1^*e_2^*}_{e_c^*}+\underbrace{e_2p_1e_2^*+ p_2}_{p_c} \\ &= e_c\ q\ e_c^*+p_c \\ \end{align}

Thus the combination of two transformations can be denoted in quaternion notation as

\left\{e_c,p_c\right\}=\left\{e_2,p_2\right\}\left\{e_1,p_1\right\}=\left\{e_2e_1\ , \ e_2p_1e_2^*+p_2\right\}

Using the knowledge about addition of quaternions and rotations and composition of rotations using quaternions this can directly be determined regarding the homogeneous transformation matrix $\mathbf{T}_c$ and its rotational and translational components $\mathbf{R}_c$ and $\vec{\mathbf{p}}_c$ .