Realization of transformations

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Overview: Quaternions

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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 $p$:

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

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 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 $p_c$:

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