Difference between revisions of "Realization of transformations"
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For a pure rotation there is no translation. Thus the translation quaternion <math>p</math> is a zero quaternion: | For a pure rotation there is no translation. Thus the translation quaternion <math>p</math> is a zero quaternion: | ||
:<math> | :<math> | ||
| − | p_r= | + | p_r=0 |
</math> | </math> | ||
Revision as of 17:01, 15 October 2015
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Quaternion notation for general transformations
Up to now transformations have been defined by homogeneous matrices combining a rotation matrix 
and a translation vector 
. Now a new notation is introduced to represent a transformation using two quaternions 
and 
:
![\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\}](/wiki/robotics/images/math/6/6/1/661d04cca4174d58de7984920f826fb3.png)
The quaternion is equivalent to and describes the rotation while is defined as 
and so equivalent to the translation.
Applying such a transformation to a quaternion 
is done by first rotating with corresponding to the rotation equation and then adding :
Assume a pure translation is to be described using the above notation. Thus the rotation quaternion describes a rotation with an angle 
which is actually no rotation at all. According to Rotations using quaternions is defined as:
The rotation axis 
can be any arbitrary unit vector as the sine of 
is zero. So of a pure translation is always just 
.
The translation quaternion has the scalar part 
and the translation vector 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](/wiki/robotics/images/math/6/4/2/642c50f36aca3936935ef715956ed74c.png)
A translation by 
on the 
-axis for example would be
For a pure rotation there is no translation. Thus the translation quaternion is a zero quaternion:
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}](/wiki/robotics/images/math/8/b/c/8bc75145f54c5ab026e992e40f2a1842.png)
But how can the two quaternions 
and 
of the quaternion notation be calculated based on the quaternions of individual transformations? The first transformation leads to
Now the second transformation is applied on 
. The resulting equation can be solved using the distributive law for quaternions to determine and :
Thus the combination of two transformations can be denoted in quaternion notation as
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 
and its rotational and translational components 
and 
.
