Difference between revisions of "Realization of transformations"
From Robotics
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= | = | ||
\left[\begin{array}{cccc} & & & \\ & \mathbf{R}_c & & \vec{\mathbf{p}}_c \\ & & & \\ 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] | ||
| + | = | ||
| + | \left\{e_c,p_c\right} | ||
\end{align} | \end{align} | ||
</math> | </math> | ||
| + | But how can the two quaternions <math>e_c</math> and <math>p_c</math> of the quaternion notation be calculated based on the individual transformations? | ||
Revision as of 16:06, 15 October 2015
| ← Back: Composition of rotations | Overview: Quaternions | Next: ??? → |
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] \equiv \left\{e,p\right\}](/wiki/robotics/images/math/3/0/1/301a5ffd539de858c0981e0180640b3f.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 :
Combination of transformations
It is known that a combination of transformations is defined as:
- Failed to parse (syntax error): \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] = \left\{e_c,p_c\right} \end{align}
But how can the two quaternions 
and 
of the quaternion notation be calculated based on the individual transformations?
