Difference between revisions of "Realization of transformations"
From Robotics
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\mathbf{T} = \left[\begin{array}{cccc} & & & \\ & \mathbf{R} & & \vec{\mathbf{p}} \\ & & & \\ 0 & 0 & 0 & 1\end{array}\right] \equiv \left\{e,p\right\} | \mathbf{T} = \left[\begin{array}{cccc} & & & \\ & \mathbf{R} & & \vec{\mathbf{p}} \\ & & & \\ 0 & 0 & 0 & 1\end{array}\right] \equiv \left\{e,p\right\} | ||
</math> | </math> | ||
| + | The quaternion <math>e</math> is equivalent to <math>\mathbf{R}</math> and describes the rotation while <math>p</math> is defined as <math>0 \oplus \vec{\mathbf{p}}</math> and so equivalent to the translation. | ||
Revision as of 10:54, 15 October 2015
| ← Back: Composition of rotations | Overview: Quaternions | Next: ??? → |
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.
