Difference between revisions of "Realization of transformations"
From Robotics
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Up to now transformations have been defined by homogeneous matrices combining a rotation matrix <math>\mathbf{R}</math> and a translation vector <math>\vec{\mathbf{p}}</math>. Now a new notation is introduced to represent a transformation using two quaternions <math>e</math> and <math>p</math>: | Up to now transformations have been defined by homogeneous matrices combining a rotation matrix <math>\mathbf{R}</math> and a translation vector <math>\vec{\mathbf{p}}</math>. Now a new notation is introduced to represent a transformation using two quaternions <math>e</math> and <math>p</math>: | ||
:<math> | :<math> | ||
| − | \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] \quad \equiv \quad \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. | 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. | ||
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\mathbf{T}_c = \mathbf{T}_2\mathbf{T}_1 = | \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}_2\mathbf{R}_1 & & \vec{\mathbf{p}}_2+\mathbf{R}_2\vec{\mathbf{p}}_1 \\ & & & \\ 0 & 0 & 0 & 1\end{array}\right] | ||
| − | + | \quad \equiv \quad | |
\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] | ||
= | = | ||
Revision as of 16:07, 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] \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 :
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]
\quad \equiv \quad
\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}](/wiki/robotics/images/math/f/4/6/f460d506a77b412034cc540fb9c38877.png)
But how can the two quaternions 
and 
of the quaternion notation be calculated based on the individual transformations?
