Difference between revisions of "Transformations"
From Robotics
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In this article general transformations used in the context of robotics and the underlying mathematics are described. All types of transformations can be achieved by [[Multiplication of matrices|matrix multiplication]]. Basis is at first the multiplication of a vector with the respective identity matrix. The identity matrix corresponds to an empty transformation. So a multiplication with an identity matrix results in the original vector, for example in two-dimensional space:<br/> | In this article general transformations used in the context of robotics and the underlying mathematics are described. All types of transformations can be achieved by [[Multiplication of matrices|matrix multiplication]]. Basis is at first the multiplication of a vector with the respective identity matrix. The identity matrix corresponds to an empty transformation. So a multiplication with an identity matrix results in the original vector, for example in two-dimensional space:<br/> | ||
| − | :<math> | + | :<math>\begin{align} |
| − | \vec{\mathbf{p}}_1=\mathbf{I}_2\cdot\vec{\mathbf{p}}_0= | + | \vec{\mathbf{p}}_1&=&\mathbf{I}_2\cdot\vec{\mathbf{p}}_0&=& |
\left[\begin{array}{cc} | \left[\begin{array}{cc} | ||
1&0\\ | 1&0\\ | ||
| Line 12: | Line 12: | ||
x_0\\ | x_0\\ | ||
y_0 | y_0 | ||
| − | \end{array}\right]= | + | \end{array}\right]&=& |
\left[\begin{array}{c} | \left[\begin{array}{c} | ||
1\cdot x_0+0\cdot y_0\\ | 1\cdot x_0+0\cdot y_0\\ | ||
0\cdot y_0+1\cdot y_0 | 0\cdot y_0+1\cdot y_0 | ||
| − | \end{array}\right]= | + | \end{array}\right]&=& |
\left[\begin{array}{c} | \left[\begin{array}{c} | ||
x_0\\ | x_0\\ | ||
y_0 | y_0 | ||
| − | \end{array}\right]= | + | \end{array}\right]&=& |
| − | \vec{\mathbf{p}}_0 | + | \vec{\mathbf{p}}_0 \\ |
| − | + | \vec{\mathbf{p}}_1&=&\mathbf{I}_3\cdot\vec{\mathbf{p}}_0&=& | |
| − | |||
| − | |||
| − | |||
| − | \vec{\mathbf{p}}_1=\mathbf{I}_3\cdot\vec{\mathbf{p}}_0= | ||
\left[\begin{array}{ccc} | \left[\begin{array}{ccc} | ||
1&0&0\\ | 1&0&0\\ | ||
| Line 37: | Line 33: | ||
y_0\\ | y_0\\ | ||
z_0 | z_0 | ||
| − | \end{array}\right]= | + | \end{array}\right]&=& |
\left[\begin{array}{c} | \left[\begin{array}{c} | ||
1\cdot x_0+0\cdot y_0+0\cdot z_0\\ | 1\cdot x_0+0\cdot y_0+0\cdot z_0\\ | ||
0\cdot y_0+1\cdot y_0+0\cdot z_0\\ | 0\cdot y_0+1\cdot y_0+0\cdot z_0\\ | ||
0\cdot y_0+0\cdot y_0+1\cdot z_0 | 0\cdot y_0+0\cdot y_0+1\cdot z_0 | ||
| − | \end{array}\right]= | + | \end{array}\right]&=& |
\left[\begin{array}{c} | \left[\begin{array}{c} | ||
x_0\\ | x_0\\ | ||
y_0\\ | y_0\\ | ||
z_0 | z_0 | ||
| − | \end{array}\right]= | + | \end{array}\right]&=& |
\vec{\mathbf{p}}_0 | \vec{\mathbf{p}}_0 | ||
| + | \end{align} | ||
</math> | </math> | ||
Revision as of 14:16, 26 May 2014
| ← Back: Adjugate Formula | Overview: Transformations | Next: Translation → |
In this article general transformations used in the context of robotics and the underlying mathematics are described. All types of transformations can be achieved by matrix multiplication. Basis is at first the multiplication of a vector with the respective identity matrix. The identity matrix corresponds to an empty transformation. So a multiplication with an identity matrix results in the original vector, for example in two-dimensional space:
![\begin{align}
\vec{\mathbf{p}}_1&=&\mathbf{I}_2\cdot\vec{\mathbf{p}}_0&=&
\left[\begin{array}{cc}
1&0\\
0&1
\end{array}\right]
\cdot
\left[\begin{array}{c}
x_0\\
y_0
\end{array}\right]&=&
\left[\begin{array}{c}
1\cdot x_0+0\cdot y_0\\
0\cdot y_0+1\cdot y_0
\end{array}\right]&=&
\left[\begin{array}{c}
x_0\\
y_0
\end{array}\right]&=&
\vec{\mathbf{p}}_0 \\
\vec{\mathbf{p}}_1&=&\mathbf{I}_3\cdot\vec{\mathbf{p}}_0&=&
\left[\begin{array}{ccc}
1&0&0\\
0&1&0\\
0&0&1
\end{array}\right]
\cdot
\left[\begin{array}{c}
x_0\\
y_0\\
z_0
\end{array}\right]&=&
\left[\begin{array}{c}
1\cdot x_0+0\cdot y_0+0\cdot z_0\\
0\cdot y_0+1\cdot y_0+0\cdot z_0\\
0\cdot y_0+0\cdot y_0+1\cdot z_0
\end{array}\right]&=&
\left[\begin{array}{c}
x_0\\
y_0\\
z_0
\end{array}\right]&=&
\vec{\mathbf{p}}_0
\end{align}](/wiki/robotics/images/math/f/b/f/fbfe5b03c247d0fc135c0473666f4e42.png)
The different types of transformations are presented in the following subarticles:
