Difference between revisions of "Transformations"
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# [[Rotation]] | # [[Rotation]] | ||
# [[Homogeneous coordinates]] | # [[Homogeneous coordinates]] | ||
| − | # [[Combinations of | + | # [[Combinations of transformations]] |
# [[Inverse transformation]] | # [[Inverse transformation]] | ||
[[Category:Article]] | [[Category:Article]] | ||
[[Category:Transformations]] | [[Category:Transformations]] | ||
Revision as of 15:33, 13 June 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.
Transformations are applied to vectors or coordinates in vector notation, respectively. All types of transformations can be achieved by matrix multiplication. To transform a vector 
to 
, has to be multilplied by the transformation matrix 
:
The basic transformation matrix is an identity matrix, which corresponds to an empty transformation. So an identity matrix as transformation matrix always results in the original vector. The following equation shows this for three dimensions, but it also holds for two, four and more dimensions:
![\vec{\mathbf{q}}_1=\mathbf{I}_3\cdot\vec{\mathbf{q}}_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{q}}_0](/wiki/robotics/images/math/7/1/e/71ecdb2116e3db5f045c9dbf77c97b97.png)
By modifying the components of the transformation matrix different types of transformations can be achieved. In n-dimensional space rotation can basically be described by an n-by-n transformation matrix. Translation however is actually a vector addition with an n-by-1 vector. So first the two basic types of transformations are explained individually. Then homogeneous coordinates are introduced to be able to apply all kinds of transformations by matrix multiplication and it is explained how transformations are combined. Conclusively the last subarticle presents a special matrix inversion method for inverse transformations.
