Difference between revisions of "Transformations"
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In this article general transformations used in the context of robotics and the underlying mathematics are described. | 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 [[Multiplication of matrices|matrix multiplication]]. To transform a vector <math>\vec{\mathbf{ | + | Transformations are applied to vectors or coordinates in vector notation, respectively. All types of transformations can be achieved by [[Multiplication of matrices|matrix multiplication]]. To transform a vector <math>\vec{\mathbf{q}}_0</math> to <math>\vec{\mathbf{q}}_1</math>, <math>\vec{\mathbf{q}}_0</math> has to be multilplied by the transformation matrix <math>\mathbf{T}</math>:<br/> |
:<math> | :<math> | ||
| − | \vec{\mathbf{ | + | \vec{\mathbf{q}}_1=\mathbf{T}\cdot\vec{\mathbf{q}}_0 |
</math> | </math> | ||
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:<br/> | 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:<br/> | ||
:<math> | :<math> | ||
| − | \vec{\mathbf{ | + | \vec{\mathbf{q}}_1=\mathbf{I}_3\cdot\vec{\mathbf{q}}_0= |
\left[\begin{array}{ccc} | \left[\begin{array}{ccc} | ||
1&0&0\\ | 1&0&0\\ | ||
Revision as of 16:45, 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.
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{p}}_0](/wiki/robotics/images/math/d/6/3/d6303e851d01fc618aa1a448fddb3637.png)
By modifying the components of the transformation matrix different types of transformations can be achieved. In n-dimensional space rotation, scaling and shearing can basically be described by n-by-n transformation matrices. Translation however is actually a vector addition with an n-by-1 vector. So first the different types of transformations are explained individually. Then homogeneous coordinates are introduced to be able to apply all kinds of transformations by matrix multiplication. Conclusively the last subarticle explains how different transformations are combined in a single transformation matrix.
