Transformations

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Overview: Transformations

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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 $\vec{\mathbf{p}}_0$ to $\vec{\mathbf{p}}_1$, $\vec{\mathbf{p}}_0$ has to be multilplied by the transformation matrix $\mathbf{T}$:

$\vec{\mathbf{p}}_1=\mathbf{T}\cdot\vec{\mathbf{p}}_0$

Basis is at first the multiplication of an identity matrix with a vector. The identity matrix corresponds to an empty transformation. So a multiplication with an identity matrix results in the original vector no matter how many dimensions the vector has. The following equations show this for two and three dimensions:

$\begin{align} \vec{\mathbf{p}}_1&=&\mathbf{I}_2\cdot\vec{\mathbf{p}}_0&=& \left[\begin{array}{cc} 1&0\\ 0&1 \end{array}\right] \cdot \left[\begin{array}{c} x_0\\ y_0 \end{array}\right]&=& \left[\begin{array}{c} 1\cdot x_0+0\cdot y_0\\ 0\cdot y_0+1\cdot y_0 \end{array}\right]&=& \left[\begin{array}{c} x_0\\ y_0 \end{array}\right]&=& \vec{\mathbf{p}}_0 \\ \vec{\mathbf{p}}_1&=&\mathbf{I}_3\cdot\vec{\mathbf{p}}_0&=& \left[\begin{array}{ccc} 1&0&0\\ 0&1&0\\ 0&0&1 \end{array}\right] \cdot \left[\begin{array}{c} x_0\\ y_0\\ z_0 \end{array}\right]&=& \left[\begin{array}{c} 1\cdot x_0+0\cdot y_0+0\cdot z_0\\ 0\cdot y_0+1\cdot y_0+0\cdot z_0\\ 0\cdot y_0+0\cdot y_0+1\cdot z_0 \end{array}\right]&=& \left[\begin{array}{c} x_0\\ y_0\\ z_0 \end{array}\right]&=& \vec{\mathbf{p}}_0 \end{align}$

By modifying the components of the The different types of transformations are presented in the following subarticles:

1. Translation
2. Rotation
3. Scaling
4. Combinations of the three transformation types