Combining translation and rotation

Suppose a rotation by $ \theta $ is performed, followed by a translation by $ x_t,y_t$. This can be used to place the robot in any desired position and orientation. Note that translations and rotations do not commute! If the operations are applied successively, each $ (x,y) \in {\cal A}$ is transformed to

$\displaystyle \begin{pmatrix}x \cos\theta - y \sin\theta + x_t x \sin\theta + y \cos\theta + y_t \end{pmatrix} .$ (3.33)

The following matrix multiplication yields the same result for the first two vector components:

$\displaystyle \begin{pmatrix}\cos\theta & -\sin\theta & x_t \sin\theta & \cos...
...- y \sin\theta + x_t x \sin\theta + y \cos\theta + y_t 1  \end{pmatrix} .$ (3.34)

This implies that the $ 3 \times 3$ matrix,

$\displaystyle T = \begin{pmatrix}\cos\theta & -\sin\theta & x_t \sin\theta & \cos\theta & y_t 0 & 0 & 1  \end{pmatrix} ,$ (3.35)

represents a rotation followed by a translation. The matrix $ T$ will be referred to as a homogeneous transformation matrix. It is important to remember that $ T$ represents a rotation followed by a translation (not the other way around). Each primitive can be transformed using the inverse of $ T$, resulting in a transformed solid model of the robot. The transformed robot is denoted by $ {\cal A}(x_t,y_t,\theta)$, and in this case there are three degrees of freedom. The homogeneous transformation matrix is a convenient representation of the combined transformations; therefore, it is frequently used in robotics, mechanics, computer graphics, and elsewhere. It is called homogeneous because over $ {\mathbb{R}}^3$ it is just a linear transformation without any translation. The trick of increasing the dimension by one to absorb the translational part is common in projective geometry [804].

Steven M LaValle 2012-04-20