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

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

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

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

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

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