Two screws

The homogeneous transformation matrix $T_i$ will be constructed by combining two simpler transformations. The transformation

$$R_i = \begin{pmatrix}\cos\theta_i & -\sin\theta_i & 0 & 0 \sin\theta_i & \cos\theta_i & 0 & 0 0 & 0 & 1 & d_i 0 & 0 & 0 & 1 \end{pmatrix}$$ (3.54)

causes a rotation of $\theta _i$ about the $z_i$-axis, and a translation of $d_i$ along the $z_i$-axis. Notice that the rotation by $\theta _i$ and translation by $d_i$ commute because both operations occur with respect to the same axis, $z_i$. The combined operation of a translation and rotation with respect to the same axis is referred to as a screw (as in the motion of a screw through a nut). The effect of $R_i$ can thus be considered as a screw about the $z_i$-axis. The second transformation is

$$Q_{i-1} = \begin{pmatrix}1 & 0 & 0 & a_{i-1} 0 & \cos\alpha_{i... ... 0 & \sin\alpha_{i-1} & \cos\alpha_{i-1} & 0 0 & 0 & 0 & 1 \end{pmatrix} ,$$ (3.55)

which can be considered as a screw about the $x_{i-1}$-axis. A rotation of $\alpha _{i-1}$ about the $x_{i-1}$-axis and a translation of $a_{i-1}$ are performed.