Matrix

j = 1

j = 2

j = 3

j = 4

j = 5

j = 6

The 4 × 4 matrix of homogeneous transformations

$A^{\text{1}}\left(x\right)=\left[\begin{array}{cc}I& p_x\\ 0^{\text{T}}& 1\end{array}\right]$

$A^{\text{2}}\left(y\right)=\left[\begin{array}{cc}I& p_y\\ 0^{\text{T}}& 1\end{array}\right]$

$A^{\text{3}}\left(z\right)=\left[\begin{array}{cc}I& p_z\\ 0^{\text{T}}& 1\end{array}\right]$

$A^{\text{4}}\left(\theta \right)=\left[\begin{array}{cc}{R}_x& 0\\ 0^{\text{T}}& 1\end{array}\right]$

$A^{\text{5}}\left(\psi \right)=\left[\begin{array}{cc}{R}_y& 0\\ 0^{\text{T}}& 1\end{array}\right]$

$A^{\text{6}}\left(\phi \right)=\left[\begin{array}{cc}{R}_z& 0\\ 0^{\text{T}}& 1\end{array}\right]$

The 6 × 6 matrix of screw transformations

$T^1\left(x\right)=\left[\begin{array}{cc}I& 0\\ T_x& I\end{array}\right]$

$T^2\left(y\right)=\left[\begin{array}{cc}I& 0\\ T_y& I\end{array}\right]$

$T^3\left(z\right)=\left[\begin{array}{cc}I& 0\\ T_z& I\end{array}\right]$

$T^4\left(\theta \right)=\left[\begin{array}{cc}{R}_x& 0\\ 0& R_x\end{array}\right]$

$T^{\text{5}}\left(\psi \right)=\left[\begin{array}{cc}{R}_y& 0\\ 0& R_y\end{array}\right]$

$T^{\text{6}}\left(\phi \right)=\left[\begin{array}{cc}{R}_z& 0\\ 0& R_z\end{array}\right]$

$T_x=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& -x_i\\ 0& x_i& 0\end{array}\right]$ ; $T_y=\left[\begin{array}{ccc}0& 0& y_i\\ 0& 0& 0\\ -y_i& 0& 0\end{array}\right]$ ; $T_z=\left[\begin{array}{ccc}0& -z_i& 0\\ z_i& 0& 0\\ 0& 0& 0\end{array}\right]$ ;

$R_x=\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos {\theta }_i& -\sin {\theta }_i\\ 0& \sin {\theta }_i& \cos {\theta }_i\end{array}\right]$ ; $R_y=\left[\begin{array}{ccc}\cos {\psi }_i& 0& \sin {\psi }_i\\ 0& 1& 0\\ -\sin {\psi }_i& 0& \cos {\psi }_i\end{array}\right]$ ; $R_z=\left[\begin{array}{ccc}\cos {\phi }_i& -\sin {\phi }_i& 0\\ \sin {\phi }_i& \cos {\phi }_i& 0\\ 0& 0& 1\end{array}\right]$