Q$2($a$)$ The frames describing the base of a robot and an object are respectively given relative to the universe frame as below. Evaluate the transformation ${?}^R{T}_H$ of the robot configuration if the hand of the robot is to be placed on the object.

$(4$ marks$)$

${?}^u{T}_R=\left[\begin{array}{cccc}0& -1& 0& 2\\ 1& 0& 0& -1\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right],{?}^U{T}_{\text{o}\text{b}\text{j}}=\left[\begin{array}{cccc}1& 0& 0& 1\\ 0& 0& -1& 4\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right]$

$($b$)$ For the industrial ABB robot shown in Figure Q$2($b$)($units in mm$)$:

$($i$)$ Show the diagram of coordinate frames based on the D$-$H $($Denavit$-$Hartenberg$)$ representation.

$(4$ marks$)$

$($ii$)$ Determine the D $-$H parameter table.

$(6$ marks$)$

$($iii$)$ Find the transformation matrices for all joints $($called $A_i$ matrices$)$ and the ${?}^{\text{B}\text{a}\text{s}\text{e}}{T}_{\text{T}\text{o}\text{o}\text{l}}$ matrix in terms of the $A_i$ matrices.

$(6$ marks$)$