Question $2($Total: $40$ marks$)$

Consider the $2-$DOF planar robotic manipulator that is conceptually depicted in Figure Q$2.$

$($a$)$ This type of robot can only move in a $2$D plane $($e$.$g$.\backslash ($ z$=0\backslash )).$ Explain how you would place the monocular camera $($specifically$,$ its orientation relative to the robot's configuration$)$ such that the projection of the end$-$effector $2$D position $\backslash (\backslash$mathbf$\{$x$\}=$f$(\backslash$mathbf$\{$q$\})\backslash )$ can be accurately represented with an affine camera model. $(10$ marks$)$

$($b$)$ Assume the affine model in $($a$),$ with the image point $\backslash (\backslash$mathbf$\{$y$\}=$h$(\backslash$mathbf$\{$x$\})\backslash )$ representing the observed end$-$effector position $\backslash (\backslash$mathbf$\{$x$\}\backslash ).$ Compute the interaction matrix $\backslash (\backslash$mathbf$\{$L$\}\backslash )$ for this visual servoing system $(15$ marks$)$

$($c$)$ Design a sensor$-$based controller $\backslash (\backslash$dot$\{\backslash$mathbf$\{$q$\}\}\backslash )$ to drive the end$-$effector image point $\backslash (\backslash$mathbf$\{$y$\}\backslash )$ towards a desired constant reference $\backslash (\backslash$mathbf$\{$y$\}\_\{$d$\}\backslash ).$ Analyse the stability of your controller. $(15$ marks$)$