Q$1$: A camera is attached to the end effector frame $T_c$ of the $3$DOF robot below as given. The robot then

makes a differential motion described as $D=[dx,dy,{}_z]=[0\mathrm{.}040\mathrm{.}03,0\mathrm{.}01].$ Find the following when ${}_1=$

$45,d_2=2,d_3=b.$

Note: $a$ is the last digit of your student ID$,b$ is the second to last digit of your student ID$.$ For example, if

your student ID is $8110165,$ then $a=5,b=6.$

$($a$)$ Find the Forward Kinematics, Inverse kinematics, and singularities for this robot.

$($b$)$ Determine the joint differential motions.

$($c$)$ Calculate the change in the end effector frame.

$($d$)$ Determine the updated position of the camera following this differential motion.

$($e$)$ Compute the necessary differential motions, measured relative to Frame $T_c,$ that would be equivalent

to moving the robot to the same new location as described in part $($e$).$

$T_c=\left[\begin{array}{cccc}0& 0& -1& a\\ 0& -1& 0& a+1\\ -1& 0& 0& a+2\\ 0& 0& 0& 1\end{array}\right]$