Shown below is a construction robot having two revolute joints and one prismatic joint. Notice that the axis of the prismatic joint has an offset of $l_1$ from the first revolute joint at the origin O $.$ Namely, the distance between point O and point A is $l_1,$ a constant, and the angle between OA and AB is $90$ degrees, a constant as well. Joint $2$ is a prismatic joint, whose displacement is given by distance $d,$ a variable. Using the geometric parameters and joint displacements shown in the figure, answer the following questions.

Figure $1.$ Construction robot with two revolute joints and one prismatic joint

$($a$)(20$ points$)$ Obtain the kinematic equations relating the end$-$effecter position and orientation to the joint displacements.

$($b$)(20$ points$)$ Joint $1$ can rotate between $45$ degrees and $135$ degrees, and joint $3$ can rotate from $-90$ degrees to $+90$ degrees, while joint $2$ can move from $l_1$ to $2l_1.$ Sketch the workspace of the end$-$effecter E within the xy plane.

$($c$)(20$ points$)$ Solve the inverse kinematics problem to find joint displacements leading the end$-$effecter to a desired position and orientation: $x_e{y}_e$ and ${}_e.$