QUESTION $1$

$[20]$

Figure $1.$ An articulated robot

Shown in Figure $1$ is an articulated robot having three revolute joints and one prismatic joint. Notice that the axis of the prismatic joint has an offset of from the first revolute joint at the origin O$.$ Namely, the distance between point $O$ and point $A,$ a constant, and the angle between $OA$ and $AB$ is $90$ degrees, a constant as well. Using the geometric parameters and joint displacements shown in the figure, answer the following questions.

$($a$)$ Obtain the kinematic equations relating the end$-$effecter position, and orientation to the joint displacements are $55,45$ and $30$ respectively.

$($b$)$ Joint $1$ can rotate between $45$ degrees and $35$ degrees, and joint $3$ can rotate from $60$ degrees to $90$ degrees, while joint $2$ moves $15$ and $35.$ Sketch the workspace of the robotic structure within the x $,$ y plane.

$(8)$

$($c$)$