Consider the RPR arm shown in the following figure.

The prismatic joint of the robot has the length $5d_2^{*}10.$ The range of

joint $1$ is $0{}_1^{*}360$ and the of the joint $3$ is $-160{}_3^{*}160.$ The

exact values of a and $b$ are not important but assume $a>b$ and

$($a$)$ For which values of ${}_3^{*}$ is the solution to the inverse kinematic problem is

always unique.

$($b$)$ Given $o_3^0$ solve the inverse kinematic problem $($i$.$e$.,$ given $x,y$ and $z$ solve

for the values of ${}_1^{*},d_2^{*}$ and ${}_3^{*})$

A joint in a manipulator has the following constraints:

$q(0)=2$

$q(4)=10$

$q^{}(0)=0$

$q^{}(4)=0$

Represent the motion of $q(t)$ as a polynomial.