Consider the following two$-$link planar RR manipulator and note the following:

End Effector

The workspace origin is fixed at the center of Joint $1.$

the length of Link$1=1,$

the length of Link$2=2,$ and

P represents the workspace coordinates of the gripper,

$P=\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}{a}_{1}cos({}_1)+a_{2}cos({}_1+{}_2)\\ a_{1}sin({}_1)+a_{2}sin({}_1+{}_2)\end{array}\right]$

For the instantaneous configuration $q=\left[\begin{array}{c}{}_1\\ {}_2\end{array}\right]=\left[\begin{array}{c}\frac{}{4}\\ -\frac{}{2}\end{array}\right]$ and joint velocities $q^{}=\left[\begin{array}{c}{}_1^{}\\ {}_2^{}\end{array}\right]=\left[\begin{array}{c}\sqrt[\phantom{2}]{2}\\ \sqrt[\phantom{2}]{2}\end{array}\right],$ evaluate the end effector workspace velocity $P^{}=\left[\begin{array}{c}{x}^{}\\ u^{}\end{array}\right]$