For the three$-$joint manipulator below, solve the following problems. $(30$ pt $)$

where $p_0=(0,0,0),p_1=(0,1,0),q_1=(0,1,1),r_1=(0,2,2)$ and $u_0=(0,0,1),u_1=(0,1,0),$

$u_2=(1,0,0).$ The angle ${}_j$ denotes the rotation angle of joint $u_j.$

$2\mathrm{.}1.$ The joint angles of the manipulator are set as ${}_0=-45,{}_1=-30$ and ${}_2=20.$

Calculate the vector $p,q,r$ after the rotation. $(10$ pt $)$

$2\mathrm{.}2.$ Plot the manipulator's configuration both before and after rotation in one figure using

MATLAB. Use different line styles for each configuration, and clearly label the zero position

to distinguish between the two. $(10pt)$

$2\mathrm{.}3.$ Find the velocity $p^{},q^{},r^{}$ under the following conditions. $(10$ pt$)$

${}_0=-45,{}_1=-30$ and ${}_2=20,{}_0^{}=20\frac{}{sec,{}_1^{}}=-10\frac{}{sec,{}_2^{}}=-15\frac{}{sec}$