PROBLEM $1(35$ Points$)$:

In the mechanism shown, the joint variables are ${}_{12},s_{14},{}_{13},{}_{15}$ and $s_{56}$ as shown. The link dimensions are $A_{0}Q=r_1=7cm,QD=a_1=5cm,A_{o}A=r_2=7cm,AC=r_3=16cm,BC=a_3=8cm$ and angle $(A_{o}QC)=90.$ Solving the equations corresponding to the loops $A_{0}ACQ{A}_o$ and $QCBDQ$ analytically, find the joint variables when $s_{14}=21cm.$

LCE$1(A_{o}ACQ{A}_o)$:$r_2{e}^{i{}_{12}}=r_{1}i+s_{14}+r_3{e}^{i{}_{13}}$

LCE$2(QCBDQ)$:$s_{14}+a_3{e}^{i{}_{13}}+s_{56}{e}^{i{}_{15}}=a_{1}i$

Find all solutions corresponding to different assembly configurations $($i$.$e$.$ different closures$)$ of the mechanism. Write the final results on the table given below.

$\backslash$table$[[,{}_{12}($deg$.),{}_{13}($deg$.),{}_{15}($deg$.),s_{56}($cm$)],[$Assembly Configuration $1,,,,],[$Assembly Configuration $2,,,,],[$Assembly Configuration $3,,,,],[$Assembly Configuration $4,,,,]]$