Problem $2$: The six$-$bar linkage shown below is used in a sofa bed. Its dimensions are listed below.

a$.$ Compute the mobility of the linkage.

b$.$ Draw the graphs representation of the linkage.

c$.$ Identify the four$-$bar linkage in it and determine its Grashof type & linkage type.

d$.$ Write the coordinates of points O$,$ A$,$ B$,$ C$,$ D$,$ E$,$ F$,$ P$1,$ P$2.$

e$.$ Find how many loops and independent loops we have in the linkage.

f$.$ Indicate a set of independent loops in the linkage.

g$.$ Write the loop equations corresponding to the independent loops.

h$.$ Use a symbolic or numeric software to solve and plot the angles of the $4$ bar linkage as a function of the input angle $.$

$\backslash$table$[[$link$,\backslash$table$[[$Length$],[($in$)]]],[$a$,4\mathrm{.}5],[$b$,3\mathrm{.}2],[$h$,7\mathrm{.}4],[$g$,8\mathrm{.}5],[$d$,1\mathrm{.}9],[$f$,2\mathrm{.}9],[e_1,4\mathrm{.}7],[e_2,2\mathrm{.}9],[p_1,30],[p_2,30]]$