PROBLEM $2$:

For the $2$ d$.$o$.$f$.$ mechanism shown below, the joint variables ${}_{12},s_{23},{}_{13}$ and $s_{34}$ are assigned by disconnecting revolute joint at C $.$ Then the LCE is

vec$(A_{o}A)+$vec$(AB)+$vec$(BD)+$vec$(DC)=$vec$(A_{o}C)$

$A_{o}C=r_1,AB=r_3,CD=r_4,$

${}_{12}$ and $s_{23}$ are the input variables. Determine the velocity variables ${}_{13}^{}$ and $s_{34}^{}$ in terms of the input velocities ${}_{12}^{}$ and $s_{23}^{}$ and the position variables.