PROBLEM $1$:

In the mechanism shown the link lengths are $AD=r_1,BC=r_3,CD=r_4$ and The joint variables are ${}_{12},s_{23}$ and ${}_{14}.{}_{12}$ is the input variable. The loop closure equation is

vec$(AB)+$vec$(BC)=$vec$(AD)+$vec$(DC)$

a$)$ Find $s_{23}^{}$ and ${}_{14}^{}$ in terms of the input velocity ${}_{12}^{}$ and the position variables.

b$)$ Find and sketch singular positions of the mechanism.

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.