PROBLEM $2(30$ Points$)$:

Consider the RPPR mechanism shown in the figure. The joint variables are ${}_{12},s_{23}$ and $s_{14}$ : The loop closure equation is

vec$(AB)+$vec$(B{C}_3)=$vec$(AQ)+$vec$(Q{C}_4)=>r_2{e}^{i{}_{12}}-s_{23}i{e}^{i{}_{12}}=r_1+s_{14}i$

where $AB=r_2,AQ=r_1.$

a$)$ Write the components of the velocity loop equation. Assuming that $s_{14}$ is the input variable, find the singular positions of the mechanism. Sketch one singular position.

b$)$ Find ${}_{12}^{},$ and $s_{23}^{}$ in terms of $s_{14}^{}$ and the position variables.

c$)$ Find ${}_{12}^{},$ and $s_{23}^{}$ in terms of $s_{14}^{},$ the velocity variables and the position variables,