Problem $3.$ A kinematic bicycle model is shown from

the top in the figure. $(x_R,y_R)$ denote the coordinates

of the rear wheel's center, $l$ is the distance of the

wheels, $$ is the yaw angle, $$ is the steering angle, $v_R$

and $v_F$ are the velocities of the rear and front wheel's

center $($i$.$e$.,$ points R and F $),v_R$ and $v_F$ are the

corresponding speeds, and $$ is the angular velocity

of the bicycle's frame.

a$)$ Express $$ using $v_R,l,$ and $.($Hint: relate $v_R$ and

$v_F$ using $(t,n)$ coordinates.$)$

b$)$ Derive the EoM, that is$,$ express $x_R^{},y_R^{},$ and ${}^{}$

using $v_R,,l,$ and $.$