For the vehicle bicycle model,

$1)$ Obtain the transfer functions from steering angle to

a$.$ yaw rate

b$.$ Lateral acceleration

c$.$ Lateral velocity

d$.$ Sideslip angle

$2)$ Based on the above transfer functions, obtain:

a$.$ Yaw rate gain functions

b$.$ Lateral acceleration gain functions

c$.$ Lateral velocity gain functions

d$.$ Sideslip angle gain functions

$3)$ By using the transfer functions you obtained in question $1,$ in Simulink, simulate the bicycle model under step steering angle input of $3$ degrees. Document your results neatly.

$4)$ For the transfer functions in $1)$;

What are the poles of the system? What are the zeros? Is the system underdamped? Overdamped?

$5)$ Change the location of CG $($the center of the gravity$)$ of the vehicle to forward and backward by changing parameters a$,$ b and calculate understeer gradient for each case. In which case is the vehicle oversteer?

Vehicle data:

a$=1\mathrm{.}2$m$,$ b$=1\mathrm{.}3$m$,$ C$\_\backslash$alpha f$=60000$N$/$rad$,$ C$\_\backslash$alpha r$=90000$N$/$rad$,$ m$=1300$kg$,$ Jz$=1200$kgm$^(2),$ u$=30$ m$/$s$.$