Figure: Three of the five reference frames used in the vehicle model. The reference frame with the I superscript represents the inertial (or ground) frame. That with the B superscript represents the noninertial vehicle body frame, centered at the CG. That with RT represents the noninertial righttire frame, centered at the wheel joint. The other tire frames are used in the 7DOF model but not pictured. Only the ground and vehicle frame were used in the kinematic bicycle model.

\[\begin{bmatrix} \dot{v}_y\\
\dot{\omega}_z \end{bmatrix} =\begin{bmatrix} \frac{2C_f+2C_r}{v_xm} & v_x+\frac{2aC_f2bC_r}{v_xm}\\ \frac{2aC_f2bC_r}{v_xI_z} & \frac{2a^2C_f+2b^2C_r}{v_xI_z} \end{bmatrix} \begin{bmatrix} v_y\\ \omega_z \end{bmatrix} +\begin{bmatrix} 2C_f\\ 2aC_f \end{bmatrix}\delta\] 
\(v_y\) and \(v_x\): lateral and longitudinal velocities
\(\omega_z\): yaw rate \(C_f\) and \(C_r\): tire stiffness (proportionality constant; slipangle to lateral force) \(m\): vehicle mass \(I_z\): vehicle moment of inertia about \(z\) \(\delta\): steerangle \(a\) and \(b\): distances, CM to front & rear of vehicle. 

Figure: Pacejka Tire Model Generated Forces.
The graphs show the lateral and longitudinal tire forces as functions of slip Angle and slip ratio, respectfully. Both graphs demonstrate nonlinear behavior. 

Figures: Simulink Layouts of .Conditional Integration PI Controller and PI Controller with Tracking Back Calculation. A variant sink is used to choose from cruise controllers.


Figure: PI Controllers with Antiwindup Measures.
