## 2 Load Sensitivity Analysis
### Vehicle Parameters  
The basic vehicle parameters for various loading conditions at the rear of the vehicle are shown in Table 1. The loaded mass, \(m_{l}\), is varied from 0 to 80 kg. Based on the governing equations for the linear two-wheel vehicle model, Equations (6) and (7), the steady-state gains from front steering angle to the yaw rate, and body side slip angle can be calculated as follows  
\[\frac{r_{st}}{\delta_{fst}}=\frac{1}{(1+AV^{2})}\frac{V}{l}, \tag{8}\]  
\[\frac{\beta_{st}}{\delta_{fst}}=\frac{1-\frac{ml_{f}}{2l_{r}K_{r}}V^{2}}{(1+AV ^{2})}\frac{l_{r}}{l} \tag{9}\]  
where, \(l=l_{f}+l_{r}\) is the vehicle wheel base and the stability factor \(A\) is defined as  
\[A=-\frac{m}{2l^{2}}\frac{l_{f}K_{f}-l_{r}K_{r}}{K_{f}K_{r}}. \tag{10}\]  
The dependence of yaw rate and the body side slip angle gains on the vehicle velocity for the loading conditions in Table 1 are shown in Figure 2a,b, respectively. Based on Figure 2, it is found that the yaw rate gain for the loaded vehicle does not increase for low velocities but increases significantly for velocities above 40 km/h. The side slip angle increases (in negative direction) for all vehicle velocities because the loading condition shifts the center of gravity to rear and, therefore, the rear tire needs more side slip angle to generate the force needed to balance the vehicle in steady-state cornering. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Description** & **Parameter** & **Unit** & \begin{tabular}{c} **1 Person** \\ **Unloaded** \\ \end{tabular} & \begin{tabular}{c} **1 Person** \\ **20 kg** \\ **Loaded** \\ \end{tabular} & \begin{tabular}{c} **1 Person** \\ **40 kg** \\ **Loaded** \\ \end{tabular} & \
\begin{tabular}{c} **1 Person** \\ **60 kg** \\ **Loaded** \\ \end{tabular} &
\begin{tabular}{c} **1 Person** \\ **80 kg** \\ **Loaded** \\ \end{tabular} \\ \hline Mass & \(m\) & kg & 570 & 590 & 610 & 630 & 650 \\ Front axle to CC Distance & \(l_{f}\) & m & 1.162 & 1.218 & 1.271 & 1.321 & 1.368 \\ Rear axle to CG Distance & \(l_{f}\) & m & 0.938 & 0.882 & 0.829 & 0.779 & 0.732 \\ Front tire cornering stiffness & \(K_{f}\) & N/rad & 10,775 & 10,541 & 10,304 & 10,064 & 9819 \\ Rear tire cornering stiffness & \(K_{r}\) & N/rad & 20,243 & 21,443 & 22,558 & 23,589 & 24,536 \\ Yaw mass moment of inertia & \(l_{z}\) & kgm\({}^{2}\) & 500 & 552 & 598 & 638 & 674 \\ Stability Factor & \(A\) & s\({}^{2}\)/m\({}^{2}\) & 0.0019 & 0.0018 & 0.0017 & 0.0015 & 0.0014 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Vehicle parameter changes against various loading conditions. The stability factor \(A\) is defined in Equation (10). This theoretical analysis shows that the driver must compensate the increase in yaw rate gain by a reduction of the steering angle. The side slip angle gain is also increasing (in the negative direction). Next, the frequency response of the vehicle yaw rate and lateral acceleration to front steering angle are determined and analyzed as shown in Figure 3. Assuming no additional direct yaw moment, the Laplace transformation of the governing equations of the linear two-wheel vehicle model gives the following transfer functions from the front steering angle to the yaw rate and the lateral acceleration, respectively:  
\[\frac{r(s)}{\delta_{f}(s)}=\frac{G_{r}^{\delta}(0)(T_{r}s+1)\omega_{n}^{2}}{s^ {2}+2\zeta\omega_{n}s+\omega_{n}^{2}}, \tag{11}\]  
\[\frac{a_{y}(s)}{\delta_{f}(s)}=\frac{G_{r}^{\delta}(0)V(T_{y1}s^{2}+T_{y2}s+1) \omega_{n}^{2}}{s^{2}+2\zeta\omega_{n}s+\omega_{n}^{2}} \tag{12}\]  
where, each parameter can be determined as follows  
\[ \begin{array}{l}G_{r}^{\delta}(0)=\frac{V}{l\left(1-\frac{1}{2l^{2}}\frac{L \xi K_{f}-\iota K_{r}}{K_{f}K_{r}}V^{2}\right)},\,T_{r}=\frac{ml_{f}V}{2K_{r}},\\ \zeta=\frac{2(l_{F}^{2}K_{f}+l_{F}^{2}K_{r})}{l_{F}V}+\frac{2(K_{f}+K_{r})}{mV },\\ \omega_{n}=\frac{2l}{V}\sqrt{\frac{K_{f}K_{r}(1+AV^{2})}{mlz}},\\ T_{y2}=\frac{l_{s}}{2K_{r}},\,T_{y1}=\frac{l_{r}}{V}.\end{array}