## 2 Load Sensitivity Analysis
### Vehicle Dynamics Modeling  
Consider the 3-DOF planar vehicle model depicted in Figure 1. Using the longitudinal velocity, \(u\), the lateral velocity, \(v\), and the yaw rate, \(r\), to describe the planar motion of the vehicle, the longitudinal, lateral, and yaw dynamics of the vehicle can be stated as  
\[m(\dot{u}-vr)=F_{x1}+F_{x2}+F_{x3}+F_{x4}, \tag{1}\]  
\[m(\dot{v}+ur)=F_{y1}+F_{y2}+F_{y3}+F_{y4}, \tag{2}\]  
\[I_{z}\dot{r}=I_{f}(F_{y1}+F_{y2})-I_{r}(F_{y3}+F_{y4})+\frac{d}{2}(F_{x2}+F_{x4 }-F_{x1}-F_{x3}) \tag{3}\]  
where, \(m\), indicates the vehicle mass, \(I_{z}\), the yaw moment of inertia, \(d\), the vehicle tread width, \(l_{f}\), the distance from front axle to CG, \(l_{r}\), the distance from rear axle to CG, while \(F_{xi}\) and \(F_{yi}\) indicate the longitudinal and lateral tire force at each tire, respectively, and the index \(i\) refers to the number of each tire (1 = front left, 2 = front right, 3 = rear left, 4 = rear right).  
Assuming low levels of lateral acceleration the lateral force characteristics of the tires can be described by a linear function of tire side slip angle. By also assuming that the influence of the tire vertical load and the longitudinal tire force are both small, the lateral tire force [25] is shown in the following equation.  
\[F_{yi}=\left(1-\frac{1}{2}\left(\frac{F_{xi}}{\mu F_{zi}}\right)^{2}\right)K_ {f,r}\alpha_{i} \tag{4}\]  
where, \(F_{zi}\) and \(\alpha_{i}\) indicate the static vertical load and the tire side slip angle of each tire, respectively, \(\mu\) is the tire-road friction coefficient, and \(K_{f,r}\) is the tire cornering stiffness of the front (rear) tire. Here, it is also assumed the tire cornering stiffness is dependent on the static vertical load on each tire according to the following equation.  
\[K_{f,r}=e_{f,r}(c_{0}-c_{1}F_{zi})F_{zi} \tag{5}\]  
where, \(e_{f,r}\) indicates the compliance steer coefficient of the front (rear) tire and \(c_{0}\) and \(c_{1}\) are the coefficients of the tire load dependent characteristics.  
Figure 1: 3-Degrees of Freedom (DOF) vehicle model in planar motion.  
For theoretical analysis and control system design, this paper employs the linear two-wheel vehicle model in planar motion. Assuming constant vehicle velocity \(V\) and small body side slip angle, the governing equations of the lateral and yaw motion can then be expressed as follows  
\[\dot{\beta}=a_{11}\beta+a_{12}r+b_{11}\delta_{f}+b_{12}M, \tag{6}\]  
\[\dot{r}=a_{21}\beta+a_{22}r+b_{21}\delta_{f}+b_{22}M \tag{7}\]  
where, \(\beta\) is the vehicle body side slip angle, \(\delta_{f}\) is the front steering angle, and \(M\) is the additional direct yaw moment by transverse distribution of longitudinal tire forces. Here, the longitudinal tire forces are not considered when determining the lateral forces (Equation (4)). Each coefficient in Equations (6) and (7) can then be expressed as follows  
\[a_{11}=-\frac{2(K_{f}+K_{r})}{mV},a_{12}=-1-\frac{2(l_{f}K_{f}-l_{r}K_{r})}{mV _{1}^{2}},b_{11}=\frac{2K_{f}}{mV},b_{12}=0,\]  
\[a_{21}=-\frac{2(l_{f}K_{f}-l_{r}K_{r})}{l_{z}},a_{22}=-\frac{2(l_{f}K_{f}+l_{r }^{2}K_{r})}{l_{z}V},b_{21}=\frac{2l_{f}K_{f}}{l_{z}},b_{22}=\frac{1}{l_{z}}.\]