## 3 Longitudinal Slip Ratio and Vertical Load of Each Wheel  
Figure 2 presents the schematic diagram of the three-degrees-of-freedom (3-DoF) vehicle model used to compute the longitudinal slip ratio and vertical load of each wheel on the ground. The notation \"CG\" is the center of gravity (COG) of the whole vehicle. The value \(h_{c}\) is the height of COG relative to the ground and the distances from the vehicle's COG to the front and rear axles are \(l_{f}\) and \(l_{r}\), respectively. The parameters \(l_{f}\) and \(l_{r}\) indicate the widths of front and rear track. The heading angle of the front tires are generated by the front steering wheel angle, \(\delta\). The side slip angle at the vehicle's COG is \(\beta\) described as the angle between the velocity vector \(V_{c}\) and the longitudinal velocity of the vehicle \(V_{x}\). Moreover, \(\dot{\psi}\) is the yaw rate of the vehicle. The lateral velocity of the vehicle is \(V_{y}\). Furthermore, \(F_{x,i}\), and \(F_{y,i}\) represent the longitudinal and lateral forces acting on each tire, where sub-notations \(i\) indicate the front-left (FL), the front-right (FR), the rear-left (RL), and the rear-right (RR) tires of the vehicle. In addition, \(F_{z,i}\) are the vertical forces of tires.  
Based on the difference between the actual longitudinal velocity at the wheel axle \(V_{wx,i}\) and the equivalent rotational velocity \(r_{eff}\omega_{i}\) of the tire, the longitudinal slip ratios can be defined as:  
\[\sigma_{i}=\frac{r_{eff}\omega_{i}-V_{wx,i}}{\max\left(r_{eff}\omega_{i},V_{wx,i}\right)}\;i=FL,FR,RL,RR \tag{2}\]  
where \(\omega_{i}\) is the rotational speed of each wheel and \(r_{eff}\) is the effective radius of the tire.  
Based on the vehicle's longitudinal (\(V_{x}\)), lateral velocity (\(V_{y}\)), and yaw rate (\(\dot{\psi}\)), the longitudinal velocity of each wheel \(V_{wx,i}\) in (2) can be obtained [22]:  
Figure 2: Schematic of 3-DOF vehicle model. (**a**) Vehicle planar model and (**b**) vehicle load transfer model.  
\[V_{wx.FL} =\left(V_{x}-\frac{\dot{\psi}\,t_{f}}{2}\right)\cos(\delta)+\left(V_{x }+\dot{\psi}\,l_{f}\right)\sin(\delta) \tag{3}\] \[V_{wx.FR} =\left(V_{x}+\frac{\dot{\psi}\,t_{f}}{2}\right)\cos(\delta)+\left(V _{x}+\dot{\psi}\,l_{f}\right)\sin(\delta)\] (4) \[V_{wx.RL} =\left(V_{x}-\frac{\dot{\psi}\,t_{r}}{2}\right)\] (5) \[V_{wx.RR} =\left(V_{x}+\frac{\dot{\psi}\,t_{r}}{2}\right) \tag{6}\]  
Considering the longitudinal and lateral acceleration of the vehicle, \(a_{x}\) and \(a_{y}\), the vertical load of each wheel \(F_{z,i}\) can be calculated as follows:  
\[F_{z.FL} =m_{v}\bigg{[}g\frac{l_{r}}{2l}-a_{x}\frac{h_{c}}{2l}-a_{y}\frac{h_{c}l_{ r}}{t_{r}l}\bigg{]} \tag{7}\] \[F_{z.FR} =m_{v}\bigg{[}g\frac{l_{r}}{2l}-a_{x}\frac{h_{c}}{2l}+a_{y}\frac{ h_{c}l_{r}}{t_{r}l}\bigg{]}\] (8) \[F_{z.RL} =m_{v}\bigg{[}g\frac{l_{f}}{2l}+a_{x}\frac{h_{c}}{2l}-a_{y}\frac{ h_{c}l_{f}}{t_{f}l}\bigg{]}\] (9) \[F_{z.RR} =m_{v}\bigg{[}g\frac{l_{f}}{2l}+a_{x}\frac{h_{c}}{2l}+a_{y}\frac{ h_{c}l_{f}}{t_{f}l}\bigg{]} \tag{10}\]  
where, \(m_{v}\) is the total vehicle mass and \(g\) is the gravity acceleration constant. Moreover, \(l=l_{f}+l_{r}\) is the wheelbase. All calculations presented here will be used for road friction and brake force estimations.