# Sliding Mode-Based Slip Control of Compact Electric Vehicle Truck for Varying Load and Yaw Rate
## 4 Analysis of Conventional Desired Yaw Rate Method  
At cornering, lateral force and slip angle are generated in the tire, and the cornering stiffness is expressed as the ratio of these two, which is constant up to a certain point. In addition, it also changes depending on the load and the radius of the tire [20], as well as according to the air pressure set in the vehicle [16]. The cornering stiffness described above can be calculated only when the slip angle and the lateral force can be empirically obtained. In particular, since the tire slip angles are calculated using an estimator, a controller with high performance and fast reaction speed is required for vehicle slip control. [PERSON] et al. proposed a method of calculating cornering stiffness using the Magic formula [6]. To compare the characteristics of vehicles using the formula, the understeer gradient values  
Figure 7: VMUG test results divided into two parts: (**a**) quadratic part of the vehicle cargo load below 350 kg; and (**b**) linear section with vehicle cargo load of 350 kg or more. used in previous methods were summarized and compared. The nominal understeer gradient values for design purposes can be expressed as the ratio of load \(W\) and the tire's cornering stiffness \(C\) on each axis. \[K_{us}=\frac{W_{f}}{C_{\alpha f}}-\frac{W_{r}}{C_{\alpha r}} \tag{17}\]  
For a detailed analysis, a comprehensive value considering the nominal understeer gradient, lateral load transfer, camber change, lateral force compliance, roll steer, and steering system compliance steer, is used as follows. \[K_{us}=K_{us0}+K_{LLT}+K_{Camber}+K_{LFCS}+K_{STRG}+K_{AT} \tag{18}\]  
In this study, our aim is to obtain an accurate value of the nominal understeer gradient for comparison using simulations, which requires calculating the cornering stiffness in real time. This can be achieved by rearranging it as follows. \[C_{\alpha f}=\frac{F_{yf}}{2\Big{(}\delta-\theta_{\alpha f}\Big{)}},\ \ C_{\alpha r}=\frac{F_{yr}}{2(\delta-\theta_{\alpha r})},\ \ \theta_{\alpha f}=\frac{v_{y}+l_{f}\gamma}{v_{x}},\ \ \theta_{\alpha r}=\frac{v_{y}+l_{r}\gamma}{v_{x}} \tag{19}\]  
However, when the wheel steer angle \(\theta_{v}\) is close to the steering angle \(\delta\) due to rapid steering of the vehicle, a notch is generated rendering the control impossible. Therefore, it is necessary to use optimal cornering stiffness, and in a number of studies, this is assumed to be constant. Alternately, a robust controller considering the nonlinearity is designed to respond to changes in cornering stiffness. In this study, the conventional constant cornering stiffness that can accurately indicate the desired yaw rate was used as a reference for comparison with VMUG. Another method of obtaining cornering stiffness is to use the Magic formula. The understeer gradient simply expressed using the cornering stiffness equation is shown as follows. \[C_{f\_Magic}=c_{1}\sin\left\{2\arctan\left(\frac{F_{zf}}{c_{2}}\right)\right\},\ \ C_{f\_Magic}=c_{1}\sin\left\{2\arctan\left(\frac{F_{zr}}{c_{2}}\right)\right\} \tag{20}\]  
The terms \(c_{1}\) and \(c_{2}\) depend on the vehicle characteristics. From Figure 8, for general road surfaces with a low-speed driving at 60 km/h or less, it can be seen that the desired yaw rate value based on the constant cornering stiffness or Magic formula is obtained without a significant error. For vigorous vehicle movement, setting the accurate desired yaw rate is difficult in practice. In Figure 8b, it can be seen that a notch occurs for high speed and low friction double lane change with the conventional estimation method, indicating that a proper control is not possible. It can be seen that for dynamic vehicle movement, the control is difficult if an appropriate desired yaw rate is not selected. When the target vehicle is a truck, the characteristics of the vehicle movement show more dynamic changes, as not only the total weight of the vehicle varies depending on the cargo weight, but the load applied to the center of mass and the vehicle shaft changes as well.