# Sliding Mode-Based Slip Control of Compact Electric Vehicle Truck for Varying Load and Yaw Rate
## 3 Proposed VMUG Strategy for Desired Yaw Rate  
In this section, reference is set for control of the desired yaw rate based on the steady states. There are two ways to set the desired yaw rate: a mathematical method and an experimental method [3]. In the mathematical method, the desired yaw rate can be expressed as the value of the velocity according to the road radius. The steering angle and the steady-state angle under the assumption of steady-state circle driving are given as follows.  
\[\gamma_{d}=\frac{V}{R} \tag{6}\]  
\[\delta_{ss}=\frac{l_{f}+l_{r}}{R}+Ka_{y} \tag{7}\]  
By rearranging with reference to the understeer gradient \(K\) and using the lateral acceleration \(a_{y}=V^{2}/R\), the desired yaw rate can be obtained as follows.  
\[\gamma_{d}=\frac{\delta}{L+V^{2}K}V \tag{8}\]  
In this study, both the experimental and mathematical methods were implemented to minimize the disadvantages of the conventional desired yaw rate based on mathematical formulation. For the experimental method, the ISO 4138 Steady State Circle test, as shown in Figure 4, was applied. Within a radius of 100 m, the lateral acceleration was increased from 0 to 0.9 g in steps of 0.3 g, the understeer gradient in steady-state section was formulated, and this was used in the equation of the desired yaw rate. The vehicle was designed as a small truck with a maximum cargo weight of 600 kg. As the value of the understeer gradient changes according to the cargo weight, the desired yaw rate also changes accordingly. The desired yaw rate using constant cornering stiffness and the yaw rate changing according to the applied load are shown in Figure 5. The yaw rate data can be obtained by increasing the cargo weight from 0 to 600 kg with the center of mass at (315 mm, 2000 mm). The dimension of the cargo is set to 0.516 \(\times\) 0.816 \(\times\) 0.816 m, and the mass inertia of the cargo is given in Table 2.  
Figure 4: ISO-4138 Steady state test path: 100 m radius circle.  
Furthermore, for control in case of overloading beyond the allowable limit, information on the desired yaw rate at the load is required. However, in the case of 40% overloading, simulation was not possible through the Steady State Circle test. Therefore, by mathematically formulating the understeer gradient that changes depending on the cargo weight, the expected understeer gradient value at overloading is estimated.  
Understeer gradient values according to lateral acceleration and cargo mass are shown in Figure 6. For lateral acceleration of 0.39 g or more, the simulation did not work properly. However, two trends of the understeer gradient could be confirmed. The proposed yaw rate strategy was divided into a linear form section and a quadratic form section through an empirical method for division according to load and lateral acceleration. Each section was formulated by polynomial fitting as shown in Figure 7.  
The section for cargo weight up to 350 kg is in quadratic form, and it is rearranged for lateral acceleration from 0.06 to 0.42 g as follows.  
\[UG_{\text{Q}}=A+B_{1}a_{y}+B_{2}a_{y}^{2} \tag{9}\]  
Rearranging the above gives the following:  
\[UG_{\text{Q}}=f_{A}(m)+f_{\text{B1}}(m)a_{y}+f_{\text{B2}}(m)a_{y}^{2} \tag{10}\]  
\[f_{A}(m)=(1.2+1.04m)10^{-3} \tag{11}\]  
\[f_{\text{B1}}(m)=(1.5924+1.3715m)10^{-3} \tag{12}\]  
\[f_{\text{B2}}(m)=(-3.9632-4.2925m)10^{-3} \tag{13}\]  
The section for cargo weight of 350 to 600 kg is in a linear form, and it is rearranged for lateral acceleration from 0.18 to 0.39 g as follows.  
\[UG_{L}=f\big{(}a_{y},m_{\text{argo}}\big{)}=c_{1}f\big{(}a_{y}\big{)}m_{\text {argo}}+c_{2} \tag{14}\]  
\[UG_{L}=\frac{1}{10^{6}}\big{(}3.0458a_{y}-3.6756\big{)}m_{\text{argo}}+0.0025 \tag{15}\]  
The VMUG equation, including cargo weight, can be rearranged and expressed as follows.  
\[VMUG=\left\{\begin{array}{l}f_{A}(m)+f_{B1}(m)a_{y}+f_{B2}(m)a_{y}^{2}\qquad (m_{\text{argo}}<350\text{kg})\\ \frac{1}{10^{6}}\big{(}3.0458a_{y}-3.6756\big{)}m_{\text{argo}}+0.0025\qquad (m_{\text{argo}}\geq 350\text{kg})\end{array}\right. \tag{16}\]