## 5 Simulation Results
### Closed-Loop Test  
The double-lane change maneuver is used in the closed-loop test to show the effectiveness of the proposed DYC system with human driver model in the loop. A 1 st-order preview-predictive driver steering model is used to conduct the lane change maneuver, see Figure 13. The driver model in lane tracking control can be expressed as follows  
\[\delta_{sw}=\frac{h_{d}}{T_{r}s+1}\big{[}y_{OL}-\big{(}y_{c}+T_{p}V\psi\big{)} \big{]} \tag{36}\]  
where, \(h_{d}\) denotes the driver corrective steering gain, \(T_{r}\) denotes the driver model steering delay time constant, \(y_{OL}\) denotes the desired preview lateral displacement, \(y_{c}\) denotes the current vehicle lateral displacement, \(T_{p}\) denotes the driver model predictive time, and \(\psi\) denotes the current vehicle yaw angle with respect to the desired lane. The driver model with the control law (36) is used to track a target course. The target course can be expressed as the following equations, and the plot of the course is shown in Figure 14.  
\[y= \begin{cases}&\frac{35}{2}\big{[}1+\tanh\big{\{}\frac{2}{30}\pi\big{(}x-X_{1 }-\frac{30}{2}\big{)}\big{\}}\big{]}(x<X_{1}+42.5)\\ &\frac{35}{2}\big{[}1-\tanh\big{\{}\frac{2}{25}\pi\big{(}x-X_{2}-\frac{25}{2} \big{)}\big{\}}\big{]}(x\geq X_{1}+42.5)\end{cases}  \tag{37}\]  
where, \(X_{1}\) and \(X_{2}\) are the positions where the lane change is conducted at the first half and the second half, respectively. Here, in the simulations, \(X_{1}\) is set to 35 m and \(X_{2}\) is set to 90 m.  
The simulations are conducted in same three conditions as in the previous section. The loading conditions, control conditions, and the parameters of the driver model are summarized in Table 3. The vehicle velocity is 80 km/h constant. The driver model parameters are set in the way that the resulted vehicle trajectory of all three conditions are almost the same, considering the adaptation behavior of the driver according to McRuer crossover model [27]. The steering gain, \(h_{d}\), and the driver steering delay time constant, \(T_{r}\), are set to smaller values in the case of loaded vehicle without DYC. According to [PERSON]'s findings [25], smaller values of driver steering delay time constant mean that the driver has to control the vehicle along the target course in a more stressful manner corresponding to the lower handling quality evaluation. Figure 13: 1 st-order preview-predictive lateral control driver model. The time history of the steering wheel angle, yaw rate, body side slip angle, lateral acceleration, and yaw moment control input are shown in Figure 15. The vehicle trajectory and Lissajous diagram of the steering wheel angle and yaw rate are shown in Figures 16 and 17, respectively. As can be noticed from Figure 15, in the case of the loaded vehicle without DYC, the overall amount of steering wheel angle required for the maneuver becomes larger, the stability of the vehicle becomes worse and, as seen in the previous section, the damping behavior of the vehicle is also worse. This implies the difficulty in vehicle course tracking control in the case of loaded vehicle. On the other hand, the proposed DYC can keep the vehicle behavior (Figure 15) as well as the vehicle trajectory, as shown in Figure 16, during the double-lane change maneuver to be almost the same as the unloaded vehicle (baseline condition), which means that the vehicle with DYC is insensitive to the loading condition. Finally, as can be noticed from the Lissajous diagram in Figure 17, in the case of loaded vehicle without DYC, a large steering wheel angle is needed to track the course. When compared with the case of unloaded vehicle and the case of loaded vehicle with DYC, the residual yaw rate of loaded vehicle without DYC when the steering wheel angle is back to the neutral position is large and there are many large whirls in the diagram of loaded vehicle without DYC, which corresponds to more difficulty in controlling the vehicle to track the target double-lane change course.