# Enhancement of Yaw Moment Control for Drivers with Excessive Steering in Emergency Lane Changes
## 5 Simulations
### Case I: The Integrated Control  
In this case, the vehicle performance under the integration control, (25), (27), and (51), is simulated. Figure 7a depicts the responses of the vehicle speed \(V_{0}\) (in black) and wheel speeds \(\omega_{j}\); Figure 7b shows the responses of the restricted front-wheel steering angle \(\delta_{f}\) (in black) and the driver steering angle \(\delta_{d}\) (in brown), along with the saturation angle \(\delta_{sat}\) (in blue) and the steering angle limit \(\tilde{\delta}_{\lim}\) (in gray). We can observe that when the vehicle speed \(V_{0}\) changes, the steering angle limit \(\tilde{\delta}_{\lim}\) is correspondingly updated. The saturation angle \(\delta_{sat}\), which mainly acts in the periods of [0.5, 1.8] and [2.1, 3.5] in seconds, can effectively make the wheel steering angle \(\delta_{f}\) behave as \(\delta_{f}\cong\delta_{d,ref}\). Figure 7c demonstrates that the yaw moment control \(M_{z}\) mostly takes place in the intervals around 0.5 s, 2 s, and 3.5 s; during these intervals, the rapid changes in the driver steering angle \(\delta_{d}\) can be seen in Figure 7b. It can be observed that the two control schemes \(\delta_{sat}\) and \(M_{z}\) carry out different tasks, that is to say, the saturation angle \(\delta_{sat}\) serves to prevent the wheel steering angle \(\delta_{f}\) from developing the unmatching disturbance \((\upsilon,\xi)\) that could have an instability effect on the vehicle in the periods of [0.5, 1.8] and [2.1, 3.5] in seconds, while the yaw moment \(M_{z}\) mainly performs the reference-following operation in the intervals around 0.5 s, 2 s, and 3.5 s. Figure 7d shows the history of the distributed wheel control torques \(T_{j}\). Figure 7e,f demonstrates that the vehicle state \((\beta,\gamma)\) remains near their reference \((\beta_{ref},\gamma_{ref})\), meaning the integration control helps to maintain the safety reference following the situation of a driver's excessive steering. The resulting vehicle acceleration \((a_{x},a_{y})\) in Figure 7g shows the lateral acceleration \(a_{y}\) operates within the safety range \(\left|a_{y}\right|\leq\widetilde{a}_{y,\text{lim}}=0.68\) g during the emergency lane change. Figure 7h depicts the responses of the combined wheel slip magnitudes \(\left\|\mathbf{S}_{j}\right\|=\sqrt{s_{Lj}^{2}+s_{Sj}^{2}}\) of the four wheels, and Figure 7i-l shows the histories of the combined wheel slips \(\mathbf{S}_{j}=\left[s_{Lj},s_{Sj}\right]^{\mathsf{T}}\) of the four wheels. We can see that all the four combined wheel slips \(\mathbf{S}_{j}\) lie in the linear region with \(\left\|\mathbf{S}_{j}\right\|\leq\widetilde{s}_{Res}\approx 0.08\) (wet asphalt).  
Figure 7: Responses with the integration control (25), (27), and (51); **(a)** the vehicle speed \(V_{0}\) and wheel speeds \(\omega_{j}\); **(b)** the steering angles \(\delta_{d}\delta_{st}\),\(\delta_{f}\) and the limit \(\widetilde{\delta}_{\text{lim}}\); **(c)** the yaw moment control \(M_{z}\); **(d)** wheel control torques \(T_{j}\); **(e)** the sideslip angle \(\beta\) and its reference \(\beta_{ref}\); **(f)** the yaw rate \(\gamma\) and its reference \(\gamma_{ref}\); **(g)** the longitudinal and lateral accelerations \((a_{x},a_{y})\); **(h)** the combined slip magnitude \(\left\|\mathbf{S}_{j}\right\|\); **(i)** the combined wheel slip \((s_{L1},s_{S1})\) of wheel 1 (front left); **(j)** the combined wheel slip \((s_{L2},s_{S2})\) of wheel 2 (front right); **(k)** the combined wheel slip \((s_{L3},s_{S3})\) of wheel 3 (rear left); **(l)** the combined wheel slip \((s_{L4},s_{S4})\) of wheel 4 (rear right).