## 4 Simulation and Discussion  
To verify the effects of the CTSA on path tracking performance under low-friction road condition, a simulation was conducted. The path tracking controllers, LQRN, LQRC, MPCN and MPCC, were implemented with MATLAB/Simulink R2019a, connected to CarSim [52]. Test scenario was given in Figure 5. In the simulation, F-segment sedan model in CarSim was selected as a test vehicle [52]. From simulation with the model under step steering or steady-state cornering, the parameters and its values of the 2-DOF bicycle model were obtained as given in the reference [25; 30; 31; 32]. In this paper, a first-order system was adopted to model actuators such as FWS and 4 WS, and 4 WIB and 4 WID, where the bandwidths of these actuators are set to 5 Hz and 2 Hz, respectively. The initial \(v_{x}\) and \(\mu\) were set to 60 km/h and 0.4, respectively. For the purpose of maintaining \(v_{x}\) as constant as possible, a built-in speed controller provided in CarSim was applied. The simulation horizon was set to 15 s.  
Generally, the larger \(\delta_{r}\), the larger \(\beta\). For the reason, it is necessary to limit \(\Delta M_{z}\) to a certain value so as to make \(\delta_{r}\) small for \(IC\#3\). \(F_{y,max}\) from F-segment sedan model in CarSim is 7500 N where \(\mu\) is 1 [52]. This will be 3000 N if \(\mu\) is set to 0.4. As shown in Equation (29), \(F_{y}\) at front wheels, i.e., \(\Delta F_{yf}\), was not used to generate \(\Delta M_{z}\) because \(\delta_{f}\) is included in the control input. From the geometric information of the test vehicle, the maximum available yaw moment at rear wheels is calculated as 3000 \(\times\) 2 \(\times\)\(l_{r}\) = 11,400 Nm. After \(\Delta M_{z}\) was obtained from LQR with \(IC\#3\), it is limited by clipping it into 11,400 Nm in this paper. \(\delta_{max}\) of front and rear wheels was set to \(30^{\circ}\). \(\alpha_{m}\) was set to \(5^{\circ}\), which is referred from CarSim tire model as given in Figure 1.  
The first simulation is done for LQRN and LQRC. Tables 2 and 3 show the simulation results of LQRN and LQRC. As mentioned earlier, the control parameters were tuned such that \(M_{Y}\) is larger than \(-0.02\) m and \(M_{OS}\) is near 0.9. These two measures are nearly the same as shown in Tables 2 and 3. For the reason, these are not used for comparison. The percentages inside the parentheses in Table 3 represent the percentage reductions of \(M_{X}\), \(M_{DX}\) and \(M_{SX}\) relative to the corresponding values given in Table 2 for each actuator combination. As shown in those tables, \(M_{X}\), \(M_{DX}\) and \(M_{SX}\) of LQRC are improved, compared to those of LQRN. This indicates that the CTSA can effectively limit \(\alpha\) to \(\alpha_{m}\), which generates \(F_{y,max}\). In case of LQRN, the actuator combination, FWS + 4 WIB of \(IC\#3\), shows the best performance in terms of \(M_{DX}\) and \(M_{SX}\). On the contrary, in case of LQRC, the actuator combination, FWS + RWS of \(IC\#3\), shows the best performance in terms of \(M_{X}\), \(M_{DX}\) and \(M_{SX}\). Moreover, this combination is superior to \(IC\#2\) of LQRC, which used the identical actuators, FWS and RWS. This means that any actuator combinations using 4 WID and 4 WIB are inferior to FWS + RWS of \(IC\#3\). The input configurations, \(IC\#1\) and \(IC\#2\), of LQR do not require a control allocation. In \(IC\#1\) and \(IC\#2\), \(\delta_{f}\) and \(\delta_{r}\), are directly computed and applied into steering actuators. However, \(IC\#3\) of LQR requires a control allocation, which is complex and time-consuming. Due to its simplicity, \(IC\#1\) and \(IC\#2\) of LQR are recommended if one do not think the difference between _IC#1_/_IC#2_ and _IC#3_ is significant. From these results, it can be concluded that CTSA is strongly recommended for better path tracking.