## 1 Introduction
Moreover, \(F_{y}\) is reduced as \(\mu\) decreases, as shown in Figure 1. On a low-friction road surface, \(F_{y}\) is more easily saturated than on high-friction one. For the above two reasons, a path tracking controller, designed on a \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Controller** & **Input Configuration** & **Actuators** & \(\mu\) & **Reference** \\ \hline LQR, MPC, SMC & [\(\delta_{f}\)], [\(\delta_{f}\)], \(\delta_{r}\)] & FWS, 4 WS & 0.4 & [30] \\ \hline \multirow{4}{*}{MPC} & \multirow{4}{*}{[\(\delta_{f}\)]} & & 0.4, 0.5 & [12] \\  & & & 0.2, 0.5 & [14] \\  & & FWS & 0.3, 0.8 & [15] \\  & & & 0.2, 0.85 & [21] \\  & & & 0.3 & [22] \\ \hline \multirow{4}{*}{MPC} & \multirow{4}{*}{[\(\delta_{f}\)]} & FWS, 4 WID & 0.4, 0.5 & [11] \\  & & FWS, 4 WID, 4 WIB & 0.6, 0.7 & [16] \\  & & FWS, 4 WID, 4 WIB & 0.4, 0.8 & [17] \\  & & FWS, 4 WIB & 0.3 & [28] \\ \hline \multirow{4}{*}{MPC} & \multirow{4}{*}{[\(\delta_{f}\)]} & FWS, 4 WID & 0.4,0.7 & [19] \\  & & FWS, 4 WID & 0.3 & [20] \\  & & FWS, 4 WID & 0.3 & [23] \\  & & FWS, 4 WID & 0.5, 0.6 & [29] \\ \hline \multirow{4}{*}{LQR} & \multirow{4}{*}{[\(\delta_{f}\)]} & 4 WS, 4 WIS & 0.4, 0.65 & [13] \\  & & 4 WS & 0.25, 0.5 & [18] \\  & & FWS & 0.34 & [24] \\ \hline \multirow{4}{*}{LQR} & \multirow{4}{*}{[\(\delta_{f}\)], [\(\delta_{f}\)], [\(\delta_{f}\)], [\(\delta_{f}\)]} & FWS, 4 WS, 4 WIB, 4 WID & 0.4 & [31; 32] \\  & & & FWS & 0.5 & [26] \\  & & FWS, 4 WID, 4 WIB & 0.3 & [27] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of path tracking controllers on low friction roads. high-friction road surface, shows a poor performance on a low-friction one. To overcome this problem, it is necessary to design a path tracking controller with the consideration of low-friction road surface and lateral tire force saturation. In this paper, a constraint on the tire slip angle (CTSA) is derived from the definition of \(\alpha\), which results in the bound constraint on steering angles. Research on vehicle motion control under low-friction road conditions has been performed in the field of vehicle stability control (VSC). In this paper, VSC means a control which tries to make a vehicle track a reference yaw rate generated by a steering angle, and lateral stability control (LSC) means a control which tries to keep the side-slip angle, \(\beta\), as small as possible [34; 35]. To date, a lot of papers have been published in the field of VSC and LSC [36; 37; 38]. Among the literature, some papers have been published to limit the tire slip angle such that \(F_{y}\) is not saturated or \(F_{y}\) is maintained at its maximum on low-friction road surface [39; 40; 41; 42; 43; 44; 45]. These papers can be classified into two categories. The first approach is to limit the slip angle of the front wheel when \(\alpha\) is over \(\alpha_{m}\) at control allocation stage [39; 40]. If \(F_{y}\) reaches its maximum, then the optimization variable \(F_{y}\) of front wheel is set to the constant \(F_{y,max}\), and an extra optimization problem is formulated with remaining variables and modified cost function. Generally, quadratic programming (QP) has been adopted for control allocation. In the research, QP is applied twice so as to limit \(\alpha\). Another method is to use a daisy chaining method (DCM) for control allocation, which is a cascade optimization under element-wise constraint [41]. The second approach is to use an extra controller for the purpose of making \(\alpha\) be \(\alpha_{m}\)[42; 43; 44; 45]. In the research, various methods have been applied to keep \(F_{y}\) at its maximum so as to generate the maximum lateral acceleration. For example, some compensation coefficients were adopted to tune the magnitude of \(F_{y,max}\). However, in these research projects, \(\alpha\) was not directly manipulated by a method. Different from the previous two research projects, CTSAs are set as the bound on steering angles in this paper.