## 2 Design of Path Tracking Controller with Constraints on Tire Slip Angles
### Control Allocation for LQR and MPC with IC#3  
As shown in Equation (10), \(\Delta M_{z}\) is calculated from \(IC\)#3. After the controllers calculate \(\Delta M_{z}\), it should be converted into the longitudinal forces applied to wheels. This force is generated by various combinations of the actuators, RWS, 4 WIB, and 4 WID. Typically, this conversion is accomplished with a process known as control allocation. In this paper, a weighted least square (WLS)-based method is employed for control allocation, which has been adopted in previous works [31; 32]. Figure 4 illustrates the tire forces at each wheel and \(\Delta M_{z}\) when \(\Delta M_{z}\) is positive [25; 30; 31; 32; 35; 39; 40]. In Figure 4, four wheels are numbered as 1, 2, 3 and 4 in the order of the front left, front right, rear left and rear right wheels, respectively. In Figure 4, \(\Delta F_{yr}\) represents \(F_{y}\) generated by the rear wheels through RWS. \(\Delta F_{x1}\), \(\Delta F_{x2}\), \(\Delta F_{x3}\) and \(\Delta F_{x4}\) denote \(F_{x}\)'s generated by 4 WID and 4 WIB. If \(\Delta F_{xi}\) is positive, it corresponds to \(T_{Di}\) generated by 4 WID; otherwise, it corresponds to \(T_{Bi}\) generated by 4 WIB. Those five tire forces need to be determined so as to generate \(\Delta M_{z}\). For the purpose, WLS-based method is selected as a control allocation method. Figure 4: Geometric information and coordinate system of tire forces and \(\Delta M_{z}\). The equilibrium condition among \(\Delta M_{z}\) and tire forces is derived as Equation (29) from the geometric information depicted in Figure 4[36; 39; 43; 46]. The components of vector \(\mathbf{z}\) are given in Equation (30). \[\underbrace{ \begin{bmatrix}z_{1}(t)&z_{2}(t)&z_{3}(t)&z_{4}(t)&z_{5}(t)\\ &\mathbf{z}(t)&\mathbf{z}(t)&z_{5}(t)\\ \end{bmatrix} }_{\mathbf{z}(t)}\underbrace{ \begin{bmatrix}\Delta F_{\pi r}(t) \\ \Delta F_{\mathrm{z}1}(t)\\ \Delta F_{\mathrm{z}2}(t)\\ \Delta F_{\mathrm{z}3}(t)\\ \Delta F_{\mathrm{z}4}(t)\end{bmatrix} }_{\mathbf{q}(t)}=\mathbf{z}(t) \mathbf{q}(t)=\Delta M_{z}(t) \tag{29}\]  
\[\left\{ \begin{array}{l}z_{1}(t)=2l_{r}\cos\delta_{r}(t),\\ z_{2}(t)=-l_{f}\sin\delta_{f}(t)+t_{f}\cos\delta_{f}(t),\;z_{3}(t)=-l_{f}\sin \delta_{f}(t)-t_{f}\cos\delta_{f}(t),\\ z_{4}(t)=l_{r}\sin\delta_{r}(t)+t_{r}\cos\delta_{r}(t),\;\;\;z_{5}(t)=l_{r}\sin \delta_{r}(t)-t_{r}\cos\delta_{r}(t)\end{array} \right. \tag{30}\]  
The quadratic cost function for WLS is given as Equation (31). Within Equation (31), \(\zeta_{i}\) is the product of \(\mu\) and \(F_{zi}\) at the wheel \(i\), i.e., \(\mu F_{zi}\), which represents the friction circle radius. The vector of virtual weights, denoted as \(\kappa\), is introduced in Equation (31) for the purpose of selecting the combination of actuators [25; 30; 31; 32; 35; 39]. The quadratic cost function, \(J_{EC}\) in Equation (32), is obtained from the constraint Equation (29). In previous research, the equilibrium condition Equation (29) should be satisfied for the purpose of generating \(\Delta M_{z}\). The cost functions, Equations (31) and (32), are summed into a single one (33) with the Lagrange multiplier, \(\eta\). In Equation (33), it is important to set \(\eta\) to a value of 1 or higher; otherwise, the equilibrium condition Equation (29) will not be met. The optimal solution for Equation (33) is algebraically calculated as Equation (34) by taking the derivative of Equation (33) with respect to \(\mathbf{q}\)[31; 32].