Design of Static Output Feedback Integrated Path Tracking Controller for Autonomous Vehicles

Abstract

This paper presents a method for designing a static output feedback integrated path tracking controller for autonomous vehicles. For path tracking, state–space model-based control methods, such as linear quadratic regulator, H_∞ control, sliding mode control, and model predictive control, have been selected as controller design methodologies. However, these methods adopt full-state feedback. Among the state variables, the lateral velocity, or the side-slip angle, is hard to measure in real vehicles. To cope with this problem, it is desirable to use a state estimator or static output feedback (SOF) control. In this paper, an SOF control is selected as the controller structure. To design the SOF controller, a linear quadratic optimal control and sliding mode control are adopted as controller design methodologies. Front wheel steering (FWS), rear wheel steering (RWS), four-wheel steering (4WS), four-wheel independent braking (4WIB), and driving (4WID) are adopted as actuators for path tracking and integrated as several actuator configurations. For better performance, a lookahead or preview function is introduced into the state–space model built for path tracking. To verify the performance of the SOF path tracking controller, simulations are conducted on vehicle simulation software. From the simulation results, it is shown that the SOF path tracking controller presented in this paper is effective for path tracking with limited sensor outputs.

1. Introduction

Since 2010, autonomous driving has been intensively studied by academia and the automotive industry due to its impacts on enhancing traffic flow and road safety [1,2,3]. In terms of road safety, it is expected that autonomous driving can decrease social problems caused by traffic accidents or car crashes. As a result, various research papers related to autonomous driving have been published to date. Autonomous driving architecture consists of perception/detection, planning, and control [3]. Among them, path tracking is used to control actuators to make a vehicle follow a target path or reference states. This paper focuses on path tracking for autonomous vehicles. In the meantime, various studies on path tracking have been conducted [4,5,6,7,8,9,10]. As a result, a huge number of papers have been published in the field of path tracking to date.

Until recently, most of the path tracking for autonomous vehicles has been proposed in the framework of state–space model and full-state feedback control. In terms of the state–space model, there are two types of state–space models. The first uses the lateral offset error, e_y, the heading error, e_ψ, and its derivatives [11,12,13,14,15,16,17]. This has been used for pure path tracking control. The second uses e_y, e_ψ, the side-slip angle, β, and the yaw rate, γ [18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38]. This has been used for path tracking, integrated with yaw-rate tracking (YRT), the vehicle stability control (VSC), or the lateral stability control (LSC). In the literature, YRT or the VSC are used to control a vehicle with independent braking for the purpose of making a vehicle follow a target yaw rate, while the LSC is used to control a vehicle with several actuators in order to maintain β as small as possible. The relationship between these two state–space models can be found in the references [13,22]. In this paper, the second type is selected because yaw-rate tracking and lateral stability are important in path tracking for collision avoidance and for driving on low-friction surfaces [21,22,24,25,28,32,33,36,37]. As a result, there are many papers on integrated path tracking and lateral stability control.

When using the state–space model with the state variables (e_y, e_ψ, β, γ), a state estimator on the lateral velocity v_y or the side-slip β at the center of gravity (CoG) is required. However, it is not easy to estimate β in real vehicles. Another approach is to use a static output feedback (SOF) control instead of a full-state one [39,40,41,42,43,44,45,46]. Following the ideas presented in the references, a SOF control is selected as a control structure, simplifying the problem by using only the available sensor outputs, without the use of any estimators, in this paper. Two sets of sensor outputs are selected for the SOF control in this paper. The first set uses (e_y, e_ψ) as sensor outputs [44]. The second set uses (e_y, e_ψ, γ) as sensor outputs because γ can be easily measured with an onboard sensor in real vehicles.

Regarding the controller design methodology, a linear quadratic regulator (LQR), H_∞ control, sliding mode control (SMC), model predictive control (MPC), backstepping, and adaptive control have been applied for the controller design. The LQR is simple and systematic when designing a path tracking controller [11,13,14,16,26,36,38]. The MPC has been widely used for path tracking since 2010 [15,16,17,19,20,22,24,25,27,28,29,30,31,32,33,34,35,47,48]. Compared to the LQR and MPC, there are a smaller number of papers on the SMC for path tracking [12,17,18,20,21,44]. Moreover, there are few papers on the SMC with the state–space model on path tracking. In this paper, the SOF controllers are designed with the LQ optimal control (LQOC) and SMC. In other words, the LQ SOF and SOF SMC controllers are designed in this paper. In the case of the MPC, it is impossible to use the MPC with an SOF control because the initial state variables for the MPC cannot be obtained from SOF. Moreover, it was shown that the path tracking performance of the MPC is nearly identical to that of the LQR [17].

From the perspective of actuators for path tracking, front wheel steering (FWS) has been primarily used to date. Recently, most papers have used FWS with other braking or driving actuators. Rear wheel steering (RWS) or four-wheel steering (4WS) has also been used for path tracking [12,14,16,17,18,19,20,23,26,36,37,38,44]. Since the mid-1990s, due to the development of electronic stability control (ESC), i.e., YRT or the VSC, 4WIB or differential braking has been installed in real vehicles. Since 2010, technological advances in electric vehicles (EVs) using multiple driving motors or in-wheel motors, 4WID, or torque vectoring function (TVF) have been selected as actuators for path tracking. As a result, 4WIB and 4WID can be used as actuators for integrated VSC and path tracking [17,36]. Among the papers using FWS for path tracking, most of them have used FWS with 4WIB or 4WID [11,15,19,20,21,22,23,24,27,28,29,30,32,33,34,35,36,37,44,47]. In this paper, it is assumed that a vehicle has FWS, RWS, 4WIB, and 4WIB because it has an in-wheel motor, and that three input configurations consisting of FWS, RWS, 4WS, 4WIB, and 4WID are selected for path tracking [37,38,39].

Generally, the steering actuators, such as FWS, RWS, and 4WS, can be modeled as a control input for the LQR [17,36,37,38]. The control inputs for FWS, RWS, and 4WS are the front steering angle δ_f, the rear steering angle δ_r, and combinations of them, i.e., 4WS. To use 4WIB and 4WID as actuators, it is necessary to include the additional control input, control yaw moment, ΔM_z, into the control input of the LQ SOF and SOF SMC controllers. In this paper, the steering angle of the RWS and the braking and tractive torque of 4WIB and 4WID are derived from ΔM_z. After calculating ΔM_z, it is derived from longitudinal and lateral control tire forces. This tire force distribution is achieved by a control allocation method [15,17,20,21,22,23,24,28,29,33,35,36,37,38,44,47]. By applying a control allocation method to ΔM_z, the rear steering angle of the RWS or the braking and driving tire forces (T_B/T_D) of 4WIB and 4WID can be determined. Because several actuators are selectively used for path tracking control, the path tracking controllers presented in this paper are called integrated ones [17,36]. With three input configurations (IPCs) consisting of FWS, RWS, 4WS, 4WIB, and 4WID, as IPC#1, IPC#2, and IPC#3, this paper will analyze the effects of those IPCs on path tracking performance. When using the steering actuators, FWS, RWS, and 4WS, the constraints related to the tire slip angle are imposed on the steering angles of those actuators in order to maximize the lateral tire force [37].

The goal of this paper is to present a method for designing an integrated path tracking controller with an SOF control. Figure 1 shows the schematic diagram of the SOF controllers and the three IPCs for path tracking presented in this paper. As shown in Figure 1, two sets of sensor outputs, i.e., (e_y, e_ψ) and (e_y, e_ψ, γ), are selected for the SOF control. The state–space model with the state variables (e_y, e_ψ, β, γ) is derived from a 2DOF bicycle model and a target path. Using the state–space model, the SOF controllers are designed. The LQOC and SMC were selected as controller design methodologies. In the SOF controllers, three input configurations, IPC#1, IPC#2, and IPC#3, are composed of δ_f, δ_r, and ΔM_z [36,37]. In Figure 1, T_B and T_D are the braking and driving torques generated by 4WIB and 4WID, respectively. As shown in Figure 1, δ_r, T_B, and T_D are generated from ΔM_z. To generate ΔM_z using RWS, 4WIB, and 4WID in IPC#3, a control allocation method is applied. The impact of the three IPCs, consisting of FWS, RWS, 4WS, 4WIB, and 4WID, on path tracking performance is analyzed in this paper. To verify path tracking performance with the LQ SOF and SOF SMC controllers, simulations were performed using CarSim. From simulation results, it is concluded which is the best controller and which is the best actuator combination, in terms of path tracking.

The key contributions of this paper are summarized as follows:

• An SOF controller that does not use a side-slip angle is designed. Two sets of sensor outputs, (e_y, e_ψ) and (e_y, e_ψ, γ), are composed from the lateral offset error, e_y, heading error, e_ψ, and yaw rate, γ. An LQOC and SMC are applied as control methods.
• Three IPCs and nine actuator combinations, consisting of FWS, RWS, 4WS, 4WIB, and 4WID, are proposed and tested for path tracking and lateral stability. The SOF controllers with the three IPCs are tested on high- and low-friction roads. From the simulation results, the effects of these IPCs on control performance are discussed.
• From the simulation, the best SOF controller, among the LQ SOF and SOF SMC, is determined, as well as the best actuator combination from various options for path tracking.

The design procedures for the LQ SOF controllers and SOF SMC with various IPCs are described in Section 2. In Section 3, a performance measurement method for path tracking is presented for the SOF controllers. Section 4 summarizes the simulation results for each controller and actuator combination. Section 5 concludes this study based on these results.

2. Design of Integrated Path Tracking Control

2.1. Derivation of State–Space Model

For path tracking, a 2-DOF bicycle model was chosen as the vehicle model, as discussed in the literature [12,14,15,16,17,18,19,20,21,22,23,24,25,26,28,30,33,35,36,37,38,48]. Figure 2 outlines the coordinates, geometry, and the parameters of the 2-DOF bicycle model. This model assumes that the longitudinal velocity (v_x) is constant [49,50]. The state variables in the model are γ and β. The equations of motion (1) are established from the free-body diagram in Figure 2 [17,36,37,38,48]. The slip angles, θ_f and θ_r, are calculated in (2) at the front and rear wheels, and the lateral tire forces, F_yf and F_yr, are calculated in (3), assuming that they are linear. From (1), (2), and (3), the linear state–space model is derived in (4).

$$\left\{\begin{array}{l}m{v}_x\left(\dot{\beta }+\gamma \right)=F_{yf}+F_{yr}\\ I_z\dot{\gamma }=l_f{F}_{yf}-l_r{F}_{yr}+\Delta M_z\end{array}\right.$$

(1)

$$\left\{\begin{array}{l}{\theta }_f={\delta }_f-{\tan }^{-1}\left(\beta +{\displaystyle \frac{l_f\gamma }{v_x}}\right)\approx {\delta }_f-\beta -{\displaystyle \frac{l_f\gamma }{v_x}}\\ {\theta }_r={\delta }_r-{\tan }^{-1}\left(\beta -{\displaystyle \frac{l_r\gamma }{v_x}}\right)\approx {\delta }_r-\beta +{\displaystyle \frac{l_r\gamma }{v_x}}\end{array}\right.$$

(2)

$$F_{yf}=C_f{\theta }_f,\hspace{1em}{F}_{yr}=C_r{\theta }_r$$

(3)

$$\begin{array}{l}\dot{\beta }=\left({\displaystyle \frac{{\varsigma }_1}{m{v}_x}}\right)\beta +\left({\displaystyle \frac{{\varsigma }_2}{m{v}_x^2}}-1\right)\gamma +{\displaystyle \frac{C_f}{m{v}_x}}{\delta }_f+{\displaystyle \frac{C_r}{m{v}_x}}{\delta }_r\\ \dot{\gamma }=\left({\displaystyle \frac{{\varsigma }_2}{I_z}}\right)\beta +\left({\displaystyle \frac{{\varsigma }_3}{I_z{v}_x}}\right)\gamma +{\displaystyle \frac{l{}_{f}C_f}{I_z}}{\delta }_f-{\displaystyle \frac{l{}_{r}C_r}{I_z}}{\delta }_r+{\displaystyle \frac{\Delta M_z}{I_z}}\\ {\varsigma }_1=-C_f-C_r,\hspace{0.33em}{\varsigma }_2=-C_f{l}_f+C_r{l}_r,\hspace{0.33em}{\varsigma }_3=-l_f^2{C}_f-l_r^2{C}_r\end{array}$$

(4)

Most of papers in the literature on path tracking have defined and used e_y and e_ψ at point C, the vehicle’s CoG, as depicted in Figure 2. To improve the performance of path tracking, a preview function has been adopted, as proposed in previous studies [12,13,17,21,22,28,36,37,38]. The preview distance, L_p, is defined in (5), where k_v is the velocity gain. In the previous work, without a preview function, the target heading angle is determined at point P. Different from the previous work, the preview point, Q, and point R on the target path are determined by using the preview distance, L_p, as shown in Figure 2. At point R, the target heading angle is defined as ψ_d. With the target heading angle and the heading one at point R, e_ψ is defined as the difference between them. Moreover, at point R, e_y is also defined [12,13,21,22,28]. The derivatives of both e_y and e_ψ are calculated, as described in (6), assuming e_ψ is small, at less than 15°. In (6), χ is the curvature of the target path at point R, which acts as a disturbance.

$$L_p=k_v\cdot v_x$$

(5)

$$\left\{\begin{array}{l}{\dot{e}}_y=v_x\sin (e_{\psi })-v_x\beta -L_p\gamma \approx v_x{e}_{\psi }-v_x\beta -L_p\gamma \\ {\dot{e}}_{\psi }={\dot{\psi }}_d-\dot{\psi }=v_x\chi -\gamma \end{array}\right.$$

(6)

With the state variables e_y, e_ψ, β, and γ, the state vector, x, the disturbance, w, and control input u are defined in (7) [36,37,38]. By combining (4), (6) and (7), the state–space model for path tracking is obtained in (8), where the matrices A, B₁, and B₂ are given in (9). As shown in A and B₂, it is assumed that v_x is constant, which is not realistic. Moreover, the preview distance, L_p, in A is time-varying because it depends on v_x. As a consequence, the state–space equation of (8) is uncertain and time-varying in real situations. To cope with the variation in L_p, an adaptive preview distance scheme was proposed in previous work [38]. The effects of the measurement error and noise in e_y and e_ψ on the path tracking performance were analyzed in reference [9]. Robust control methodologies, such as guaranteed cost control or quadratic stabilization, can be applied to cope with uncertain and time-varying parameters in A and B₂. However, this is out of scope in this paper.

$$x={\left[\begin{array}{cccc}{e}_y& e_{\psi }& \beta & \gamma \end{array}\right]}^T,\hspace{1em}w=\chi ,\hspace{1em}u={\left[\begin{array}{ccc}{\delta }_f& {\delta }_r& \Delta M_z\end{array}\right]}^T$$

(7)

$$\dot{x}=Ax+B_{1}w+B{}_{2}u$$

(8)

$$A=\left[\begin{array}{cccc}0& v_x& -v_x& -L_p\\ 0& 0& 0& -1\\ 0& 0& {\displaystyle \frac{{\varsigma }_1}{m{v}_x}}& {\displaystyle \frac{{\varsigma }_2}{m{v}_x^2}}-1\\ 0& 0& {\displaystyle \frac{{\varsigma }_2}{I_z}}& {\displaystyle \frac{{\varsigma }_3}{I_z{v}_x}}\end{array}\right],\hspace{1em}{B}_1=\left[\begin{array}{c}0\\ v_x\\ 0\\ 0\end{array}\right],\hspace{1em}{B}_2=\left[\begin{array}{lll}0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill \\ {\displaystyle \frac{C_f}{m{v}_x}}\hfill & {\displaystyle \frac{C_r}{m{v}_x}}\hfill & 0\hfill \\ {\displaystyle \frac{l{}_{f}C_f}{I_z}}\hfill & -{\displaystyle \frac{l{}_{r}C_r}{I_z}}\hfill & {\displaystyle \frac{1}{I_z}}\hfill \end{array}\right]$$

(9)

The vector of the control inputs, u, in (7) consists of three elements: δ_f, δ_r, and ΔM_z. These elements are used to define three distinct vectors of the control inputs, u₁, u₂, and u_3, which correspond to the input configurations, IPC#1, IPC#2, and IPC#3, as given in references [36,37]. Derived from (9), the input matrices B₂₁, B₂₂, and B₂₃ are determined in (10), where the matrix B_2,k represents the k-th column of B₂ in (9). In the case of IPC#3, ΔM_z is generated by RWS, 4WIB, 4WID, and its seven combinations, i.e., RWS, 4WIB, 4WID, 4WIB+4WID, RWS+4WIB, RWS+4WID, and RWS+4WIB+4WID. For this reason, it is necessary to convert ΔM_z into the rear steering angle, δ_r, and the braking and tractive torque, T_Bi and T_Di, at each wheel. In this paper, T_Bi and T_Di at wheel i are generated by 4WIB and 4WID, respectively.

$$\left\{\begin{array}{lll}{u}_1={\delta }_f\hfill & B_{21}=\left[B_{2,1}\right]\hfill & \mathrm{for}\hspace{0.33em}IPC\#1\hfill \\ u_2={\left[\begin{array}{cc}{\delta }_f& {\delta }_r\end{array}\right]}^T\hfill & B_{22}=\left[\begin{array}{cc}{B}_{2,1}& B_{2,2}\end{array}\right]\hfill & \mathrm{for}\hspace{0.33em}IPC\#2\hfill \\ u_3={\left[\begin{array}{cc}{\delta }_f& \Delta M_z\end{array}\right]}^T\hfill & B_{23}=\left[\begin{array}{cc}{B}_{2,1}& B_{2,3}\end{array}\right]\hfill & \mathrm{for}\hspace{0.33em}IPC\#3\hfill \end{array}\right.$$

(10)

2.2. Design of LQR for Path Tracking

In the case of designing the LQR, an LQ cost function should be defined. The input configurations IPC#1, IPC#2, and IPC#3 have associated LQ cost functions, denoted as J₁, J₂, and J₃, which are given in (11). These can be transformed into a vector–matrix form, as given in (12). The weighting matrices Q and R_i in (12) can be derived from (11), as given in (13). The weight, ρ_i, is determined through Bryson’s rule, as outlined in (14), where ξ_i represents the maximum allowable value (MAV) for the respective term [17,36,37,38,51]. By adjusting ξ_i, the path tracking performance can be tuned. For IPC#i, the control input, u_i, of the LQR is calculated using formula (15), where P_i is the solution to the Riccati equation for IPC#i.

$$\begin{array}{l}{J}_0={\displaystyle {\int }_0^{\infty }\left\{{\rho }_1{e}_y^2+{\rho }_2{e}_{\phi }^2+{\rho }_3{\beta }^2+{\rho }_4{\gamma }^2\right\}dt}\hspace{1em}\\ \left\{\begin{array}{ll}{J}_1=J_0+{\displaystyle {\int }_0^{\infty }\left\{{\rho }_5{\delta }_f^2\right\}dt}\hfill & \mathrm{for}\hspace{0.33em}\hspace{0.33em}IPC\#1\hfill \\ J_2=J_0+{\displaystyle {\int }_0^{\infty }\left\{{\rho }_5{\delta }_f^2+{\rho }_6{\delta }_r^2\right\}dt}\hfill & \mathrm{for}\hspace{0.33em}\hspace{0.33em}IPC\#2\hfill \\ J_3=J_0+{\displaystyle {\int }_0^{\infty }\left\{{\rho }_5{\delta }_f^2+{\rho }_7\Delta M_z^2\right\}dt}\hfill & \mathrm{for}\hspace{0.33em}\hspace{0.33em}IPC\#3\hfill \end{array}\right.\end{array}$$

(11)

$$J_i=={\displaystyle {\int }_0^{\infty }{\left[\begin{array}{c}x\\ u_i\end{array}\right]}^T\left[\begin{array}{cc}Q& 0\\ 0& R_i\end{array}\right]\left[\begin{array}{c}x\\ u_i\end{array}\right]dt},\hspace{0.33em}i=1,2,3$$

(12)

$$Q=diag\left({\rho }_1,{\rho }_2,{\rho }_3,{\rho }_4\right),\hspace{1em}\left\{\begin{array}{ll}{R}_1={\rho }_5\hfill & \mathrm{for}\hspace{0.33em}\hspace{0.33em}IPC\#1\hfill \\ R_2=diag\left({\rho }_5,{\rho }_6\right)\hfill & \mathrm{for}\hspace{0.33em}\hspace{0.33em}IPC\#2\hfill \\ R_3=diag\left({\rho }_5,{\rho }_7\right)\hfill & \mathrm{for}\hspace{0.33em}\hspace{0.33em}IPC\#3\hfill \end{array}\right.$$

(13)

$${\rho }_i={\displaystyle \frac{1}{{\xi }_i^2}}$$

(14)

$$u_i=-K_{i}x=-R_i^{-1}{B}_{2i}^T{P}_{i}x,\hspace{0.33em}i=1,2,3$$

(15)

2.3. Design of LQ SOF Controllers for Path Tracking

The sensor outputs for the SOF controller are defined in (16). As shown in (16), the vectors of sensor outputs, y₂ and y₃, use the lateral offset, heading errors, and yaw rate, respectively. The vectors, y₂ and y₃, have been used in previous works [44,47]. Using these vectors, the SOF controllers are defined in (17).

$$\left\{\begin{array}{l}{y}_2=\left[\begin{array}{c}{e}_y\\ e_{\psi }\end{array}\right]=\left[\begin{array}{llll}1\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 1\hfill & 0\hfill & 0\hfill \end{array}\right]x=C_{2}x\\ y_3=\left[\begin{array}{c}{e}_y\\ e_{\psi }\\ \gamma \end{array}\right]=\left[\begin{array}{llll}1\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 1\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 1\hfill \end{array}\right]x=C_{3}x\end{array}\right.$$

(16)

$$u_{r,i}=K_r{y}_r=K_r{C}_{r}x,\hspace{1em}i=1,2,3,\hspace{0.33em}\hspace{0.33em}r=2,3$$

(17)

It is known that there are no analytical methods to find the optimal gain matrices K₂ and K₃ for the SOF controllers, which minimize the LQ cost functions [39]. For this reason, the optimization problem is formulated to find the K₂ and K₃, as in (18). In (18), K_r is either K₂ or K₃. The second row of the constraints in (18) means that the closed-loop system should be stable for the given K_r. The optimization problem (18) is known as non-convex, where a global optimum cannot be guaranteed [39]. To solve this problem, meta-heuristic methods, such as the genetic algorithm (GA), the evolutionary strategy (ES), and particle swarm optimization (PSO), have been proposed and successfully applied to date [52,53]. Among them, a meta-heuristic method, CMS-ES, is applied in this paper [54]. A detailed description of CMA-ES can be found in references [54,55]. Let the resulting SOF controllers using y₂ and y₃ from (18) be denoted as LQSOF2 and LQSOF3, respectively.

$$\begin{array}{ll}\underset{K_r}{\min }\hfill & J_i=\mathrm{trace}\left(P_i\right)\hspace{1em}\hfill \\ s.t.\hfill & \left\{\begin{array}{l}{P}_i=P_i^T>0\hfill \\ \max \left(\mathrm{Re}\left[A+B_{2i}{K}_r{C}_r\right]\right)<0\hfill \\ \begin{array}{l}{\left(A+B_{2i}{K}_r{C}_r\right)}^T{P}_i+P_i\left(A+B_{2i}{K}_r{C}_r\right)\hspace{0.33em}\\ \hspace{1em}\hspace{1em}\hspace{1em}+Q+C_r^T{K}_r^T{N}_i^T+N_i{K}_r{C}_r+C_r^T{K}_r^T{R}_i{K}_r{C}_r=0\end{array}\hfill \end{array}\right.\hfill \end{array}\hspace{0.33em}\left\{\begin{array}{l}i=1,2,3\hfill \\ r=2,3\hfill \end{array}\right.\hspace{0.33em}$$

(18)

2.4. Design of SOF SMC

To date, the SMC has been designed with a full-state feedback structure for path tracking [17,21]. With the vectors of outputs, y₂ and y₃, the SOF SMC is designed in this paper. The SMC also uses the state–space model (8). The sliding surface or error surface, s, to be minimized is defined in (19), where M is the matrix representing the weights on the state variables. Note that s is scalar. Generally, M is set to the row vector corresponding to the state vector, x. The convergence condition of the sliding surface is given in (20). In (20), K_SMC is a parameter that is used to tune the convergence speed of s. Naturally, the larger the K_SMC, the faster the convergence speed of s. By combining (19), (20) and (8), (21) is obtained. From (21), the control input of the full-state feedback SMC is derived in (22) [17,21]. In (22), (•)⁺ is the pseudo-inverse of a matrix (•). In this paper, the disturbance feedforward term, i.e., the second term on the right side of (22), is neglected. The design parameters of the SMC are M and K_SMC.

$$s=Mx$$

(19)

$$\dot{s}=-K_{SMC}s\hspace{1em}\left(K_{SMC}>0\right)$$

(20)

$$\dot{s}=M\dot{x}=M\left(Ax+B_{1}w+B{}_{2i}u_i\right)=-K_{SMC}Mx,\hspace{0.33em}\hspace{0.33em}i=1,2$$

(21)

$$u_i=-{\left(MB_{2i}\right)}^{+}M\left(A+K_{SMC}I\right)x-{\left(MB_{2i}\right)}^{+}M{B}_{1}w,\hspace{0.33em}\hspace{0.33em}i=1,2$$

(22)

When designing the SMC for IPC#3, ΔM_z is included in the control input, u₃. In u₃, there are large differences between the gains for FWS and ΔM_z. For this reason, the control inputs, u₁ and u₃, for FWS and ΔM_z, are separately designed, as given in (23) and (24), if IPC#3 is selected. Then, these control inputs are combined into u₃, as given in (25).

$$\dot{s}=M\dot{x}=M\left(Ax+B_{1}w+B{}_{2,i}u_i\right)=-K_{SMC}Mx,\hspace{0.33em}\hspace{0.33em}i=1,3$$

(23)

$$u_i=-{\left(MB_{2,i}\right)}^{+}M\left(A+K_{SMC}I\right)x-{\left(MB_{2,i}\right)}^{+}M{B}_{1}w,\hspace{0.33em}\hspace{0.33em}i=1,3$$

(24)

$$u_3=\left[\begin{array}{c}{u}_1\\ u_3\end{array}\right]=\left[\begin{array}{c}-{\left(MB_{2,1}\right)}^{+}M\left(A+K_{SMC}I\right)\\ -{\left(MB_{2,3}\right)}^{+}M\left(A+K_{SMC}I\right)\end{array}\right]x$$

(25)

In (22) and (25), the SMC has the full-state feedback form. For this reason, it should be transformed into the SOF form. By combining (17) with (22) or (25), (26) is obtained. By comparing the coefficients of both sides in (26), (27) is obtained. From (27), the gain matrix of the SOF SMC is obtained in (28). Let the resulting SOF controllers using y₂ and y₃ from (28) be denoted as the SOFSMC2 and SOFSMC3, respectively.

$$u_{i,r}=K_r{y}_r=K_r{C}_{r}x=-{\left(MB_2\right)}^{+}M\left(A+K_{SMC}I\right)x,\hspace{0.33em}\hspace{0.33em}i=1,2,3,\hspace{0.33em}\hspace{0.33em}r=2,3$$

(26)

$$K_r{C}_r=-{\left(MB_{2i}\right)}^{+}M\left(A+K_{SMC}I\right),\hspace{0.33em}\hspace{0.33em}i=1,2,3,\hspace{0.33em}\hspace{0.33em}r=2,3$$

(27)

$$K_r=-\left\{{\left(MB_{2i}\right)}^{+}\left(MA+K_{SMC}M\right)\right\}{C}_r^{+},\hspace{0.33em}\hspace{0.33em}i=1,2,3,\hspace{0.33em}\hspace{0.33em}r=2,3$$

(28)

2.5. Constraints Related to Tire Slip Angle

In previous work, the constraints related to the tire slip angle were proposed to avoid the saturation of lateral tire force, F_y [37]. F_yf and F_yr are easily saturated by excessive steering as θ goes over θ_m, which is a value that gives the maximum F_y. As a result, as F_y decreases, path tracking performance deteriorates. The idea of the constraints related to the tire slip angle is simple, where δ_f and δ_r are constrained such that θ cannot exceed θ_m.

It is assumed that F_y has its maximum, F_y,max, at the tire slip angle θ_m. If θ is larger than θ_m, F_y is saturated and smaller than F_y,max [49,50]. In the case of this condition, a path tracking controller cannot provide the maximum performance. For this reason, θ should be constrained to be smaller than θ_max, as given in (29). θ_max is not necessarily identical to θ_m, where θ_m is determined by a particular tire model or carpet plot data, and θ_max is set by a designer. By combining (29) with (2), (30) is derived. From (30), the constraints on δ_f and δ_r are obtained in (31). The constraints, (31), are applied to the steering angles obtained from the LQR, LQ SOF controller, and SOF SMC. In previous work, θ_max was set to 5°.

As shown in (31), it is essential to measure or estimate β. However, it is hard to measure β with an onboard sensor in real vehicles. Moreover, the SOF controllers presented in (17) do not use β. For this reason, (31) cannot be used for the SOF controllers. However, from the experiences in the simulations, the front steering angles did not go over 8° on low-μ surfaces [17,36,37,38]. Following this fact, δ_f and δ_r are constrained, as in (32), where σ_max is the maximum steering angle used to avoid the saturation of the lateral tire forces [37].

$$\left|\theta \right|\le {\theta }_{max}$$

(29)

$$\left|{\theta }_f\right|=\left|{\delta }_f-\beta -{\displaystyle \frac{l_f\gamma }{v_x}}\right|\le {\theta }_{max},\hspace{1em}\left|{\theta }_r\right|=\left|{\delta }_r-\beta +{\displaystyle \frac{l_r\gamma }{v_x}}\right|\le {\theta }_{max}$$

(30)

$$\left\{\begin{array}{l}-{\theta }_{max}+\beta +{\displaystyle \frac{l_f\gamma }{v_x}}\le {\delta }_f\le {\theta }_{max}+\beta +{\displaystyle \frac{l_f\gamma }{v_x}}\hspace{1em}\\ -{\theta }_{max}+\beta -{\displaystyle \frac{l_r\gamma }{v_x}}\le {\delta }_r\le {\theta }_{max}+\beta -{\displaystyle \frac{l_r\gamma }{v_x}}\end{array}\right.$$

(31)

$$-{\sigma }_{max}\le {\delta }_f,{\delta }_r\le {\sigma }_{max}$$

(32)

2.6. Control Allocation for LQR, LQ SOF Controllers, and SOF SMC

In accordance with (10), ΔM_z is derived from IPC#3. After the controller obtains ΔM_z, it should be transformed into additives, ΔF_x and ΔF_y, acting on the wheels. These forces are generated by seven actuator combinations consisting of RWS, 4WIB, and 4WID. Typically, this is accomplished by control allocation [15,17,20,21,22,23,24,28,29,33,35,36,37,38,44,47]. Following previous work, a weighted least squares-based method is employed for control allocation in this paper [20,21,23,29,35,44]. After obtaining the ΔF_x and ΔF_y acting on the wheels, they are transformed into the rear steering angle and braking and driving torques of the wheels. Figure 3 summarizes this procedure.

Figure 4 illustrates the geometric relationship among the tire forces, ΔF_yr, ΔF_x1, ΔF_x2, ΔF_x3, ΔF_x4, and ΔM_z, when ΔM_z is positive [36,37,38]. In Figure 4, the wheel indices are 1, 2, 3, and 4, in order from the front left, front right, rear left, and rear right wheels, respectively. Within Figure 4, ΔF_yr is generated by RWS. ΔF_x1, ΔF_x2, ΔF_x3, and ΔF_x4 are generated by 4WIB and 4WID. According to the sign of ΔF_xi, it corresponds to the tractive torque T_Di generated by 4WID or the braking torque T_Bi generated by 4WIB. These five tire forces need to be determined to generate ΔM_z, and the weighted least squares (WLSs)-based method is utilized for this purpose.

The vectors for the five tire forces to generate ΔM_z are shown in Figure 4, as expressed in (33) [36,37,38]. The components of vector z are given in (34).

$$\underset{z}{\underbrace{\left[\begin{array}{ccccc}{c}_1& c_2& c_3& c_4& c_5\end{array}\right]}}\underset{q}{\underbrace{\left[\begin{array}{c}\Delta F_{yr}\\ \Delta F_{x1}\\ \Delta F_{x2}\\ \Delta F_{x3}\\ \Delta F_{x4}\end{array}\right]}}=\Delta M_z$$

(33)

$$\left\{\begin{array}{l}{c}_1=2l_r\cos {\delta }_r,\\ c_2=-l_f\sin {\delta }_f+0.5\cdot t_f\cos {\delta }_f,\hspace{1em}{c}_3=-l_f\sin {\delta }_f-0.5\cdot t_f\cos {\delta }_f,\\ c_4=l_r\sin {\delta }_r+0.5\cdot t_r\cos {\delta }_r,\hspace{1em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{c}_5=l_r\sin {\delta }_r-0.5\cdot t_r\cos {\delta }_r\end{array}\right.$$

(34)

The quadratic form defines the cost function for weighted least squares, as in (35). In (35), ζ_i is the product of μ and F_zi at wheel i, i.e., μF_zi, which represents the friction circle radius. In (35), the radii of the friction circles, μF_zi, must be estimated. For the purpose of this paper, F_zi is estimated using longitudinal and lateral accelerations [56]. In addition to this, μ is assumed to be estimated using the observer proposed in reference [57]. The vector of virtual weights, denoted as κ, is introduced in (35) for the purpose of selecting the combination of actuators [36,37,38]. The equality constraint (33) is transformed into a quadratic form, as shown in (36). In previous research, the satisfaction of (33) was essential for generating ΔM_z. The cost function (35) and the constraint (36) are unified into a single cost function (37) through the utilization of the Lagrange multiplier, η. In (37), it is imperative to set η to a value of 1 or higher; otherwise, the constraint (36) will not be met. The optimal solution for (37) is algebraically derived in (38) by taking the derivative of (37) with respect to q [36,37,38].

$$\begin{array}{l}{J}_Q={\displaystyle \frac{{\kappa }_2\Delta F_{x1}^2}{{\zeta }_1^2}}+{\displaystyle \frac{{\kappa }_3\Delta F_{x2}^2}{{\zeta }_2^2}}+{\displaystyle \frac{{\kappa }_1\Delta F_{yr}^2+{\kappa }_4\Delta F_{x3}^2}{{\zeta }_3^2}}+{\displaystyle \frac{{\kappa }_1\Delta F_{yr}^2+{\kappa }_5\Delta F_{x4}^2}{{\zeta }_4^2}}\hspace{0.33em}=q^T\Theta q\\ \Theta =\mathrm{diag}\left[{\displaystyle \frac{1}{{\zeta }_3^2}}+{\displaystyle \frac{1}{{\zeta }_4^2}},{\displaystyle \frac{1}{{\zeta }_1^2}},{\displaystyle \frac{1}{{\zeta }_2^2}},{\displaystyle \frac{1}{{\zeta }_3^2}},{\displaystyle \frac{1}{{\zeta }_4^2}}\right]\kappa \hspace{1em}\\ \kappa =\mathrm{diag}\left[\begin{array}{ccccc}{\kappa }_1& {\kappa }_2& {\kappa }_3& {\kappa }_4& {\kappa }_5\end{array}\right]\end{array}$$

(35)

$$J_{EC}={\left(zq-\Delta M_z\right)}^T\left(zq-\Delta M_z\right)$$

(36)

$$J_{CA}=J_Q+\eta \cdot J_{EC}=q^T\Theta q+\eta {\left(zq-\Delta M_z\right)}^T\left(zq-\Delta M_z\right)$$

(37)

$$q_{opt}=\eta {\left(\Theta +\eta z^{T}z\right)}^{-1}{z}^T\Delta M_z$$

(38)

When the weighted least squares are applied to the control allocations involving ΔM_z, an arbitrary combination of actuators comprising RWS, 4WIB, and 4WID can be configured based on a specific input configuration. To account for this variability, the virtual weights, denoted as κ_i, can be customized to match a particular combination selected from RWS, 4WIB, 4WID, 4WIB+4WID, RWS+4WIB, RWS+4WID, and RWS+4WIB+4WID [36,37,38]. The vectors of the virtual weights regarding RWS are outlined in (39). Initially, all the virtual weights are set to 1. In (39), the value of ε is 10⁻⁴, and the terms marked with • insignificantly impact the steering actuator. The first term of κ in (39) corresponds to ΔF_yr, which is converted into δ_r. When the first term of κ is adjusted to ε, ΔF_yr is generated through the weighted least squares process, as depicted in the first row of (39).

$$\left\{\begin{array}{l}\begin{array}{ll}\mathrm{RWS}:& \hspace{1em} \kappa =\mathrm{diag}\left[\begin{array}{ccccc}\epsilon & •& •& •& •\end{array}\right]\end{array}\\ \begin{array}{ll}\mathrm{No}\hspace{0.33em}\mathrm{RWS}:& \kappa =\mathrm{diag}\left[\begin{array}{ccccc}1& •& •& •& •\end{array}\right]\end{array}\end{array}\right.$$

(39)

The vectors of the virtual weights regarding 4WIB and 4WID are provided in (40). As depicted in Figure 4, it becomes evident that only ΔF_x2 and ΔF_x4 should be generated in the event of a negative ΔM_z and the availability of 4WIB. This scenario is illustrated by the second row of the first term, 4WID, in (40). If both 4WIB and 4WID are accessible for ΔM_z generation, no restrictions are imposed on ΔF_x’s, as indicated in the last row in (40). The virtual weights, as presented in (39) and (40), can be configured in accordance with the available combinations of actuators, RWS, 4WIB, 4WID, 4WIB+4WID, RWS+4WIB, RWS+4WID, and RWS+4WIB+4WID, intended for ΔM_z generation.

$$\left\{\begin{array}{ll}4\mathrm{WID}\hfill & :\hspace{1em}\left\{\begin{array}{l}\kappa =\mathrm{diag}\left[\begin{array}{ccccc}•& 1& \epsilon & 1& \epsilon \end{array}\right]\hspace{1em}\mathrm{if}\hspace{0.33em}\Delta M_z>0\\ \kappa =\mathrm{diag}\left[\begin{array}{ccccc}•& \epsilon & 1& \epsilon & 1\end{array}\right]\hspace{1em}\mathrm{if}\hspace{0.33em}\Delta M_z<0\end{array}\right.\hfill \\ 4\mathrm{WIB}\hfill & :\hspace{1em}\left\{\begin{array}{l}\kappa =\mathrm{diag}\left[\begin{array}{ccccc}•& \epsilon & 1& \epsilon & 1\end{array}\right]\hspace{1em}\mathrm{if}\hspace{0.33em}\Delta M_z>0\\ \kappa =\mathrm{diag}\left[\begin{array}{ccccc}•& 1& \epsilon & 1& \epsilon \end{array}\right]\hspace{1em}\mathrm{if}\hspace{0.33em}\Delta M_z<0\end{array}\right.\hfill \\ 4\mathrm{WID}/4\mathrm{WIB}\hfill & :\hspace{1em}\kappa =\mathrm{diag}\left[\begin{array}{ccccc}•& \epsilon & \epsilon & \epsilon & \epsilon \end{array}\right]\hfill \end{array}\right.$$

(40)

The sets of virtual weights corresponding to actuator combinations in IPC#3 are summarized in

Table 1. Sets of virtual weights in control allocation method.

	Actuator Combinations		Vector of Virtual Weights
IPC#1	AC#1	FWS
IPC#2	AC#2	4WS
IPC#3 FWS	AC#3	+RWS	$\kappa =\mathrm{diag}\left[\begin{array}{ccccc}\epsilon & 1& 1& 1& 1\end{array}\right]$
	AC#4	+RWS+4WID	$\left\{\begin{array}{l}\kappa =\mathrm{diag}\left[\begin{array}{ccccc}\epsilon & 1& \epsilon & 1& \epsilon \end{array}\right]\hspace{1em}\mathrm{if}\hspace{0.33em}\Delta M_z>0\\ \kappa =\mathrm{diag}\left[\begin{array}{ccccc}\epsilon & \epsilon & 1& \epsilon & 1\end{array}\right]\hspace{1em}\mathrm{if}\hspace{0.33em}\Delta M_z<0\end{array}\right.$
	AC#5	+RWS+4WIB	$\left\{\begin{array}{l}\kappa =\mathrm{diag}\left[\begin{array}{ccccc}\epsilon & \epsilon & 1& \epsilon & 1\end{array}\right]\hspace{1em}\mathrm{if}\hspace{0.33em}\Delta M_z>0\\ \kappa =\mathrm{diag}\left[\begin{array}{ccccc}\epsilon & 1& \epsilon & 1& \epsilon \end{array}\right]\hspace{1em}\mathrm{if}\hspace{0.33em}\Delta M_z<0\end{array}\right.$
	AC#6	+RWS+4WID+4WIB	$\kappa =\mathrm{diag}\left[\begin{array}{ccccc}\epsilon & \epsilon & \epsilon & \epsilon & \epsilon \end{array}\right]$
	AC#7	+4WID	$\left\{\begin{array}{l}\kappa =\mathrm{diag}\left[\begin{array}{ccccc}1& 1& \epsilon & 1& \epsilon \end{array}\right]\hspace{1em}\mathrm{if}\hspace{0.33em}\Delta M_z>0\\ \kappa =\mathrm{diag}\left[\begin{array}{ccccc}1& \epsilon & 1& \epsilon & 1\end{array}\right]\hspace{1em}\mathrm{if}\hspace{0.33em}\Delta M_z<0\end{array}\right.$
	AC#8	+4WIB	$\left\{\begin{array}{l}\kappa =\mathrm{diag}\left[\begin{array}{ccccc}1& \epsilon & 1& \epsilon & 1\end{array}\right]\hspace{1em}\mathrm{if}\hspace{0.33em}\Delta M_z>0\\ \kappa =\mathrm{diag}\left[\begin{array}{ccccc}1& 1& \epsilon & 1& \epsilon \end{array}\right]\hspace{1em}\mathrm{if}\hspace{0.33em}\Delta M_z<0\end{array}\right.$
	AC#9	+4WID+4WIB	$\kappa =\mathrm{diag}\left[\begin{array}{ccccc}1& \epsilon & \epsilon & \epsilon & \epsilon \end{array}\right]$

. As shown in Table 1, seven actuator combinations, AC#3, …, AC#8 and AC#9 in IPC#3, are selected by setting the vector of the virtual weights. For example, AC#6 or (FWS+RWS+4WID+4WIB) can be selected by setting κ to diag[ε ε ε ε ε].

In this paper, T_Bi and T_Di for wheel i are computed based on the values of ΔF_x1, ΔF_x2, ΔF_x3, and ΔF_x4, which are determined through the weighted least squares-based method, specifically utilizing q_opt, as presented in (41). Depending on the sign of ΔF_xi, the calculations for T_Bi and T_Di at wheel i are carried out as in (41), respectively [36,37,38].

$$\left.\left\{\begin{array}{l}{T}_{Bi}\hspace{0.33em}\hspace{0.33em}\mathrm{if}\hspace{0.33em}\Delta F_{xi}<0\hfill \\ T_{Di}\hspace{0.33em}\hspace{0.33em}\mathrm{if}\hspace{0.33em}\Delta F_{xi}>0\hfill \end{array}\hspace{0.33em}\right.\right\}=h\left({\displaystyle \frac{r_{wi}\Delta F_{xi}}{{\upsilon }_i}},{\omega }_i\right),\hspace{0.33em}\hspace{0.33em}i=1,\cdots ,4$$

(41)

The calculation of δ_r is performed by utilizing the value of ΔF_yr, acquired from q_opt and the definitions found in (2) and (3). ΔF_yr is transformed into the rear slip angle, θ_r, using the second term in (3). Through the utilization of (2), δ_r is subsequently determined, as outlined in (42) [45]. The SOF controllers using y₂ and y₃ cannot use β. For this reason, (43) is used instead of (42).

$${\delta }_r=-{\alpha }_r+\beta -{\displaystyle \frac{l_r\gamma }{v_x}}\approx -{\displaystyle \frac{\Delta F_{yr}}{2C_r}}+\beta -{\displaystyle \frac{l_r\gamma }{v_x}}$$

(42)

$${\delta }_r\approx -{\displaystyle \frac{\Delta F_{yr}}{2C_r}}-{\displaystyle \frac{l_r\gamma }{v_x}}$$

(43)

3. Performance Evaluation for Path Tracking Control

In the domain of path tracking, e_y and e_ψ have traditionally served as measures for assessing path tracking performance. Different from these measures, recent papers have proposed other measures by adopting a double lane change maneuver for collision avoidance as the target path [11,16,17,19,20,26,27,29,30,32,38,47]. This path is selected to assess the reachability and agility of path tracking [17,36,37,38]. Figure 5 presents the path to track and the actual trajectory of the vehicle. Five metrics, from D₁ to D₄ and D_OS, are defined in (44), based on the points, from P₁ to P₃, on the path, and from A₁ to A₄ on the actual trajectory, as shown in Figure 5. In (44), D₁ and D₂ stand for the longitudinal and lateral offsets of the peak, respectively. D₃ and D₄ can be considered as the response delay and settling delay, both of which have units of meters, respectively. D_OS is a measure of how much overshoot occurred. In (44), x and y, in the parentheses, correspond to the x- and y-positions of point •, respectively. Naturally, the smaller these metrics, the better the path tracking performance, except for D₂.

$$\left\{\begin{array}{l}{D}_1=x\left(A_1\right)-x\left(P_1\right)=x\left(A_1\right)-73.2\\ D_2=y\left(A_1\right)-y\left(P_1\right)=y\left(A_1\right)-3.53\\ D_3=x\left(A_2\right)-x\left(P_2\right)=x\left(A_2\right)-91.5\\ D_4=x\left(A_4\right)-x\left(P_3\right)=x\left(A_4\right)-190.0\\ D_{OS}={\displaystyle \frac{\left|y\left(A_3\right)-1.65\right|}{1.65+3.53}}\times 100\end{array}\right.$$

(44)

In Figure 5, the agility and reachability of the controller are measured by D₁ and D₂, respectively. In this paper, if D₂ exceeds −0.05 m, the controller’s path tracking performance is regarded as satisfactory. D_OS indicates the lateral damping capability, standing for its agility. In this paper, the performance is satisfactory if D_OS remains below 16%. D₃ represents the response delay, standing for the agility along the longitudinal motion. D₄ can be interpreted as the settling time in transient responses, or the settling distance needed to converge to A₄, indicating the convergence speed of the vehicle’s lateral motion toward the target y-position, i.e., −1.65 m. In this research, the performance is regarded as satisfactory if D₄ is less than 16 m on low-friction surfaces. Generally, the best performance is attained when D_OS is around 1%, provided that D₂ exceeds −0.05 m. In this paper, the weights and k_v in J₁, J₂, and J₃ are tuned such that D₂ is maintained near −0.02 m. In this case, D_OS is always less than 1%. More comprehensive explanations of these measures can be found in references [17,36,37,38].

Generally, lateral stability is represented by β. In a real vehicle, it is regarded that lateral stability is maintained if β is controlled not to exceed 3° [36]. Generally, the smaller the β, the better. In this paper, let $\max \left|\beta \right|$ denote the maximum absolute value of β. In this way, the maximum absolute value of the angular rate of β is defined as $\max \left|\dot{\beta }\right|$.

4. Simulation and Discussion

To verify the effects of the SOF controllers and input configurations on path tracking performance, simulations were performed. The six path tracking controllers, i.e., two full-state feedback controllers, i.e., a full-state feedback LQR and a full-state feedback SMC, and four SOF controllers, i.e., LQSOF2, LQSOF3, SOFSMC2, and SOFSMC3, were implemented with MATLAB/Simulink 2019a, connected to CarSim 8.0 [58]. The test scenario was given in Figure 5. As a test vehicle, the F-segment sedan model in CarSim was selected [17,36,37,38]. This model is a nonlinear 27-DOF model with a single sprung mass, four wheels, suspensions, and a steering mechanism. In the model, nonlinear tire models, represented by a carpet plot, were used. The spring and the damper in this model are nonlinear, represented by piece-wise linear interpolation [56]. The parameters of the F-segment sedan are given in

Table 2. Parameters of F-segment sedan in CarSim.

Parameter	Value	Parameter	Value
ms	1823 kg	Iz	6286 kg-m2
lf	1.27 m	lr	1.90 m
tf	42,000 N/rad	tr	62,000 N/rad
Cf	1.6 m	Cr	1.6 m

.

In this paper, it was assumed that the target vehicle had the actuators, FWS, RWS, 4WIB, and 4WID, which were modeled as a first-order system. The bandwidths for the steering actuators were set to 5 Hz, and the bandwidths for 4WIB and 4WID were set to 2 Hz, respectively. The initial speed was set to 60 km/h, which was maintained as a constant using a built-in speed controller provided in CarSim. The simulations were performed on high- and low-friction surfaces, where μ was 0.8 and 0.5, respectively. In this paper, speed variation is not considered because the path tracking performance of a controller at low speeds is better than that at high speeds. The simulation horizon was set to 10 s.

As pointed out in previous work, ΔM_z should be limited to a smaller δ_r and β. From the maximum lateral tire forces and geometric information from CarSim, it was limited to 11,400 Nm [36]. When using full-state feedback controllers, θ_max was set to 5°, which was obtained from CarSim tire data. When using the SOF controllers, σ_max was set to 8°.

In the simulations, the weights in the LQ cost function, the gain of the SMC, and the preview distance were tuned for each controller and actuator combination until a satisfactory result was obtained. For this reason, the tuning parameters are different from one another, according to the selected controller and the actuator combination.

In the simulations, the LQR and SMC are used as a baseline. The other path tracking controllers, such as the pure pursuit method, Stanley method, PID control, LQR, MPC, and SMC, were compared, as in previous work [9,17]. As mentioned earlier, it is impossible to use MPC with an SOF control because the initial state variables for MPC cannot be obtained using SOF. Moreover, it was shown that the path tracking performance of the MPC is nearly identical to that of the LQR [17].

4.1. Simulation on High-Friction Surfaces

The first simulation was performed for six controllers on the road where μ was 0.8. The velocity gain, k_v, was set to 0.18. The weights in the LQ cost function were tuned such that D₂ was near −0.020 m and $\max \left|\beta \right|$ was less than 2°. As a result, D₂ and D_OS can be neglected because they were nearly identical values over several actuator combinations.

Table 3. Simulation results for LQR on high-friction surface.

	Actuator Combinations		D1 (m)	D2 (m)	DOS (%)	D3 (m)	D4 (m)	$\mathbf{max}\left|\mathit{\beta }\right|$ (deg)	$\mathbf{max}\left|\dot{\mathit{\beta }}\right|$ (deg/s)
IPC#1	AC#1	FWS	0.31	−0.021	0.3	0.91	−2.69	0.95	7.3
IPC#2	AC#2	4WS	0.33	−0.020	0.3	0.93	−2.77	1.01	7.61
IPC#3 FWS	AC#3	+RWS	0.46	−0.021	0.6	1.07	−2.31	1.44	8.02
	AC#4	+RWS+4WID	0.47	−0.021	0.6	1.07	−2.32	1.43	8.01
	AC#5	+RWS+4WIB	0.47	−0.020	0.6	1.13	−2.21	1.42	7.99
	AC#6	+RWS+4WID+4WIB	0.48	−0.020	0.6	1.08	−2.36	1.42	8.01
	AC#7	+4WID	0.32	−0.020	0.4	1.30	−3.28	0.80	7.28
	AC#8	+4WIB	0.18	−0.021	0.5	1.19	−2.47	1.09	7.60
	AC#9	+4WID+4WIB	0.18	−0.020	0.6	1.15	−2.80	1.03	7.60

,

Table 4. Simulation results for LQSOF2 on high-friction surface.

	Actuator Combinations		D1 (m)	D2 (m)	DOS (%)	D3 (m)	D4 (m)	$\mathbf{max}\left|\mathit{\beta }\right|$ (deg)	$\mathbf{max}\left|\dot{\mathit{\beta }}\right|$ (deg/s)
IPC#1	AC#1	FWS	−0.15	−0.021	0.0	−0.08	0.77	1.23	9.57
IPC#2	AC#2	4WS	−0.13	−0.021	0.0	−0.05	0.47	1.44	9.32
IPC#3 FWS	AC#3	+RWS	−0.30	−0.021	0.0	−0.18	2.29	1.79	8.65
	AC#4	+RWS+4WID	−0.15	−0.021	0.0	−0.06	0.74	1.42	9.57
	AC#5	+RWS+4WIB	−0.30	−0.020	0.1	−0.18	2.45	1.89	8.79
	AC#6	+RWS+4WID+4WIB	−0.15	−0.021	0.0	−0.06	0.77	1.41	9.51
	AC#7	+4WID	−0.16	−0.020	0.0	−0.07	0.79	1.25	9.61
	AC#8	+4WIB	−0.15	−0.021	0.0	−0.08	0.77	1.23	9.57
	AC#9	+4WID+4WIB	−0.13	−0.021	0.0	−0.05	0.47	1.44	9.32

,

Table 5. Simulation results for LQSOF3 on high-friction surface.

	Actuator Combinations		D1 (m)	D2 (m)	DOS (%)	D3 (m)	D4 (m)	$\mathbf{max}\left|\mathit{\beta }\right|$ (deg)	$\mathbf{max}\left|\dot{\mathit{\beta }}\right|$ (deg/s)
IPC#1	AC#1	FWS	0.57	−0.021	0.9	0.98	−3.81	0.98	4.28
IPC#2	AC#2	4WS	0.58	−0.020	0.9	0.99	−3.15	1.75	5.76
IPC#3 FWS	AC#3	+RWS	0.30	−0.021	0.2	0.52	−3.30	1.44	8.36
	AC#4	+RWS+4WID	0.30	−0.020	0.2	0.45	−3.26	1.79	9.19
	AC#5	+RWS+4WIB	0.29	−0.020	0.2	0.55	−2.89	1.10	4.01
	AC#6	+RWS+4WID+4WIB	0.30	−0.020	0.2	0.57	−3.32	1.07	4.38
	AC#7	+4WID	0.47	−0.020	0.7	1.07	−3.71	0.88	7.12
	AC#8	+4WIB	0.41	−0.021	0.5	0.73	−3.37	1.16	4.71
	AC#9	+4WID+4WIB	0.40	−0.021	0.6	0.73	−3.63	1.07	4.91

,

Table 6. Simulation results for SMC on high-friction surface.

	Actuator Combinations		D1 (m)	D2 (m)	DOS (%)	D3 (m)	D4 (m)	$\mathbf{max}\left|\mathit{\beta }\right|$ (deg)	$\mathbf{max}\left|\dot{\mathit{\beta }}\right|$ (deg/s)
IPC#1	AC#1	FWS	−0.18	−0.020	0.0	0.31	−3.82	0.97	7.50
IPC#2	AC#2	4WS	−0.06	−0.020	0.0	0.62	−3.24	1.51	12.55
IPC#3 FWS	AC#3	+RWS	0.07	−0.020	0.1	0.67	−1.36	1.07	6.77
	AC#4	+RWS+4WID	0.19	−0.020	0.9	1.17	−3.75	1.09	8.29
	AC#5	+RWS+4WIB	0.07	−0.020	0.1	0.67	−1.36	1.07	6.77
	AC#6	+RWS+4WID+4WIB	0.06	−0.020	0.0	0.68	−1.22	1.12	6.68
	AC#7	+4WID	0.25	−0.021	0.7	1.49	−3.91	1.05	11.27
	AC#8	+4WIB	0.10	−0.020	0.0	0.73	−2.73	0.81	6.68
	AC#9	+4WID+4WIB	0.11	−0.020	0.0	0.73	−3.03	0.97	7.17

,

Table 7. Simulation results for SOFSMC2 on high-friction surface.

	Actuator Combinations		D1 (m)	D2 (m)	DOS (%)	D3 (m)	D4 (m)	$\mathbf{max}\left|\mathit{\beta }\right|$ (deg)	$\mathbf{max}\left|\dot{\mathit{\beta }}\right|$ (deg/s)
IPC#1	AC#1	FWS	−0.26	−0.020	0.0	0.02	2.26	1.02	6.73
IPC#2	AC#2	4WS	−0.25	−0.020	0.0	0.05	1.65	1.13	6.19
IPC#3 FWS	AC#3	+RWS	−0.42	−0.020	0.0	−0.15	4.56	1.72	6.41
	AC#4	+RWS+4WID	−0.42	−0.021	0.0	−0.06	−5.52	1.85	7.87
	AC#5	+RWS+4WIB	−0.36	−0.021	0.0	−0.14	3.98	1.17	6.50
	AC#6	+RWS+4WID+4WIB	−0.49	−0.020	0.3	−0.29	0.37	1.43	10.05
	AC#7	+4WID	−0.23	−0.020	0.2	0.20	−6.13	1.41	9.13
	AC#8	+4WIB	−0.27	−0.020	0.0	0.03	3.21	1.06	7.10
	AC#9	+4WID+4WIB	−0.26	−0.020	0.2	0.02	1.97	1.02	6.61

and

Table 8. Simulation results for SOFSMC3 on high-friction surface.

	Actuator Combinations		D1 (m)	D2 (m)	DOS (%)	D3 (m)	D4 (m)	$\mathbf{max}\left|\mathit{\beta }\right|$ (deg)	$\mathbf{max}\left|\dot{\mathit{\beta }}\right|$ (deg/s)
IPC#1	AC#1	FWS	0.08	−0.020	0.0	0.37	−1.44	0.99	5.97
IPC#2	AC#2	4WS	−0.04	−0.020	0.0	0.29	0.06	1.01	5.48
IPC#3 FWS	AC#3	+RWS	−0.13	−0.020	0.0	0.16	0.22	1.30	5.92
	AC#4	+RWS+4WID	−0.15	−0.020	0.0	0.21	−2.43	1.69	8.11
	AC#5	+RWS+4WIB	−0.17	−0.020	0.0	0.11	1.24	1.36	6.08
	AC#6	+RWS+4WID+4WIB	−0.70	−0.021	0.7	−0.06	−2.72	1.31	12.89
	AC#7	+4WID	0.02	−0.021	0.0	0.52	−4.15	1.36	9.13
	AC#8	+4WIB	−0.04	−0.020	0.0	0.29	−0.06	0.95	6.71
	AC#9	+4WID+4WIB	−0.03	−0.020	0.1	0.28	−0.61	1.00	6.15

show the simulation results for the LQR, LQSOF2, LQSOF3, SMC, SOFSMC2, and SOFSMC3, respectively. In these tables, AC#1, …, AC#8, and AC#9 in the second column represent the actuator combinations, FWS, 4WS, FWS+RWS, FWS+RWS+4WID, FWS+RWS+4WIB, FWS+RWS+4WID+4WIB, FWS+4WID, FWS+4WIB, and FWS+4WIB+4WID, respectively.

In terms of the LQOC, including the LQR and LQ SOF controllers, as given in Table 3, Table 4 and Table 5, the LQSOF2 is better than the LQR in terms of D₁ and D₃, and the LQSOF3 is better than the LQR in terms of D₃ and D₄. Moreover, when the LQSOF2 is applied, D₁ and $\max \left|\beta \right|$ are inversely proportional to each other, as given in Table 4. A notable feature of the LQSOF2 is that it gives the smallest D₁ and the largest $\max \left|\beta \right|$ among the three controllers. This is caused by the fact that β is not used in the LQSOF2. On the contrary, this fact does not hold for the LQR and LQSOF3. Compared to the LQSOF2, there are few differences between the LQR and LQSOF3. For this reason, the LQSOF2 is preferred over the LQSOF3. In other words, if the yaw rate, γ, is used for the LQSOF3, it shows a nearly identical performance to the LQR. For LQR, AC#7 is the best actuator combination. For LQSOF2, AC#1, AC#7, and AC#8 show the best performances. For the LQSOF3, AC#5 is the best actuator combination. These facts can be identified in Table 3, Table 4 and Table 5.

In view of the SMC, SOFSMC2 and SOFSMC3 are better than the SMC in terms of D₁. On the contrary, there are few differences among the SMC, SOFSMC2, and SOFSMC3 in terms of D₃ and D₄. Similar to the LQSOF2, the SMCSOF2 gives the smallest D₁ and the largest $\max \left|\beta \right|$ among the three controllers, and D₁ and $\max \left|\beta \right|$ are inversely proportional to each other when using the SMCSOF2, as given in Table 7. This is caused by the fact that β is not used in the SMCSOF2. On the contrary, if the yaw rate, γ, is used for the SMCSOF3, it shows nearly identical performance to the SMC. For the SMC, AC#1 is the best actuator combination. For the SMCSOF2, AC#5 shows the best performance. For SMCSOF3, AC#8 is the best actuator combination.

As shown in Table 3, Table 4, Table 5, Table 6, Table 7 and Table 8, there are slight differences among the nine actuator combinations, AC#1, …, AC#8, and AC#9, given in IPC#1, IPC#2, and IPC#3 for a single controller. This is a natural consequence, as each tire can generate larger forces on high-friction surfaces. These results mean that the simplest actuator, i.e., AC#1, FWS, is recommended on high-friction surfaces. If collision avoidance is critical, then the LQSOF2 and SOFSMC2 are recommended because they give the smallest D₁.

Figure 6 and Figure 7 show the simulation results for the LQSOF2 and SOFSMC2 for each actuator combination on high-friction surface. As shown in Figure 6 and Figure 7, there are few differences in vehicle trajectories and heading angles with the LQSOF2 and SMCSOF2 for the nine actuator combinations, AC#1, …, AC#8, and AC#9. This is caused by the fact that path tracking controllers show good performance on high-friction surfaces, regardless of controller design methodologies or control structures, and that the tuning parameters of the LQSOF and SOFSMC were finely tuned for each actuator combination. In other words, D₁ and D_OS were tuned to show nearly identical values to one another for actuator combinations, which resulted in nearly the same responses.

4.2. Simulation on Low-Friction Surfaces

The second simulation was performed for six controllers on the road, where μ was 0.4. The velocity gain, k_v, was set to 0.25. The weights in the LQ cost function were tuned such that D₂ was near −0.020 m and $\max \left|\beta \right|$ was less than 2°. As a result, D₂ and D_OS can be neglected because they were nearly identical values over several actuator combinations.

Table 9. Simulation results for LQR on low-friction surface.

	Actuator Combinations		D1 (m)	D2 (m)	DOS (%)	D3 (m)	D4 (m)	$\mathbf{max}\left|\mathit{\beta }\right|$ (deg)	$\mathbf{max}\left|\dot{\mathit{\beta }}\right|$ (deg/s)
IPC#1	AC#1	FWS	1.82	−0.021	0.5	8.05	4.61	0.58	12.66
IPC#2	AC#2	4WS	1.82	−0.020	0.9	7.99	3.89	1.64	13.46
IPC#3 FWS	AC#3	+RWS	1.74	−0.021	0.5	7.90	3.93	1.65	12.16
	AC#4	+RWS+4WID	1.72	−0.021	0.5	7.94	4.14	1.46	12.04
	AC#5	+RWS+4WIB	1.87	−0.021	0.8	8.60	4.29	1.53	12.13
	AC#6	+RWS+4WID+4WIB	1.79	−0.020	0.6	8.21	4.28	1.32	12.11
	AC#7	+4WID	1.91	−0.020	0.9	8.90	5.23	0.63	12.52
	AC#8	+4WIB	2.10	−0.021	0.7	8.91	4.23	0.66	13.06
	AC#9	+4WID+4WIB	1.99	−0.020	0.9	8.45	4.44	0.63	12.84

,

Table 10. Simulation results for LQSOF3 on low-friction surface.

	Actuator Combinations		D1 (m)	D2 (m)	DOS (%)	D3 (m)	D4 (m)	$\mathbf{max}\left|\mathit{\beta }\right|$ (deg)	$\mathbf{max}\left|\dot{\mathit{\beta }}\right|$ (deg/s)
IPC#1	AC#1	FWS	2.22	−0.021	0.9	9.11	5.35	0.57	13.19
IPC#2	AC#2	4WS	2.27	−0.020	0.9	9.11	5.37	1.11	13.34
IPC#3 FWS	AC#3	+RWS	2.17	−0.021	0.4	8.52	4.39	1.07	8.25
	AC#4	+RWS+4WID	2.30	−0.021	0.6	9.11	5.09	0.85	9.93
	AC#5	+RWS+4WIB	2.04	−0.021	0.8	8.24	3.67	1.69	10.74
	AC#6	+RWS+4WID+4WIB	2.12	−0.020	0.6	9.15	5.54	1.89	11.64
	AC#7	+4WID	2.38	−0.021	0.7	10.35	7.51	0.60	12.97
	AC#8	+4WIB	2.51	−0.021	0.8	10.29	6.31	0.66	13.40
	AC#9	+4WID+4WIB	2.37	−0.021	0.9	9.57	6.13	0.62	13.34

,

Table 11. Simulation results for SMC on low-friction surface.

	Actuator Combinations		D1 (m)	D2 (m)	DOS (%)	D3 (m)	D4 (m)	$\mathbf{max}\left|\mathit{\beta }\right|$ (deg)	$\mathbf{max}\left|\dot{\mathit{\beta }}\right|$ (deg/s)
IPC#1	AC#1	FWS	1.54	−0.021	0.4	7.76	4.36	0.58	12.46
IPC#2	AC#2	4WS	1.74	−0.020	0.9	8.11	4.33	1.56	15.92
IPC#3 FWS	AC#3	+RWS	1.52	−0.021	0.7	7.59	3.28	1.00	12.38
	AC#4	+RWS+4WID	1.74	−0.021	0.6	9.00	5.46	0.76	11.59
	AC#5	+RWS+4WIB	1.42	−0.021	0.0	7.39	4.76	1.58	12.51
	AC#6	+RWS+4WID+4WIB	1.72	−0.020	0.8	9.36	6.66	1.50	11.42
	AC#7	+4WID	2.22	−0.021	1.0	10.07	6.95	0.57	12.31
	AC#8	+4WIB	1.34	−0.020	0.8	7.18	2.61	0.73	11.57
	AC#9	+4WID+4WIB	1.41	−0.021	0.6	7.60	3.37	0.62	12.13

,

Table 12. Simulation results for SOFSMC2 on low-friction surface.

	Actuator Combinations		D1 (m)	D2 (m)	DOS (%)	D3 (m)	D4 (m)	$\mathbf{max}\left|\mathit{\beta }\right|$ (deg)	$\mathbf{max}\left|\dot{\mathit{\beta }}\right|$ (deg/s)
IPC#1	AC#1	FWS	0.81	−0.020	0.4	7.69	3.53	0.59	10.89
IPC#2	AC#2	4WS	0.65	−0.021	0.7	7.60	3.25	1.20	13.16
IPC#3 FWS	AC#3	+RWS	0.51	−0.021	0.2	7.04	3.03	0.57	10.59
	AC#4	+RWS+4WID	0.37	−0.021	0.9	7.56	3.58	0.87	10.68
	AC#5	+RWS+4WIB	−0.32	−0.020	0.0	5.79	2.13	0.88	11.61
	AC#6	+RWS+4WID+4WIB	−0.17	−0.021	0.9	7.13	3.47	0.94	11.29
	AC#7	+4WID	0.65	−0.020	1.0	7.09	2.41	0.60	11.09
	AC#8	+4WIB	0.63	−0.020	0.6	6.82	2.14	0.64	10.85
	AC#9	+4WID+4WIB	0.58	−0.021	0.6	6.92	2.46	0.64	11.12

and

Table 13. Simulation results for SOFSMC3 on low-friction surface.

	Actuator Combinations		D1 (m)	D2 (m)	DOS (%)	D3 (m)	D4 (m)	$\mathbf{max}\left|\mathit{\beta }\right|$ (deg)	$\mathbf{max}\left|\dot{\mathit{\beta }}\right|$
IPC#1	AC#1	FWS	2.32	−0.020	0.8	9.23	5.70	0.56	13.33
IPC#2	AC#2	4WS	2.02	−0.020	0.9	8.55	4.68	1.02	14.38
IPC#3 FWS	AC#3	+RWS	1.61	−0.021	0.8	7.99	3.48	0.83	12.01
	AC#4	+RWS+4WID	1.66	−0.020	0.6	8.88	5.43	0.87	11.53
	AC#5	+RWS+4WIB	1.44	−0.021	0.0	7.56	4.90	0.79	11.89
	AC#6	+RWS+4WID+4WIB	1.49	−0.021	0.4	8.45	5.43	0.90	11.42
	AC#7	+4WID	1.69	−0.021	0.3	8.17	4.97	0.58	12.47
	AC#8	+4WIB	1.55	−0.020	0.5	7.54	3.61	0.73	11.70
	AC#9	+4WID+4WIB	1.82	−0.021	0.3	8.73	5.32	0.62	12.38

show the simulation results for the LQR, LQSOF3, SMC, SOFSMC2, and SOFSMC3, respectively. In the simulation, for the LQSOF2, it was impossible to find a controller gain that gave a convergent trajectory. In other words, it is not recommended that the LQ SOF control using (e_y, e_ψ) be used for path tracking on low-friction surfaces. As shown in this case, the deviation between the controllers increases on low-friction surfaces. In these results, the LQR and SMC are used as a baseline.

Table 9 and Table 10 show the simulation results for the LQR and LQSOF3, respectively. As shown in these tables, the LQSOF3 shows slightly worse performance than the LQR. For example, the D₁, D₃, and D₄ for the LQSOF3 were slightly increased, compared to those for the LQR. The D₃ of the LQR varies from 7.90 to 8.91, while the D₃ of the LQSOF3 varies from 8.52 to 10.35. The D₄ of the LQR varies from 3.89 to 5.23 while the D₃ of the LQSOF3 varies from 3.67 to 7.51. This is caused by the fact that β is not used for control. For this reason, the LQR is preferred over the LQSOF3 on low-friction surfaces. In view of the actuator combination, AC#2 and AC#3 show the best performance for the LQR, and AC#3 and AC#5 show the best performance for the LQSOF3.

Table 11, Table 12 and Table 13 show the simulation results for the SMC, SOFSMC2, and SOFSMC3, respectively. As shown in these tables, the SOFSMC2 outperforms the SMC and SOFSMC3 in terms of path tracking performance. Moreover, the SOFSMC2 outperforms the LQR. On the other hand, the SMC and SMCSOF3 show almost the same results as each other. In other words, if the yaw rate, γ, is used for the SMCSOF3, it shows nearly identical performance to the SMC. For this reason, the SMCSOF2 is preferred over the SMC and SMCSOF3 on low-friction surfaces.

In view of the actuator combination, the best actuator combinations for the SMC, SOFSMC2, and SOFSMC3 are AC#8, AC#5, and AC#8, respectively. From those results, it is recommended to use the SOFSMC2 using (e_y, e_ψ) on low-friction surfaces and to select the actuator combination AC#8.

Figure 8 and Figure 9 show the simulation results for the LQSOF3 and SOFSMC3 for each actuator combination on low-friction surfaces. As shown in Figure 8 and Figure 9, there are few differences in vehicle trajectories and heading angles for the LQSOF3 and SMCSOF3 using the actuator combinations in spite of low-friction surfaces. It is natural that every actuator combination is tuned to give the best result. In other words, D₁ and D_OS were tuned to show nearly identical values to one another for actuator combinations, which resulted in nearly the same responses.

4.3. Discussion on Simulation Results

In Section 4.1 and Section 4.2, the proposed LQ SOF and SOF SMC controllers were verified through simulations using the vehicle simulation package. As a result, it was shown that both SOF controllers are effective in path tracking with multiple actuators. When implementing those controllers, there are obstacles, such as uncertain and time-varying parameters, measurement error, and noise. The most critical problem is to tune the velocity gain, k_v, according to the tire–road friction. This was investigated in reference [38]. Another problem in the real-world implementation of these controllers is caused by parameter uncertainties, measurement error, and noise. The parameter uncertainty can be overcome by using a robust controller [5,6,8,9]. The measurement noise can be cancelled by a filter, which causes a delay in the output signals. The delay caused by a filter can severely deteriorate path tracking performance [8,9,59]. The last implementation issue is caused by controller fragility. The controller fragility of the LQ SOF and SOF SMC controllers presented in this paper can deteriorate path tracking performance. To date, there have been few approaches for non-fragile path tracking control. In the area of robot trajectory tracking control, a robust non-fragile controller has been designed [60,61].

5. Conclusions

This paper presented a method for designing a path tracking controller using a static output feedback control structure. From a 2-DOF bicycle model and a target path, a state–space model was derived. In the state–space model, three input configurations, representing nine actuator combinations, were defined. With the state-space model, the LQ cost functions, full-state feedback LQR, and full-state feedback SMC were designed. Two sets of sensor outputs were defined for the SOF control. Using these sets, the LQ SOF and SOF SMC controllers were designed by using LQOC and SMC methodology. To distribute ΔM_z in IPC#3, the weighted least squares-based method was applied as a control allocation. To maximize the performance, the constraints related to the tire slip angle were imposed on the steering angles. To verify the performance of the path tracking controllers using the SOF control structure, simulations were performed in the vehicle simulation package, CarSim. From the simulation results, the following points were identified:

• The LQ SOF and SOF SMC controllers, using three sensor outputs, can give equivalent performances as the full-state feedback controls, LQR, and SMC. In other words, the LQ SOF and SOF SMC show nearly identical performance to the LQR and SMC if the yaw rate is used for the control, respectively. On the contrary, the SOF controllers using two sensor signals show the smallest D₁ and the largest side-slip angle because the side-slip angle is not used in the control. From the simulation results, small differences in terms of path tracking performance can be identified. This means that the side-slip angle and the yaw-rate signals are not needed for path tracking. In other words, the LQ SOF and SOF SMC controllers with two sensor signals, i.e., the LQSOF2 and SMCSOF2, are recommended for path tracking.
• In view of the actuator combination, it was shown that there are small differences among actuator combinations derived from three input configurations. This is caused by the fact that each tire can generate a larger lateral force on high-friction surfaces, and that each tire can generate a very small lateral force on low-friction surfaces. On high-friction surfaces, the simplest actuator, i.e., FWS, is recommended among the actuator combinations. If collision avoidance is critical, then the LQSOF2 and SOFSMC2 are recommended because they can reduce D₁. On low-friction surfaces, IPC#3 (FWS+RWS) is recommended for the LQR and LQSOF3, and IPC#3 (FWS+4WIB) is recommended for the SOFSMC2.

Based on the above results, it is desirable to use the LQSOF2 or SMCSOF2 with the actuator combinations FWS, FWS+RWS, or FWS+4WIB. Among the three actuator combinations, the third is preferred because 4WIB is mandatory to be installed on a real vehicle.

As mentioned earlier, parameter uncertainty, measurement error, and noise can deteriorate path tracking performance. Moreover, the controller gains of the LQ SOF and SOF SMC controllers are sensitive to their variations. To cope with these problems, a robust and non-fragile controller will be designed in future research.