Path Tracking Control with Constraint on Tire Slip Angles under Low-Friction Road Conditions

Featured Application

Path Tracking Control for Autonomous Driving.

Abstract

This paper presents a method to design a path tracking controller with a constraint on tire slip angles under low-friction road conditions. On a low-friction road surface, a lateral tire force is easily saturated and decreases as a tire slip angle increases by a large steering angle. Under this situation, a path tracking controller cannot achieve its maximum performance. To cope with this problem, it is necessary to limit tire slip angles to a value where the maximum lateral tire force is achieved. The most commonly used controllers for path tracking, linear quadratic regulator (LQR) and model predictive control (MPC), are adopted as a controller design methodology. The control inputs of LQR and MPC are front and rear steering angles and control yaw moment, which have been widely used for path tracking. The constraint derived from tire slip angles is imposed on the steering angles of LQR and MPC. To fully verify the performance of the path tracking controller with the constraint on tire slip angles, a simulation is conducted on vehicle simulation software. From the simulation results, it is shown that the path tracking controller with the constraint on tire slip angles presented in this paper is quite effective for path tracking on low-friction road surface.

1. Introduction

Over the past decade, research groups and the automobile industry have intensively explored autonomous driving because it is predicted to have impacts on several aspects of driving such as traffic flow, ride comfort, and road safety [1,2,3]. With respect to safety, autonomous driving has been expected to reduce social costs and damages incurred from drivers by preventing traffic accidents. As a result, a huge amount of papers on autonomous driving have been published in the meantime. According to the survey, it has been widely known that autonomous driving operations include perception, localization, planning, control and navigation, and system management [3]. Among them, planning or path planning generates a reference or target path based on information obtained from environments, and path tracking controls a vehicle, following the reference states with several actuators. This paper focuses on path tracking control (PTC). For the last decade, PTC has been intensively explored [4,5,6]. Consequently, a huge deal of papers have been published to date in the field of PTC [4,5,6,7,8,9,10].

To date, most path tracking controllers for autonomous driving have been designed on a high-friction road surface where the tire-road friction coefficient, μ, was set between 0.8 and 1.0. Since 2019, some papers concentrating on PTC under low-friction road conditions have been published [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]. These papers are classified according to controller design methodologies, input configurations, actuator combinations and tire-road friction coefficient, as given in

Table 1. Summary of path tracking controllers on low friction roads.

Controller	Input Configuration	Actuators	μ	Reference
LQR, MPC, SMC	[δf], [δf δr]	FWS, 4WS	0.4	[30]
MPC	[δf]	FWS	0.4, 0.5	[12]
			0.2, 0.5	[14]
			0.3, 0.8	[15]
			0.2, 0.85	[21]
			0.3	[22]
MPC	[δf ∆Fx]	FWS, 4WID	0.4, 0.5	[11]
		FWS, 4WID, 4WIB	0.6, 0.7	[16]
		FWS, 4WID, 4WIB	0.4, 0.8	[17]
		FWS, 4WIB	0.3	[28]
MPC	[δf]	FWS, 4WID	0.4,0.7	[19]
	[δf ∆Fx ∆Mz]	FWS, 4WID	0.3	[20]
	[δf], [∆Mz]	FWS, 4WID, 4WIB	0.3	[23]
	[δf ∆Mz]	FWS, 4WID	0.5, 0.6	[29]
LQR	[δf δr]	4WS, 4WIS	0.4, 0.65	[13]
	[δf δr]	4WS	0.25, 0.5	[18]
	[δf]	FWS	0.34	[24]
LQR	[δf], [δf δr], [δf δr ∆Mz]	FWS, 4WS, 4WIB, 4WID	0.4	[31,32]
SMC	[∆Mz]	4WIS, 4WID, 4WIB	0.4	[25]
	[δf]	FWS	0.5	[26]
	[δf ∆Mz]	FWS, 4WID, 4WIB	0.3	[27]

. As shown in Table 1, the lower limits of μ were set between 0.25 and 0.4. In this paper, a low friction road condition means that μ is 0.4 or less.

In Table 1, δ_f, δ_r, ∆F_x and ∆M_z are the control inputs, which represent the front and rear steering angles, the control longitudinal force, and the control yaw moment, respectively. An input configuration is composed of those control inputs. In Table 1, seven input configurations were given. With those input configurations, a linear quadratic regulator (LQR), model predictive control (MPC), and sliding mode control (SMC) were adopted as a controller design methodology.

From the point of view of an actuator for PTC, front wheel steering (FWS), rear wheel steering (RWS) and 4-wheel steering (4WS) have been selected as an actuator for PTC to date. For the last decade, by virtue of development of in-wheel motor or e-corner module systems, 4-wheel independent steering/braking/driving (4WIS/4WIB/4WID) has been available for vehicle stability control (VSC) and PTC. Several actuator combinations can be composed of those actuators, as shown in Table 1. If ∆M_z or ∆F_x is selected as a control input, a control allocation is needed to distribute it the control tire forces generated by RWS, 4WID and 4WIB.

Naturally, on a low-friction road surface, the magnitudes of longitudinal and lateral tire forces, F_x and F_y, become small because μ is low [33,34]. Among them, F_y’s generated at front wheels have significant effects on path tracking performance in cornering. Figure 1 shows F_y with respect to the tire slip angle, α, and μ. In Figure 1, the values of the tire slip angles, α_m and α_∞, are 5° and 20°, which give the maximum and saturated lateral tire forces, F_y,max and F_y,∞, respectively. As shown in Figure 1, F_y is saturated and decreased to F_y,∞ as α increases to α_∞ over α_m where F_y,max is generated. In other words, F_y becomes F_y,∞, which is smaller than F_y,max due to larger slip angle α_∞ over α_m. For example in Figure 1, 16% of F_y,max is reduced at α_∞ when μ is 0.6. Moreover, F_y is reduced as μ 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 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 α, 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, β, 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 α is over α_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 α. 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 α be α_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, α 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.

The aim of this paper is to present a method to impose a CTSA for the purpose of making a path tracking controller better under low friction conditions. From the definition of tire slip angle, CTSA is derived. To consider CTSA in controller design procedure, LQR and MPC are adopted as a controller design methodology. δ_f and δ_r and ∆M_z are selected as a control input, which make three input configurations. When imposing CTSA in LQR, the control inputs, δ_f and δ_r, are clipped by the constraint. In MPC, the constraint on steering angle is simply added into the optimization procedure. To distribute ∆M_z into tire forces generated by various actuator combinations, a control allocation is applied. Through the control allocation, the effects of various actuator combinations composed from FWS, 4WS, 4WID and 4WIB, on path tracking performance is analyzed. A simulation is done on a vehicle simulation software, CarSim 8.02, to verify path tracking performance with CTSAs. From simulation results, it is concluded that a path tracking controller with CTSA shows quite a better performance in terms of path tracking than one without CTSA on low-friction road surface.

The contributions of this paper can be summarized as follows:

• This paper presents a method to impose a CTSA on steering angles in order to generate the maximum lateral force on a low-friction road surface. The constraint is simply derived from the definition of tire slip angle and easily applied into steering angles, i.e., control inputs of LQR and MPC. From simulation results, it is verified that the CTSA can greatly improve the path tracking performance on low-friction surface.
• The proposed CTSA makes the most of the lateral tire force. This means that there are little tire forces left to improve the control performance with extra actuators. Under that condition, the effects of various combinations composed from the actuators such as RWS, 4WID, and 4WIB on control performance are discussed from simulation results.
• From the discussion on simulation results, it is recommended that LQR is better than MPC in terms of computation time and simplicity of controller structure, and that the actuator combination, FWS + RWS, shows the best performance under CTSA.

This paper consists of five sections. In Section 2, design procedures for LQR and MPC with various actuator combinations are presented. Moreover, the derivation procedure for CTSAs is also presented. In Section 3, performance measures on path tracking are presented for the purpose of tuning LQR and MPC. In Section 4, simulation is conducted and simulation results are discussed in view of the performance measures. The conclusion of this paper is drawn in Section 5.

2. Design of Path Tracking Controller with Constraints on Tire Slip Angles

2.1. Vehicle Model

In the context of PTC, a 2-DOF bicycle model has been widely employed to represent yaw and lateral behaviors of a vehicle, as discussed in the literature [25,30,31,32,33,34,35,39,40]. Figure 2 shows the coordinates, geometry, and variables of the bicycle model. This models the lateral and yaw motions of a vehicle under the assumption of a constant longitudinal velocity. The state variables for this model are the yaw rate, γ, and the side-slip angle, β. With these variables, the equations of motion (1) are obtained [32]. The tire slip angles, denoted as α_f and α_r, are defined as Equation (2) for the front and rear wheels. The linear lateral tire forces, F_yf and F_yr, are calculated as Equation (3). By combining Equations (1)–(3), the state-space equation for the bicycle model is derived as Equation (4).

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

(1)

$${\alpha }_f\left(t\right)={\delta }_f\left(t\right)-\beta \left(t\right)-\frac{l_f\gamma \left(t\right)}{v_x},\hspace{1em}{\alpha }_r\left(t\right)={\delta }_r\left(t\right)-\beta \left(t\right)+\frac{l_r\gamma \left(t\right)}{v_x}$$

(2)

$$F_{yf}\left(t\right)=C_f{\alpha }_f\left(t\right),\hspace{1em}{F}_{yr}\left(t\right)=C_r{\alpha }_r\left(t\right)$$

(3)

$$\begin{array}{l}\dot{\beta }\left(t\right)=\left(-\frac{C_f+C_r}{m{v}_x}\right)\beta \left(t\right)+\left(\frac{-C_f{l}_f+C_r{l}_r}{m{v}_x^2}-1\right)\gamma \left(t\right)+\frac{C_f}{m{v}_x}{\delta }_f\left(t\right)+\frac{C_r}{m{v}_x}{\delta }_r\left(t\right)\\ \dot{\gamma }\left(t\right)=\left(\frac{-C_f{l}_f+C_r{l}_r}{I_z}\right)\beta \left(t\right)+\left(-\frac{l_f^2{C}_f+l_r^2{C}_r}{I_z{v}_x}\right)\gamma \left(t\right)+\frac{l_f{C}_f}{I_z}{\delta }_f\left(t\right)-\frac{l_r{C}_r}{I_z}{\delta }_r\left(t\right)+\frac{1}{I_z}\mathrm{\Delta }{M}_z\left(t\right)\end{array}$$

(4)

Figure 3 shows the coordinates and variables derived from the target path for path tracking. The literature on PTC introduced the concepts of lateral offset and heading errors, denoted as e_y and e_φ, at a specific point C as depicted in Figure 3 [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]. To enhance path tracking performance, this paper introduces a preview function, as used in the previous works [24,26,30,31,32]. The preview distance, L_p, is calculated using Equation (5), where k_v is the velocity gain representing the preview interval. In Figure 3, the point C is located at the vehicle’s center of gravity. The preview point, P, and the point Q on the target path are determined with L_p. At the point Q, the time derivatives of e_y and e_φ are calculated as Equation (6), under the assumption that e_φ is limited to an angle less than 10 degrees and, consequently, can be approximated as sine_φ ≈ e_φ.

$$L_p=k_v\cdot v_x$$

(5)

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

(6)

The vectors x, w, and u of the states, the disturbances, and control inputs are defined as Equation (7), respectively [11,12,14,15,16,17,19,20,21,22,23,28,29,31,32]. With those vectors and Equations (4), (6) and (7), the state–space equation for PTC is derived as Equation (8), where the matrices A, B₁ and B₂ are defined as Equation (9) [32].

$$\left\{\begin{array}{l}x\left(t\right)={\left[\begin{array}{cccc}{e}_y\left(t\right)& e_{\phi }\left(t\right)& \beta \left(t\right)& \gamma \left(t\right)\end{array}\right]}^T\hfill \\ w\left(t\right)=\chi \left(t\right)\hfill \\ u\left(t\right)={\left[\begin{array}{ccc}{\delta }_f\left(t\right)& {\delta }_r\left(t\right)& \mathrm{\Delta }{M}_z\left(t\right)\end{array}\right]}^T\hfill \end{array}\right.$$

(7)

$$\dot{x}\left(t\right)=Ax\left(t\right)+B_{1}w\left(t\right)+B_{2}u\left(t\right)$$

(8)

$$A=\left[\begin{array}{cccc}0& v_x& -v_x& -L_p\\ 0& 0& 0& -1\\ 0& 0& -\frac{C_f+C_r}{m{v}_x}& \frac{-C_f{l}_f+C_r{l}_r}{m{v}_x^2}-1\\ 0& 0& \frac{-C_f{l}_f+C_r{l}_r}{I_z}& -\frac{l_f^2{C}_f+l_r^2{C}_r}{I_z{v}_x}\end{array}\right],\hspace{1em}{B}_1=\left[\begin{array}{c}0\\ v_x\\ 0\\ 0\end{array}\right],\hspace{1em}{B}_2=\left[\begin{array}{lll}0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill \\ \frac{C_f}{m{v}_x}\hfill & \frac{C_r}{m{v}_x}\hfill & 0\hfill \\ \frac{l_f{C}_f}{I_z}\hfill & -\frac{l_r{C}_r}{I_z}\hfill & \frac{1}{I_z}\hfill \end{array}\right]$$

(9)

Three types of control input, named u₁, u₂ and u_3, corresponding to input configurations IC#1, IC#2 and IC#3, are composed of three elements, δ_f, δ_r and ∆M_z, as given in Equation (10) [31,32]. In Equation (10), B₂{k} representing the k-th column of the matrix B₂. In Equation (10), δ_f is available in all input configurations, while δ_r is exclusively selected in IC#2 and IC#3. In the case of IC#3, RWS, 4WIB and 4WID are used to generate ∆M_z. With those actuators, δ_r and the braking and traction torques (T_Bi and T_Di) for each wheel are obtained from ∆M_z. This paper assumes that T_Bi and T_Di are generated through the use of 4WIB and 4WID, respectively.

$$\left\{\begin{array}{ccc}{u}_1\left(t\right)={\delta }_f\left(t\right)\hfill & B_{21}=\left[B_2\left\{1\right\}\right]\hfill & \mathrm{for}\hspace{0.33em}IC\#1\hfill \\ u_2\left(t\right)={\left[{\delta }_f\left(t\right)\hspace{0.33em}{\delta }_r\left(t\right)\right]}^T\hfill & B_{22}=\left[\begin{array}{cc}{B}_2\left\{1\right\}& B_2\left\{2\right\}\end{array}\right]\hfill & \mathrm{for}\hspace{0.33em}IC\#2\hfill \\ u_3\left(t\right)={\left[{\delta }_f\left(t\right)\hspace{0.33em}\mathrm{\Delta }{M}_z\left(t\right)\right]}^T\hfill & B_{23}=\left[\begin{array}{cc}{B}_2\left\{1\right\}& B_2\left\{3\right\}\end{array}\right]\hfill & \mathrm{for}\hspace{0.33em}IC\#3\hfill \end{array}\right.$$

(10)

2.2. Design of LQR

When designing LQR, a linear quadratic cost function (LQCF) is to be defined. LQCFs, J₁, J₂ and J₃, for PTC with three input configurations, IC#1, IC#2, and IC#3, are defined as Equation (11), respectively. These LQCFs are transformed into the vector-matrix form as Equation (12). The weighting matrices Q and R_i in Equation (12) are defined in Equation (13). The weight ρ_i in LQCFs is determined with Bryson’s rule, as given in Equation (14), where ξ_i represents the maximum allowable value for the associated terms [31,32,46]. For better path tracking performance, ξ_i and k_v are tuned. The control inputs u₁, u₂ and u₃ for IC#1, IC#2 and IC#3 are obtained as Equation (15). In Equation (15), P_i stands for the solution to the Riccati equation for IC#i.

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

(11)

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

(12)

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

(13)

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

(14)

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

(15)

2.3. Constraint on Tire Slip Angles

Generally, there are physical limits on the front and rear steering angles, δ_f and δ_r. For example, δ_f and δ_r are limited to 30° and 5°, respectively. Under the limitations, δ_f is easily saturated on a low-friction road surface [33,34]. Moreover, as shown in Figure 1, F_yf is also saturated and decreased as α_f is over α_m where F_y,max is generated. In case of that condition, a path tracking controller cannot achieve the maximum performance. For the reason, α should be limited to α_m, as given in Equation (16). By combining Equation (2) and Equation (16), Equation (17) is obtained. From Equation (17), the constraints on α_f and α_r are obtained as Equation (18). Let this denote the constraint on the tire slip angle, Equation (18), as CTSA.

$$\left|{\alpha }_f\left(t\right)\right|\le {\alpha }_m,\hspace{0.33em}\left|{\alpha }_r\left(t\right)\right|\le {\alpha }_m$$

(16)

$$\left|{\alpha }_f\left(t\right)\right|=\left|{\delta }_f\left(t\right)-\beta \left(t\right)-\frac{l_f\gamma \left(t\right)}{v_x}\right|\le {\alpha }_m,\hspace{1em}\left|{\alpha }_r\left(t\right)\right|=\left|{\delta }_r\left(t\right)-\beta \left(t\right)+\frac{l_r\gamma \left(t\right)}{v_x}\right|\le {\alpha }_m$$

(17)

$$\left\{\begin{array}{l}-{\alpha }_m+\beta \left(t\right)+\frac{l_f\gamma \left(t\right)}{v_x}\le {\delta }_f\left(t\right)\le {\alpha }_m+\beta \left(t\right)+\frac{l_f\gamma \left(t\right)}{v_x}\hspace{1em}\\ -{\alpha }_m+\beta \left(t\right)-\frac{l_r\gamma \left(t\right)}{v_x}\le {\delta }_r\left(t\right)\le {\alpha }_m+\beta \left(t\right)-\frac{l_r\gamma \left(t\right)}{v_x}\end{array}\right.$$

(18)

In order to calculate the lower and upper bounds of Equation (18), it is essential to measure or estimate β. However, β is hard to measure with an on-board sensor. For this reason, β should be estimated with an observer. For the purpose, an extended Kalman Filter is employed in this paper [47].

From the point of view of LQR, Equation (18) is the constraint on the control inputs, δ_f and δ_r. For this reason, it is necessary to design LQR with the consideration of the constraints. The simplest way to impose the constraints, Equation (18), is to clip the control inputs obtained from LQR with the constraints. This does not need any additional procedures. The second method is to design with LQR under the constraints, Equation (18). The third method is to use MPC because it can explicitly handle any constraints in optimization procedure. The second method is not considered in this paper. Let this denote LQR without and with the CTSA as LQRN and LQRC, respectively.

2.4. Design of MPC with Constraints on Tire Slip Angles

As shown in Table 1, MPC has been widely adopted as a path tracking controller for the last decade because it can handle nonlinear or time-varying systems and nonlinear constraints besides linear time-invariant ones [11,12,14,16,17,19,20,21,22,27,28,29,30]. Generally, MPC is designed in the discrete-time domain. For the reason, the first step of MPC is to discretize Equation (8) with the sampling time, T_s. As a result, the discrete–time version of Equation (8) is obtained as Equation (19). In Equation (19), the subscript k represents a particular time instant. If k is 0, it means the current time instant. The matrices F, G and H in Equation (19) are obtained as an approximated form of Equation (20). In this paper, it is assumed that the disturbance w_k in Equation (19) is set to 0. The discrete–time version of LQCF, Equation (11), is obtained as Equation (21). In Equation (21), N is the prediction horizon. Let this denote the current state vector as x₀. The state vector at the time instant k is obtained as Equation (22) from Equation (19). In Equation (22), x₀ is the current state. By expanding Equation (22) from the first to N-th time instant, new vectors and matrices are defined as Equation (23). With these definitions, the state–space equation and LQCF from the first to N-th time instant are calculated as Equation (24) and Equation (25), respectively. By replacing X of Equation (25) with Equation (24), the LQCF, Equation (25), is obtained by Equation (26). In Equation (27), the bound on steering angle, δ_max, is imposed on the control input U. Moreover, instead of Equation (27), CTSAs can be imposed on the control input, as set in Equation (28). In Equation (28), U_f and U_r are the elements of U corresponding to δ_f and δ_r, respectively. β₀ and γ₀ are the values of β and γ, obtained from the current state x₀. With this manner, the quadratic programming with the optimization variable U, the cost function J_MPC, and the bound constraints of Equation (27) and Equation (28) is formulated [7]. In this paper, the solver, quadprog(), provided in MATLAB R2019a optimization toolbox, was applied to find the optimum U at every time instant k. After finding the optimum, MPC usually selects the first element of the optimum U as the control input. In this paper, T_s and N are set to, respectively, 0.01 s and 30.

$$x_{k+1}=F{x}_k+G{w}_k+H{u}_k$$

(19)

$$F=I+A\cdot T_s,\hspace{1em}G=B_1\cdot T_s,\hspace{1em}H=B_2\cdot T_s$$

(20)

$$J_{MPC}={\displaystyle \sum _{k=0}^{N-1}\left(x_k^{T}Q{x}_k+u_k^{T}R{u}_k\right)}$$

(21)

$$x_{k+1}=F^k{x}_0+{\displaystyle \sum _{i=0}^{k-1}{F}^{i-1}}H{u}_{k-i-1}$$

(22)

$$\begin{gathered} X=\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\\ ⋮\\ x_N\end{array}\right],\hspace{1em}\overline{F}=\left[\begin{array}{c}F\\ F^2\\ F^3\\ ⋮\\ F^N\end{array}\right],\hspace{1em}\overline{H}=\left[\begin{array}{ccccc}H& 0& 0& 0& 0\\ FH& H& 0& 0& 0\\ F^{2}H& FH& H& 0& 0\\ ⋮& ⋮& ⋮& \ddots & 0\\ F^{N-1}H& F^{N-2}H& F^{N-3}H& \cdots & H\end{array}\right],\hspace{1em}U=\left[\begin{array}{c}{u}_0\\ u_1\\ u_2\\ ⋮\\ u_{N-1}\end{array}\right] \\ \overline{Q}=diag\left(Q,\cdots ,Q\right),\hspace{1em}\overline{R}=diag\left(R,\cdots ,R\right) \end{gathered}$$

(23)

$$X=\overline{F}{x}_0+\overline{H}U$$

(24)

$$J_{MPC}=x_0^{T}Q{x}_0+X^T\overline{Q}X+U^T\overline{R}U$$

(25)

$$J_{MPC}=U^T\left({\overline{H}}^T\overline{Q}\overline{H}+\overline{R}\right)U+2x_0^T{\overline{F}}^T\overline{Q}\overline{H}U$$

(26)

$$-{\delta }_{\max }\le U\le {\delta }_{\max }$$

(27)

$$\left\{\begin{array}{l}-{\alpha }_m+{\beta }_0+\frac{l_f{\gamma }_0}{v_x}\le U_f\le {\alpha }_m+{\beta }_0+\frac{l_f{\gamma }_0}{v_x}\hspace{1em}\\ -{\alpha }_m+{\beta }_0-\frac{l_r{\gamma }_0}{v_x}\le U_r\le {\alpha }_m+{\beta }_0-\frac{l_r{\gamma }_0}{v_x}\end{array}\right.$$

(28)

Different from LQR, MPC can directly handle the constraints on the control inputs, i.e., δ_f and δ_r, as given in Equation (27) and Equation (28). However, MPC is to solve the optimization problem every time step, which requires a large amount of computations, compared to LQR with clipping. For the reason, there are hard limitations in implementing MPC on real vehicles. Let this denote MPC without and with the CTSA as MPCN and MPCC, respectively.

2.5. Control Allocation for LQR and MPC with IC#3

As shown in Equation (10), ΔM_z is calculated from IC#3. After the controllers calculate ΔM_z, it should be converted into the longitudinal forces applied to wheels. This force is generated by various combinations of the actuators, RWS, 4WIB, and 4WID. 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 ΔM_z when ΔM_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, ∆F_yr represents F_y generated by the rear wheels through RWS. ∆F_x1, ∆F_x2, ∆F_x3 and ∆F_x4 denote F_x‘s generated by 4WID and 4WIB. If ∆F_xi is positive, it corresponds to T_Di generated by 4WID; otherwise, it corresponds to T_Bi generated by 4WIB. Those five tire forces need to be determined so as to generate ΔM_z. For the purpose, WLS-based method is selected as a control allocation method.

The equilibrium condition among ∆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 z are given in Equation (30).

$$\underset{z\left(t\right)}{\underbrace{\left[\begin{array}{ccccc}{z}_1\left(t\right)& z_2\left(t\right)& z_3\left(t\right)& z_4\left(t\right)& z_5\left(t\right)\end{array}\right]}}\underset{q\left(t\right)}{\underbrace{\left[\begin{array}{c}\mathrm{\Delta }{F}_{yr}\left(t\right)\\ \mathrm{\Delta }{F}_{x1}\left(t\right)\\ \mathrm{\Delta }{F}_{x2}\left(t\right)\\ \mathrm{\Delta }{F}_{x3}\left(t\right)\\ \mathrm{\Delta }{F}_{x4}\left(t\right)\end{array}\right]}}=z\left(t\right)q\left(t\right)=\mathrm{\Delta }{M}_z\left(t\right)$$

(29)

$$\left\{\begin{array}{l}{z}_1\left(t\right)=2l_r\cos {\delta }_r\left(t\right),\\ z_2\left(t\right)=-l_f\sin {\delta }_f\left(t\right)+t_f\cos {\delta }_f\left(t\right),\hspace{1em}{z}_3\left(t\right)=-l_f\sin {\delta }_f\left(t\right)-t_f\cos {\delta }_f\left(t\right),\\ z_4\left(t\right)=l_r\sin {\delta }_r\left(t\right)+t_r\cos {\delta }_r\left(t\right),\hspace{1em}\hspace{0.33em}\hspace{0.33em}{z}_5\left(t\right)=l_r\sin {\delta }_r\left(t\right)-t_r\cos {\delta }_r\left(t\right)\end{array}\right.$$

(30)

The quadratic cost function for WLS is given as Equation (31). Within Equation (31), ζ_i is the product of μ and F_zi at the wheel i, i.e., μF_zi, which represents the friction circle radius. The vector of virtual weights, denoted as κ, 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 ∆M_z. The cost functions, Equations (31) and (32), are summed into a single one (33) with the Lagrange multiplier, η. In Equation (33), it is important to set η 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 q [31,32].

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

(31)

$$J_{EC}={\left\{z\left(t\right)q\left(t\right)-\mathrm{\Delta }{M}_z\left(t\right)\right\}}^T\left\{z\left(t\right)q\left(t\right)-\mathrm{\Delta }{M}_z\left(t\right)\right\}$$

(32)

$$J_{CA}=J_Q+\eta \cdot J_{EC}=q^T\left(t\right)\mathrm{\Theta }q\left(t\right)+\eta {\left\{z\left(t\right)q\left(t\right)-\mathrm{\Delta }{M}_z\left(t\right)\right\}}^T\left\{z\left(t\right)q\left(t\right)-\mathrm{\Delta }{M}_z\left(t\right)\right\}$$

(33)

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

(34)

When WLS is applied to control allocation for IC#3, an arbitrary combination of actuators comprising RWS, 4WID, and 4WIB can be configured. To account for several actuator combinations, the virtual weights, denoted as κ_i, can be set to match a particular combination of those actuators [31,32]. The detailed usage of the virtual weights and the procedure to derive δ_r, T_Di and T_Bi of each wheel can be also found in the literature [31,32].

In the field of VSC or PTC, the friction circle constraint, as defined in Equation (35), has been included into the optimization procedure for control allocation [48,49,50]. This constraint can be transformed to define the tire force margin (TFM), as defined in Equation (36). TFM indicates how much extra tire force can be generated under VSC or PTC. If TFM is small, F_x and F_y are near its maximum, which indicates the fact that an extra actuator selected to generate these forces becomes useless. If TFM is zero, this indicates the fact that both F_x and F_y are saturated and that there are no margins to generate extra tire force for VSC or PTC. As shown in Equation (36), TFM becomes much smaller under low-friction road conditions. If CTSA is imposed, then it is expected that TFM easily becomes zero.

$${\left\{\mu F_z\left(t\right)\right\}}^2\ge F_x^2\left(t\right)+F_y^2\left(t\right)$$

(35)

$$\mathrm{TFM}\left(t\right)=\mu F_z\left(t\right)-\sqrt{F_x^2\left(t\right)+F_y^2\left(t\right)}$$

(36)

3. Performance Measures for Path Tracking Control

In the domain of PTC, e_y and e_φ have been widely used as indicators for assessing path tracking performance. However, different from conventional PTC literature, this paper adopts a double lane change maneuver as the target path for collision avoidance [13,16,17,20,21,31,32]. This maneuver was employed to evaluate the reachability and agility of a path tracking controller [31,32]. Figure 5 shows both the target path and the trajectory of a vehicle. In Figure 5, the points A, B and C are on the target path, which represent the peak, zero-crossing and convergence points, respectively. The points D, E, F and G are on the vehicle trajectory, which represent the peak, zero-crossing, over-shoot and convergence points, respectively. Five metrics, M_X, M_Y, M_OS, M_DX and M_SX, representing path tracking performance, are defined as Equation (37), based on the points along the target path and vehicle trajectory in Figure 5. In Equation (37), the subscripts X and Y correspond to the x- and y-positions of the corresponding point, respectively.

$$\left\{\begin{array}{l}{M}_X=D_X-A_X=D_X-73.20\\ M_Y=D_Y-A_Y=D_Y-3.53\\ M_{DX}=E_X-B_X=E_X-91.50\\ M_{SX}=G_X-C_X=G_X-190.00\\ M_{OS}=\frac{\left|F_Y\right|-1.65}{1.65+3.53}\times 100\end{array}\right.$$

(37)

M_X and M_Y represent the peak’s center and lateral offsets, respectively. In this study, agility and reachability are represented by M_X and M_Y, respectively. If M_Y exceeds −0.02 m, the controller’s path tracking performance can be regarded as satisfactory. M_OS represents the percentage of overshoot, which is identical to that of automatic control theory. This indicates lateral damping, reflecting agility. In this paper, a satisfactory performance is achieved if M_OS remains below 16%, which is equivalent to an overshoot of 0.85 m. M_DX and M_SX represent the response delay and the settling delay, respectively. M_DX corresponds to a delay, standing for the agility of longitudinal motion. M_DX highly depends on μ. M_SX can be interpreted as the settling time, indicating the convergence speed of the vehicle’s lateral motion toward a target y-position. M_SX is closely related to M_OS. In essence, smaller absolute values of these metrics correspond to improved path tracking performance.

As mentioned previously, VSC and LSC aim for yaw rate tracking and lateral stability, respectively. Typically, yaw rate tracking performance is assessed by the difference between real and reference yaw rates, while lateral stability performance is evaluated by the magnitude of β. It has been considered satisfactory if β is less than 3 degrees [51]. Let the maximum absolute value of β be referred to as MASSA. The side-slip angular rate, when combined with β, has been also employed as a measure of lateral stability [30]. Let this denote the maximum absolute value of side-slip angular rate as MASSAR.

In this paper, the tuning parameters are the maximum allowable value ξ_i given in Equation (14) and the velocity gain k_v given in Equation (5). There are three tuning objectives when tuning these parameters in this paper. The first objective is collision avoidance, which is achieved by tuning parameters such that M_Y is larger than −0.02 m. Generally, the best M_SX is attained when M_OS is around 0.9, provided that M_Y exceeds −0.02 m. The second objective is path tracking, which is achieved by tuning parameters such that M_X, M_DX and M_SX are maintained as small as possible. Among them, the minimum of M_DX clearly depends on μ. The third objective is lateral stability, which is achieved by tuning parameters such that MASSA is less than 2°.

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 4WS, and 4WIB and 4WID, where the bandwidths of these actuators are set to 5 Hz and 2 Hz, respectively. The initial v_x and μ 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 δ_r, the larger β. For the reason, it is necessary to limit ∆M_z to a certain value so as to make δ_r small for IC#3. F_y,max from F-segment sedan model in CarSim is 7500 N where μ is 1 [52]. This will be 3000 N if μ is set to 0.4. As shown in Equation (29), F_y at front wheels, i.e., ∆F_yf, was not used to generate ∆M_z because δ_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 × 2 × l_r = 11,400 Nm. After ∆M_z was obtained from LQR with IC#3, it is limited by clipping it into 11,400 Nm in this paper. δ_max of front and rear wheels was set to 30°. α_m was set to 5°, which is referred from CarSim tire model as given in Figure 1.

The first simulation is done for LQRN and LQRC.

Table 2. Summary of the simulation results with LQRN for IC#1, IC#2, IC#3.

	Actuator Combinations	MX (m)	MY (m)	MOS	MDX (m)	MSX (m)	MASSA (deg)	MASSAR (deg/s)
IC#1	FWS	1.25	−0.018	1.0	8.58	4.39	0.58	13.19
IC#2	4WS	1.58	−0.017	0.9	8.94	4.97	1.26	13.99
IC#3 FWS	+RWS	1.15	−0.019	0.9	8.51	4.31	0.79	13.16
	+RWS + 4WID	1.24	−0.018	0.9	8.47	4.54	0.77	13.13
	+RWS + 4WIB	1.62	−0.018	0.9	8.66	4.11	0.80	12.97
	+RWS + 4WID + 4WIB	1.47	−0.018	0.9	9.44	5.52	0.80	13.25
	+4WID	1.67	−0.018	0.9	9.04	5.10	0.60	13.38
	+4WIB	1.69	−0.017	0.9	8.14	3.57	0.68	12.71
	+4WID + 4WIB	1.66	−0.017	0.9	9.64	5.71	0.63	13.47

and

Table 3. Summary of the simulation results with LQRC for IC#1, IC#2, IC#3.

	Actuator Combinations	MX (m)	MY (m)	MOS	MDX (m)	MSX (m)	MASSA (deg)	MASSAR (deg/s)
IC#1	FWS	0.77 (38%)	−0.017	0.9	5.95 (31%)	1.67 (62%)	0.87	13.14
IC#2	4WS	1.01 (36%)	−0.017	0.9	6.03 (33%)	1.42 (71%)	0.70	12.90
IC#3 FWS	+RWS	0.29 (75%)	−0.017	0.9	5.28 (38%)	0.28 (94%)	0.83	13.73
	+RWS + 4WID	0.36 (71%)	−0.018	0.9	5.78 (32%)	1.24 (73%)	0.79	13.52
	+RWS + 4WIB	1.07 (34%)	−0.018	0.9	6.36 (27%)	1.71 (58%)	0.91	13.76
	+RWS + 4WID + 4WIB	0.46 (69%)	−0.018	0.9	5.57 (41%)	0.50 (91%)	0.87	14.48
	+4WID	1.13 (32%)	−0.018	0.9	7.63 (16%)	4.30 (16%)	0.96	13.36
	+4WIB	1.44 (15%)	−0.018	0.9	7.38 (9%)	3.50 (2%)	1.63	13.61
	+4WID + 4WIB	0.99 (40%)	−0.018	0.9	6.16 (36%)	1.40 (76%)	0.83	13.23

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 Table 2 and Table 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 α to α_m, which generates F_y,max.

In case of LQRN, the actuator combination, FWS + 4WIB 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 4WID and 4WIB 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, δ_f and δ_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. Moreover, it is also concluded that FWS + RWS of IC#3 is recommended as an actuator combination instead of IC#2 in terms of path tracking performance, and that IC#1 or IC#2 is recommended as an input configuration instead of IC#3 in terms of simplicity and ease of implementation.

Figure 6 shows the plots drawn from the simulation results of IC#1 of LQRN and LQRC. In Figure 6d,e, the legends, FL, FR, RL and RR, stand for front left, front right, rear left and rear right wheels, respectively. As shown in Figure 6a, δ_f was limited to a certain value near α_m. Compared to the limits on the steering angles, ±30°, α_m, i.e., 5°, is quite small. As a result, F_y‘s at front wheels were not saturated and were generated to its own maximum, as shown in Figure 6d. For the reason, TFMs of front wheels become 0, as shown in Figure 6e, and the path tracking performance of LQRC was improved, compared to LQRN, as shown in Figure 6b. Because the maximum F_y‘s are generated by FWS with CTSA of LQRC, TFMs of LQRC at front wheels became zero, which indicates the other actuators are not needed for FWS to generate extra F_y’s. This can be checked in Figure 6e. On the other hand, as shown in Figure 6d, TFMs of LQRN at front wheels did not become zero, which means the other actuators can generate extra F_y‘s at front wheels.

The second simulation is done for MPCN and MPCC.

Table 4. Summary of the simulation results with MPCN for IC#1, IC#2, IC#3.

	Actuator Combinations	MX (m)	MY (m)	MOS	MDX (m)	MSX (m)	MASSA (deg)	MASSAR (deg/s)
IC#1	FWS	2.10	−0.018	0.9	9.42	5.67	0.57	12.92
IC#2	4WS	2.17	−0.019	0.9	9.42	5.95	1.48	13.32
IC#3 FWS	+RWS	2.07	−0.018	0.9	9.44	5.36	0.78	13.23
	+RWS + 4WID	2.45	−0.018	0.9	9.78	5.84	0.80	13.25
	+RWS + 4WIB	2.35	−0.019	0.9	9.42	4.85	0.86	12.82
	+RWS + 4WID + 4WIB	2.17	−0.019	0.9	10.16	6.34	0.83	13.01
	+4WID	2.35	−0.019	0.9	9.80	5.83	0.61	13.38
	+4WIB	2.05	−0.019	0.9	8.25	3.77	0.66	11.77
	+4WID + 4WIB	2.17	−0.019	0.9	10.16	6.34	0.83	13.01

and

Table 5. Summary of the simulation results with MPCC for IC#1, IC#2, IC#3.

	Actuator Combinations	MX (m)	MY (m)	MOS	MDX (m)	MSX (m)	MASSA (deg)	MASSAR (deg/s)
IC#1	FWS	1.54 (27%)	−0.018	0.9	6.72 (29%)	2.69 (53%)	0.83	12.83
IC#2	4WS	1.54 (29%)	−0.019	0.9	6.68 (29%)	2.49 (58%)	1.33	13.45
IC#3 FWS	+RWS	1.22 (41%)	−0.018	0.9	6.20 (34%)	1.32 (75%)	0.83	13.84
	+RWS + 4WID	1.45 (41%)	−0.018	0.9	7.12 (27%)	2.88 (51%)	0.80	13.15
	+RWS + 4WIB	1.85 (21%)	−0.018	0.9	6.87 (27%)	2.63 (46%)	0.84	12.57
	+RWS + 4WID + 4WIB	1.41 (35%)	−0.019	0.9	6.47 (36%)	1.57 (75%)	0.84	13.73
	+4WID	1.85 (21%)	−0.019	0.9	8.53 (13%)	5.33 (9%)	0.90	13.26
	+4WIB	2.08 (2%)	−0.018	0.9	7.73 (6%)	3.40 (10%)	1.88	11.31
	+4WID + 4WIB	1.41 (35%)	−0.019	0.9	6.47 (36%)	1.57 (75%)	0.84	13.73

show the simulation results of MPCN and MPCC. As shown in those tables, M_X, M_DX and M_SX of MPCC are improved, compared to those of MPCN. This coincides with the results of LQRC. This means that CTSA can effectively limit α to α_m, which cause the maximum F_y.

Figure 7 show the plots drawn from the simulation results of IC#1 of MPCN and MPCC. In Figure 7d,e, the legends, FL, FR, RL and RR, stand for front left, front right, rear left and rear right wheels, respectively. As shown in Figure 7c, F_y’s of MPCN were saturated due to the large α. On the contrary, F_y’s of MPCC were not saturated due to the CTSA, as shown in Figure 7c. Because the maximum F_y’s are generated by FWS with the CTSA of MPCC, TFM of MPCC at front wheels became zero, which means the other actuators are not needed for FWS to generate extra F_y’s. This can be checked in Figure 7d. On the other hand, as shown in Figure 6d, TFM of MPCN at front wheels did not become zero, which means the other actuators can generate extra F_y’s at front wheels.

5. Conclusions

This paper presented a path tracking controller with CTSA on low-friction road surface. The contents of this paper are summarized as follows:

• This paper presented a method to impose a CTSA such that the maximum lateral force is generated on low-friction road surface. The constraint is derived simply from the definition of tire slip angle and easily applied into LQR and MPC. From simulation results, the CTSA can greatly improve the path tracking performance on low-friction surface.
• The proposed CTSA makes the most of the lateral tire force. This means that there are little tire forces left to improve the control performance. Under that condition, it was shown that various combinations composed from RWS, 4WID, and 4WIB have little effects on control performance.
• From the simulation results, it is recommended that LQR is better than MPC in terms of computation time and simplicity of controller structure. Moreover, RWS calculated from the control yaw moment in IC#3 provided the best path tracking result. However, IC#3 requires a control allocation, which is complex and time-consuming, compared to IC#1 and IC#2. For the reason, IC#1 and IC#2 are preferred to IC#3 due to its simplicity and ease of implementation if one do not think the difference between IC#1/IC#2 and IC#3 is significant.

The CTSA adopted in this paper are easy to derive and simple to apply to LQR or MPC. Moreover, a procedure to design LQR with bound constraints on the control inputs are not needed. For the reason, it is recommended that a path tracking controller should include the CTSA.

The limitations of the proposed method are to measure the yaw rate and velocities and to estimate the side-slip angle with available sensors. To overcome these issues, sensor specification and estimation schemes are to be investigated. In further research, experiments on a scaled vehicle will be done with the proposed method. Moreover, extra experiments will be performed on real vehicles on low friction conditions.