A Lateral Control Method of Intelligent Vehicles Based on Shared Control

Abstract

This paper studies the lateral control problem for intelligent vehicles based on the concept of shared control. Considering the participation of drivers in the control loop, a shared control-based lateral controller is designed, which consists of two differed controllers: one is an LQR-based autonomous driving controller and the other is a driver’s intention-based fuzzy controller. For the vehicle dynamic model with two-degrees of freedom, an autonomous driving controller based on LQR and a driver’s intention-based fuzzy controller are designed. Then, the lateral controller based on shared control is constructed, which integrates the aforementioned two controllers. Finally, the co-simulation by MATLAB/Simulink and Carsim is presented. Furthermore, simulation results show that the designed lateral controller can track the desired path with better performance than the LQR-based autonomous driving controller.

1. Introduction

In recent years, with the rapid development of new techniques such as 5G, big data, cloud computing and AI, the intelligence of the automotive industry and the traditional cars are gradually developing into intelligent vehicles [1,2,3,4]. Since intelligent vehicles have great potential to improve road safety, passenger comfort, energy conservation, and emission reduction [5], they have become research hotspots. Furthermore, remarkable results have been achieved [6]. Network advanced communication techniques support the connectivity of intelligent vehicles despite their reduced perception ability, which improve the environmental perception of self-driving cars [7,8,9].

To measure the level of autonomous driving, the Society of Automotive Engineers International has released a classification system based on the level of driver intervention and attention, these include levels of autonomy for self-driving cars ranging from L0 to L5, and research on self-driving cars is generally focused on L3 level [10]. Currently, the intelligent vehicles with partial or fully autonomous driving functions are becoming increasingly mature, and have been used in unmanned shuttle, courier delivery and unmanned ferry. Many cities allow autonomous vehicles to test and operate. For example, the Toyota e-Palette self-driving minibus served as a ferry for athletes in the Olympic Village during the Tokyo Olympic and Paralympic Games; Baidu Apollo self-driving cars and driverless micro-circulation buses have been operating for a long time in the Shougang Park, the office of the Beijing Winter Olympic Organizing Committee. As pointed out in [11], autonomous vehicles were permissible in several states in the United States, even without the necessity of drivers.

However, the self-driving still faces challenges. Although Waymo announced plans in March 2022 to remove safety drivers for fully autonomous rides in [12], only a few intelligent transportation systems in specific scenarios can achieve a high-level autonomy in the actual traffic environment. In addition, the sensing networks deployed by intelligent vehicles cannot cover complex traffic environments. As stated in the White Book on Traffic Safety of Self-Driving Cars released by CAC and Baidu, the road testing permits were issued by the California Department of Motor Vehicles in April 2014. The statistical results showed that 149 accidents occurred in real road testing of self-driving cars and 105 accidents occurred during driver takeover between 2019 and mid-2020. Especially, an e-Palette self-driving bus struck and injured a blind athlete while turning right into a crosswalk during the Tokyo Paralympic Games. Consequently, the self-driving bus was taken out of service.

The above description illustrates that there are still many issues to be resolved to achieve a breakthrough in high-level autonomous driving technique without human involvement. As is known to all, it is not only limited by the technological development, but also by the constraints of relevant legal imperfections. At present, most autonomous vehicles still need the participation of drivers, which provides guidance and assess the potential hazards in the presence of the failed algorithm. The human–machine cooperative control uses the asymmetric advantages of human intelligence and machine intelligence, for achieving the hybrid enhancement of human–machine intelligence and the two-way information exchange. Therefore, the active safety assisted control or shared control technology with human involvement is expected to reduce traffic accidents [13]. Notely, the human–machine shared control method has been widely used in auxiliary robots [14,15], remote operation robots [16,17] and intelligent vehicles [18], which balances the advantages of driver and automatic vehicles to achieve the best cooperation [19,20,21]. Shared control provides a solution to the problem for driving controller conversion of intelligent vehicles [22].

Therefore, this paper designs a shared control-based method to achieve the lateral motion control of intelligent vehicles with driver’s participation. The main contributions of this paper are summarized as follows.

• To achieve the flexible shared control, a model-based LQR controller and a driver’s intention-based fuzzy controller are designed. The weight of the two controllers can be adjusted according to the needs of designers.
• The co-simulation of MATLAB and Carsim is executed, and the lateral motion controller based on shared control can better track the prescribed trajectory with a permissible deviation.

2. T-FVDM

The driving control of intelligent vehicles mainly includes the longitudinal and lateral control. The longitudinal control means that the vehicles adjust the speed to maintain enough space between vehicles, via uses the least amount of braking to ensures a relatively constant speed. The lateral control refers to path tracking with driving safety and ride comfort, which enables vehicles to follow the desired path through automatic steering control.

When vehicles travel on good roads with a low speed, there is generally no need to consider the stability control and other dynamic problems. Hence the kinematic model-based design of the path tracking controller has a reliable control performance. However, intelligent vehicles sometimes travel at high speed in complex traffic environments, and a more accurate vehicle dynamic model is necessary to design for improving the reliable controller. Accordingly, the lateral dynamic model is the basis of the shared control system [23].

The T-FVDM describes the vehicle dynamic system with some simplifications and assumptions, which simplifies the analysis process and retains the most basic vehicle dynamics. Therefore, it has a wide range of applications in vehicle dynamic research and vehicle control. Although the T-FVDM is simple, it quantifies the key parameters affecting the lateral motion of the vehicle, including the position of the vehicle’s center of mass and the lateral deflection characteristics of the tires. Clearly, these are the basis for studying the manipulating stability of vehicles [24]. In this paper, according to the principles of vehicle dynamics, a T-FVDM is established, which takes into account two freedoms of vehicles, i.e., lateral and yaw.

As shown in Figure 1, $l_{\mathrm{f}}$ is the front axle wheelbase, $l_{\mathrm{r}}$ is the rear axle wheelbase, $a_{\mathrm{f}}$ is the vehicle sideslip angle of the front wheel, and $a_{\mathrm{r}}$ is the vehicle sideslip angle of the rear wheel; $v_x$ is the longitudinal speed, and $v_y$ is the lateral speed; $\omega$, $\phi$, ${\delta }_{\mathrm{f}}$ are the yaw velocity, the yaw angle, and the front wheel angle, respectively.

When the longitudinal speed of the vehicle is determined, only the motion of the vehicle in both lateral and yaw dimensions are focused, and the lateral forces $F_{\mathrm{f}}$ and $F_{\mathrm{r}}$ on the front and rear wheels can be expressed as:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{F}_{\mathrm{f}}=C_{\mathrm{f}}\left[{\delta }_{\mathrm{f}}-{\displaystyle \frac{\left(v_{\mathrm{y}}+l_{\mathrm{f}}\phi \right)}{v_{\mathrm{x}}}}\right]\hfill \\ F_{\mathrm{r}}=-{\displaystyle \frac{C_{\mathrm{r}}\left(v_{\mathrm{y}}-l_{\mathrm{r}}\phi \right)}{v_{\mathrm{x}}}},\hfill \end{array}\right.\end{array}$$

(1)

where $C_{\mathrm{f}}$ is the cornering stiffness of the front tire, and $C_{\mathrm{r}}$ is the cornering stiffness of the rear tire.

The dynamic differential equations of the lateral and yaw motion are

$$\begin{array}{c}\hfill \left\{\begin{array}{c}2F_{\mathrm{f}}cos{\delta }_{\mathrm{f}}+2F_{\mathrm{r}}=m\left({\dot{v}}_{\mathrm{y}}-v_{\mathrm{x}}\dot{\phi }\right)\hfill \\ 2l_{\mathrm{r}}{F}_{\mathrm{f}}cos{\delta }_{\mathrm{f}}-2l_{\mathrm{f}}{F}_{\mathrm{r}}=I\ddot{\phi },\hfill \end{array}\right.\end{array}$$

(2)

where m and I are the known mass and rotational inertia with the established vehicle, respectively, and other parameters have been explained thereinbefore. It is noted that the lateral force on the front and rear is twice as much as the lateral force of the corresponding tire.

The conversion between the coordinate system of vehicles and the inertial coordinate system, is

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{P}_{\mathrm{X}}={\dot{p}}_{\mathrm{x}}cos\phi -{\dot{p}}_{\mathrm{y}}sin\phi ,\\ P_{\mathrm{Y}}={\dot{p}}_{\mathrm{x}}sin\phi -{\dot{p}}_{\mathrm{y}}cos\phi .\end{array}\right.\end{array}\end{array}$$

(3)

where $P_{\mathrm{X}}$ and $P_{\mathrm{Y}}$ are the positions in terms of X and Y directions in the global coordinate system, respectively; $p_{\mathrm{x}}$ and $p_{\mathrm{y}}$ are the positions in terms of x and y directions in the vehicle local coordinate system. Furthermore, the linear equation of state is simplified as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left(\begin{array}{c}{\ddot{v}}_{\mathrm{y}}\\ \ddot{\phi }\end{array}\right)=\left(\begin{array}{cc}\frac{C_{\mathrm{f}}+C_{\mathrm{r}}}{m{v}_{\mathrm{x}}}& \frac{l_{\mathrm{r}}{C}_{\mathrm{f}}-l_{\mathrm{f}}{C}_{\mathrm{r}}}{m{v}_{\mathrm{x}}}-v_{\mathrm{x}}\\ \frac{l_{\mathrm{r}}{C}_{\mathrm{f}}-l_{\mathrm{f}}{C}_{\mathrm{r}}}{I{v}_{\mathrm{x}}}& \frac{{l_{\mathrm{r}}}^2{C}_{\mathrm{f}}+{l_{\mathrm{f}}}^2{C}_{\mathrm{r}}}{I{v}_{\mathrm{x}}}\end{array}\right)\left(\begin{array}{c}{\dot{v}}_{\mathrm{y}}\\ \dot{\phi }\end{array}\right)+\left(\begin{array}{c}-\frac{C_{\mathrm{f}}}{m{v}_{\mathrm{x}}}\\ -\frac{l_{\mathrm{r}}{C}_{\mathrm{f}}}{I}\end{array}\right){\delta }_{\mathrm{f}}\end{array}\end{array}$$

(4)

3. Shared Control

Shared control was first proposed by Sheridan and Verplank, and was defined as “a situation where both automation and humans work on the same task at the same time” [25]. It is generally believed that intelligent control systems are good at the execution of fine tasks, while humans are skilled in the judgment and upper-level planning. Moreover, humans can monitor the operating status of the robotic system in real time and correct some errors. Therefore, in many cases, it is necessary to rely on humans and intelligent control systems to complete complex tasks [22]. For intelligent vehicles, the movement is completely determined by the intelligent control system. However, if a driver intervenes, the movement is determined by the driver. For the existing problem, the human–machine shared control is an acceptable solution, and it can determine the movement behavior of the vehicle together with the driver [19,26,27,28].

In shared control system of vehicles as shown in Figure 2, a weighting operation is needed to coordinate the control behavior of the driver and the intelligent control system, it is

$$\begin{array}{c}\hfill \begin{array}{c}\hfill u_{\mathrm{s}}=\lambda u_{\mathrm{a}}+(1-\lambda )u_{\mathrm{h}},0\le \lambda \le 1,\end{array}\end{array}$$

(5)

where $u_{\mathrm{s}}$ is the shared control input, $u_{\mathrm{a}}$ is the control command of the intelligent vehicle autopilot system, $u_{\mathrm{h}}$ is the driver’s intention-based control input, and $\lambda$ is the adjustable weight coefficient.

Remark 1.

In the proposed shared control scheme, it is assumed that both the intelligent control system and the driver have a perception of the external environment. The autonomous driving system controls the steering angle through the control input of an LQR controller (see Equation (10)). Accordingly, the driver controls the steering angle of the vehicle through a fuzzy controller, which uses fuzzy rules to deduce control commands in the Driver Intent Recognition part.

4. LQR Algorithm for Intelligent Vehicles

LQR optimal control has the ability to achieve the desired performance index with low cost, and to tune the unstable system simultaneously [29,30,31]. When the intelligent vehicle tracks the target path, the distance and heading deviations need to be considered. To be clearer, they are described in Figure 3. The distance deviation $e_{\mathrm{d}}$ is the distance from the centroid of the vehicle to the road centerline, and the heading deviation $e_{\phi }$ is the deviation between the vehicle heading and the road direction; ${\phi }_{\mathrm{des}}$ is the ideal heading angle.

For the deviations $e_{\mathrm{d}}$ and $e_{\phi }$, the equations are obtained as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\ddot{e}}_{\mathrm{d}}=-\frac{C_{\mathrm{f}}+C_{\mathrm{r}}}{m{v}_{\mathrm{x}}}{\dot{e}}_{\mathrm{d}}+\frac{C_{\mathrm{f}}+C_{\mathrm{r}}}{m}{e}_{\phi }+\frac{l_{\mathrm{f}}{C}_{\mathrm{f}}-l_{\mathrm{r}}{C}_{\mathrm{r}}}{m{v}_{\mathrm{x}}}{\dot{e}}_{\phi }+\frac{C_{\mathrm{f}}}{m}{\delta }_{\mathrm{f}}-\left({\displaystyle \frac{l_{\mathrm{f}}{C}_{\mathrm{f}}-l_{\mathrm{r}}{C}_{\mathrm{r}}}{m{v}_{\mathrm{x}}}}-v_{\mathrm{x}}\right){\dot{\Psi }}_{\mathrm{des}}.\end{array}\end{array}$$

(6)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\ddot{e}}_{\phi }=-\frac{l_{\mathrm{f}}{C}_{\mathrm{f}}-l_{\mathrm{r}}{C}_{\mathrm{r}}}{I{v}_{\mathrm{x}}}{\dot{e}}_{\mathrm{d}}\frac{l_{\mathrm{f}}{C}_{\mathrm{f}}-l_{\mathrm{r}}{C}_{\mathrm{r}}}{I}{e}_{\phi }-\frac{{l_{\mathrm{r}}}^2{C}_{\mathrm{r}}+{l_{\mathrm{f}}}^2{C}_{\mathrm{f}}}{I{v}_{\mathrm{x}}}{\dot{e}}_{\phi }+\frac{l_{\mathrm{f}}{C}_{\mathrm{f}}}{I{v}_{\mathrm{x}}}{\delta }_{\mathrm{f}}-\left({\displaystyle \frac{l_{\mathrm{f}}^2{C}_{\mathrm{f}}-l_{\mathrm{r}}^2{C}_{\mathrm{r}}}{I{v}_{\mathrm{x}}}}\right){\dot{\Psi }}_{\mathrm{des}}.\end{array}\end{array}$$

(7)

where ${\Psi }_{\mathrm{des}}={\displaystyle \frac{v_{\mathrm{x}}}{R_{\mathrm{r}}}}$ is the yaw rate reference, and $R_{\mathrm{r}}$ is the turning radius of the planned path.

Let $x={\left[\begin{array}{cccc}{e^{\mathrm{T}}}_{\mathrm{d}}& {{\dot{e}}^{\mathrm{T}}}_{\mathrm{d}}& {e^{\mathrm{T}}}_{\phi }& {{\dot{e}}^{\mathrm{T}}}_{\phi }\end{array}\right]}^{\mathrm{T}}$, we have

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \dot{x}=Ax+B_1{u}_{\mathrm{a}}+B_2{\dot{\Psi }}_{\mathrm{des}}.\end{array}\end{array}$$

(8)

where A, $B_1$ and $B_2$ represent the state transition matrix, control matrix and feedforward matrix of the continuous-time vehicle model, respectively, and the details are

$$\begin{array}{c}\hfill A=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& -\frac{C_{\mathrm{f}}+C_{\mathrm{r}}}{m{v}_{\mathrm{x}}}& \frac{C_{\mathrm{f}}+C_{\mathrm{r}}}{m}& \frac{-l_{\mathrm{f}}{C}_{\mathrm{f}}+l_{\mathrm{r}}{C}_{\mathrm{r}}}{m{v}_{\mathrm{x}}}\\ 0& 0& 0& 1\\ 0& -\frac{l_{\mathrm{f}}{C}_{\mathrm{f}}-l_r{C}_{\mathrm{r}}}{I{v}_{\mathrm{x}}}& \frac{l_{\mathrm{f}}{C}_{\mathrm{f}}-l_r{C}_{\mathrm{r}}}{I}& -\frac{{l_{\mathrm{f}}}^2{C}_{\mathrm{f}}+{l_{\mathrm{f}}}^2{C}_{\mathrm{f}}}{I{v}_{\mathrm{x}}}\end{array}\right],\\ \hfill B_1=\left[\begin{array}{c}0\\ \frac{C_{\mathrm{f}}}{m}\\ 0\\ \frac{l_{\mathrm{f}}{C}_{\mathrm{f}}}{I{v}_{\mathrm{x}}}\end{array}\right],B_2=\left[\begin{array}{c}0\\ -\frac{l_{\mathrm{f}}{C}_{\mathrm{f}}-l_{\mathrm{r}}{C}_{\mathrm{r}}}{m{v}_{\mathrm{x}}}-v_{\mathrm{x}}\\ 0\\ -\frac{l_{\mathrm{f}}^2{C}_{\mathrm{f}}+l_{\mathrm{r}}^2{C}_{\mathrm{r}}}{I{v}_{\mathrm{x}}}\end{array}\right].\end{array}$$

Remark 2.

In [32], the yaw rate reference was introduced and used to obtain the corresponding error. Similarly, it has been also used in Equations (6) and (7) (see the terms $e_{\phi }$ and ${\dot{e}}_{\phi }$).

Using the LQR control theory to design a lateral controller for autonomous driving, the control system performance indicator is

$$\begin{array}{c}\hfill J={\int }_0^{\infty }\left[x^{T}Qx+u_{\mathrm{a}}^{T}R{u}_{\mathrm{a}}\right]dt,\end{array}$$

(9)

where Q and R are the positive definite weighting matrixs. The control law of the LQR controller is designed as

$$\begin{array}{c}\hfill u_{\mathrm{a}}=-Kx+{\delta }_{ff}=-R^{-1}{B}_1^{T}Px+{\delta }_{ff},\end{array}$$

(10)

$$\begin{array}{c}\hfill {\delta }_{ff}=\frac{m{v_{\mathrm{x}}}^2}{R_{\mathrm{r}}L}\left[\frac{l_{\mathrm{r}}}{2C_{\mathrm{f}}}-\frac{l_{\mathrm{f}}}{2C_{\mathrm{r}}}+\frac{l_{\mathrm{f}}}{2C_{\mathrm{r}}}{k}_3\right]+\frac{L}{R_{\mathrm{r}}}-\frac{l_{\mathrm{r}}}{R_{\mathrm{r}}}{k}_3,\end{array}$$

(11)

where $K=\left[k_1,k_2,k_3,k_4\right]$, ${\delta }_{ff}$ is the feedforward component of steering angle, $L=(l_{\mathrm{f}}+l_{\mathrm{r}})$ is the total wheel base, and P is the positive definite solution of the following well-known algebraic Riccati equation:

$$\begin{array}{c}\hfill A^{T}P+PA+Q-P{B}_1{R}^{-1}{B}_1^{T}P=0.\end{array}$$

(12)

5. Driver Intent Recognition

At present, the driver’s intent is crucial and has been widely used in intelligent driver assistance systems and autonomous driving systems, such as driving intent for lane changing, steering, starting, parking and braking. Based on the GHMM/GGAP-RBFNN hybrid model, the driver’s braking intention recognition model was studied to improve the accuracy of normal braking and light braking in [33]. The prediction of steering intention was given via a novel hybrid algorithm of the time-series model based on deep learning in [34]. In [35], driving intention recognition was extracted and dynamically identified based on the rough set theory and back-propagation artificial neural networks.

Considering the complexity of the lateral control, such as nonlinearity of the driving process and the difficulty of obtaining the accurate model, the fuzzy control method is introduced to transform the driver’s input into the control input of intelligent vehicles. The angular deviation $e_{\phi }$ and the angular deviation rate ${\dot{e}}_{\phi }$ are the input variables, and the angle increment $u_{\mathrm{h}}$ of the front wheel is the output variable. $\{\mathrm{NB},\mathrm{NM},\mathrm{NS},\mathrm{ZO},\mathrm{PS},\mathrm{PM},\mathrm{PB}\}$ are the fuzzy sets of $e_{\phi }$, ${\dot{e}}_{\phi }$ and $u_{\mathrm{h}}$. Both input and output membership functions are taken as triangular membership functions. Based on the summary of skilled drivers’ experience and control engineering knowledge, 49 rules were formulated in

Table 1. Fuzzy control rules.

$u_{\mathrm{h}}$		${\dot{e}}_{\phi }$
		NB	NM	NS	ZO	PS	PM	PB
$e_{\phi }$	NB	NB	NB	NM	NM	NS	NS	ZO
	NM	NB	NM	NM	NS	NS	ZO	PS
	NS	NM	NM	NS	NS	ZO	PS	PS
	ZO	NM	NS	NS	ZO	PS	PS	PM
	PS	NS	NS	ZO	PS	PS	PM	PM
	PM	NS	ZO	PS	PS	PM	PM	PB
	PB	ZO	PS	PS	PM	PM	PB	PB

. According to the designed fuzzy rules, the fuzzy output is obtained by fuzzy inference with the driver’s input. Next, the centroid method is used for defuzzification, and the control input of the front wheel angle is obtained.

6. Simulation Results

In order to verify the proposed control scheme, a Co-Simulation of CarSim and MATLAB is performed in this section. CarSim is a simulation software for vehicle dynamics, which can simulate the response of a vehicle to the road and aerodynamic inputs. It is mainly used to predict and simulate the handling stability, braking, smoothness, power and economy of the whole vehicle [33].

In CarSim, the C-class passenger car model is adopted, and the main parameters are shown in

Table 2. Main parameters of the vehicle.

Name (Unit)	Symbol	Numerical
vehicle quality (kg)	m	1412
moment of inertia of the vehicle (kg$·{\mathrm{m}}^2$)	I	1536.7
cornering stiffness of the front wheel (N/rad)	$C_{\mathrm{f}}$	−110,000
cornering stiffness of the rear wheel (N/rad)	$C_{\mathrm{r}}$	−110,000
front wheelbase (m)	$l_{\mathrm{f}}$	1.015
rear wheelbase (m)	$l_{\mathrm{r}}$	1.895

. The fixed vehicle speed is controlled by the controller in Carsim, and the front-wheel turning angle is calculated by the controller in the Simulink.

In Figure 4, the shared control is achieved by combining the driver’s intention-based fuzzy control algorithm and the LQR-based automatic driving algorithm.

The simulation results are shown in Figure 5 and Figure 6. It can be seen that the LQR and shared controllers can track the planned path well, as shown in in Figure 5. For a comparison, the distance deviations of LQR and shared controllers are given in Figure 6, The tracking performance of the lateral controller based on shared control is better than that of the LQR-based autonomous driving controller. To quantify the results of the LQR control and the shared control, we introduce $e_{\mathrm{L}x}$ and $e_{\mathrm{L}y}$, which are the trajectory tracking errors of the LQR control in terms of x and y directions, respectively. Accordingly, $e_{\mathrm{S}x}$ and $e_{\mathrm{S}y}$ are the trajectory tracking errors of the share control in terms of x and y directions, respectively. For comparison, the performance index $E={\sum }_{t=0}^{30}(\parallel e_{.x}{\parallel }_1+\parallel e_{.y}{\parallel }_1)$ is calculated for the LQR control and shared control cases, respectively, i.e., $3.8395\times {10}^6$ and $3.8198\times {10}^6$. Clearly, the superiority of the proposed shared control method is verified.

7. Conclusions

For the intelligent vehicle control system with the driver’s participation, this paper proposed a control scheme for the lateral motion based on shared control. With the constructed T-FVDM, an LQR controller for autonomous driving system and a fuzzy control-based controller have been designed. Based on these controllers, the lateral controller hwas constructed according to the shared control method. Through the co-simulation of MATLAB/Simulink and Carsim, the simulation results showed that the lateral controller based on shared control has provided better tracking performance than the LQR-based controller. In our future work, the shared control of intelligent vehicles using data-driven techniques will be further investigated [36,37,38], even considering non-cooperative behaviors [39,40,41,42,43,44].