Research on a Multimode Adaptive Cruise Control Strategy with Emergency Lane-Changing Function

Abstract

In emergency situations, it is difficult to meet the requirements of safe driving only by relying on the braking system, and the probability of accidents can be reduced by employing an emergency lane-changing mode. To improve the adaptability of the distributed electric vehicle adaptive cruise control (ACC) strategy to complicated and volatile conditions, a multimode ACC strategy with emergency lane-changing function is proposed. Firstly, the ACC is divided into four modes aimed at the problem of complex conditions, and a switching strategy is designed to control the switching of them. Simultaneously, the car-following mode is divided in greater detail based on time to collision (TTC), and the acceleration weighted average algorithm is adopted for accuracy and output continuity during switching. Then, the ACC is established with a hierarchical control framework, in which a PID-based cruise mode and a multi-objective optimized car-following mode based on model predictive control (MPC) are devised. The target brake wheel cylinder pressure is selected as the emergency brake pressure in takeover mode. In addition to the MPC-based system, the emergency lane-changing mode incorporates a yaw moment controller in the upper-level controller to improve body stability during emergency lane changing in the upper-level controller. In the lower-level controller, the upper-level output is converted into driving torque, wheel cylinder pressure, and front wheel angle to control vehicle travel and generate additional yaw moment. Finally, the results indicate that the presented multimode switching strategy can adapt to complex and instable transportation environments. In the cruise control scenario, the host vehicle can rapidly reach cruising speed within 5 s. In the car-following scenario, the host vehicle can stably follow the preceding vehicle with an acceleration of −5–3.5 m/s² and a jerk of −2–2 m/s³ throughout the entire process, maintaining a safe distance from the preceding vehicle. In emergency lane-changing scenarios, vehicles with body stability control can better follow the lane-changing trajectory, and tracking accuracy is improved by 65%. Simultaneously, parameters such as front wheel angle, yaw rate, sideslip rate, and lateral acceleration remain within the normal range. In mixed switching scenarios, each mode can be correctly switched according to diverse operating conditions, and obstacle avoidance can be accomplished through horizontal and vertical strategies, which also verify the effectiveness and rationality of the control strategy proposed.

1. Introduction

By referring to the Handbook of Intelligent Vehicles, when speed is greater than about 35 km/h, the distance required for collision avoidance by steering is less than the distance required for collision avoidance by braking [1]. That is, for some emergency conditions, the collision avoidance mode of emergency steering may be more effective at protecting occupants than collision avoidance by braking. The value of the road adhesion coefficient also significantly affects the lateral and longitudinal switching strategy. Handbook of Intelligent Vehicles shows that when the road adhesion coefficient decreases or speed increases too much, the tires move into a nonlinear region, which causes the vehicle to lose control easily and increases avoidance distance [1].

From the perspective of longitudinal control, ACC controllers are generally designed hierarchically, including upper-level and lower-level controllers [2,3], and manifold algorithms have been employed to devise upper-level controllers: PD feedback controller, model predictive control, sliding mode control (SMC), linear quadratic regulator (LQR) [4], reinforcement learning, etc. Domestic and foreign researchers have conducted extensive research on the subject. The construction of the lower-level controller is also an essential element of the ACC system, and the powertrain, braking system, and switching strategies between driving/braking are the main research areas. The inverse model has long been mainly utilized to track and control the outputs of upper-level controllers [5,6], and the relative velocity and distance between two vehicles have been adopted to partition the switching strategy in the lower-level controller [7,8]. Amid the above control algorithms, PID [9,10,11], LQR [12,13,14], and MPC [15,16,17] have been comprehensively researched and applied in ACC systems.

In lateral control, its requisite is to control the vehicle to steadily follow the lane-changing trajectory. In this regard, there are more forms of lane-changing trajectories, which are primarily classified as the following: local path (polynomial curve, B-spline curve, Bezier curve, etc.) or global path (Dijkstra algorithm, ant colony algorithm, A* algorithm, etc.). Among them, the polynomial curve [18,19,20] and the Bezier curve [21,22] have been widely used and studied. For improving the stability and safety of the vehicle during lane changing, direct yaw moment control (DYC) and active front steering (AFS) have been applied to construct the yaw moment controller, and the literature [23,24,25,26] describes an integrated DYC and AFS controller, making full use of the advantages of AFS and DYC, which ensures the body stability of the vehicle on curved roads or during lane changing.

Nevertheless, the current ACC system seldom considers the emergency conditions ahead and how to avoid collisions, and its function is comparatively unitary, and the incorporation and division of the lane-changing function is rarely considered in mode switching. Conventional Delphi control has been compartmentalized into two modes: one was the cruise mode and the other was the car-following mode, where the appropriate control mode was opted according to a variety of diverse situations [27]. The ACC system exploited by the German BMW Group was based on the Delphi control mode with the addition of a cornering mode, which was chiefly divided into three modalities and permitted the vehicle to follow the preceding vehicle on a curved road [28]. Inliterature [29], the driver’s lane-change intention was incorporated in the classical ACC mode, which was classified into four modes, but the paper did not consider emergency lane changing for collision avoidance. As the conditions became more unpredictable, more modes were excogitated, and references [30,31,32] divided the control modes into five, six, and eight, respectively, for the sake of preferably accommodating manifold conditions.

Based on the above background, a multimode ACC strategy with an emergency collision avoidance function is proposed in this paper, taking into consideration the emergency conditions that may momentarily occur. Using combined longitudinal and lateral collision avoidance, this article adopts a hierarchical structure to implement the ACC control strategy. In the upper-level controller, the multi-objective optimization MPC algorithm is used to design the car-following mode, the PID algorithm is used to design the cruise mode, MPC is used to design the emergency lane-change mode, and a vehicle stability control system is designed based on LQR to ensure the safety of the vehicle during lane changing. The outputs of the lower-level controller are the motor torque and total brake torque demands for driving and braking, which are sent to the vehicle model so that the host vehicle can accurately track the desired acceleration. It expands the application range of adaptive cruise control systems and improves the performance index of the adaptive cruise control system. Compared with the current work, the main contributions of this paper are as follows. This article integrates the emergency lane change function on the basis of the existing ACC, dividing the vehicle mode into more reasonable main mode and sub-mode, using a weighted average algorithm to design the switching strategy between modes, ensuring driving safety and improving driving comfort. At the same time, a vehicle stability control system is designed to improve the accuracy of the vehicle following the target trajectory. The ACC control strategy proposed in this article can adapt to complex and ever-changing traffic environments with high safety, robustness, and portability. This paper is organized as follows. Section 2 defines the ACC framework. The repartition of overall mode and sub-mode and the design of switching strategy are carried out in Section 3. Section 4 adopts hierarchical control in the upper controller, devises a PID-based cruise mode and an MPC-based optimized car-following mode, opts for target wheel cylinder pressure as the emergency braking pressure for takeover mode, and designs the MPC-based emergency lane-changing mode, incorporating DYC; in addition, the lower controller is designed. The performance of the ACC strategy is tested based on the HIL simulation platform; concurrently, the validity and feasibility of the strategy are verified in Section 5. Finally, conclusions and future use of the paper are presented in Section 6.

2. ACC Framework Definition

ACC automatically can adjust its velocity along with the driving conditions of the preceding vehicle. Currently, the multi-objective coordinated ACC strategy, which integrates safety, fuel economy, and comfort, has been introduced [33]. Yet, owing to the unpredictable driving conditions and the movement status of surrounding vehicles, the ACC strategy suffers from frequent acceleration, deceleration, and unstable velocity control. The design of multimode ACC is conducive to ameliorating the safety and comfort of the vehicle under complex traffic environments, the core of which lies in the reasonable mode division method and switching strategy. In emergency conditions, making full use of longitudinal and lateral emergency collision avoidance strategies to avoid obstacles is conducive to defending the personal safety of occupants. Based on this background and the aforementioned summary, the framework of ACC in this paper is illustrated in Figure 1. It is noteworthy that the cruise mode is controlled by velocity, while the other three modes are controlled by distance.

3. Design of Mode Switching Strategy

In this paper, the ACC system is repartitioned into four main modes and four sub-modes. However, the division of the four following modes, steady following, acceleration, heavy deceleration, and light deceleration, needs to conform to the driving characteristics of most drivers.

3.1. Four Main Modes of Division

Safe distance design is the basis for dividing the four main modes, and switching conditions directly affects the smoothness and logic of the whole model. Distance, $TTC$, and critical velocity $V_s$ are innovatively selected as switching criteria in this paper, which have the preponderance of low computational effort and lesser decision time.

(1)

Cruising distance

When there is no preceding vehicle or the preceding vehicle is too remote or pulls out the current lane, the host vehicle implements the cruise mode. Considering the controller’s adaptability to unpredictable scenarios, this paper calculates the cruising distance based on the time headway and velocity, which can be calculated as follows [34]:

$$S=T_h{v}_f+d_0,$$

(1)

where parameter $S$ is the cruising distance; parameter $T_h$ is time headway, the general value range of which is 5–8 s (in this paper, 6 s is selected as the cruising time); parameter $v_f$ is the host vehicle velocity; parameter $d_0$ is the minimal fixed safety inter-distance.

(2)

Takeover distance

When the preceding vehicle urgently decelerates or the obstacle drops from it and the switching conditions meet, the host vehicle enters takeover mode and brakes with the maximum brake wheel cylinder pressure to avoid collision. $S_f$ is defined as the braking displacement of the host vehicle, and $S_d$ as the displacement of the preceding vehicle; therefore, the takeover distance can be defined by

$$S=S_f+d_0-S_d.$$

(2)

(3)

Car-following distance

When the immediate inter-distance is between the cruising distance and the takeover distance, the host vehicle enters the following mode. Considering that velocity variation affects the time headway, a variable time headway (VTH) strategy that can be expediently predicted is devised for estimating the practical conditions, and its expression is as follows:

$$t_h=\{\begin{array}{c} t_{h\_\max }\\ t=t_0-c_v{v}_{rel}-c_a{a}_p\\ t_{h\_\min }\end{array}\begin{array}{c}if t>t_{h\_\max }\\ if t_{h\_\min }<t<t_{h\_\max }\\ if t<t_{h\_\min }\end{array},$$

(3)

where parameter $t_h$ refers to time headway; parameters $t_0$ and $c_v$ stand for constants greater than 0, where $t_0=1.5$ s and $c_v=0.05$; parameter $v_{rel}$ represents that the relative velocity and its value can be detected by an on-board radar; parameters $t_{h\_\max }$ and $t_{h\_\min }$ are the upper and lower limits of the time headway, respectively, where $t_{h\_\max }=2.2$ s, $t_{h\_\min }=0.2$ s; parameter $a_p$ denotes the acceleration of the preceding vehicle; parameter $c_a$ is a constant greater than 0, where $c_a=0.3$.

Combined with the variable time headway, the desired car-following distance is given by

$$S=t_{h}v+d_0.$$

(4)

(4)

Emergency lane change criteria

When the longitudinal emergency braking cannot avoid collision yet meets the lane-changing requirement, the host vehicle enters emergency lane-changing mode. In trajectory planning, steering safety distance S represents the longitudinal displacement from the host vehicle to the preceding vehicle at critical collision, which can be expressed as follows:

$$S=v_{rel}(t_c-t_0)+{\displaystyle {\int }_0^{t_c}{\displaystyle {\int }_0^{\sigma }{a}_{rel}}}dtd\sigma ,$$

(5)

where parameter $v_{rel}$ denotes relative velocity; parameter $a_{rel}$ represents relative acceleration; parameters $t_0$ and $t_c$ are the turning start time and the collision time, respectively. In this paper, assuming no transformation in the host vehicle’s velocity after switching to emergency lane-changing mode occurs, the above equation can be simplified to the following expression:

$$S=v_{rel}(t_c-t_0).$$

(6)

The relative velocity is equal to 63 km/h, which represents the braking/steering boundary point, that is, the judgment condition.

3.2. Four Sub-Mode Division

The car-following mode focuses on the steady following, acceleration, strong deceleration, and weak deceleration, so sub-conditions need to be divided. This paper is based on the inverse of $TT{C}^{-1}$ to devise sub-mode, which is defined as follows:

$$TT{C}^{-1}=\frac{\Delta v}{d},$$

(7)

where parameter $\Delta v$ denotes relative velocity; parameter $d$ stands for inter-distance between two vehicles.

When $TT{C}^{-1}$ is close to 0, the host vehicle and the preceding vehicle are safer; conversely, vehicles are prone to collision. Therefore, the sub-mode is divided into more explicit sections as follows:

• When $-0.05\le TT{C}^{-1}\le 0.05$ [35], the vehicle is in steady following condition, and its acceleration range is −0.6–0.6 m/s²;
• When $-0.12\le TT{C}^{-1}<-0.05$, the vehicle is in weak deceleration condition, and its maximum deceleration constraint is −2 m/s²;
• When $TT{C}^{-1}<-0.12$, the vehicle is in strong deceleration condition, and its maximum deceleration constraint is −5 m/s²;
• When $TT{C}^{-1}>0.05$, the vehicle is in the acceleration condition, and its acceleration constraint range is 0.6–3.5 m/s².

3.3. Overall Mode Switching Strategy

Based on the above chapters, the control logic of overall mode is designed (Figure 2).

The $S_0$, $S_1$, $S_3$, $TTCs$, and the critical relative velocity $V_s$ in Figure 2 are the switching conditions of various modes, respectively, where $S_0$ is defined as the cruising distance, $S_1$ is the takeover distance, $S_3$ is the lane-changing distance, and $TTCs=1.3$ s [36].

In order to ensure that the switching process does not lead to abrupt changes in vehicle acceleration, the weighted average algorithm is used in this paper to test the validity of mode switching, and the output acceleration is processed continuously, as shown in Equation (8):

$$a_{\mathrm{wd}}=w_1{a}_{\mathrm{w}1}+w_2{a}_{\mathrm{wn}},$$

(8)

where $a_{\mathrm{w}1}$ and $a_{\mathrm{wn}}$ are the expected acceleration calculated by the existing control mode and the control mode to be entered, respectively, and $a_{\mathrm{wd}}$ is the output of the upper controller in the transition region. $w_1$ and $w_2$ are the weight coefficients, and their values are shown in Figure 3.

Figure 3 shows that the size of the weight coefficient $w$ depends on the number of times $x$ of the new recognition pattern, and $w_1+w_2=1$. When continuously entering the new mode for less than five times and $w_2=0$, the output value of the previous mode is still maintained to avoid frequent switching between modes. When $x$ exceeds 10 times and $w_1=0$, the control mode spans the transition zone and enters a new mode. When $5<x<10$, the output of the two modes is linearly weighted and averaged to avoid severe fluctuations in vehicle acceleration. In particular, for cruise and cut in/cut out conditions, the effectiveness of mode switching has been verified in the radar algorithm, so the weighted average is directly started from $x>5$.

4. Design of Hierarchical Control Strategy for the ACC Mode

4.1. Upper-Level Controller Design

The cruise mode adhibits the classical PID algorithm to power the vehicle to reach cruising velocity quickly and steadily. The car-following mode is devised with the MPC algorithm, which can dispose of the steady following, strong deceleration, weak deceleration, and acceleration conditions, while taking into account the optimization indicators. Target wheel cylinder pressure is employed as braking pressure in the takeover mode. In the emergency lane-changing mode, the MPC-based trajectory tracking controller is devised, incorporating DYC to heighten body stability.

4.1.1. Cruising Mode Design

Generally, PID control is calculated by inputting the error between the current state value and the set value and calculating the variable output in the form of

$$u(t)=k_{p}e(t)+k_i{\displaystyle {\int }_0^{t}e(t)dt+k_d}\frac{de(t)}{dt},$$

(9)

where parameter $u(t)$ is the output variable of the controller; parameter $e(t)$ is the error between the current value and the set value; parameters $k_p$, $k_i$ and $k_d$ are the proportional, integral, and differential coefficients, respectively, which are determined by the trial and error method, where $k_p=10$, $k_i=0.001$, and $k_d=1$.

4.1.2. Car-Following Mode Design

The exploitation of the car-following mode directly affects the performance of the overall ACC controller; consequently, a multi-objective ACC controller for MPC is exploited. It includes the three parts described below.

(1)

Longitudinal dynamic model design

Referring to the kinematic relationship between the two vehicles (Figure 4), the kinematic equation for the host and preceding vehicle, considering the acceleration and jerk of the host vehicle, can be obtained as presented in Equation (10)

$$\{\begin{array}{c}\Delta x(k+1)=\Delta x(k)+v_{rel}(k)T_s+\frac{1}{2}{a}_p(k)T_s^2-\frac{1}{2}a(k)T_s^2\\ v(k+1)=v(k)+a(k)T_s\\ v_{rel}(k+1)=v_{rel}(k)+a_p(k)T_s-a(k)T_s\\ a(k+1)=(1-\frac{T_s}{\tau })a(k)+\frac{T_s}{\tau }u(k)\\ j(k+1)=-\frac{1}{\tau }a(k)+\frac{1}{\tau }u(k)\end{array},$$

(10)

where parameter $\Delta x(k)$ is the actual inter-distance between the host and the preceding vehicle; parameter $a_p(k)$ is the acceleration of the preceding vehicle; parameter $v_{rel}(k)$ is the velocity of the host vehicle; parameter $u(k)$ is the desired acceleration; parameter $j(k)$ is the jerk; parameter $T_s$ is the sampling time; parameter $\tau$ is the control time of the lower-level controller.

$\mathit{x}(k)={[\Delta x(k),v(k),v_{rel}(k),a(k),j(k)]}^T$ are opted as the state quantities in this paper. The defining parameter $x(k)$ is the position of the host vehicle, and the preceding vehicle’s acceleration $w(k)$ is chosen as the system disturbance parameter. Likewise, $\mathit{y}(k)={[\delta (k),v_{rel}(k),a(k),j(k)]}^T$ is selected as the output variable, and the state equation of longitudinal kinematic is formulated as follows:

$$\{\begin{array}{l}x(k+1)=Ax(k)+Bu(k)+Gw(k)\\ y(k)=Cx(k)-Z\end{array},$$

(11)

where $A$ and $C$ are the system matrices; $B$ is the input matrix; $G$ is the perturbation matrix; $w(k)$ is the front vehicle’s acceleration; $Z={\left[\begin{array}{cccc}{d}_0& 0& 0& 0\end{array}\right]}^T$; parameter $t_h$ denotes variable time headway.

(2)

Control purpose and constraint analysis

The security of the two vehicles is actualized by pledging that the vehicles are in a safe inter-distance. We assume that $x_p(k)$ is the preceding vehicle’s position and $d_c$ is the minimum safe distance, for which the constraints are as follows:

$$x_p(k)-x(k)\ge d_c.$$

(12)

The error between the actual and desired distance between the vehicles should be guaranteed to be as small as possible, with the following constraints:

$$\delta (k)\to 0, v_{rel}(k)\to 0 , \mathrm{as} k\to \infty .$$

(13)

The principal factors influencing comfort are the host vehicle’s acceleration and jerk; therefore, the two parameters demand to be optimally controlled as follows:

$$\{\begin{array}{l}\min |a(k)|\\ \min |j(k)|\end{array}.$$

(14)

In addition, it needs to be considered that the performance of the host vehicle, the velocity, acceleration, jerk, and desired acceleration are constrained by

$$v_{\min }\le v(k)\le v_{\max }, a_{\min }\le a(k)\le a_{\max }, j_{\min }\le j(k)\le j_{\max }, u_{\min }\le u(k)\le u_{\max }.$$

(15)

(3)

MPC-based acc control algorithm design

According to Equation (11), the prediction equation within the prediction horizon is established, $e_x(k)$ defined as the error between the actual state quantity and the predicted value and $W(k+p)$ defined as the set of disturbance quantities. Assuming equal values at each moment, the expression is rectified by

$$\begin{array}{l}{\widehat{X}}_p(k+p|k)=\overline{A}x(k)+\overline{B}U(k+n)+\overline{G}W(k+p)+\overline{H}{e}_x(k)\\ {\widehat{Y}}_p(k+p|k)=\overline{C}\mathit{x}(k)+\overline{D}U(k+n)+\overline{E}W(k+p)+\overline{F}{e}_x(k)-\overline{Z}\end{array},$$

(16)

where ${\widehat{X}}_p(k+p|k)$ is the set of predicted state variables, ${\widehat{Y}}_p(k+p|k)$ is the predicted output variable, $U(k+n)$ is the sequence of desired acceleration output variables. In the prediction model, each prediction matrix is involved: $\overline{A}$, $\overline{B}$, $\overline{G}$, $\overline{H}$, $\overline{C}$, $\overline{D}$, $\overline{E}$, $\overline{F}$, $\overline{Z}$, where $\overline{A}$ and $\overline{C}$ are the system matrices; $\overline{B}$ and $\overline{D}$ are the input matrices; $\overline{G}$ and $\overline{E}$ are the perturbation matrices; $\overline{H}$ and $\overline{F}$ are the error matrices; $w(k)$ is the front vehicle’s acceleration; $\overline{Z}={\left[\begin{array}{cccc}Z& Z& Z& Z\end{array}\right]}^T$.

The weight matrices $Q$ and coefficient $R$ are added, where $Q$ stands for followability and stands $R$ for safety. The following objective function is rectified and written:

$$J={\displaystyle \sum _{i=1}^p{[{\widehat{y}}_p(k+i|k)-y_r(k+i)]}^T}Q[{\widehat{y}}_p(k+i|k)-y_r(k+i)]+{\displaystyle \sum _{i=0}^{n-1}u{(k+i)}^T}Ru(k+i)$$

(17)

where matrix $Q=diag(q_{\delta },q_v,q_a,q_j)$ and set $u(k+i)$ is the desired acceleration.

According to the regulations for solving Quadratic Programming (QP) problems, the preceding constraints are organized to obtain

$$\Omega U(k+n)\le T$$

(18)

with

$$\Omega =\left[\begin{array}{c}\overline{L}\overline{B}\\ -\overline{L}\overline{B}\\ I\\ I\end{array}\right], T=\left[\begin{array}{c}\overline{N}-\overline{L}\overline{G}W(k+p)-\overline{L}\overline{A}x(k)-\overline{L}\overline{H}{e}_x(k)\\ -\overline{M}+\overline{L}\overline{G}W(k+p)+\overline{L}\overline{A}x(k)+\overline{L}\overline{H}{e}_x(k)\\ U_{\max }\\ -U_{\min }\end{array}\right],$$

where

$$M=\left[\begin{array}{c}{d}_0\\ v_{\min }\\ a_{\min }\\ j_{\min }\end{array}\right], N=\left[\begin{array}{c}Inf\\ v_{\max }\\ a_{\max }\\ j_{\max }\end{array}\right], L=\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right], \overline{M}=\left[\begin{array}{c}M\\ M\\ ⋮\\ M\end{array}\right],$$

$$\overline{N}=\left[\begin{array}{c}N\\ N\\ ⋮\\ N\end{array}\right], \overline{L}=\left[\begin{array}{cccc}L& & & \\ & L& & \\ & & \ddots & \\ & & & L\end{array}\right], U_{\max }=\left[\begin{array}{l}{u}_{\max }\\ ⋮\\ u_{\max }\end{array}\right], U_{\max }=\left[\begin{array}{l}{u}_{\min }\\ ⋮\\ u_{\min }\end{array}\right],$$

where $Inf$ stands for infinity, i.e., the value of the inter-distance can be infinite.

Invoking the active set method for Matlab to solve, the complete problem formulation is summarized by

$$\begin{array}{l}\underset{U(k+n)}{\min }\{U{(k+n)}^T{K}_{1}U(k+n)+2K_{2}U(k+n)\}\\ s.t. \Omega U(k+n)\le T\end{array}.$$

(19)

In this paper, the first control variable in the output sequence is taken as the input of the lower-level controller, thus controlling the vehicle driving.

4.1.3. Takeover Mode Design

Referring to the experiment requirements in the standard Euro-NACP, the representative velocity nodes of 80 km/h, 100 km/h, and 120 km/h are selected as the initial velocity in CarSim, and then the vehicle coasts of 0.5 s, with $\mu =0.85$, and then the brake with specific wheel cylinder pressure $P_b$ to obtain the corresponding deceleration $a_e$, as shown in

Table 1. Deceleration under each velocity and wheel cylinder pressure.

${\mathit{P}}_{\mathit{b}}$ (Mpa)	${\mathit{a}}_{\mathit{e}}$ (m·s−2)
	80 km/h	100 km/h	120 km/h
3.85	5.88	5.97	6
4	6.1	6.17	6.25
5	7.5	7.64	7.7
6	7.6	7.9	7.98
7	8	8.1	8.18
8	8.23	8.28	8.33

. After analysis and comparison, 7 Mpa is finally adopted as the emergency brake wheel cylinder pressure.

4.1.4. Emergency Lane-Change Mode Design

(1)

Path planning

In this paper, the quintic polynomial function is adopted as the lane-changing path that satisfies the conditions of continuous curvature and zero lateral velocity and acceleration at the starting point:

$$y(x)=a_5{x}_d{}^5+a_4{x}_d{}^4+a_3{x}_d{}^3+a_2{x}_d{}^2+a_1{x}_d+a_0,$$

(20)

where parameter $a_0$–$a_5$ is the polynomial coefficient.

(2)

Path tracking controller design

By analyzing the 3-DOF vehicle dynamics model and the dynamic equations [37], simplifying the tire model, and transforming the vehicle coordinate into world coordinate, the nonlinear vehicle dynamic model can be established.

$$\{\begin{array}{c}m{\dot{v}}_y=-m{v}_x\dot{\phi }+2\left[C_{cf}\left({\delta }_f-\frac{v_y+a\dot{\phi }}{v_x}\right)+C_{cr}\frac{b\dot{\phi }-v_y}{v_x}\right]\\ m{\dot{v}}_x=m{v}_y\dot{\phi }+2\left[C_{cf}\left({\delta }_f-\frac{v_y+a\dot{\phi }}{v_x}\right){\delta }_f+C_{lf}{s}_f+C_{lr}{s}_r\right]\\ I_z\ddot{\phi }=2\left[a{C}_{cf}\left({\delta }_f-\frac{v_y+a\dot{\phi }}{v_x}\right)-b{C}_{cr}\frac{b\dot{\phi }-v_y}{v_x}\right]\\ \dot{Y}=v_x\sin \phi +v_y\cos \phi \\ \dot{X}=v_x\cos \phi -v_y\sin \phi \end{array},$$

(21)

where parameters $v_x$, $v_y$, ${\dot{v}}_x$, ${\dot{v}}_y$ are the vehicle longitudinal velocity, the lateral velocity, the longitudinal acceleration, and the lateral acceleration, respectively; parameter $\dot{\phi }$ is the yaw rate; parameters $a$ and $b$ are the distance from the centroid to the front and rear axle, respectively; parameter $I_z$ is the rotational inertia; parameter ${\delta }_f$ is the front wheel angle; parameter $C$ is tire stiffness; parameters $f$, $r$, $l$, $c$ are the front and rear wheel and the longitudinal and lateral direction; parameter $s$ is the slip ratio.

A momentous segment of the MPC theory is prediction of the output of the system. Combining control quantities and system state variables, an equation form can be obtained by linearizing and discretizing:

$$\{\begin{array}{l}\tilde{\xi }\left(k+1|t\right)={\tilde{A}}_{k,t}\tilde{\xi }\left(k|t\right)+{\tilde{B}}_{k,t}\Delta u\left(k|t\right)\\ \eta \left(k|t\right)={\tilde{C}}_{k,t}\tilde{\xi }\left(k|t\right)\end{array},$$

(22)

where ${\tilde{A}}_{k,t}=\left[\begin{array}{cc}{A}_c& B_c\\ 0_{m\times n}& I_m\end{array}\right], {\tilde{B}}_{k,t}=\left[\begin{array}{l}{B}_c\\ I_m\end{array}\right], {\tilde{C}}_{k,t}={\left[\begin{array}{c}{C}_c\\ 0\end{array}\right]}^T$; $\tilde{\mathit{\xi }}\left(k|t\right)$ is the state quantity of the system; $\Delta u\left(k|t\right)$ is the system control increment; $0_{m\times n}$ is the $m\times n$ dimensional 0 matrix; $I_m$ is the $m$ dimensional identity matrix.

In the design of the objective function, relevant constraints are introduced, and for preventing the situation where the controller is without optimal solution, a relaxation factor is introduced, so $J$ can be expressed as

$$J(\tilde{\xi }(t),\Delta U(t))={\displaystyle \sum _{i=1}^P\Vert \eta (t+i|t)-{\eta }_{ref}(t+i|t)\Vert }\begin{array}{c}2\\ Q\end{array}+{\displaystyle \sum _{i=1}^{N-1}\Vert \Delta u(t+i|t)\Vert }\begin{array}{c}2\\ R\end{array}+\rho {\epsilon }^2,$$

(23)

where ${\eta }_{ref}(t+i|t),i=1,2,\dots ,P$ is the reference output; matrices $Q$ and $R$ stand for weight; parameter $\rho$ is the weight coefficient; parameter $\epsilon$ is the relaxation factor; $P$ and $N$ are the prediction and the control horizon, respectively.

The ultimate control quantities and control increment are constrained by

$$\begin{array}{l}{u}_{\min }(t+k)\le u(t+k)\le u_{\max }(t+k), k=0,1,\dots ,N-1\\ \Delta u_{\min }(t+k)\le \Delta u(t+k)\le \Delta u_{\max }(t+k), k=0,1,\dots ,N-1\end{array}.$$

(24)

Next, the objective function is transformed into thr QP problem and the solution of MPC is evolved into the below optimization problem:

$$J(t)=E^T\mathrm{QE}+\Delta U^T{\Theta }^{T}Q\Theta \Delta U+\Delta U^{T}R\Delta U+\rho {\epsilon }^2,$$

(25)

where

$$E=Y_{ref}(t)-{\mathsf{\Psi }}_t\tilde{\xi }(t|t),Y_{ref}(t)=[{\eta }_{ref}(t+1|t),\cdots {\eta }_{ref}(t+P|t)],$$

$${\Theta }_t=\left[\begin{array}{cccc}{\tilde{C}}_{t,t}{\tilde{B}}_{t,t}& 0& \cdots & 0\\ {\tilde{C}}_{t,t}{\tilde{A}}_{t,t}{\tilde{B}}_{t,t}& {\tilde{C}}_{t,t}{\tilde{B}}_{t,t}& \cdots & 0\\ \cdots & \cdots & \cdots & \cdots \\ {\tilde{C}}_{t,t}{\tilde{A}}_{t,t}{}^N{\tilde{B}}_{t,t}& {\tilde{C}}_{t,t}{\tilde{A}}_{t,t}{}^{N-1}{\tilde{B}}_{t,t}& \cdots & {\tilde{C}}_{t,t}{\tilde{A}}_{t,t}{\tilde{B}}_{t,t}\\ \cdots & \cdots & \cdots & \cdots \\ {\tilde{C}}_{t,t}{\tilde{A}}_{t,t}{}^{P-1}{\tilde{B}}_{t,t}& {\tilde{C}}_{t,t}{\tilde{A}}_{t,t}{}^{P-2}{\tilde{B}}_{t,t}& \cdots & {\tilde{C}}_{t,t}{\tilde{A}}_{t,t}{}^{P-N}{\tilde{B}}_{t,t}\end{array}\right],\Delta U(t)=\left[\begin{array}{c}\Delta u(t|t)\\ \Delta u(t+1|t)\\ ⋮\\ \Delta u(t+N-1|t)\end{array}\right],{\Psi }_t=\left[\begin{array}{c}{\tilde{C}}_{t,t}{\tilde{A}}_{t,t}\\ {\tilde{C}}_{t,t}{\tilde{A}}^2{}_{t,t}\\ ⋮\\ {\tilde{C}}_{t,t}{\tilde{A}}^N{}_{t,t}\\ ⋮\\ {\tilde{C}}_{t,t}{\tilde{A}}^P{}_{t,t}\end{array}\right].$$

The solution method is identical to the car-following mode; the unique difference is the control quantity. Conclusively, the optimal control sequence can be obtained by taking the first value $\Delta u^{*}(k)$ in the sequence as the control increment at this moment.

$$\Delta u(k)=u(k-1)+\Delta u^{*}(k).$$

(26)

(3)

Yaw moment calculation

By constraining the vehicle’s yaw rate and sideslip angle, the torque is redistributed considering tire load rate so that the vehicle can drive normally while generating additional yaw moment to preserve body stability. By analyzing the 2-DOF vehicle dynamics model and the dynamic equations [37], the following representations can be collated:

$$\{\begin{array}{l}\dot{\beta }=\frac{C_{cf}+C_{cr}}{m{v}_x}\beta +\left(\frac{a{C}_{cf}-b{C}_{cr}}{m{v}_x{}^2}-1\right)\gamma -\frac{C_{cf}}{m{v}_x}\delta \\ \dot{\gamma }=\frac{a{C}_{cf}-b{C}_{cr}}{I_z}\beta +\frac{a^2{C}_{cf}+b^2{C}_{cr}}{I_z{v}_x}\gamma -\frac{a{C}_{cf}}{I_z}\delta \end{array},$$

(27)

where parameters $C_{cf}$ and $C_{cr}$ are front and rear wheel cornering stiffness, respectively.

When the steering wheel input is certain and the vehicle is traveling steadily, $\dot{\beta }=0$, $\dot{\gamma }=0$, substituting these parameters into above equation, the ideal state can be expressed as:

$$\{\begin{array}{l}{{\beta }^{\prime }}_d=\frac{b+ma{v}_x^2/(C_{cr}(a+b))}{(a+b)(1+K{v}_x^2)}\\ {{\gamma }^{\prime }}_d=\frac{v_x}{(a+b)(1+K{v}_x^2)}\delta \end{array}$$

(28)

where: understeer gradient $K=\frac{m}{{(a+b)}^2}(\frac{a}{C_{cr}}-\frac{b}{C_{cf}})$.

Aiming at acquiring the relationship between yaw moment and movement error, the yaw moment can be calculated, adopting LQR as follows:

$$J=\frac{1}{2}\cdot {\displaystyle {\int }_0^{\infty }\left[x^T(t)\cdot Q\cdot x(t)+u^T(t)\cdot R\cdot u(t)\right]}dt,$$

(29)

where matrix $x$ is the error between $\gamma$, $\beta$, and desired ones, that is, $x={\left[\begin{array}{cc}\Delta \beta & \Delta \gamma \end{array}\right]}^T$; parameter $u$ is the additional yaw moment required to restore vehicle to ideal conditions, with $u=\left[\Delta M\right]$.

The optimal yaw moment is calculated by solving the algebraic Riccati equation online.

$$\Delta M^{*}=c_1\cdot \Delta \beta +c_2\Delta \gamma ,$$

(30)

where $c_1$ and $c_2$ are the optimal control rates.

(4)

Torque distribution strategy

The torque is redistributed taking into account the tire loading rate. The required longitudinal force $F_x$ and yaw moment $\Delta M$ expressions can be obtained by analyzing the tire forces applied to each wheel as follows:

$$\{\begin{array}{l}{F}_x=F_{xfl}\cos {\delta }_f+F_{xfr}\cos {\delta }_f+F_{xrl}+F_{xrr}\\ \Delta M=\left(-\frac{d}{2}\cos {\delta }_f+a\sin {\delta }_f\right)F_{xfl}+\left(\frac{d}{2}\cos {\delta }_f+a\sin {\delta }_f\right)F_{xfr}-\frac{d}{2}{F}_{xrl}+\frac{d}{2}{F}_{xrr}\end{array},$$

(31)

where parameter $d$ is the wheel distance; corner marks $fl$, $fr$, $rl$, and $rr$ represent the front left, front right, rear left, and rear right wheels of the vehicle, respectively.

Combining the necessary constraints, the tire longitudinal force distribution can be calculated as

$$\begin{array}{c}\min J={\displaystyle \sum _{i=1}^4\frac{F_{xi}{}^2}{{\left({\mu }_i{F}_{zi}\right)}^2}}\\ s.t. |F_{xi}|\le \sqrt{{\mu }_{\mathrm{i}}^2{F}_{zi}^2-F_{yi}^2}\le {\mu }_i{F}_{zi},-T_{\max }/r\le F_{xi}\le T_{\max }/r\end{array},$$

(32)

where parameter $\mu$ is the road friction coefficient; parameter $F_z$ the vertical load of the tire; parameter $T_{\max }$ is the maximum torque; parameter $r$ is the tire rolling radius.

The above two expressions are transformed into the norm form and the QP problem; then, they are solved to obtain the optimal solution.

4.2. Lower-Level Controller Design

A hub motor is utilized as the drive source [37] in this paper; accordingly, the lower-level controller then converts the desired acceleration $a_{des}$ calculated by the upper-level controller into the drive torque $T_m$ and wheel cylinder pressure $P_b$ that control the whole vehicle model. It contains the sections described below.

(1)

Vehicle driving model

Under the consideration of longitudinal resistance only, the driving torque can be given by

$$T_m=\frac{r}{i_0{\eta }_t}\left[\frac{C_{D}A{v}^2}{21.15}+mg(f\cos \theta +\sin \theta +\delta ma)\right],$$

(33)

where parameter $i_0$ is the main gearbox ratio; parameter ${\eta }_t$ is the transmission efficiency of the driveline; parameter $C_D$ is the air resistance coefficient; parameter $A$ is the longitudinal wind area; parameter $v$ is the velocity; parameter $m$ is the mass; parameter $f$ is the rolling resistance coefficient; parameter $\theta$ is the road slope; parameter $\delta$ is the rotating mass conversion coefficient; parameter $a$ is the acceleration; parameter 21.15 is obtained by unit conversion.

(2)

Vehicle braking model

In the braking process, without considering the acceleration resistance and driving force, and assuming that vehicle is driving on smooth road, the braking force $F_b$ can be collated:

$$F_b=-ma-\frac{C_{D}A{v}^2}{21.15}-mgf\cos \theta .$$

(34)

Within maximum braking force $F_b$, braking force $F_b$ varies linearly with oil pressure $P_b$, and $K_b$ is defined as the braking coefficient [38], that is, $K_b=1286$. Consequently, $P_b$ is obtained by

$$P_b=\frac{F_b}{K_b}.$$

(35)

(3)

Driving/Braking switching strategy and control method

Conducting coasting experiments in CarSim to acquire acceleration threshold $a$ at each velocity (10–120 km/h),and to assure smooth switching between driving/braking, intercalating the transition zone $\Delta h$, $\Delta h=0.1$ m/s² [39], above and below the original curve benchmark is necessary; this can compose the driving/braking switching logic. When $a_{des}>a-\Delta h$, the vehicle is driving; when $a_{des}<a-\Delta h$, the vehicle is braking; when $a-\Delta h<a_{des}<a+\Delta h$, the vehicle remains in its current mode.

To improve the control accuracy of the system, the PID method is used to quickly converge the deviation value between the expected longitudinal acceleration and the actual longitudinal acceleration to 0, achieving efficient control of ACC vehicles. The trial-and-error method is used to set the control parameter of the PID controller, where $k_p=0.2$, $k_i=1.8$.

5. Simulation and Results

To verify the proposed mode switching strategy and the controller, the hardware-in-the-loop simulation platform is built by using the Ni-PXI real-time system, and the ACC control algorithm is compiled and downloaded to the raspberry controller.

The designed vehicle model needs to be deployed to the real-time system through VeriStand software, and then configure the CAN board and map the input and output variables. The HIL test platform is shown in Figure 5. Concurrently, other simulation parameters are aggregated in

Table 2. Parameters in the simulation.

Para/Unit	Value	Para/Unit	Value	Para/Unit	Value
$m$/kg	1280	$d_0$/m	5	$d_c$/(m)	7
$h$/m	540	$v_{\min }$/(m/s)	0	$v_{\max }$/(m/s)	40
$r$/m	302	$a_{\min }$/(m/s2)	−5.5	$a_{\max }$/(m/s2)	3.5
$v_{\max }$/(m/s)	120	$u_{\min }$/(m/s2)	−5.5	$u_{\max }$(m/s2)	3.5
$T_{\max }$/(N·m)	200	$j_{\min }$/(m/s3)	−3	$j_{\max }$(m/s3)	3
$C_D$	0.3	$\tau$/(s)	0.5	${\rho }_{\delta }={\rho }_v={\rho }_a={\rho }_j$	0.96
$A$/m2	2.2	$p$	16	$n$	5
$\delta$	1.09	$A$	26	$N$	3
$i_0$	4.1	$T_s$/(s)	0.2	$\epsilon$	1000
${\eta }_t$	0.92	$Q$ (ACC)	Diag (40, 150, 2, 2)	$R$ (ACC)	10
$d$/m	1880	$Q$ (Lane Change)	Diag (1000, 50)	$R$ (Lane Change)	5 × 105
$\theta$/°	0	$Q$ (DYC)	[1, 0; 0, 100]	$R$ (DYC)	[1, 0; 0, 1]

.

5.1. Simulation Setup

Considering the mode switching and the sub-mode repartition, four traffic scenarios are erected: cruise, car following, emergency braking, emergency lane changing, and mixed switching for simulation. Then, the vehicle’s following, safety, and comfort are analyzed.

• Cruise control scenario: The vehicle’s initial velocity is 90 km/h, the cruise velocity is 120 km/h; there is no vehicle ahead of the host vehicle.
• Car-following scenario: The preceding vehicle performs acceleration, weak deceleration, strong deceleration, and steady-state following actions between 0 and 180 s.
• Emergency lane-changing scenario: A stationary obstacle is placed and the host vehicle’s initial velocity is 120 km/h in CarSim. Two types of controllers are used to verify the control effect, without stability and with stability.
• Mixed switching scenario: The preceding vehicle performs acceleration, weak deceleration, strong deceleration, and steady following actions between 0 and 180 s. Notice that its cruise velocity is 120 km/h and a stationary obstacle abruptly appears at 160 s.

5.2. Results

(1)

Cruise control scenario: The results are revealed in Figure 6 and Figure 7.

It can be seen in Figure 6 that when the vehicle enters the cruise mode, it can respond quickly to the controller’s decision and accelerate from 90 km/h to 120 km/h in approximately 4.8 s with practically zero velocity overshoot. In Figure 7, it can be seen that the acceleration of the host vehicle fluctuates marginally over 0–1.2 s; after that, the actual acceleration is basically stable at 1.85 m/s², and after 6.5 s, the host vehicle’ acceleration can be maintained at 0 m/s², which allows the occupants to feel comfortable and verifies the effectiveness of the designed cruise function. Compared to reference [9], the PID controller proposed in this paper can satisfy the corresponding requirements faster and has a smaller acceleration overshoot due to the appropriate gains.

(2)

Car-following scenario: The results are exhibited in Figure 8, Figure 9, Figure 10 and Figure 11.

Figure 8 and Figure 10 show that the vehicle performed favorably in velocity following and velocity response under diverse scenarios with a velocity overshoot of 5 km/h at 90 s and 134 s. Despite the MPC velocity control, there exists the overshoot phenomenon; the vehicle is ultimately able to steadily follow the preceding vehicle. During the deceleration, the vehicle does not collide with the preceding vehicle and invariably preserves a secure distance; palpably, when the preceding vehicle is driving steadily, the host vehicle follows the preceding vehicle with lesser inter-distance error.

In Figure 9, it can be seen that the actual acceleration is within the acceleration constraints of the sub-mode, which is preserved at −5–3.5 m/s² throughout. In Figure 11, the jerk is basically maintained at −2–2 m/s³, ensuring the comfort of the occupants in whole period except for a transient discomfort during the strong deceleration and acceleration. The main reason for this is that we pay more attention to the driving safety in the optimization design, so the jerk exceeds 6 m/s³. Compared with traditional MPC algorithms [17], the ACC method based on a real-time optimization control algorithm can balance the driver safety and the riding comfort.

(3)

Emergency lane-changing scenario: The results are illustrated in Figure 12, Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17.

In Figure 12 and Figure 13, the vehicles with both with and without stability controller can achieve emergency collision avoidance. The lateral error without the stability controller ups to 0.42 m, while the vehicle with stability control can better track the quintic polynomial. Its trajectory appears smooth and without sudden change and its maximum lateral error is 0.145 m, and the trajectory tracking accuracy is heightened by 65%. Figure 14 illustrates that the vehicle with the stability controller has a smaller deflection amplitude of the front-wheel steering angle, which fleetly reverts to zero after changing lane successfully, while the one without the stability controller exhibits a modicum of fluctuation after changing the lane and recovers stability after 2.5 s. Figure 15 and Figure 16 present the evaluation parameters of vehicle stability and comfort simultaneously, directly affecting vehicle maneuverability. The magnitude with the stability controller is distinctly smaller and converges more speedily than that without the stability controller, and its maximum $\gamma$ and $\beta$ are within the stability range, indicating that the vehicle has good body stability performance. Figure 17 shows the results of lateral acceleration, which, with stability control, can be restricted within 3.5 m/s², indicating that the tires operate in the linear region, while the lateral acceleration without the stability controller exceeds 0.4 g yet does not prevent the vehicle from tracking the quintic polynomial and achieving lateral collision avoidance. It is obvious that vehicles without body control are more arduous to maintain body stability during lane changing and following the trajectory, mainly due to the fact that lane changing can easily lead to a drift, causing the loss of control of the vehicle.

(4)

Mixed switching scenario: The results are represented in Figure 18, Figure 19, Figure 20 and Figure 21.

In Figure 18 and Figure 19, it can be seen that throughout the simulation process, the host vehicle’s velocity can better track the velocity of the preceding vehicle as well as the cruising velocity, and the velocity and acceleration do not engender irregular changes during the process of cutting out and cutting in. During the process of emergency braking, the host vehicle fleetly brakes with the maximum deceleration of 7.9 m/s² while ensuring safety, but it also rapidly coincides with the preceding vehicle’s velocity. Except for the emergency deceleration, which exceeds 5 m/s², the rest periods can be maintained at −5–3.5 m/s². The overall mode-switching process is from car-following to cruise to car-following to takeover to car-following to cruise to lane changing to cruise, with a steady switch between modes. Figure 20 shows that the host vehicle enters the cruise mode as the radar detects no vehicle ahead; the detection distance is 0 m. When the driving time amounts to 99 s, the vehicle brakes urgently, yet it is finally able to maintain an inter-distance of 5.6 m and does not collide with the preceding vehicle; at 160 s, the host vehicle takes an emergency lane changing to avoid collision with an undesired obstacle ahead. The vehicle can essentially maintain −2–2 m/s³ under other working conditions, except for the ephemeral discomfort during acceleration and emergency braking (see Figure 21). Compared to a single ACC longitudinal control strategy [11,12,13,14,15,16], the designed control strategy can avoid collisions in both vertical and horizontal directions, greatly improving the vehicle’s ability to avoid danger.

6. Conclusions

In this study, an adaptive cruise control strategy with emergency lane change function is presented. In view of the unpredictable working conditions, especially the emergency conditions, the overall control mode is divided into four modes along with the more micromesh car-following mode, while the mode switching strategy is devised. Then PID and MPC algorithms are utilized for the four modes, and considering the high velocity during the lane change, the stability controller is incorporated to improve body stability. Ultimately, we construct the HIL simulation platform. The results of the miscellaneous simulation conditions show that the devised ACC controller can fleetly and steadily track the preceding vehicle’s velocity or cruise velocity with practically no overshoot under the PID algorithm and within 5 km/h under the MPC algorithm, meanwhile quickly recovering stability. Except for the emergency condition, the acceleration under other conditions can be maintained at −5–3.5 m/s². The inter-distance graph reveals that its minimum distance is about 5.5 m, and the inter-distance can adjust dynamically according to the relative parameters under other conditions. Compared with the vehicle with no stability controller, the vehicle with the stability controller has better control performance, and all indicators are within the range of comfort and safety; in addition, the tracking accuracy is improved by 67%. In the mixed switching scenario, four modes switch rapidly and smoothly according to the complex and volatile traffic scenarios, and vehicle acceleration does not engender frequent jumps, which improves the controller’s adaptability to the complex traffic environment.

In summary, the ACC control strategy devised in this paper satisfies the requirements of followability, safety, comfort, and adaptability. In addition, the switching modes switch effectively and smoothly. Above all, collision avoidance can be actualized by longitudinal and lateral collision avoidance strategies under emergency conditions.

The strategy verification in this paper is only actualized through HIL simulation, which lacks an in-depth consideration of the actual working conditions and does not cover all the speed ranges. Regarding the directions of our future research, we will handle more realistic traffic scenarios and adjust mode switching strategies for different traffic scenarios in all speed ranges. A thorough sensitivity analysis using appropriate methods will be conducted in order to clarify the effectiveness of each of the parameters of the different modes adopted.