Emergency steering control of autonomous vehicle for

Vehicle System Dynamics International Journal of Vehicle Mechanics and Mobility

ISSN: 0042-3114 (Print) 1744-5159 (Online) Journal homepage: http://www.tandfonline.com/loi/nvsd20

Emergency steering control of autonomous vehicle for collision avoidance and stabilisation Xiangkun He, Yulong Liu, Chen Lv, Xuewu Ji & Yahui Liu To cite this article: Xiangkun He, Yulong Liu, Chen Lv, Xuewu Ji & Yahui Liu (2018): Emergency steering control of autonomous vehicle for collision avoidance and stabilisation, Vehicle System Dynamics, DOI: 10.1080/00423114.2018.1537494 To link to this article: https://doi.org/10.1080/00423114.2018.1537494

Published online: 01 Nov 2018.

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VEHICLE SYSTEM DYNAMICS https://doi.org/10.1080/00423114.2018.1537494

Emergency steering control of autonomous vehicle for collision avoidance and stabilisation Xiangkun Hea , Yulong Liua , Chen Lvb , Xuewu Jia and Yahui Liua a The State Key Laboratory of Automotive Safety and Energy, Tsinghua University, Beijing, People’s Republic

of China; b The School of Mechanical and Aerospace Engineering and the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, Singapore ABSTRACT

ARTICLE HISTORY

Collision avoidance and stabilisation are two of the most crucial concerns when an autonomous vehicle finds itself in emergency situations, which usually occur in a short time horizon and require large actuator inputs, together with highly nonlinear tyre cornering response. In order to avoid collision while stabilising autonomous vehicle under dynamic driving situations at handling limits, this paper proposes a novel emergency steering control strategy based on hierarchical control architecture consisting of decision-making layer and motion control layer. In decision-making layer, a dynamic threat assessment model continuously evaluates the risk associated with collision and destabilisation, and a path planner based on kinematics and dynamics of vehicle system determines a collision-free path when it suddenly encounters emergency scenarios. In motion control layer, a lateral motion controller considering nonlinearity of tyre cornering response and unknown external disturbance is designed using tyre lateral force estimation-based backstepping sliding-mode control to track a collision-free path, and to ensure the robustness and stability of the closed-loop system. Both simulation and experiment results show that the proposed control scheme can effectively perform an emergency collision avoidance manoeuvre while maintaining the stability of autonomous vehicle in different running conditions.

Received 5 February 2018 Revised 28 September 2018 Accepted 2 October 2018 KEYWORDS

Autonomous vehicle; emergency steering control; collision avoidance; vehicle dynamics; driving limits

1. Introduction Motor vehicles have become an indispensable means of transportation in the present-day world, but the mobility brought by vehicles comes at a price [1–3]. In 2015, about 1.3 million people around the world were killed in traffic accidents, ranking tenth on the World Health Organisation’s list of top causes of death [4]. Ninety-three per cent of the motor vehicle traffic crashes can be traced to human error [5]. With the rapid development of artificial intelligence and automobile technology, autonomous vehicle is expected to take more burden and stress from human driver, thus enhancing safety and reducing driver’s

CONTACT Yahui Liu

[email protected]; Xuewu Ji

© 2018 Informa UK Limited, trading as Taylor & Francis Group

[email protected]

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X. HE ET AL.

workload, etc [6–8]. Autonomous vehicle is a product of multi-disciplinary knowledge and theories, in which environment recognition system, decision-making system, motion control system are the three main components of the software system [9,10]. Many researchers have reported the progress made on overall architecture and feasibility of autonomous vehicle technology [11–14]. This paper mainly focuses on the decision-making and motion control of autonomous vehicle. Accident analysis shows that collision accounts for 97.8% of all traffic accidents [15]. There are two main reasons for them. First, misjudgment of vehicle travelling risk by driver. Second, driver’s operational error or delayed reaction. Therefore, collision avoidance technology for autonomous vehicle has become a hot topic for researchers. A collision avoidance strategy for autonomous vehicle was proposed using hierarchical frameworks, in which a high-level MPC algorithm communicates collision-free paths to a low-level MPC algorithm responsible for path tracking [16]. A collision avoidance system for autonomous vehicle was developed using a motion planner and MPC-based active vehicle steering and active wheel torque control [17]. An additional feature of an MPC-based strategy for collision avoidance is that it continuously optimises the performance index by receiving information about vehicle position, heading angle and obstacles in the environment [18]. A rear-end collision avoidance system of autonomous vehicle was designed using hierarchical scheme consisting of linear threat assessment, projected escape path planning with nonzero initial condition, reference path generator and linear state feedback controller [19]. A shared control method of semiautonomous vehicle was proposed for obstacle avoidance and stability control using two safe driving envelopes [20]. In this method, one of the envelopes was defined by vehicle driving limits, and the other by spatial limitations imposed by lane boundaries and obstacles. In addition, an MPC-based strategy determined at each time step whether the current driver’s command allowed for a safe vehicle path within these two envelopes, intervening only when such a path did not exist. A dualenvelop-oriented path-tracking scheme for autonomous vehicle was described in [21], in which the shape of vehicle was considered as inner-envelop and the feasible road region was described as outer-envelop. Then an implicit linear MPC algorithm was proposed to design moving horizon path-tracking controller to handle dangerous scenarios that may lead to collision or running out of road. A two-stage control approach was proposed for autonomous vehicle obstacle avoidance in highway cruise conditions [22]. In this scheme, an outer-loop nonlinear nonconvex model predictive control was adopted to generate the collision-free path, and an inner-loop linear controller with preview information was used to track collision-free trajectory. A collision avoidance system for autonomous vehicle is developed, which mainly consisted of a path planner and a robust tracking controller [23]. In this system, the path planner was designed based on polynomial parameterisation optimised by simulated annealing algorithm, and the robust tracking controller was designed to resist external disturbances and follow planning path. A path-planning method based on the theory of virtual potential field and a path-tracking strategy using multiconstrained MPC were proposed for autonomous vehicle, which seeks to minimise the incidence for collision on roads [24]. An improved reinforcement learning algorithm was applied to develop an obstacle avoidance control scheme so that autonomous vehicle could execute continuous actions [25]. Moreover, the vehicle dynamics constraints and traffic rule constraints were added to the control strategy, which makes vehicle motion more effective.

VEHICLE SYSTEM DYNAMICS

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However, when autonomous vehicles leave research laboratory and enter public traffic, they must be able to deal with emergency situations, some of which may necessitate manoeuverings, such as emergency collision avoidance, which happens in a short time horizon and requires large actuator inputs, together with high yaw rates. Tyres will be highly saturated and begin to sideslip. In this situation, the characteristics of tyre force become highly nonlinear, making it difficult to stabilise the vehicle. Meanwhile, path following can no longer be executed effectively, as this may jeopardise vehicle stability. For these reasons, emergency collision avoidance technology of autonomous vehicle has attracted wide attention from academia and industry in recent years. Biral et al. proposed a four-wheel optimal steering control scheme of emergency collision avoidance for semiautonomous vehicle based on optimal control, in order to fully exploit vehicle’s manoeuvrability limits [26]. Seewald et al. proposed an emergency collision avoidance strategy for semiautonomous vehicle by path planner using a 5th order polynomial and nonlinear vehicle lateral controller [27,28]. Cao et al. designed a comprehensive architecture of emergency collision avoidance system for autonomous vehicle, which integrated a decision-making module, a path planning module, an MPC-based lateral motion control module and a fuzzy logic-based longitudinal motion control module to deal with potential hazards on road [29]. To deal with the coupled and nonlinear features of visionbased autonomous vehicle under the conditions of emergency avoidance of obstacles, Guo et al. developed a coordinated steering and braking control scheme based on nonlinear backstepping control framework and adaptive fuzzy sliding-mode control technique [30]. Funke et al. proposed a control strategy that integrates path tracking, vehicle stabilisation and collision avoidance and mediates among these sometimes conflicting objectives by giving priority to emergency collision avoidance [31]. In addition, the framework was implemented using a feedback-feedforward-based longitudinal motion controller and an MPC-based lateral motion controller. Although the above research achievements were successful to some extent, there is still room for improvement and perfection. Firstly, most of the studies did not discuss threat assessment associated with collision and destabilisation of autonomous vehicle. Secondly, some researchers, in attempting to solve problems connected with emergency collision avoidance for autonomous vehicle, tend to focus on obstacle avoidance technology within or close to the linear region of tyres. Thirdly, due to the fact that tyre operating at or close to its physical limits of friction exhibits highly nonlinear cornering response and the fact that unknown external disturbances can be caused by changing driving conditions, most of the control schemes mentioned above would be insufficient to ensure path-tracking capability and stability of autonomous vehicle in emergency collision avoidance. Based on above analysis, it is necessary to investigate how to effectively conduct threat assessment associated with collision and destabilisation, and to reduce the adverse effects of nonlinearity of tyre cornering response and unknown external disturbance on emergency collision avoidance for autonomous vehicle at or near the physical limits of tyre friction. For threat assessment, compared with time to collision (TTC) scheme, dynamic safety distance model was found to be better at considering the risk of collision and destabilisation for autonomous vehicle [19,28]. For tyre nonlinearity and unknown external disturbance, the combined scheme of dynamic states estimation and nonlinear robust control could be an effective solution. With higher lateral accelerations or lower road adhesion, the characteristics of tyre force would become highly nonlinear or saturated, making it difficult to

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Figure 1. Architecture of emergency collision avoidance control scheme.

control the vehicle. Therefore, estimation of tyre lateral force can effectively compensate for the nonlinearity and uncertainty of system, thus improving vehicle control performance at or close to the driving limits. The backstepping sliding-mode control is a specific type of robust control, which combines the merits of both backstepping control and sliding-mode control, and has shown its effectiveness in dealing with multiple dynamics, nonlinearity and uncertainty [32,33]. The central idea of backstepping sliding-mode control is that some appropriate functions of state variables are selected recursively as pseudocontrol inputs for lower dimension subsystems of the overall system [34]. Moreover, the Lyapunov function is used to guarantee the asymptotic stability of each subsystem. Nevertheless, this type of approach is designed on the basis of an assumed mathematical model, whose imperfections can lead to the lowered performance of the controller. Hence, the need to design some kind of compensator is clear. Based on these considerations, for emergency scenarios on expressway, this paper proposes a novel emergency steering control strategy which can maintain autonomous vehicle’s stabilisation while avoiding collision in dynamic driving situations at handling limits. A block diagram of architecture for the emergency steering control scheme is shown in Figure 1, which consists of decision-making layer and motion control layer. In decisionmaking layer, a dynamic threat assessment model continuously analyses the risk associated with collision and destabilisation, and a path planner based on kinematics and dynamics of vehicle system determines a collision-free path when vehicle suddenly enters emergency situations. In motion control layer, a lateral motion controller considering nonlinearity of tyre cornering response and unknown external disturbance is developed using tyre lateral force estimation-based backstepping sliding-mode control to track the collision-free path, and to guarantee the robustness and stability of the closed-loop system. Finally, a Matlab/Simulink-CarSim co-simulation and a test in hardware-in-the-loop (HIL) system were conducted to verify the effectiveness of the proposed control scheme. This paper is organised as follows: in Section 2 the decision-making layer is designed, including a dynamic threat assessment model and a path planner. In Section 3, the motion


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control layer is illustrated. In Section 4 and Section 5, the simulation results and experiment results are analysed. Finally, the conclusion of this paper is made in Section 6.

2. Decision-making layer design The decision-making layer consists of a dynamic threat assessment model and a path planner. The former continuously analyses the risk metrics associated with collision and destabilisation. The latter, based on fifth-order polynomial equation, calculates a collision-free trajectory considering the constraints of kinematics and dynamics when an autonomous vehicle encounters an emergency situation. As shown in Figure 2, a single lane change manoeuvre is adopted to avoid collision. A fifth-order polynomial equation can be used to describe the path of lane change: y = AT X, with

A = a0

a1

X= 1

x

a2 x2

a3 x3

(1)

a4 x4

a5 x5

T

T

,

,

where x, y are the longitudinal and lateral coordinates for the escape path, and an (n = 1,2, . . . ) is polynomial coefficients. The boundary restraint conditions of the fifth-order polynomial are defined as: ⎧ ⎪ ⎨y(x0 ) = 0, y(xT ) = yT (2) y˙ (x0 ) = 0, y˙ (xT ) = 0 , ⎪ ⎩ K(x0 ) = 0, K(xT ) = 0 with K=

d2 y/dx2 3/2

[1 + (dy/dx)2 ]

,

where x0 and y0 are longitudinal and lateral coordinates of vehicle centroid at the beginning moment of collision avoidance manoeuvre respectively, and x0 = 0, xT and yT are longitudinal and lateral terminal point coordinates for the collision-free trajectory respectively, K is the curvature of the path.

Figure 2. Illustration of the collision-free trajectory.

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Substitute Equation (1) into the boundary restraint conditions, the following relation can be obtained: T BA = 0 yT 0 0 0 0 , (3) where

⎡ 1 ⎢1 ⎢ ⎢0 B=⎢ ⎢0 ⎢ ⎣0 0

x0 xT 1 1 0 0

x02 xT2 2x0 2xT 2 2

x03 xT3 3x02 3xT2 6x0 6xT

x04 xT4 4x03 4xT3 12x02 12xT2

⎤ x05 xT5 ⎥ ⎥ 5x04 ⎥ ⎥. 5xT4 ⎥ ⎥ 20x03 ⎦ 20xT3

with Equation (3), the coefficients A can be derived from: yT yT yT T A = 0 0 0 10 x3 −15 x4 6 x5 . T

T

T

(4)

Combining Equation (1) and Equation (4), the expression of collision-free trajectory can be obtained: 3 4 5 x x x y(x) = 10yT − 15yT + 6yT . (5) xT xT xT So far the path planning process has only considered kinematic constraint conditions. However, in an emergency collision avoidance, tyre could be highly saturated and begin to sideslip, which makes it difficult to stabilise the vehicle. Therefore, in the decisionmaking layer, vehicle dynamics constraint conditions need to be taken into consideration, and a dynamic threat assessment model has to be designed to continuously assess the risk associated with collision and destabilisation. The lateral acceleration at the barycentre of vehicle can be defined as: ay = vx γ + v˙y ,

(6)

with vy = vx tan(β), where vx is longitudinal velocity, vy is lateral velocity, γ is yaw rate of vehicle body, β is sideslip angle of vehicle body. By Equation (6), the lateral acceleration is described as: vx β˙ ay = vx γ + v˙x tan(β) + . 1 + tan2 (β)

(7)

The lateral acceleration must be bounded by tyre–road friction coefficient, and then the following relation can be given: vx γ + ac ≤ μg, with vx β˙ ac = v˙x tan(β) + , 1 + tan2 (β) where μ is tyre–road friction coefficient, g is gravitational acceleration.

(8)


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Define the following relation as: ac = (1 − k)μg,

(9)

where k (0 < k < 1) is dynamic factor. Combining Equation (8) and Equation (9), the following relation can be obtained: γ ≤k

μg . vx

(10)

According to the kinematic principle, desired yaw rate can be given: γd = Kvx .

(11)

With Equation (2) and Equation (5), Equation (11) can be written as:

γd =

T

1 + 30yT =

2

3

T

T

60yT xx3 − 180yT xx4 + 120yT xx5 x2 xT3

− 60yT

x3 xT4

+ 30yT

x4 xT5

60yT U(U − 1)(2U − 1) 3 vx , 2 30yT 2 4 4 2 xT 1 + xT U (U − 1)

2 32

vx

(12)

where U=

x . xT

Considering that genetic algorithm has the ability to search a global optimal, and it can solve any kind of continuous or discrete optimisation problem [35], this paper adopts the optimisation scheme as shown in Figure 3. The population size Np, the chromosome length Lc and the termination generation Gt are designed to be 100, 20 and 500, respectively. The crossover probability Pc and the mutation probability Pm are selected to be 0.8 and 0.1, respectively.

Figure 3. Illustration of the genetic algorithm scheme.

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As a result, when U equals about 0.20, the equivalent maximum value of desired yaw rate can be obtained: p1 yT γde_ max = (13) 3 vx , p2 yT2 2 2 xT 1 + x 2 T

where p1 , p2 are gain parameters, and the approximate values are 5.76 and 0.59, respectively. Since the desired yaw rate must satisfy vehicle dynamics constraint conditions, the following relation can be given: |γde_ max | ≤ k

μg , vx

(14)

Combining Equation (13) and Equation (14), the following relation can be obtained: f (vx , k, μ, xT, yT ) ≤ 0,

(15)

with f =

vx2 kμg

xT2

p1 y T

1+

p2 yT2 xT2

32 − 1,

where f is risk assessment function. During collision avoidance, in order to more effectively assess risks and further explore the limits of safe driving, real-time distance between following vehicle and lead vehicle should be adopted. Design the following relations as: xT = nxfl , (16) yT = myfl where n, m are positive proportional coefficients. The adopted emergency collision avoidance scheme is shown in Figure 4, where xfl is the vehicle-to-vehicle longitudinal distance from radar or camera, and yfl is lateral displacement of following vehicle when driving distance is xfl , respectively. The assumption that (xfl , yfl ) is a point on the collision-free trajectory, and combining Equation (5) and Equation (16), the following equation can be obtained: m=

n5 . 10n2 − 15n + 6

(17)

Substitute Equation (16) and Equation (17) into the risk assessment function f, the improved risk assessment function E is derived: E=

vx2 kμg

p1 yfl

xfl2

1+

p2 yfl2 xfl2

n8 (10n2 −15n+6)2

n3 3 10n2 − 15n + 6 − 1. 2

(18)

Let the improved risk assessment function E be zero, that is E = 0. The longitudinal distance between following vehicle and lead vehicle xfl and the positive proportional coefficient n are regarded as dependent variable and independent variable, respectively. Other


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Figure 4. Illustration of emergency collision avoidance scheme.

Figure 5. Illustration of the critical dynamic factor at different vehicle-to-vehicle distance, different velocity and different tyre–road friction coefficient.

terms in Equation (18) are regarded as constants. The minimum of xfl can be obtained when n approximately equals 2.2. In addition, by Equation (17), when n equals 2, m equals 2. Therefore, in order to simplify the design process, n = 2 is selected in this section, and Equation (18) can be written as: E=

1 vx2 2 kμg

p1 yfl

xfl2

1+

p2 yfl2

3 − 1.

(19)

3 ,

(20)

2

xfl2

Let E = 0, the critical dynamic factor is derived: kc =

1 vx2 2 μg

p1 yfl

xfl2

1+

p2 yfl2

2

xfl2

The critical dynamic factor kc can predict and analyse the emergency degree of collision avoidance process. Passenger vehicle width is about 1.8 m, and safety margin of lateral displacement is set to 0.4 m in this section, and yfl is designed to be 2.2 m. As shown in Figure 5, the critical dynamic factor kc increases with vx , with the decrease of xfl and μ. Moreover, it can be seen that the kc on the domain (0.8, 1) increases sharply with the decrease of xfl and the increase of vx .

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Figure 6. Logic diagram of decision-making for emergency collision avoidance.

Figure 7. Observation for minimum lateral distance from lead vehicle during emergency collision avoidance.

Therefore, Equation (20) is adopted as dynamic threat assessment model in this paper. It can be seen from Figure 6 that, when the kc is greater than threat threshold Th , an emergency collision avoidance manoeuvre is triggered, else it is not activated. Considering that the kc on the domain (0.8, 1) increases sharply with the decrease of xfl and the increase of vx , and this work focuses on emergency collision avoidance situation, the Th is set to 0.85. In order to evaluate the effectiveness of the proposed emergency collision avoidance scheme, minimum lateral distance from lead vehicle needs to be observed. As shown in Figure 7, G is barycentre of following vehicle, ψ is the vehicle heading, D0 is the distance between middle point for the back of lead vehicle and the medial axis of following vehicle, Wl and Wf are lead vehicle width and following vehicle width respectively. Therefore, in this paper, the minimum lateral distance from lead vehicle can be described as: 1 DL = [yf cot(ψ) + xl0 − xf ] sin(ψ) − (Wf + Wl ), 2

(21)

where (xf , yf ) is the barycentric coordinate of following vehicle, (xl0 ,0) is middle point coordinate for the back bumper of lead vehicle. Because tyre–road information and vehicle states estimation schemes have already been discussed in detail in [36–38], it is assumed that tyre–road friction coefficient, vehicle velocity and sideslip angle can be estimated directly, and other control data can be measured by environment recognition system or sensing system.


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3. Motion control layer design In this section, the design of lateral motion controller is shown, which adopts a tyre lateral force estimation-based backstepping sliding-mode control strategy. To focus on the study of emergency collision avoidance for autonomous vehicle, the control of actuator is not discussed in this study.

3.1. System models for control design To design control law, the system models are established in this section, which consist of vehicle kinematics model and vehicle dynamics model. 3.1.1. Vehicle dynamics model In designing vehicle steering controller, a widely used simplified vehicle model with two-degrees-of-freedom (2DOF) is employed to capture the vehicle’s essential lateral dynamics: ⎧ (FLfl + FLfr ) + (FLrl + FLrr ) ⎪ ⎪ −γ ⎨β˙ = mvx a(FLfl + FLfr ) − b(FLrl + FLrr ) ⎪ ⎪ ⎩γ˙ = Jz

,

(22)

where m is the vehicle total mass, Jz is yaw moment of inertia, a and b are distance from the centre of gravity to front and rear-axle, respectively, FLf and FLr are tyre lateral forces of front-axle and rear-axle, FLfl and FLfr are tyre lateral forces on the left and right of front-axle, FLrl and FLrr are tyre lateral forces on the left and right of rear-axle, respectively. 3.1.2. Vehicle kinematics model To focus on path-tracking ability, the state variables of vehicle dynamics are transformed into state variables relevant to the collision-free path. Generally, it is desirable to eliminate both lateral error e and heading error ψ. But only one of them can be reduced with single input steering angle δ f . In this paper, the projected error ep is adopted to combine the lateral error e and the heading error ψ. The vehicle kinematics model for the states in Figure 8 is given by: ⎧ ⎪ e˙ = vx sin(ψ) + vy cos(ψ) ⎪ ⎪ ⎪ ⎨s˙ = v cos(ψ) − v sin(ψ) x y ⎪ep = e + xp sin(ψ) ⎪ ⎪ ⎪ ⎩ψ = ψ − ψ r

,

(23)

where s is the distance along the reference path, ψ r is the heading of the reference path and xp is the constant projected distance.

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Figure 8. Vehicle system model used for motion control.

3.2. Realisation of tyre lateral force estimation-based backstepping sliding-mode control scheme According to small angle approximation for ψ, and by differentiating ep and ψ in Equation (23), the following relations can be obtained: ⎧ ⎪ e˙ = vy + vx ψ ⎪ ⎪ ⎪ ⎨e = e + x ψ p p . (24) ˙ ⎪ ˙ ˙ e = e + x ψ p p ⎪ ⎪ ⎪ ⎩ψ˙ = γ − K˙s In the case of relatively high lateral acceleration or low adhesion, the closed-loop steering response becomes underdamped, and the result will be significant oscillation of yaw rate [39]. In order to eliminate the projected error and the oscillation of yaw rate, with Equation (24), the following relations can be derived: ˙ s − K¨s, ψ¨ = γ˙ − K˙

(25)

˙ e¨ = v˙y + v˙x ψ + vx ψ,

(26)

˙ s − K¨s). e¨ p = v˙y + v˙x ψ + vx (γ − K˙s) + xp (γ˙ − K˙

(27)

Substitute Equation (22) into Equation (27), the following equation can be obtained:

a(FLfl + FLfr ) − b(FLrl + FLrr ) ˙ s − K¨s . e¨ p = v˙y + v˙x ψ + vx (γ − K˙s) + xp − K˙ Jz (28) In practical situation, measuring tyre lateral forces is difficult for technical, physical and economic reasons. Therefore, these important data need to be observed or estimated. In


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order to fully represent the tyre cornering characteristics, the following estimation method of nonlinear tyre lateral force is adopted: ˆ ˆ ˆ ˆ ˆFL = FLfl FLfr = μCfl αf μCfr αf , (29) Fˆ Lrl Fˆ Lrr μCˆ rl αr μCˆ rr αr where α f and α r are slip angles of front-axle tyres and rear-axle tyres, Cfl and Cfr are tyre cornering stiffnesses on the left and right of front-axle, Crl and Crr are tyre cornering stiffnesses on the left and right of rear-axle respectively. In addition, using small angle approximations, the tyre slip angles of front-axle and rear-axle can be described as: ⎧ v sin(β) + aγ ⎪ ⎪ α − δf ≈ β + a vγx − δf = arctan ⎨ f v cos(β) , (30) vsin(β) − bγ ⎪ ⎪ ⎩αr = arctan ≈ β − b vγx v cos(β) where δ f is front wheel steering angle, v is the velocity of vehicle. In order to fully represent the tyre cornering characteristics, the following Pacejka nonlinear cornering stiffnesses are adopted [40]: ⎧ ˆ Fˆ ⎪ ˆ fr = Cf 0 sin 2 arctan FZfr ⎨Cˆ fl = Cf 0 sin 2 arctan ZZfl , C Z f0 f0 (31) , ⎪ ⎩Cˆ = C sin 2 arctan Fˆ Zrl , Cˆ = C sin 2 arctan Fˆ Zrr r0 rr r0 rl Zr0 Zr0 where Cf 0 and Cr0 are normal cornering stiffnesses of front-axle and rear-axle, Zf 0 and Zr0 are load factors of front-axle and rear-axle respectively. Considering the longitudinal and lateral acceleration, the vertical load of each wheel can be estimated by [41]: ⎧ h2 m ⎪ Fˆ Zfl = (g b2 − v˙x h2 − v˙y bh ⎪ c + v˙ x v˙ y gc ) L ⎪ ⎪ ⎪ ⎪ ⎨Fˆ Zfr = (g b − v˙x h + v˙y bh − v˙x v˙y h2 ) m 2 2 c gc L , (32) h ah h2 m a ⎪ ˆ ⎪ F = (g + v ˙ − v ˙ − v ˙ v ˙ ) x y x y Zrl ⎪ 2 2 c gc L ⎪ ⎪ ⎪ ⎩Fˆ = (g a + v˙ h + v˙ ah + v˙ v˙ h2 ) m Zrr x y x y 2

2

c

gc L

where h is the height of the centre of gravity, c is track width, L is wheelbase. Combining Equations (28), (29) and (30), the following relations can be obtained: ⎧ e¨ p = p1 + p2 + p3 δf ⎪ ⎪ ⎪ ⎪ ⎪ p1 = v˙y + v˙x ψ + vx (γ − K˙s) ⎪ ⎪ ⎪ ⎨ μa(vx β + aγ )(Cˆ fl + Cˆ fr ) μbαr (Cˆ rl + Cˆ rr ) . (33) ˙ s − K¨s − − K˙ p2 = xp ⎪ ⎪ vx Jz Jz ⎪ ⎪ ⎪ ⎪ ⎪ μa(Cˆ fl + Cˆ fr ) ⎪ ⎩p3 = −xp Jz In order to take into account random road input’s influence on lateral motion control performance of autonomous vehicle, the equivalent random disturbance acting on front-axle

14

X. HE ET AL.

is defined as [42]: dfe (·) = p3

de (·) , i

(34)

with |dfe (·)| ≤ D, where de (·) is equivalent random disturbance acting on steering actuator, i is steering ratio, D is a positive constant. Hence, under equivalent random disturbance, Equation (33) can be written as: de (·) e¨ p = p1 + p2 + p3 δf + i = p1 + p2 + p3 δf + dfe (·).

(35)

In addition, for the design of the proposed control scheme, the nonlinear vehicle-road system with the disturbance can be transformed into the following form: x˙ 1 = x2 , (36) x˙ 2 = p1 + p2 + p3 δf + dfe (·) where x1 = ep . In the following part, the design procedure of backstepping sliding-mode control approach for system (36) is given. Step 1. With Equation (36), the tracking error e1 and its corresponding first order derivative are defined as: ⎧ ⎪ ⎨xd = 0 (37) e1 = x1 − xd = x1 , ⎪ ⎩ e˙ 1 = x˙ 1 − x˙ d = x2 where xd is the desired value. To make the tracking error e1 converge to zero, select Lyapunov function as: 1 L1 = e21 . 2

(38)

Hence, the differentiation of Equation (38) is derived: L˙ 1 = e1 e˙ 1 .

(39)

In order to realise L˙ 1 ≤ 0, the sliding-mode surface function is defined as: sf = x2 + c1 x1 = e˙ 1 + c1 e1 , where c1 is a strictly positive gain parameter.

(40)


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With Equation (40), Equation (39) can be described as: L˙ 1 = e1 sf − c1 e21 .

(41)

If sf = 0, then L˙ 1 ≤ 0. Hence, in order to meet the Lyapunov stability theory, the design of next step is required. Step 2. Design Lyapunov function as: 1 L2 = L1 + s2f . (42) 2 Combining Equation (36) and Equation (40), the differentiation of the sliding-mode surface function can be written as: s˙f = x˙ 2 + c1 x˙ 1 = p1 + p2 + p3 δf + dfe (·) + c1 e˙ 1 .

(43)

Therefore, the derivative of L2 becomes: L˙ 2 = L˙ 1 + sf s˙f = e1 sf − c1 e21 + sf [p1 + p2 + p3 δf + dfe (·) + c1 e˙ 1 ].

(44)

In order to achieve L˙ 2 ≤ 0, a stabilising control law is designed: δf = −

p1 + p2 + c2 sf + e1 + c1 e˙ 1 + η tanh(sf ) , p3

(45)

where c2 and η are the strictly positive gain parameters. Substitute Equation (45) into Equation (44), with Equation (34) and according to Appendix A in [42], the following relation can be derived: L˙ 2 = −c1 e21 − c2 s2f − ηsf tanh(sf ) + sf dfe (·) ≤ −c1 e21 − c2 s2f − ηsf tanh(sf ) + |sf dfe (·)| = −c1 e21 − c2 s2f − ηsf tanh(sf ) + |dfe (·)| · |sf | ≤ −c1 e21 − c2 s2f − ηsf tanh(sf ) + D|sf |.

(46)

The η (or c1 , or c2 ) can be designed, for instance, the following relation: η D.

(47)

Therefore, the following relation can be obtained: L˙ 2 ≤ −c1 e21 − c2 s2f − ηsf tanh(sf ) + D|sf | ≤ 0.

(48)

Accordingly, x1 → 0, x2 → 0 and sf → 0 as t → ∞, in which case the closed-loop system is globally asymptotically stable. With Equation (45), the control input for steering actuator can be obtained: δsc = iδf .

(49)

The numerical values of main parameters for the lateral motion controller are given in Table 1.

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Table 1. The main parameters of the controller. Symbol a b c h xp m Jz Cf 0 Cr0 Zf 0 Zr0 η c1 c2 i

Parameters

Value

Units

Distance from front-axle to gravity centre Distance from rear-axle to gravity centre Track width Height of gravity centre Projected distance Vehicle mass Yaw inertia Normal cornering stiffness of front-axle Normal cornering stiffness of rear-axle Load factor of front-axle Load factor of rear-axle Strictly positive gain parameter Strictly positive gain parameter Strictly positive gain parameter Steering ratio

1.192 1.598 1.565 0.506 10 1528.13 2280 23,000 38,000 6000 6500 1 20 20 18.5

m m m m m kg kg m2 N/rad N/rad N N null null null null

4. Simulation and discussions In our previous work [42,43], the experimental verification of vehicle model was implemented to obtain an accurate simulation model, and the parameters of the vehicle model were given. In this section, two simulation cases are presented to verify the effectiveness of the proposed emergency steering control strategy for autonomous vehicle. The cosimulations are conducted based on Matlab/Simulink – CarSim with a high-fidelity and full-vehicle model.

4.1. Test on a low adhesion-coefficient road (µ = 0.3) Considering that the vehicle can easily become unstable when it makes a sharp turn on a low adhesion road, an emergency collision avoidance manoeuvre is conducted on an ice snow covered pavement with road adhesion coefficient μ = 0.3 and velocity v = 54 km/h to verify the effectiveness of the emergency steering control scheme for autonomous vehicle. As shown in Figure 9(e), the peak lateral accelerations of vehicle controlled by a slidingmode control strategy (SMC) based on nominal model [44], by the proposed solutions without de (·) and with de (·) are about 2.57 –2.45 and –2.47 m/s2 , respectively. Hence, it can be inferred that, during the emergency collision avoidance, the tyres of vehicle using the different methods run in the situation which is very close to tyre–road friction limit (μg). The results in Figure 9(a) show that, compared with the SMC strategy, the proposed solutions without de (·) and with de (·), both show superior performance in tracking collision-free trajectory. It can be found from Figure 9(b) that, peak values of lateral trajectory tracking error of vehicle controlled by the SMC scheme, by the proposed solutions without de (·) and with de (·) are about –0.19, –0.07 and –0.07 m, respectively. Figure 9(c) shows the peak values of heading error for vehicle controlled by the SMC strategy, by the proposed solutions without de (·) and with de (·) are about 1.81°, 0.44° and 0.52°, respectively.


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Figure 9. The results of emergency collision avoidance manoeuvre on an ice snow covered pavement (µ = 0.3). (a) Global trajectory of the path following, (b) path-tracking error, (c) heading error, (d) observation of minimum lateral distance from lead vehicle, (e) vehicle lateral acceleration, (f) vehicle yaw rate, (g) vehicle sideslip angle, (h) tyre lateral forces from Carsim and estimated tyre lateral forces for front-axle and rear-axle, (i) control input of steering actuator and equivalent random disturbance, (j) angle rate of steering actuator.

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It can be seen from Figure 9(d) that, after the vehicle-to-vehicle longitudinal distance is equal to zero, the minimum lateral distances from lead vehicle DL by the SMC strategy, by the proposed solutions without de (·) and with de (·) are about 0.54, 0.60 and 0.60 m, respectively. Hence, the vehicle controlled by any of the three schemes can avoid collision. In the case of low adhesion coefficient, the closed-loop steering response becomes underdamped, which results in some oscillation in vehicle lateral motion. Figure 9(e–g) illustrate that, compared with the SMC strategy, the proposed solution shows more satisfactory dynamics control performance during emergency collision avoidance. As shown in Figure 9(a–g), the proposed solution can effectively resist against unknown external disturbance. In Figure 9(h), it can easily be found that, the estimated tyre lateral forces can converge consistently to the tyre lateral forces from Carsim. As shown in Figure 9(i), the peak values of steering angle in the proposed solutions without de (·) and with de (·) are smaller than that of the SMC strategy, and the peak value of the disturbance is about 16.76° which is roughly a quarter of the peak steering angle controlled by the proposed scheme. It can also be found from Figure 9(j) that, the steering angle rate controlled by the proposed scheme has faster convergence than that of the SMC strategy. 4.2. Test on a high adhesion-coefficient road (µ = 1.0) In order to further evaluate the performance of the proposed control method, an emergency collision avoidance manoeuvre is conducted on a dry asphalt pavement with road adhesion coefficient μ = 1.0 and velocity v = 90 km/h. The results in Figure 10(e) show that the peak lateral accelerations of vehicle controlled by the SMC strategy, by the proposed solutions without de (·) and with de (·) are about 7.76, –7.76 and –7.70 m/s2 , respectively. Meanwhile, it can also be seen from Figure 10(g) that, the tyres are highly saturated. Hence, it can be inferred that, during emergency collision avoidance, the tyres of vehicle using the different methods work in their nonlinear region, and the autonomous vehicle operates at or close to its driving limits. In Figure 10(a), it can easily be seen that, compared with the SMC strategy, the proposed solutions without de (·) and with de (·), both exhibit satisfactory performance in tracking collision-free trajectory. As shown in Figure 10(b), peak values of lateral trajectory tracking error of vehicle controlled by the SMC strategy, by the proposed solutions without de (·) and with de (·) are about –0.86, 0.49 and 0.49 m, respectively. Figure 10(c) shows the peak values of heading error for vehicle controlled by the SMC strategy, by the proposed solutions without de (·) and with de (·) are about 4.95°, –2.87° and 2.86°, respectively. It can be found from Figure 10(d) that, after the vehicle-to-vehicle longitudinal distance is equal to zero, the minimum lateral distances from lead vehicle DL by the SMC strategy, by the proposed solutions without de (·) and with de (·) are about –0.48, 0.10 and 0.10 m, respectively. Therefore, the vehicle controlled by the SMC strategy cannot avoid collision on a high adhesion road, and the vehicle controlled by the proposed scheme effectively avoids collision. It can also be seen from Figure 10(d) that, there appears to be some second order oscillation in the steering response bringing DL quite close to zero. This is because in the case of high lateral acceleration, the closed-loop steering response becomes underdamped, which results in some oscillation in vehicle lateral motion. As shown in Figure 10(e–g), compared with the SMC strategy, the proposed solution shows more superior dynamics control performance during emergency collision


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Figure 10. The results of emergency collision avoidance manoeuvre on a dry asphalt pavement (µ = 1.0). (a) Global trajectory of the path following, (b) path-tracking error, (c) heading error, (d) observation of minimum lateral distance from lead vehicle, (e) vehicle lateral acceleration, (f) vehicle yaw rate, (g) vehicle sideslip angle, (h) tyre lateral forces from Carsim and estimated tyre lateral forces for front-axle and rear-axle, (i) control input of steering actuator and equivalent random disturbance, (j) angle rate of steering actuator.

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Figure 11. The HIL system for autonomous driving experiment.

Figure 12. Diagram of architecture for the HIL system.

avoidance. As shown in Figure 10(a–g), the proposed solution can effectively resist against unknown external disturbance. In Figure 10(h), it can be found that the estimated tyre lateral forces can converge consistently to the tyre lateral forces from Carsim. As shown in Figure 10(i), the steering angle in the proposed solutions without de (·) and with de (·) are fewer oscillations than that of the SMC strategy, and the peak value of the disturbance is about –41.30° which is roughly a quarter of the peak steering angle controlled by the proposed solution. It can also be seen from Figure 10(j) that, the steering angle rate controlled by the proposed scheme has faster convergence than that of the SMC strategy.

5. Experiment results with HIL system An HIL experiment is conducted to test the real-time performance of the proposed emergency steering control method for autonomous vehicle. Figure 11 shows the testing facilities, which consists mainly of a DS1501 MicroAutoBox from dSPACE, a Freescale G36 actuator driver, a steer-by-wire system, a EXLAR electric servo cylinder for steering resistance loading, a BOSCH steering angular sensor, a NI® PXI hardware, a monitor, a power supply, a terminal block, two host PCs and two displays. Figure 12 illustrates the implementation architecture of the HIL system, in which ® vehicle-road system model of CarSim platform is encoded into NI PXI hardware. After ® obtaining the vehicle motion states from NI PXI hardware, the proposed strategy, which


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Figure 13. The experiment results of emergency collision avoidance manoeuvre on an ice snow covered pavement (µ = 0.3). (a) path-tracking error, (b) heading error, (c) observation of minimum lateral distance from lead vehicle, (d) vehicle lateral acceleration, (e) vehicle yaw rate, (f) vehicle sideslip angle, (g) output of steer-by-wire system and equivalent random disturbance, (h) tyre lateral forces from Carsim and estimated tyre lateral forces for front-axle and rear-axle.

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is implemented in the MicroAutoBox, calculates the steering angle command of the steerby-wire system. Then the angle command is sent to the actuator driver to operate the steer-by-wire system, and the angular sensor feeds the values of corresponding steering column angle and angular speed back to nonlinear vehicle model of CarSim platform. The steering resistance torque, which is obtained from CarSim platform, acts on the rack of the steer-by-wire system by the EXLAR electric servo cylinder. The sampling rate of the experiments is 100 Hz. In the HIL system, an emergency collision avoidance manoeuvre is performed on a low adhesion-coefficient road with friction coefficient μ = 0.3 and velocity v = 54 km/h. The test results are shown in Figure 13. It can be seen that the results are similar to but not exactly the same as those given by CarSim-Simulink co-simulation. This is mainly because of communication delay and real characteristics of the electro-mechanical system. The results in Figure 13(d) show that the peak lateral accelerations of vehicle controlled by the SMC strategy, by the proposed solutions without de (·) and with de (·) are about – 2.63, –2.32 and 2.18 m/s2 , respectively. Hence, it can be inferred that, during the emergency collision avoidance, the vehicle controlled by any of these methods operates in the situation which is very close to the tyre–road friction limit (μg). As shown in Figure 13(a), peak values of lateral path-tracking error of vehicle controlled by the SMC strategy, by the proposed solutions without de (·) and with de (·) are about – 0.39, –0.13 and 0.16 m, respectively. Figure 13(b) shows the peak values of heading error for vehicle controlled by the SMC strategy, by the proposed solutions without de (·) and with de (·) are about 3.25°, 0.58° and 0.95°, respectively. It can be seen from Figure 13(c) that, after the vehicle-to-vehicle longitudinal distance is equal to zero, the minimum lateral distances from lead vehicle DL by the SMC strategy, by the proposed solutions without de (·) and with de (·) are about 0.30, 0.54 and 0.51 m, respectively. Therefore, the vehicle controlled by any of the three schemes can avoid collision. Figure 13(d–f) illustrate that, compared with the SMC strategy, the proposed solution shows more superior dynamics control performance during emergency collision avoidance. As shown in Figure 13(a–f), the proposed solution can effectively resist against unknown external disturbance. As shown in Figure 13(g), the peak values of steering angle in the proposed solutions without de (·) and with de (·) are smaller than that of the SMC strategy, and the peak value of the disturbance is about 16.76° which is roughly a quarter of the peak steering angle controlled by the proposed method. As shown in Figure 13(h), the estimated tyre lateral force of rear-axle can acceptably converge to the tyre lateral force from Carsim, and there is a significant mismatch between the estimated tyre lateral force of front-axle and the tyre lateral force from Carsim. However, according to above analysis, the proposed solution shows good lateral motion control performance, that is, the proposed method exhibits superior robustness to model mismatch.

6. Conclusion In this paper, a novel emergency steering control strategy is proposed to realise collision avoidance and ensure the stability of autonomous vehicle at the same time under dynamic driving situations at handling limits. The proposed scheme adopts a hierarchical


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control architecture consisting of two layers. In the decision-making layer, a dynamic threat assessment model continuously evaluates the risk associated with collision and destabilisation, and a path planner based on kinematics and dynamics of vehicle system determines a collision-free path when it suddenly enters emergency situations. In the motion control layer, a lateral motion controller considering nonlinear tyre cornering response and unknown external disturbance is developed using tyre lateral force estimation-based backstepping sliding-mode control approach. The results of co-simulation and experiment in HIL system show that, in different running conditions, both the proposed schemes without de (·) and with de (·) can provide sufficient collision avoidance capability as well as yaw stability for autonomous vehicle at or close to the driving limits. Compared with the SMC scheme, the proposed strategy shows superior lateral motion control. In addition, the proposed scheme exhibits satisfactory robustness to unknown external disturbances in emergency collision avoidance. In future work, behaviour decision-making considering steering and braking will be investigated to avoid collision while maintaining stabilisation of autonomous vehicle in emergency situations.

Disclosure statement No potential conflict of interest was reported by the authors.

Funding This work was supported by the National Natural Science Foundation of China [grant numbers U1664263, 51875302], the Independent Research Program of Tsinghua University [grant number 2015Z09006].

References [1] Eskandarian AE. Handbook of intelligent vehicles. London: Springer-Verlag; 2012. [2] He Z, Ji X. Nonlinear robust control of integrated vehicle dynamics. Veh Syst Dyn. 2012;50(2):247–280. [3] Wu J, Wang X, Li L, et al. Hierarchical control strategy with battery aging consideration for hybrid electric vehicle regenerative braking control. Energy. 2017;145:301–312. [4] World Health Organization. Top 10 causes of death worldwide. 2015. Available from: http://www.who.int/mediacentre/factsheets/fs310/en/ [5] Maddox J. Improving driving safety through automation. National Highway Traffic Safety Administration; 2012. [6] Petrov P, Nashashibi F. Modeling and nonlinear adaptive control for autonomous vehicle overtaking. IEEE Trans Intell Transp Syst. 2014;15(4):1643–1656. [7] Hu C, Jing H, Wang R, et al. Robust H∞ output-feedback control for path following of autonomous ground vehicles. Mech Syst Signal Process. 2016;70:414–427. [8] Brown M, Funke J, Erlien S, et al. Safe driving envelopes for path tracking in autonomous vehicles. Control Eng Pract. 2017;61:307–316. [9] González D, Pérez J, Milanés V, et al. A review of motion planning techniques for automated vehicles. IEEE Trans Intell Transp Syst. 2016;17(4):1135–1145. [10] Ni J, Hu J. Dynamics control of autonomous vehicle at driving limits and experiment on an autonomous formula racing car. Mech Syst Signal Process. 2017;90:154–174. [11] Kritayakirana K, Gerdes JC. Using the centre of percussion to design a steering controller for an autonomous race car. Veh Syst Dyn. 2012;50(suppl. 1):33–51.

24

X. HE ET AL.

[12] Gao Y, Gray A, Tseng HE, et al. A tube-based robust nonlinear predictive control approach to semiautonomous ground vehicles. Veh Syst Dyn. 2014;52(6):802–823. [13] Liu Y, Fan X, Lv C, et al. An innovative information fusion method with adaptive Kalman filter for integrated INS/GPS navigation of autonomous vehicles. Mech Syst Signal Process. 2018;100:605–616. [14] Hu X, Chen L, Tang B, et al. Dynamic path planning for autonomous driving on various roads with avoidance of static and moving obstacles. Mech Syst Signal Process. 2018;100:482–500. [15] National Highway Traffic Safety Administration. Traffic safety facts 2013. US Department of Transportation HS 811-870; 2015. [16] Gao Y, Lin T, Borrelli F, et al. Predictive control of autonomous ground vehicles with obstacle avoidance on slippery roads. ASME 2010 dynamic systems and control conference; Cambridge, Massachusetts, USA; 2010. p. 265–272. [17] Shim T, Adireddy G, Yuan H. Autonomous vehicle collision avoidance system using path planning and model-predictive-control-based active front steering and wheel torque control. Proceedings of the institution of mechanical engineers. Part D: J Automobile Eng. 2012;226(6):767–778. [18] Kim W, Kim D, Yi K, et al. Development of a path-tracking control system based on model predictive control using infrastructure sensors. Veh Syst Dyn. 2012;50(6):1001–1023. [19] Shah J, Best M, Benmimoun A, et al. Autonomous rear-end collision avoidance using an electric power steering system. Proc Inst Mech Eng Part D: J Automobile Eng. 2015;229(12):1638–1655. [20] Erlien SM, Fujita S, Gerdes JC. Shared steering control using safe envelopes for obstacle avoidance and vehicle stability. IEEE Trans Intell Transp Syst. 2016;17(2):441–451. [21] Guo H, Liu J, Cao D, et al. Dual-envelop-oriented moving horizon path tracking control for fully automated vehicles. Mechatronics. 2018;50:422–433. [22] Rosolia U, De Bruyne S, Alleyne AG. Autonomous vehicle control: a nonconvex approach for obstacle avoidance. IEEE Trans Control Syst Technol. 2017;25(2):469–484. [23] Wnag CY, Zhao WZ, Xu ZJ, et al. Path planning and stability control of collision avoidance system based on active front steering. Sci China Technol Sci. 2017;60(8):1231–1243. [24] Ji J, Khajepour A, Melek WW, et al. Path planning and tracking for vehicle collision avoidance based on model predictive control with multiconstraints. IEEE Trans Veh Technol. 2017;66(2):952–964. [25] Zong X, Xu G, Yu G, et al. Obstacle avoidance for self-driving vehicle with reinforcement learning. SAE Int J Passenger Cars-Electron Electr Syst. 2018;11(1):30–39. [26] Galvani M, Biral F, Nguyen BM, et al. Four wheel optimal autonomous steering for improving safety in emergency collision avoidance manoeuvres. 2014 IEEE 13th international workshop on advanced motion control (AMC); Yokohama, Japan; 2014. p. 362–367. [27] Keller M, Hass C, Seewald A, et al. Driving simulator study on an emergency steering assist. 2014 IEEE international conference on systems, man and cybernetics (SMC); San Diego, CA, USA; 2014. p. 3008–3013. [28] Seewald A, Haß C, Keller M, et al. Emergency steering assist for collision avoidance. ATZ Worldwide. 2015;117(1):14–19. [29] Cao H, Song X, Huang Z, et al. Simulation research on emergency path planning of an active collision avoidance system combined with longitudinal control for an autonomous vehicle. Proc Inst Mech Eng, Part D: J Automobile Eng. 2016;230(12):1624–1653. [30] Guo J, Hu P, Wang R. Nonlinear coordinated steering and braking control of visionbased autonomous vehicles in emergency obstacle avoidance. IEEE Trans Intell Transp Syst. 2016;17(11):3230–3240. [31] Funke J, Brown M, Erlien SM, et al. Collision avoidance and stabilization for autonomous vehicles in emergency scenarios. IEEE Trans Control Syst Technol. 2017;25(4):1204–1216. [32] Piltan F, Mansoorzadeh M, Zare S, et al. Artificial tune of fuel ratio: design a novel siso fuzzy backstepping adaptive variable structure control. Int J Electr Comput Eng. 2013;3(2):171–185. [33] Su Z, Wang H, Li N, et al. Exact docking flight controller for autonomous aerial refueling with back-stepping based high order sliding mode. Mech Syst Signal Process. 2018;101:338–360.


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[34] Liu J. Radial basis function (RBF) neural network control for mechanical systems: design, analysis and MATLAB simulation. Beijing: Springer-Verlag; 2013. [35] Sivanandam SN, Deepa SN. Introduction to genetic algorithms. New York: Springer Science & Business Media; 2007. [36] Liu YH, Li T, Yang YY, et al. Estimation of tire-road friction coefficient based on combined APF-IEKF and iteration algorithm. Mech Syst Signal Process. 2017;88:25–35. [37] Hashemi E, Khosravani S, Khajepour A, et al. Longitudinal vehicle state estimation using nonlinear and parameter-varying observers. Mechatronics (Oxf). 2017;43:28–39. [38] Chen T, Xu X, Chen L, et al. Estimation of longitudinal force, lateral vehicle speed and yaw rate for four-wheel independent driven electric vehicles. Mech Syst Signal Process. 2018;101:377–388. [39] Kapania NR, Gerdes JC. Design of a feedback-feedforward steering controller for accurate path tracking and stability at the limits of handling. Veh Syst Dyn. 2015;53(12):1687–1704. [40] Pacejka H. Tire and vehicle dynamics. 3rd ed. Oxford: Elsevier; 2012. [41] Doumiati M, Victorino A, Lechner D, et al. Observers for vehicle tyre/road forces estimation: experimental validation. Veh Syst Dyn. 2010;48(11):1345–1378. [42] Ji X, He X, Lv C, et al. Adaptive-neural-network-based robust lateral motion control for autonomous vehicle at driving limits. Control Eng Pract. 2018;76:41–53. [43] Ji X, He X, Lv C, et al. A vehicle stability control strategy with adaptive neural network sliding mode theory based on system uncertainty approximation. Veh Syst Dyn. 2018;56(6):923–946. [44] Liu J, Wang X. Advanced sliding mode control for mechanical systems: design, analysis and MATLAB simulation. Beijing: Springer-Verlag; 2012.