Expert Systems With Applications 121 (2019) 38–48

Contents lists available at ScienceDirect

Expert Systems With Applications

journal homepage: www.elsevier.com/locate/eswa

Autonomous path tracking control of intelligent electric vehicles based

on lane detection and optimal preview method

Xizheng Zhang

a , b , ∗, Xiaolin Zhu

a

a The Innovative Center of Wind Equipments and Energy Conversion, Hunan Institute of Engineering, Xiangtan 411104, China b School of Electrical and Information Engineering , Hunan University, Changsha 410082, China

a r t i c l e i n f o

Article history:

Received 3 September 2018

Revised 10 November 2018

Accepted 2 December 2018

Available online 3 December 2018

Keywords:

Intelligent electric vehicles

Land marking detection

Optimal preview control

Linear quadratic regulator (LQR)

a b s t r a c t

In this paper, a novel autonomous tracking control (ATC) of intelligent electric vehicles (IEVs) based on

lane detection and sliding-mode control (SMC) is innovatively developed to implement accurate path

tracking and optimal torque distribution between the motors of IEVs. Initially, the road image was cap-

tured by the camera and was processed to extract the lane markings and to calculate the desired steering

angle by the lateral trajectory tracking error and the head tracking error. Then, to accurately track the de-

sired path, an optimal preview linear quadratic regulator (OP_LQR) based on SMC approach with 2-DOF

vehicle model was proposed. To prove the effectiveness of the proposed OP_LQR scheme, the marking

recognition analysis and the optimization results of the traditional three controllers are obtained and

compared. Results show that the lane marking identification algorithm has high accuracy. Moreover, the

actual path with the proposed method can better track the desired trajectory and appropriate differential

braking torques are allocated into four wheels.

© 2018 Elsevier Ltd. All rights reserved.

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1. Introduction

Nowadays, the gas-powered internal combustion engine (ICE)

vehicles with high fuel consumption and emissions, to a large ex-

tent, cause the air pollution problem and harm human health. Hy-

brid electric vehicles (HEVs) and Intelligent electric vehicles (IEVs)

are one of the most promising solutions to achieve energy saving

and environmental protection ( García, Torreglosa, Fernández, & Ju-

rado, 2013; Li, Chen, Luo, & Wang, 2012 ). Especially, IEVs are the

research frontier in the field of vehicle engineering and the trend

of the automobile industry development. IEVs are an increasingly

important part of intelligent transportation systems ( Li, 2017; Max-

emchuk, Lin, & Gu, 2015 ), whose intelligence is concentrated in

smart and safe driving. The two important components of IEVs are

the environmental perception system and the autonomous track-

ing control system (ATC). They are not only related but also inter-

active: the environmental perception is the prerequisite, and the

ATC control is the purpose. IEVs have the characteristics of param-

eter uncertainty, time delay, and highly nonlinear dynamics, and is

a typical complex coupled system. How to use the environmental

perception information to realize ATC is key for IEVs. Based on the

surrounding environment and the vehicle states, IEVs is controlled

∗ Corresponding author at: The Innovative Center of Wind Equipments and En-

ergy Conversion, Hunan Institute of Engineering, Xiangtan 411104, China.

E-mail address: z_x_z20 0 0@163.com (X. Zhang).

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https://doi.org/10.1016/j.eswa.2018.12.005

0957-4174/© 2018 Elsevier Ltd. All rights reserved.

y a certain logic. In practical applications, ATC mainly includes the

ateral control and the longitudinal control ( Ganzelmeier, Helbig, &

chnieder, 2001; Park, Lee, Jin, & Kwak, 2014; Zhang, 2013 ). The

ormer mainly studies the path tracking ability of IEVs; the latter

ainly studies the speed tracking ability of IEVs.

Fig. 1 shows the basic structure of the ATC system of IEVs, and

t can see that in the lateral control system, cameras, radars and

ther sensors are responsible for collecting the tracking error sig-

als and feeding them back to the upper-layer controller, where

he desired steering wheel angle is calculated by the lateral tra-

ectory error and the lateral direction error ( Jeffrey, 2010; Zhang,

013 ). Then the real-time control of the steering system is realized

hrough the steering actuator, the tracking of the desired trajectory

s achieved.

Due to the complexity, nonlinearity, time-varying, and uncer-

ainty of IEVs, an accurate mathematical model cannot generally

e obtained. Many different vehicle models have been used in the

ehicle ATC in the past. The vehicle models used in ATC can be

lassified as the steering geometry models ( Jeffrey, 2010 ), the kine-

atics models ( Ataur, 2009 ) and the dynamics models ( Han, Yim,

ee, & Kim, 2009 ). The steering geometry model is the earliest and

ost widely used model, which using a simple formula to repre-

ent the geometric relationship between the steering angle of the

ront wheel and the desired path trajectory of the vehicle. The

odel contains two categories: non-previewing model and pre-

iewing model.

X. Zhang and X. Zhu / Expert Systems With Applications 121 (2019) 38–48 39

Controller

Vision sensors

Upper-layer

Actuators Steering IEVs

Lower-layer Vehicleec(t)

θe(t)

xd(t)

Fig. 1. The control structure of the ATC system.

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ROI

Fig. 2. The image segmentation into ROI and ROUI.

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Moreover, the ATC becomes a nonlinear, complex and time-

arying control problem. Modern control theory such as nonlinear

ontrol, robust control and intelligent control methods, can sig-

ificantly improve the control performance. In recent years, with

he development of control theory, more and more modern control

echnologies such as nonlinear control, robust control and intelli-

ent control have been applied to autonomous vehicle autonomous

racking control to improve the tracking performance ( Guo, Li, Li,

nd Wang, 2013; Wang, Jing, Hu, Chadli, & Yan, 2016b; Han et al.,

009; Ma, Li, Gao, Guo, & Lian, 2006; Tan, Bu, & Bougler, 2007; Tu,

010; R. Wang, Ma, & Shi, 2004 ).

Based on the modern control theory, Ganzelmeie proposed a

obust H ∞ lateral control strategy for intelligent vehicles that

an effectively handle the parameter uncertainty of the lateral

ontrol model ( Ganzelmeier et al., 2001 ). Tan et al. (2007) de-

igned the automatic steering control strategy by using a hybrid

2/H ∞ synthesis method for low-speed driving conditions of IEVs.

. Wang et al. (2004) proposed a sliding mode control (SMC)

ethod for the lateral path tracking under high-speed driving con-

itions and adopted a quadratic optimal design to construct a

witched hyper-plane. To deal with the parametric uncertainty of

he previewing kinematic model, a lateral adaptive fuzzy SMC was

roposed, compared with the traditional SMC, it can improve the

racking accuracy and the response characteristics. Kayacan, Ra-

on, Kaynak, and Saeys (2015) constructed a lateral Type-2 fuzzy

ontroller, which adopted an adaptive SMC to adjust the fuzzy

ules in real time. To solve the path following problem, Wang, Hu,

an, and Chadli (2016a) successfully developed a modified com-

osite nonlinear feedback control; its main contribution is to indi-

ectly control the lateral offset and the heading angle error of the

ehicle, making them converge to zero by directly tracking control

he vehicle yaw rate and the lateral speed. However, these control

chemes have not used the vision system to sense the road infor-

ation and are mainly focused on vehicle dynamics control.

With the rapid development of IEVs, the computer vision tech-

ology has been widely utilized into ATC, among which lane mark-

ng detection technology provides conditions for vision-based IEVs

avigation and automatic driving, and is one of the core technolo-

ies of intelligent transportation systems. Based on the road color

nformation, Cheng, Jeng, Tseng, and Fan (2006) proposed a lane

ine extraction method, and the size, shape and motion informa-

ion are used to distinguish the vehicle with the same color as

he lane line from the real lane, then the left and right bound-

ry lines of the lane are accurately located. Wang, Lin, and Chen

2010) integrated self-clustering algorithm, fuzzy clustering algo-

ithm and fuzzy logic to process spatial information and detect the

ane lines by using Canny operator based edge extraction. These

lgorithms are suitable for scenes with good light intensity and

lear lane lines, but the detection performance is not satisfied un-

er the actual scenes under vehicle occlusion or weak illumina-

ion. Y. Wang, Teoh, and Shen (2004) proposed a B-spline based

ane detection and tracking scheme, where the CHEVP algorithm

as used to determine the initial point and the minimal mean

quare error method was used to fit the lane line position; this

cheme is very strong for noise, shadow and illumination changes

nd can be applied to unstructured roads. Li, Chen, Li, Shaw, and

uchter (2014) combined the laser radar, the visual data and the

ptimal selection strategy to detect the lane line and the optimal

riving area. These algorithms can effectively avoid the influence of

ther road surface interferences, and has higher robustness; how-

ver, they requires complex iterative calculation and cumbersome

arameter optimization, which is difficult to meet the real-time re-

uirements of lane detection.

Although plentiful control strategies and various land marking

etection as mentioned earlier have been proposed and applied

n IEVs, a combination of the two and the related application into

ath tracking control of IEVs are really few and expectant. By com-

ining with the vehicle-state information, Ma et al. (2006) pro-

osed an improved limited-time optimal preview lateral control

trategy to fulfill the need for higher real-time performance re-

uirements for the lane keeping control systems. Dahmani, Chadli,

abhi, and El Hajjaji (2013) proposed a nonlinear vehicle model de-

uced from the lateral dynamics and a road curvature estimator for

ane departure detection based on a vision system. Nevertheless,

o the best of our knowledge, there lacks a vehicle-road lateral dy-

amics model and its control to derive the lateral path offset and

he head tracking error through the lane markings detection.

In this paper, through the analyses above, a novel autonomous

ath tracking control of IEVs based on lane marking detection and

ptimal preview linear quadratic regulator (LQR) is innovatively

eveloped. The main contributions include the following: (i) de-

elopment of an universal structure of vision-based ATC control

or IEVs without need of measurements of path/location informa-

ion; (ii) derivation of the vehicle-road lateral dynamics model as

ell as the lateral path offset and the head tracking error through

he lane markings detection; and (iii) design of an optimal pre-

iew LQR control based on SMC approach to achieve high accuracy

f tracking.

. Lane marking detection

There are a lot of interference factors in the image of real

oads, such as noise, light unevenness, water and stains, shadows

f buildings and green belts on roads, vehicle interference, reduced

ear and tear of lane lines, and interference signs. These influences

ring great difficulties to the extraction of lane marking features

Tan et al., 2007; Diaz-Cabrera, Cerri, & Medici, 2015; Wang et al.,

010 ). A proper choice of a lane marking detection heavily depends

n the type of system and environment in which the lane mark-

ng detection is to be performed. Since the proposed autonomous

racking control is used only on certain types of roads including

rban and highway scenarios, it might not be necessary to detect

ll possible lane markings at all as long as a safe path or lead ve-

icle to follow can be specified; that is to say, the detection object

ocuses on the left and the right boundaries of the lane in this pa-

er.

The lane image is firstly divided into two parts as shown in

ig. 2 : region of unrelated information area (ROUI) in the top and

egion of interested area (ROI) in the bottom. ROI contains enough

elated information about the lane marking. To handle the adverse

mpact of noise, road shadows, road line wear, etc., the overall flow

f the lane image process and lane marking detection in this work

40 X. Zhang and X. Zhu / Expert Systems With Applications 121 (2019) 38–48

Car camera Grayscaling Binaryzation

Computation of lateral tracking errorand vehicle head error

Comparison between desired path andlane markings

Modified Houghtranformation

Lane markingdetection and tracking

Desiredpath plan

Filterenhancement

Imagesegmentation

Edgeenhancement

Fig. 3. The overall flow of the lane image process.

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is depicted as Fig. 3 , which includes the following steps: gray scal-

ing, filter enhancement, binarization and edge detection for the

ROI of the lane image.

2.1. Gray scaling of the lane image

By using a color CCD camera on the vehicle, the lane images

are photographed and collected real-time. Since the color image

contains a large amount of color information, which is not neces-

sary for the lane marking detection, it is usually to grayscale color

images to reduce the amount of data. Grayscale image contains

enough information to distinguish the lane marking from the back-

ground. The original image collected by the camera is an RGB color

model. Therefore, the color of each pixel in the image is jointly de-

cided by three components: R, G and B.

The gray scaling process is implemented by using the weighted

average method defined as following:

I g = w r R + w g G + w b B (1)

where w r , w g and w b represent the weights of R, G and B compo-

nents, respectively. According to previous studies, it is found that

these coefficients depend on the importance of each component

and the grayscale image is most suitable for human visual charac-

teristics with w r = 0.3, w g = 0.59 and w b = 0.11 ( Umbaugh, 2010 ).

2.2. Filter enhancement of the lane image

The conditions of the actual road environment are relatively

complex and affected by external factors such as light, shadow, and

road surface water stains. In order to efficiently reduce the noise

interference, the median filtering based image enhancement is in-

troduced in this work. The median filtering method usually uses a

two-dimensional window as a sliding window, and the pixel value

of the middle point of the window is replaced with the median

value of pixels in the coverage area of the two-dimensional win-

dow.

The median of a series pixels is defined as following ( Umbaugh,

2010 ):

p c (m, n ) = Median

i, j∈ W

[ p ( i, j ) ] (2)

where p ( i, j ) denotes the pixels series, the notation ( m, n ) means

the centroidal coordinate of the image block, W is the filtering

window.

By continuously sliding the window, the filtering process is

done in a new sampling area until the entire image is sampled.

2.3. Binaryzation of the lane image

After the median filtering process, the amount of information

contained in the 256-level grayscale image is still relatively abun-

dant. In order to improve the efficiency of lane marking detection,

reduce the data processing burden and highlight the contour fea-

tures of the lane marking, the binaryzation of the lane image is

needed. The key issue of this process is to dealt with the threshold

egmentation of the image. In this work, the Otsu threshold seg-

entation method ( Sezgin and Sankur, 2004 ) is adopted. The algo-

ithm splits the image by selecting an optimal threshold that max-

mizes the variance between the target object and the background.

ince its decision criterion is based on the statistical principle of

he gray histogram, the operation speed is relatively fast and it is

uitable for occasions requiring high real-time performance.

.4. Edge enhancement of the lane image

The image edge detection algorithm can achieve the extraction

f the cross-connection between the target object and the back-

round. Based on the characteristics of the edge detection algo-

ithm, the lane edge features are extracted for lane marking de-

ection. There are several edge detection operators in the previous

tudies, such as Roberts operators, Canny operators and Sobel op-

rators.

The traditional Sobel operator can enhance the horizontal and

ertical edge features of the image, and effectively suppress the

nterference of the edges in other directions. Meanwhile, the pro-

essing speed of the algorithm is fast. However, the main defect of

raditional Sobel operator comes from the likelihood of detecting

he false edges ( Meuter et al., 2009 ). According to the directional

eatures of lane lines, an improved Sobel algorithm is proposed in

his work to extract the edge features of lane marking. In this pa-

er, based on the directional characteristics of the lane line, the

mproved Sobel operator is set in the 45 ° and the 135 °, and by de-

ecting the edge features of the image in the oblique direction the

alse edges detection can be avoided. The calculations of the edges

ased on the improved Sobel operators are as follows

e [ p ( i, j ) ] =

((s π/ 4

)2 +

(s 3 π/ 4

)2 )1 / 2

(3)

here s π /4 and s 3 π /4 denote the Sobel operators of the angles of

/4 and 3 π /4 rad, respectively.

.5. Model of the lane line

After extracting the detection features of the lane image, a lane

ine model is established and a modified Hough transform is used

o globally extract the lane markings based on prior knowledge.

he mathematical model of lane lines can be generally divided into

wo types: straight lines and curves. Since the linear model of the

traight lines has few parameters and is simple to calculate, it’s

ommonly used and can be expressed as follows

( t ) = kx ( t ) + b (4)

here x ( t ) and y ( t ) represent the ordinate of the lane image plane,

represents the slope of the line, and b is the intercept of the line.

he linear model is determined by the parameters k and b .

Since this work aims to detect on the structural road with bet-

er road conditions such as urban and highway road, the linear

odel that meets the requirements of real-time system is selected

s the lane line model. Although the actual road line is not a stan-

ard straight line, the straight-line model is suitable for assisted

riving when the vehicle is driving at a low speed.

.6. Modified Hough transform based on prior knowledge

The Hough transform and least squares are the most commonly

sed line detection methods. However, the traditional Hough trans-

orm has certain shortcomings ( Milanés et al., 2012; Thrun, Mon-

emerlo, Dahlkamp, & Stavens, 2009; Y. Wang et al., 2004 ): (i) each

hite point in the road image needs to be spatially transform, re-

ulting in high computation cost and is not conducive to real-time

pplication; (ii) the criterion for judging the existence of a straight

X. Zhang and X. Zhu / Expert Systems With Applications 121 (2019) 38–48 41

Fig. 4. x —y plane.

Fig. 5. ρ—θ plane.

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Fig. 6. The preview steering geometry model.

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ine is relatively simple, especially on a given threshold, and may

ause wrong extraction. To perform preliminary detection of lane

arkers, the traditional Hough transform algorithm is improved

y using the prior knowledge of road image, which makes it more

uitable for the extraction of lane line parameters.

As shown in Figs. 4 and 5 , by using the Hough transforma-

ion, the pixels P a , P b in the image plane can be projected into

he curves C a , C b in the polar coordinate space with parameters

ρ , θ ). Generally, if the target object satisfies certain function rela-

ionship such as linear or circular, then it can be easily detected in

he ρ—θ parameter space. Figs. 4 and 5 gives the space transform

rom x —y plane into ρ—θ plane, and the transformation relation is

= x ∗cos θ + y ∗sin θ .

In this work, to implement path tracking control of the vehicle,

he detection target is set to be the left and the right markers of

he current lane. Since there may be multiple lane lines, we ex-

ract the lane line parameters N ( ρ , θ ) in the left and the right half

mages separately, and then find the minimal value θ l in (0, π /2)

nd the maximal value θ r in ( −π / 2, 0). Finally, the corresponding

traight line slope and intercept are calculated according to the ( ρ ,

) value, and the left and right straight lines taken out are the cur-

ent lane lines. In order to reduce the size of the accumulation ma-

rix N ( ρ , θ ) and to improve the efficiency of the algorithm, we set

he polar angle θ l in the range [ −4 π /9, −π /9], and θ r in the range

π /9, 4 π /9]. Based on the acquired structured road image infor-

ation, the modified Hough transform based on prior knowledge

ill initially detect and locate the lane lines in the static image of

he road image. Then, the least square method is used to fit the

ane line feature points in the local interest search area, and the

ptimal lane line parameters are extracted. The steps for extract-

ng lane lines by the modified Hough transform modified based on

rior knowledge are shown in Table 1 .

. Autonomous path tracking based on preview LQR control

In this section, the autonomous tracking control of the intelli-

ent vehicle is to minimize the lateral tracking error between the

ehicle and the desired path by adjusting the steering wheel an-

le. Moreover, the angle error between the vehicle motion direc-

ion and the tangential direction of the desired path is minimized

oth to ensure the tracking accuracy and to improve the smooth-

ess and comfort of the vehicle. Three kinds of vehicle models

re firstly introduced, and based on which different lateral control

ethods are proposed to achieve the ATC.

.1. Lateral control with non-preview steering geometry model

The IEVs can be modeled by a linear two-degree-of-freedom

2DOF) two-wheel model. A simple geometric relation between the

ront wheel angle of the vehicle and the track to be driven by

he rear wheel is satisfied, which is the vehicle steering geometry

odel ( Park et al., 2014; Zhang, 2013 ). The relationship between

he front steering angle and the radius of curvature of the road

rack is expressed as:

an δ f = L/R (5)

here δf is the steering angle of the front wheel, L is the axis dis-

ance of the vehicle, and R is the curvature radius of the desire

ath.

The lateral control method for non-predictive vehicle steering

eometry model uses the lateral tracking error e f at the front

heel. The angle tracking error θψ

at the front wheel is defined

s:

ψ

= ψ − ψ d (6)

here θψ

is the angle error of the tracking path, ψ is the actual

aw angle of the vehicle, and ψ d is the desired yaw angle.

The purpose of the tracking controller is to adjust the steering

ngle δf to force the angle error, the lateral error to be zero, thus

he control law is designed as follows:

f = θψ

+ arctan

(μe f / v x

)(7)

here μ is the adjustable coefficient, v x is the longitudinal vehicle

peed.

The lateral control with non-preview steering geometry model

s suitable for the low-speed tracking conditions and for the path

racking situations where a single curve appears on a straight road.

.2. Lateral control with preview steering geometry model

When the desired cornering angle is quite large for the

ow speed vehicle driving condition, the control accuracy of the

on-predictive vehicle steering geometry model is low and the

ontrol performance isn’t satisfied. In this situation, the lateral con-

rol with preview steering geometry model of the intelligent vehi-

le should be designed for the path tracking control. The steering

eometry model with the single-point preview curvature is shown

n Fig. 6 . The lateral path tracking error e p can be determined by

he front steering angle δf of the vehicle and the desired path at

he preview point P ( p x , p y ) ( Cui, Zhang, & Wang, 2012 ), which is

head of the vehicle with the distance l p .

42 X. Zhang and X. Zhu / Expert Systems With Applications 121 (2019) 38–48

Table 1

Modified Hough transform algorithm.

Algorithm 1: Solution of the Modified Hough transform algorithm

Step 1: Establish the polar coordinate system with the given parameter space ( ρ , θ ), initialize the array N ( ρ , θ ) with all zero elements, where θ is in [ −π /9,

4 π /9], and ρ is in [ −R, R], R is the distance between the corners in the image.

Step 2: Do Hough transform on the image. Find the non-zero pixels ( x, y ) in the left and right half of the image; let the angle increment �θ = 1 °, traverse all

θ , then calculate ρ = x ∗cos θ + y ∗sin θ .

Step 3: Let N ( ρ , θ ) = N ( ρ , θ ) + 1 and go to Step 2 until all pixels have been transformed. The Hough transform matrix is then obtained.

Step 4: Search the peak point in the parameter space. For a given threshold M , if N ≥ M , it is considered the peak point.

Step 5: Detect the lane marking. If N > M holds, for the left and right half of the image, find the minimal θ l in the range [ −4 π /9, −π /9], and the maximal θ r

in [ π /9, 4 π /9]. The corresponding parameters ( ρ , θ ) of these two elements are the feature parameters of the line.

Step 6: Calculate ρ l = x ∗cos θ l + y ∗sin θ l and ρr = x ∗cos θ r + y ∗sin θ l , then the corresponding parameters ( ρ l , θ l ) and ( ρr , θ r ) are the feature parameters of the

line.

END

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From Fig. 6 , the curvature radius at the preview point and the

desired steering angle of the front wheel can be derived as follows:{R = l p / ( 2 sin 0 . 5 θp ) δd = arctan ( L/R )

(8)

where θp is the preview angle between the preview point and the

center of the rear wheel and is determined as:

θp = arcsin ( e p / l p ) / 2 (9)

where the preview distance l p is proportional to the longitudinal

vehicle speed as l p = v x k p , k p is the preview coefficient.

By combining (8) and (9) , the lateral control with preview steer-

ing geometry model is designed as

δd = θψ

+ arctan

(2 L e p

k 2 p v 2 x

)(10)

By calibration of the camera, the points in the image can be

converted into those in the real world coordinate system. As show

in Fig. 6 , the lateral path tracking error e p can be obtained directly

from the image, so that the relationship among the lateral angle

tracking error θ e , the distance from the center of mass (COM) l c to

the preview point and the lateral path tracking error at COM e c is

expressed as:

θe = arcsin ( e p / l p ) e c = e p l p / l c

(11)

3.3. Optimal preview LQR control based on sliding mode approach

3.3.1. Nonlinear vehicle model

The body model of the vehicle in this work assumes that the

wheel is always in contact with the road, the roll axis is on the

center plane of the vehicle, and the vehicle pitch, vertical move-

ments car are ignored. There are seven degrees of freedom (DOF):

vertical, lateral, yaw and the own rotation of each wheel. Thus, an

7-DOF nonlinear vehicle model for stability analysis and design are

established.

In general, the dynamics in the left side and in the right side

of the vehicle are assumed to be symmetrical, thus the vehicle dy-

namic equations can be expressed as following.

Vehicle longitudinal dynamics: ∑

F x = m ( v x − γ · v y ) = F xf cos δf − F yf sin δf + F xr (12)

Vehicle lateral dynamics: ∑

F y = m ( v y + γ · v x ) = F yf cos δf − F xf sin δf + F yr (13)

Vehicle yaw dynamics: ∑

M z = I z ˙ γ = l f (F yf cos δf − F xf sin δf

)− l r F yr (14)

Wheel kinematic equation:

J ω ω i = T d,i − T b,i − F x,i R w

(15)

here F x , F y and M z represent the total longitudinal force, lateral

orce and yaw moment, respectively; V, v x , v y represent the cen-

roidal, longitudinal and lateral vehicle speed, respectively; m is

he total weight of the vehicle; γ is the yaw rate; δf is the front

teering angle; F xi , F yi are the longitudinal and lateral tire force

omponents ( i = f, r for the front and rear wheels), respectively; I z s the yaw moment of inertia; T d,i , T b,i are the wheel driving torque

nd braking torque, respectively; l f and l r are the horizontal dis-

ance between the front and rear axles and the COM; J w

, ω i , R w

re the wheel rolling moment of inertia, speed, and roll radius, re-

pectively.

When the vehicle is traveling in a curve, the tire exerts a corre-

ponding lateral force on the ground due to the centrifugal force.

ecause the tire has lateral elasticity, the vehicle will have a certain

ide deviation in the traveling direction. When the vehicle is trav-

ling normally, the lateral acceleration is generally not more than

.4 g, and the tire side angle will not exceed 4 ∼5 °, which makes

hat the lateral force and the side angle is approximately linear as

ollowing:

F yf = μC f α f

F yr = μC r αr (16)

here C f , C r denotes the tire cornering stiffness, αf , αr is the

ideslip angle of the front tires and the rear tires, respectively; μs the tire adhesion coefficient, which is limited by the approxi-

ate friction circle, and the maximum tire longitudinal force can

e calculated as follows

x , max =

√

| μF 2 z − F 2 y | (17)

From the 2-DOF model of the vehicle dynamics, with the as-

umption that the side angle at COM is small, the sideslip angles

an be calculated as:

α f = δ f − β − l f γ / v x αr = δr − β + l r γ / v x

(18)

here αf , αr is the sideslip angle of the front tires and the rear

ires, respectively; β is the centroid sideslip angle.

For the front wheel steering vehicle, the longitudinal speed is

lways set to be constant in path tracking; thus, by combining

12) –(14) and (18) , the linearized vehicle dynamics can be obtained

s:

˙ v y = −C f v y + C r l f γm v x +

C f m

δ f +

l r C r γ −C r v y m v x − v x γ

˙ γ = − l f C f v y + l 2 f C f γI z v x +

l f C f I z

δ f +

l r C r v y −l 2 r C r γI z v x

(19)

From the curvature of the desired trajectory, the desired yaw

ate and the desired lateral acceleration of the car can be calcu-

ated:

d = κ(s ) v x (20)

here κ( s ) is the curvature of the desired path; γ d is desired yaw

peed.

X. Zhang and X. Zhu / Expert Systems With Applications 121 (2019) 38–48 43

Fig. 7. The lateral tracking error.

3

m

e

v

fi

l

s

m

t

d

t

C{

{

x

w

u

f

t

q

u

w

J

w

w

u

w

P

d

s

w

D

t

t

t

t

p

w

B

w

B

t

i

o

c

b

o

m

s

w

u

m

f

e

i

w

c

t

o

c

d[

w

n

s

t

o

m

s

3

s

i

t

l

s

f{

.3.2. Optimal LQR control

In order to accurately track the desired path, a 2-DOF control

odel to design is adopted in this work. Then, a vehicle-road lat-

ral dynamics model is innovatively established based on 2-DOF

ehicle model and vehicle-road kinematic. In Fig. 7 , the camera is

xed on the longitudinal center-line of the vehicle, so the center-

ine of the captured image is the longitudinal axis of the vehicle as

hown by the green line, while the road trajectory obtained by lane

arking detection is as shown by the blue line. After determining

he preview point P ( p x , p y ) in the image, the lateral offset can be

etermined accordingly. If the path tracking error is small enough,

he lateral path error e c ( t ) and the head tracking error θ e ( t ) at the

OM can be derived as following

e c =

˙ v y + v x ˙ θe

θe = ˙ γ − ˙ γd

(21)

By combine Eq. (19) with Eq. (21) , one can obtain:

e c = −C f + C r m v x e c +

C f + C r m

θe − l f C f −l r C r m v x

˙ θe − (1 +

l f C f −l r C r

m v 2 x ) γd +

C f δ f

m v x

θe = − l f C f −l r C r I z v x ˙ e c − l f C f −l r C r

I z θe − l 2

f C f + l 2 r C r

I z v x ( ˙ θe + γd ) +

C f δ f

m v x − ˙ γd

Rewrite Eq. (21) in the matrix form as following:

˙ = A x + B 1 u δ + B 2 γd (22)

here the states x = [ e c , ˙ e c , θe , ˙ θe ] T and the matrices

A = [

0 1 0 0

0 C 0

m v x C 0 m

C 1 m v x

0 0 0 1

0 −C 1 I z v x

C 1 I z

−C 2 I z v x

] B 1 = [

0 0

1 0

0 0

0 1

] B 2 = [

0

− C 1 I z v x

−v x 0

− C 2 I z v x

] ,

δ = [ u e u γ ] T , where u e , u γ are the equivalent control input

or the path tracking error and the head tracking error, respec-

ively; the coefficients C 0 = C f + C r , C 1 = l f C f − l r C r , C 2 = l 2 f C f + l 2 r C r .

The state feedback control law is designed by using the linear

uadratic regulator (LQR) method:

δ = −K · x (t) (23)

here K is the feedback gain to be determined as following.

Define the cost function:

=

1

2

∫ ∞

0

[ x T (t) Q 1 (t) x (t) + u

T δ (t) R(t) u δ(t)] dt (24)

here the weighting matrices Q 1 is diagonal, R is with proper

eights.

The solely optimal control is deduced as:

∗δ = −R

−1 B

T 1 P x (t) (25)

here the matrix P is determined by solving the Riccati equation

A − P B 1 R

−1 B

T 1

P + A

T P + Q 1 = 0 .

The calculated equivalent control needs to be distributed among

ifferent wheel, that is, according to the recognition of the driving

tate and the detection of the steering wheel, it is judged on which

heel or on which wheels the vehicle distributes the yaw moment.

rive or brake commands are issued to four hub motors to make

he wheel torque ultimately translate into the interaction between

he tire and the ground to achieve both the vehicle stability and

he path tracking control. Since the IEVs is driven independently,

he system has a variety of flexible input combinations to accom-

lish the distribution of the equivalent yaw moment between the

heels. Consider the following allocation:

· u =

[u e u γ

]T (26)

here the matrix B and the control input u are defined as:

= [

C f m

−C 1 m v x

l f C r + l r C f m v x

l f C f I z

− t f 2 I z

t r 2 I z

] , u = [ δf , F xf , F xr ] T , where t f and t r are

he front and rear track distances.

There is actuator redundancy in Eq. (26) . The control distributor

s to complete the optimal allocation of control under the restraints

f ground attachment and actuator position/rate constraints. It is a

onstrained multivariate optimization problem. The control distri-

ution of Eq. (26) can be converted into a constrained quadratic

ptimization problem as following

in

u

1

2

u

T Q 2 u + C T u (27)

. t . B u = [ u e , u γ ] T , u lb ≤ u ≤ u ub

here Q 2 is a positive symmetric matrix, C is the coefficient, and

lb , u ub are the upper and lower bound of the actuator restriction.

The optimization problem of Eq. (27) includes both the require-

ents for the tracking error in path control and the requirements

or the yaw moment in independent braking. After checking the el-

ments of the control matrix B, one can easily find that the control

nput δf directly acts and determines the lateral path error along

ith the head tracking error and has greater control effect. The

oupling between the two states may cause conflicts between con-

rol targets during the optimization process, especially in the case

f actuator failures, which need to prioritize the vehicle stability

ontrol. To solve this problem, a virtual control effort τ is intro-

uced and the modified equation is:

e c θe

]= −

[

C f + C r m v x

l f C f −l r C r m v x

l r C r −l f C f I z v x

−l 2 f C f −l 2 r C r

I z v x

]

·[

˙ e c ˙ θe

]+

[B κ

]·[

u

τ

](28)

here the virtual control matrix κ = [ κe , κγ ] T ; by introducing the

ew virtual control term τ , the equality constraint on the lateral

lip angle of the centroid is relaxed without affecting the yaw rate

racking. Thus, the choice of κ is determined by making the dc gain

f γ → τ zero.

The optimal model in (18) is transformed as

in

u

1

2

u

T Q 2 u + C T u , u = [ u , τ ] T (29)

. t . [ B, κ] · u = [ u e , u γ ] T , u lb ≤ u ≤ u ub .

.3.3. Sliding-mode control for wheel slip

The longitudinal force of the wheel is directly related to the tire

lip rate. For a given desired slip ratio, a slip rate controller (SRC)

s designed to adjust the longitudinal force of each wheel. Under

he influence of vehicle steering and yaw motion, the wheel center

ongitudinal speed v i ( i = f, r for the front and rear wheels) and the

lip ratio s i along the wheel coordinate system are calculated as

ollows:

v i , L = ( v x − 0 . 5 t f γ ) cos δf + ( v y + aγ ) sin δf

v i , R = ( v x + 0 . 5 t f γ ) cos δf + ( v y + aγ ) sin δf (30)

44 X. Zhang and X. Zhu / Expert Systems With Applications 121 (2019) 38–48

V

Fig. 8. The gray scaling of the lane image.

Fig. 9. The filtered lane image.

Fig. 10. The edge detection of the filtered image.

4

a

i

d

o

n

c

n

a

t

w

e

t

fi

d

s i =

{R w

ω i / v i − 1 , under barking condition

1 − v i / ( R w

ω i ) , under driving condition

(31)

Under the fixed side slip angle, the longitudinal force of the tire

will increase rapidly with the increase of the longitudinal slip ratio

until reaching the peak at the critical slip rate s ∗, then will de-

crease after the slip rate continues to increase. When the actual

slip ratio is small, the wheel motor torque T i is proportional to the

desired longitudinal force F xi as

T i = R w

F x , i (32)

However, under a certain vertical load, due to the restriction of

wheel adhesion, when the desired tire longitudinal force is exces-

sive, the wheel torque will cause the actual slip ratio to exceed the

critical slip ratio, resulting in the wheel locking and the longitu-

dinal forces reduced. Therefore, when the actual slip ratio is large,

it is necessary to adopt a slip rate control strategy to adjust the

motor torque, thereby limiting the wheel slip ratio.

Under vehicle braking, a sliding mode function with slip error

as a variable is constructed:

s e = s − s ∗ s ∗ < 0 (33)

Consider the Lyapunov function:

=

1 2 s

2 e ≥ 0 (34)

The longitudinal force of the tire is the F x,i = C T s i + �F x,i , where

C T is the longitudinal cornering stiffness of the tire in linear region,

�F x,i is a non-linear term and satisfies 0 ≤ �F x,i ≤ F x,i , thus,

˙ s i =

R w

I ω v i ( T i − R w

C T s i ) −R

2 w

I ω v i �F x,i − ( 1 + s i )

˙ v i v i

(35)

Thus, under barking the wheel torque is designed as following:

T i = R w

C T s ∗ +

I ω v i R w

( 1 − s i ) − I ω v i

R w

( 1 − s i ) 2

k s sgn ( s e ) (36)

where the gain k s > R 2 w ( I ω v i ) −1 F xi + ηs , the notation ‘ sgn ’ denotes

sign function, and the sliding-mode coefficient ηs > 0. From Eq.

(30) , we have ˙ V ≤ −ηs | s e | . Once s e = 0 , we have ˙ V < 0 . Since V > 0

holds, this means that the error of the entire slip rate is uniformly

asymptotically stable, and the zero is a stable point of the system.

Similarly, under driving condition the wheel torque is designed

as following:

T i = R w

C T s ∗ + I ω v i ( 1 + s i ) − I ω v i k s sgn ( s e ) / R w

(37)

Compared with the existing works, the proposed optimal LQR

control has the following difference or advantages: (i) the model

in (22) directly uses the tracking errors instead of the vehicle yaw

rate and the lateral velocity as the state variables; (ii) a new virtual

control is introduced into the model to decouple the two states;

(iii) yaw moment and driving torque are distributed on each wheel

in an optimal way.

4. Results and analysis

In order to prove the effectiveness of the proposed control

scheme, the lane marking detection analysis and the optimization

results of optimal preview LQR and conventional methods are com-

pared. The live videos of Xiangtan city’s multiple urban roads and

high-speed sections are firstly collected for lane line recognition

experiments. Then, the controller performance is validated in 3

simulation tests.

.1. Lane marking detection analysis

The results of gray-scaling the lane image using weighted aver-

ge method are shown in Fig. 8 . It can be seen that this method

s more suitable for this work, and has less influence on the image

epth information.

For the lane grayscale, Fig. 9 gives the performance comparison

f classical image de-noising algorithms. Fig. 9 (a), (b) are the de-

oising results by using the mean filter and the median filter. By

omparison, it can be known that the median filtering method can

ot only eliminate the top peak signal of the triangular signal, but

lso well eliminate the salt and pepper noise, and is insensitive to

he signal jump. It can be seen that the median filter can perform

ell when dealing with the vehicle noise as well as retains enough

dge details for the lane line detection.

By using the improved Sobel operator to perform edge detec-

ion on the filtered image, results are shown in Fig. 10 . From the

gure, it’s found that the improved Sobel edge detection algorithm

oes not detect the false edges. Moreover, the edge feature of the

X. Zhang and X. Zhu / Expert Systems With Applications 121 (2019) 38–48 45

Table 2

Statistical results of the lane marking detection under 3 scenarios.

Scenario Total frames Correct detection frame Correct detection rate

Normal illumination 300 290 96.6%

Vehicle occlusion 300 284 94.6%

Weak illumination 300 271 90.1%

Fig. 11. Hough transformation of the image.

Fig. 12. The lane markings extraction under typical road conditions.

l

s

p

a

t

u

t

t

m

b

p

b

e

a

o

c

a

e

a

S

Fig. 13. The lane markings extraction under different scenarios: (a) normal illumi-

nation; (b) vehicle occlusion; (c) weak illumination.

o

d

T

4

d

t

4

w

n

c

o

a

t

J

f

c

fi

ane marking is also enhanced, and it is applicable for the occa-

ions with high real-time requirements.

As can be seen from Fig. 11 , the Hough transform based on

rior knowledge can accurately detect the lane line, and the left

nd right lane lines intersect at the blanking point. Fig. 11 shows

he detection performance of the improved Hough transformation

nder typical road conditions.

Fig. 12 shows the experimental results of lane marking detec-

ion under typical road conditions. Fig. 13 shows the experimen-

al results of lane marking detection under three conditions: nor-

al illumination, weak illumination and vehicle occlusion. It can

e seen that the lane line identification algorithm proposed in this

aper not only can accurately identify dashed, solid, straight roads,

ut also can adapt to the interference, such as interference lines,

xtra signs, vehicle occlusions, building shadows, zebra crossings

nd other road environments. Table 2 gives the statistical results

f the lane marking detection by using the proposed scheme. It

an be seen that under good road condition, such as in Scenario

: the illumination in the video sequence is normal, the interfer-

nce factors such as the same direction vehicle and zebra crossing

re less, then the recognition rate can reach more than 96.6%; in

cenario b: despite of more vehicles on the roadway and vehicle

cclusion, the recognition rate is still about 94.6%; in Scenario c:

ue to the weak illumination, the recognition rate falls to 90.1%.

he average time to process each frame of captured image is about

2 ms, which meets the requirements of real-time application.

Through experimental tests, it can be seen that under the con-

ition of non-high speed driving, based on the straight line model,

he lane markings can be detected ideally.

.2. Autonomous control results

The vehicle model was simulated and the ATC performances

ith the present LQR optimization controller, the preview and

on-preview controller, were analyzed and compared under three

ases. The steering system is simplified as an inertial link with one

rder.

The vehicle parameters of the simulated IEV are listed

s m = 1.401 kg, I z = 2677.2 kg.m

2 , l f = 1.013 m, l r = 1.702 m,

f = 1.554 m, t r = 1.534 m, C f = 113.2 kN.m/rad, C r = 90.2 kN.m/rad,

w

= 0.9 kg.m

2 , R w

= 0.31 m.

The optimization parameters of the three controllers are set as

ollows: the tire adhesion coefficient of the dry road is μ= 0.8; the

urvature of the desired path is set as κ( s ) = 1; the preview coef-

cient is as k p = 0.5; the scale coefficient of non-preview steering

46 X. Zhang and X. Zhu / Expert Systems With Applications 121 (2019) 38–48

Fig. 14. The transient response under step steering. (a) lateral tracking offsets; (b)

head tracking errors;(c) steering angle; (d) lateral tire forces; (e) path trajectories.

Fig. 14. Continued

l

t

s

r

d

t

t

t

b

m

t

v

p

d

r

w

i

s

c

v

a

s

f

a

g

t

h

t

b

c

t

geometry is ηf = 3 ; the sliding-mode coefficient ηs = 0.1 ; the nomi-

nal weights of the optimization controllers are used as:

Q 1 = diag ( [ 0 . 5 , 0 , 0 , 0 ] ) , R = 1

Q 1 = diag ([

10 , 1 , 1 , 10

3 ])

, C = 1

the virtual control matrix κ = [ κe , κγ ] T = [1 , C 1 mV I z C 0

] T , and the LQR

feedback gain is: K = [97.9709, −0.4 802, 0.4 808, −0.0083; 7.1913,

98.5335, 101.4661, 0.9268].

Case 1: Transient response under step steering

Fig. 14 shows the transient state responses with constant speed

and a step input (rotation angle of 2 π /3) on the steering wheel.

For comparison with the desired trajectory tracking error e c_d , the

ateral tracking errors e c_LQR for LQR optimization control, e c_pre for

he preview control and e c_non_pre without preview control are pre-

ented. The simulation results show that the trajectory tracking er-

or and the angle tracking error of the vehicle can reach extremely

angerous critical values immediately under the non-preview con-

rol, which indicates that the vehicle is nearly out of control at

his time, and some corresponding control strategies are needed

o ensure the desired steering of the vehicle. From Fig. 14 , it can

e seen that after adopting the preview control and the LQR opti-

ization control, both the trajectory tracking errors and the angle

racking errors of the two controllers can well track the desired

alue and enable the vehicle to steer steadily. Compared with the

review control, the LQR control method applies the appropriate

ifferential braking torque in four wheels, thus the lateral tracking

esponse of the vehicle can better track the desired value. Mean-

hile, the vehicle angle tracking error is also well suppressed, and

s stable within the domain of zero error. Fig. 14 (e) shows the de-

ired trajectory, the actual trajectory and the heading of the vehi-

le’s centroid under the OP_LQR method. It can be seen that the

ehicle can track the ideal trajectory well.

Case 2: Steady response under sinusoidal steering

Fig. 15 shows the steady state responses with constant speed

nd a sinusoidal input (rotation angle of ±π /2, 0.25 Hz) on the

teering wheel. The results show that the vehicle trajectory can’t

ollow the ideal single-shift line under the non-preview control,

nd the angle tracking error quickly reaches the extremely dan-

erous value. This means that the vehicle has lost the control at

his time, and some necessary control is needed to ensure the ve-

icle’s single shift line driving. As can be seen from Fig. 15 , af-

er adopting the preview control and the LQR optimization control,

oth controllers can better track the ideal model, and the vehi-

le lane change error is smaller. Compared with the preview con-

rol, the OP_LQR proposed in this work can better track the ideal

X. Zhang and X. Zhu / Expert Systems With Applications 121 (2019) 38–48 47

Fig. 15. The transient response under sinusoidal steering. (a) lateral tracking off-

sets; (b) steering angle; (c) path trajectories.

l

s

F

t

f

l

fl

i

Fig. 16. The transient response under steering failure. (a) lateral tracking offsets;

(b) steering angle; (c) path trajectories.

u

w

a

a

w

t

p

i

s

ateral trajectory and the vehicle angle tracking error is also well

uppressed; thus, the vehicle can complete the lane change well.

ig. 15 (c) shows the desired trajectory, the actual trajectory, and

he heading of the COM.

Case 3: Steady response under steering failure

Under the same settings as Case 2, the steering failure of the

ront wheels was set at t = 2.25–3.05 s. Fig. 16 shows the simu-

ation results. From Fig. 16 (a), it is found that, despite the large

uctuation, the lateral tracking offset of the proposed controller

s still smaller than 0.02 m during the duration of steering fail-

re, while the preview control has a large tracking error. In fact,

hen steering breaks down, the control allocation must compute

nd distribute control effort s again in the available actuators to

dapt to the new control circumstances. Fig. 16 (b) gives the front

heel steering angle, and it is found that during the failure period,

he steering angle adjustment is disabled and locked. Fig. 16 (c)

resents the desired trajectory and the actual trajectory, indicat-

ng that the vehicle can steer according to the ideal trajectory in

pite of slight error after the failure occurs.

48 X. Zhang and X. Zhu / Expert Systems With Applications 121 (2019) 38–48

Y

D

G

G

H

J

K

L

L

M

M

M

M

P

S

T

T

U

W

W

W

5. Conclusions and future work

In order to improve the control performance of the autonomous

tracking control of IEVs under complex road conditions, a novel

ATC control strategy was proposed, which mainly including the fol-

lowing work:

(1) A lane marking detection algorithm was developed to ex-

tract the road line. The lateral tracking error and the angle

tracking error are obtained by the processed road image and

the markings.

(2) Based on the preview model, an optimal LQR scheme with

sliding-mode approach was proposed to implement the ATC

control of IEVS.

(3) The optimal preview control strategy and other strategies

were compared and analyzed under two cases. Results ver-

ifies the feasibility of the marking recognition approach

and the effectiveness of the proposed control strategy in

improving both the tracking accuracy and the response

time.

In the future, this project should focus on embedding the pro-

posed tracking strategy into the instrumented-vehicle, which is op-

erated under actual road tests with uncertainties, measuring errors,

and environment variations for verifying the effectiveness of the

strategy.

Author contributions

Prof. ZHANG is in charge of the whole section of the paper, and

has completed the general structure design, the controller design

and the tests.

Mr. ZHU is mainly majored in the lane marking detection.

Acknowledgements

The authors are grateful to the anonymous reviewers and to

the support of the National Natural Science Foundation of China

( 61673164 ), the Key Project of Hunan Educational Department (no.

14A032) and the Xiangtan Science and Technology Project (CXY-

B20181008).

Conflict of interest

Declarations of interest: None.

Supplementary materials

Supplementary material associated with this article can be

found, in the online version, at doi: 10.1016/j.eswa.2018.12.005 .

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