Passive actuator-fault-tolerant path following control of autonomous ground electric vehicle with in-wheel motors

Advances in Engineering Software 134 (2019) 22–30

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Advances in Engineering Software journal homepage: www.elsevier.com/locate/advengsoft

Research paper

Passive actuator-fault-tolerant path following control of autonomous ground electric vehicle with in-wheel motors

T

Te Chena, Long Chena,b, , Xing Xua,b, , Yingfeng Caia,b, Haobin Jianga,b, Xiaoqiang Suna,b ⁎

a b

⁎

School of Automotive and Traffic Engineering, Jiangsu University, Zhenjiang 212013, China Automotive Engineering Research Institute, Jiangsu University, Zhenjiang 212013, China

ARTICLE INFO

ABSTRACT

Keywords: Autonomous ground vehicles Path following Vehicle stability Actuator fault Fault-tolerant control Orientated tire force allocation

This paper investigates the fault-tolerant path following control problem of autonomous ground electric vehicles with in-wheel motors through hierarchical control strategy. The sliding mode observer with boundary layer is designed to estimate the vehicle state, and the time-delay estimation method is used to compute the actuator fault. Considering the actuator fault, the fault-tolerant path following control strategy is proposed, in which the upper layer controller is developed to achieve path following control and guarantee vehicle stability simultaneously by sliding mode control method, and the lower layer controller is presented to achieve the control efforts of upper layer controller by adaptive orientated tire force allocation method. The simulations are implemented in the CarSim-Simulink co-simulation platform, and the simulation results have verified the effectiveness of proposed fault-tolerant path following control method.

1. Introduction Recent decades have witnessed the booming development of technology and expansion of production scale in automobile industry [1–3]. Nowadays, with the growing maturity of control technology and increasing demand of drivers for safety, maneuverability and riding pleasure, the research of vehicle intelligentization has received extensive attention [4–6]. Owning to accurate and flexible torque response capability of in-wheel motors, the distributed drive electric vehicle provides more degree of freedom in dynamic control of vehicle [7–11], which is propitious to the intellectualization and stability control of the autonomous ground vehicle. In the intelligent driving process of autonomous vehicle, the path following control is a relatively common intelligent driving scenario, in which the control target is to achieve the minimization of lateral offset and heading error, and many fruitful results have been presented in recent years [12–15]. In [13], a path following control method was developed for autonomous electric vehicles with in-wheel motors based on potential field approach, in which the designed controller can produce a steering corridor with a referenced tracking error tolerance. In [14], the longitudinal, lateral, yaw and quasi-static roll motions of intelligent vehicles are considered, and the path following control system was designed on the basis of model predictive control with the active safety steering control method of vehicle being presented.

⁎

In the existing studies, in order to achieve the path following of autonomous ground vehicle, many advanced control theories have been applied to promote the design path following controller, including model predictive control [16], robust control [17–19], sliding mode control [20–23] and so on. Furthermore, some researchers have tried to unite more than one control method in designing the vehicle path following controller, so as to fuse the advantages of multiple control methods and improve the overall control performance [24–25]. In [24], in order to guarantee the stability of the autonomous vehicle in the path following process, the unscented Kalman filter is designed to estimate vehicle running state, and the multi-constraints model predictive control method was applied to tracking ideal path. In [25], to solve the limitation of the simple bicycle model, the actuator dynamics of the steering system is considered in the controller design, and the model predictive control method and quadratic programming optimization was combined obtain the precise and smooth tracking effectiveness. In the current studies, some works begin to spend their efforts on the study of joint path following and vehicle stability control [26–30]. In [26], a steering controller is presented for autonomous ground vehicle to ensure the stability of vehicle at the limits of handling and achieve the path following control at the same time. With the in-depth research, researchers are paying more attention to the path following control problem of autonomous vehicles under complex and extreme driving conditions [18,31–35]. Timely and fast information flow is one of the

Corresponding authors. E-mail addresses: [email protected] (L. Chen), [email protected] (X. Xu).

https://doi.org/10.1016/j.advengsoft.2019.05.003 Received 5 September 2018; Received in revised form 30 March 2019; Accepted 19 May 2019 Available online 28 May 2019 0965-9978/ © 2019 Elsevier Ltd. All rights reserved.

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important factors that affect controller performance. If the time delay or information lost happens to the path following controller, the effectiveness of path following will be affected by the poor information source. In [18], the delays and data dropouts of networked control system are modeled, the uncertainty of the tire cornering stiffness and the external disturbances are also considered, and a robust path following control method is presented to ensure the path following accuracy and vehicle stability. In the course of real vehicle driving, some vehicle parameters like vehicle speed, tire cornering stiffness are timevarying. Moreover, some unknown interferences also cause the deviation to path following controller. In [31–33], the uncertainty of tire cornering stiffness, varying vehicle speed, and some external disturbances are considered in the process of designing the path following controller, and the robust H∞ output-feedback control strategy and the terminal sliding mode control are applied to guarantee the control effect. The ultimate development goal of autonomous vehicle is to achieve complete driverless running [36–40]. In the path following process, once an actuator fails, the control performance of the entire control system may decrease or even fail [41–45]. For example, if the steering system of vehicle fails, the vehicle steering movement will be affected or even the vehicle will lose the steering capability. At this time, the vehicle is incapable to follow the instructions given by the controller to turn and track the ideal path. Highly intelligent autonomous vehicle should have the ability to deal with burst actuator failures, and synchronously ensure the path following effectiveness and vehicle stability when fault occurs. Therefore, the research of fault-tolerant path following control for more intelligentized autonomous ground vehicle is very essential and meaningful. In this paper, a novel fault-tolerant path following control strategy of autonomous ground electric vehicle with in-wheel motors is proposed by sliding mode control with the actuator fault of vehicle being considered. In order to estimate the vehicle state, a sliding mode observer with boundary layer is developed and the convergence proof of the sliding mode observer is provided. With the vehicle state being estimated, the time-delay estimation method is introduced to estimate the actuator fault. The hierarchical control method is used to design the overall fault-tolerant path following system. The upper layer controller is designed to obtain the path following control and vehicle stability control at the same time when actuator fault occurs based on the sliding mode control method, and the stability condition of the sliding mode controller is proved. The lower layer controller is used to achieve the control efforts of upper layer controller by the proposed orientated tire force allocation method. The rest of this paper is organized as follows. The vehicle model is developed is Section 2. The fault-tolerant path following control strategy is designed in Section 3. The simulation results are provided in Section 4, followed by the conclusion in Section 5.

Fig. 1. Single-track vehicle model.

=

1 (lf Fyf Iz

lr Fyr + Mz )

(2)

where β represents the sideslip angle of vehicle, γ is the yaw rate of vehicle, m is the vehicle mass, Iz stands for the moment of inertia, lf and lr are the distances from vehicle gravity center to the front and rear axle, respectively. ΔMz is the external yaw moment generated by the driving torques of four in-wheel motors and it can be expressed as 4

Fxi [( 1)ibs cos

Mz =

f

+ lf sin f ]

(3)

i=1

where bs is the half tread of wheel base, Fxi (i = 1,2,3,4) represents the longitudinal force of the ith tire, and the serial numbers 1, 2, 3, and 4 of the wheels are respectively corresponding to the front-right, the frontleft, the rear-left and the rear-right wheel. Fyf and Fyr are the lateral forces of front and rear tires and can be written as

Fyf = 2Cf

Fyr = 2Cr

f

(4)

r

where Cf and Cr express the cornering stiffness of the front and rear tires, αf and αr represent the slip angles of front and rear tires which can be given by f

=

f

lf

+ vy vx

r

=

lr

vy (5)

vx

where δf is the steering angle of front wheels. By assuming that the sideslip angle is small, we have β = vy/vx. Combining above equations, the single-track vehicle model can be derived as

=

=

Cf + Cr mvx2

Cr lr

Cf lf Iz

1+

Cf lf

Cr lr

mvx2

Cf lf2 + Cr lr2 Iz vx

+

+

Cf lf Iz

Cf mvx

f

+

f

1 Mz Iz

In this section, as shown in Fig. 2, the path following model is derived with both lateral offset and heading error being considered. In Fig. 2, ρ is the curvature of the desired path, ψh represents the actual vehicle heading angle and we have h = , ψd is the desired vehicle heading angle, e expresses the lateral offset from the vehicle center of

2.1. Single-track vehicle model In this paper, the origin of dynamic coordinate system xoy is fixed on the vehicle coincides with the vehicle gravity center, the x axis is the longitudinal axis of the vehicle (the forward direction is positive), the y axis is the lateral axis of the vehicle (the right-to-left direction is positive). The pitch, roll, vertical motions and the suspension system of the vehicle are ignored. It is assumed that the mechanical properties of each tire are same. The schematic diagram of the vehicle model is shown in Fig. 1, assuming that the front-wheel angle is small, the dynamics equation of single-track vehicle model with two degree of freedom can be expressed as

1 (Fyf + Fyr ) m

(7)

2.2. Path following model

2. Vehicle model

=

(6)

(1)

Fig. 2. Path following model. 23

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gravity to the closest point P on the desired path, the heading error ψ represents the difference between ψh and ψd and we have ψ = ψh − ψd. The the derivative of heading error can be expressed as

=

h

d

=

=

(9)

+ vy cos

Assuming that the heading error ψ is small, Eq. (9) is rewritten as

e = vx

H

Cf l 2f + Cr lr2

, a2 = Cf l f

Cf

(1 +

Cf l f

Cr lr

mvx2

) , a3 =

Cr lr

Cf l f Iz

,

1 , Iz

a4 = , b1 = mv , b2 = I , b3 = combining Eqs. (6)–(10), Iz v x z x and with the actuator fault being considered, the state space equation for fault-tolerant path following controller design can be obtained as (11)

x = f (x , t ) + Bu + g (t ) y = Cx where the state variable is x = [

e ] , the control output is T

T

b1 0 y=[ the control input is u = [ f b2 b3 , 02 × 2 vx a + a 1 2 C = [ 03× 1 I3× 3 ], f (x , t ) = [ f1 f2 ]T , f1 = , f2 = , vx ( + ) a3 + a4 g(t) represents the dynamic fault input. The fault input g(t) consists of unknown system interferences, un-modeled perturbations and actuator fault. Even when actuator is fault-free, both the interferences and the un-modeled perturbations of system will influence the effect of path following controller, so they can't be ignored. When the failure of actuator occurs, these three factors are regarded as the unknown dynamic inputs of the path following system. It should be noticed that the actuator failure is not specifically assigned to a certain local actuator, and the overall vehicle dynamics system is considered as one actuator in Eq. (11). When a part of the vehicle dynamics system fails, the unknown dynamic inputg(t)actually represents the impact or damage of actuator failure (also including the interferences and the un-modeled perturbations) to the path following control system. In order to eliminate the influence ofg(t), the design objective of the fault-tolerant path following controller is to synchronously ensure the path following accuracy and the vehicle stability even when actuator fault occurs. e ]T ,

Mz ]T , B =

C

f (x , t ) ex

2

g )=exT ( f

HCex + e ( µ

×sgn{( eyi, k (t )

g

HCex +

g) (14)

f )

0

0 ) × ( e yi, k

1 (t )

(15)

0 )}

where i = 1, 2, 3,0 < ‖ξ0‖ ≤ ξ1, Ki is the gain matrix of μi and used to regulate the convergence rate of iterative algorithm. It should be noticed that µ = [ µ1 µ 2 µ3 ]T and μi is corresponding to the yaw rate, heading error and lateral offset, respectively. According to Eq. (15), it can be found that, if the observer is not at the sliding mode boundary layer, the gain matrix of the switching term will be regulated to be larger, at this time, the calculation result of sign function is 1. If the observer enters into the sliding mode boundary layer, the calculation result of sign function is −1 and the gain matrix will decrease. 3.2. Fault estimation When actuator fails, the fault and system interference are regarded as the unknown dynamic input of vehicle path following system. With the system states being estimated in Section 3.1, the unknown dynamic fault input can be estimated and then used to provide the information to later fault-tolerant controller. Choose a small enough time step Δt and assume that it is an iteration cycle of vehicle path following system, the estimation equation of unknown dynamic input can be given by

3.1. System states estimation For the vehicle path following system in Eq. (11), in order to design the fault-tolerant controller, the fault estimation should be carried out firstly. Since the sideslip angle of vehicle is hard and costly to be measured directly, and the measurements of system (11) such as yaw rate, heading error, lateral offset are also affected by measurement noise, so, before the fault estimation, a sliding mode observer is developed for reliable estimation of system states and can be expressed as

y^ ) + y^ = Cx^

(13)

1

µi, k + 1 (t ) = µi, k (t ) + Ki eyi, k (t )

3. Fault-tolerant path following control method

x^ = f (x^ , t ) + Bu + H (y

ey

1

where f = f (x^ , t ) f (x , t ) . It is known that the wider the boundary layer, the better the smoothness of the control signal. The boundary layer can effectively eliminate buffeting phenomena, but drive the sliding mode deviating from the steady state at the same time. According to Eq. (14), to ensure the existence of sliding mode and the asymptotic stability of observer synchronously, the gain matrix μ should be chosen with the condition ‖μ‖ > ‖g‖ + ‖Δf‖ being satisfied to ensure V1 < 0 hold. In the process of path following control, the actuator fault and system interferences are unpredictable, so the superior limit value of ‖g‖ is unknown and time-varying. Therefore, in the process of μ design, in order to prevent the possible error, theμ should be conservatively selected for a larger amount. However, excessive amount of μ will lead to unnecessary high-frequency buffeting. In order to solve this problem, an iterative algorithm of switching gain matrix is presented, in which the switching gain matrix is adjusted in real time along with the variation of unknown dynamic input. It means that the estimation performance of sliding model observer can be ensured even when the actuator fails. The iterative algorithm of switching gain matrix is

2.3. Problem formulation of actuator fault Defining that

µ ,

ey >

V1 = exT (f (x^ , t ) (10)

+ vy = vx ( + )

,

ey

where ey = y^ y is the output error, ξ1 is the boundary layer thickness of sliding mode observer, μ is the gain matrix of the switching term. The Lyapunov function of sliding mode observer is chosen as 1 V1 = 2 exT ex , where ex = x^ x is the system state error. The differential equation of Lyapunov function is obtained as

where d = vx . Based on the Serret-Frenet equation, the derivative of lateral offset can be expressed as

e = vx sin

ey 1

(8)

vx

ey

µ

g^ (t ) = g^ (t

t ) =x ( t ^ f (x , t t)

(12)

where matrix H is the observer gain, λ is the switching term of sliding mode observer. Owning to the existence of switching term, chattering phenomena are very common in sliding mode observer. In order to weaken the high-frequency buffeting, a boundary layer around the sliding mode is introduced and the switching termλis given by

Bu (t

t )=

Bu (t

t)

t) 1 ^ (x (t t

f (x , t

t)

t) x^ (t

Bu (t

2 t ))

t )=x^ (t f (x^ , t

t)

t) (16)

It should be noticed that, in the actual vehicle driving process, both in-wheel motors, vehicle steering system and even some unpredictable parts of actuators, all have the possibility of a failure, so it is hard to judge which part of the vehicle executor has broken down in time. The presented fault estimation method can evaluate the damage caused by 24

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unknown faults and interferences to the control system without knowing where the fault occurs. That is to say, the estimated fault is not the size of the fault itself, but the disturbance level brought by the fault to the path following system at the control level. This idea is more in line with the actual situations of engineering practice and helps to improve the application scope of the next fault-tolerant controller. 3.3. Design of fault-tolerant controller In the process of vehicle driving and path following control, a sudden actuator failure will lead inaccuracy or even serious deviation to vehicle stability and the performance of path following control. Consequently, the path following controller should be fault-tolerant or fault-insensitive, which can ensure the vehicle stability and path following accuracy at the same time with the influences of actuator fault and some unknown interferences. Decouple the path following system (11), we have

x1 = f1 (x1, x2) + B1 u + g1 (t ) x2 = f2 (x1, x2 ) + B2 u + g2 (t )

Fig. 3. Global fault-tolerant vehicle path following control strategy.

fault-tolerant SMC can be achieved by optimal orientated tire force allocation method. The cost function to achieve the external yaw moment ΔMz can be defined as

(17)

]T , x2 = [ e ]T . In the path following model, it can be where x1 = [ found that the whole vehicle is actually regarded as a moving point and b1 0 the actuator input is not included. Therefore, we have B1 = , b2 b3 B2 = 0, g2(t) = 0. The sliding mode surface is defined as

s = (x1

x1r ) + k (x2

J = kf FxT WFx + (1

where Fx = [ Fx1 Fx 2 Fx 3 Fx 4 ] represents the allocated tire forces, W = diag [ w1 w2 w3 w4 ] is the control allocation weight matrix and used to adjust the magnitude of longitudinal tire forces,kf (0 < kf < 1) is the positive weight coefficient used to coordinate the first item and second item in cost function. The matrix Bx represents the control effectiveness matrix and is written as follows according to Eq. (3).

where k is the control matrix, x1r = [ 0 d ] is the referenced vehicle steady state, in which the referenced vehicle sideslip angle is 0 and the referenced vehicle yaw rate is γd, x2r = [0 0]T is the referenced path following control target which means that the heading error and lateral offset should be regulated as close to 0 as possible. The control law of sliding mode controller is designed as T

Bx = [

u = u1 + B1 u2

(20)

kf2 )

u2 =

s 2

, ,

s > s

(1 2J 2F x

f

cos cos

f bs f bs

+ lf sin + lf sin

f

(24)

f]

kf ) BxT (Bx Fx

Mz ) = (kf W + (1

kf ) BxT Bx ) Fx

kf ) BxT Mz

(25)

kf ) BxT Bx

= kf W + (1

(26)

With W > 0 and BxT Bx > 0 being satisfied, the inequality 2 > 0 holds. Fx Therefore, the objective function J has the global minimum with the Fx being chosen as

2

(21)

where τ is switching control gain of sliding mode controller, = g^1 (t ) + , and ε is a selected positive number. The Lyapunov function of sliding mode controller is chosen as 1 V2 = 2 sT s , the differential equation of Lyapunov function is calculated as

Fx = (kf W + (1

Fx1 F F F = x 2 = x3 = x 4 Fz1 Fz 2 Fz 3 Fz 4

kf2 ) + B1 1 u2 ) + g1 + kf2 )=sT (u2 + g1) 0

kf ) BxT Mz

(27)

= 0. which can be obtained by solving The allocation result of longitudinal tire forces is involved with the tire load transfer, and in order to maximize the tire forces usage, the longitudinal tire force should be allocated to be proportional to the normal tire vertical loadFzi(i = 1, 2, 3, 4).

= sT (x1 + kx2)=sT (f1 + B1 u + g1 + kf2 ) + g1 ) = s ( ( g^1 (t ) + ) + g1 )

kf ) BxT Bx ) 1 (1

J Fx

V2

s (

+ lf sin

f

2J

2

=sT (f1 + B1 (B1 1 ( f1

f bs

+ lf sin

J = kf WFx + (1 Fx

and u2 represents the switching control item. Same as the design of sliding model observer, in order to solve the problem of buffeting, a boundary layer thickness is defined as ξ2 and applied to the controller design process. Therefore, the switching control item and can be given by s s

f bs

The objective of tire forces allocation is to figure out the minimal control efforts. On the basis of Eq. (23), we have

where u1 represents the equivalent control item and can be given by

u1 = B1 1 ( f1

cos cos

(19)

1

(23)

Mz )

T

(18)

x2r )

Mz )T (Bx Fx

kf )(Bx Fx

(22)

(28)

where the normal tire vertical load can be expressed as

Accordingly, the presented control law is able to ensure the convergence of sliding mode controller, and achieve the fault-tolerant path following and vehicle stability control simultaneously.

mg

Fz1 = lr ( 2l +

The global fault-tolerant vehicle path following control strategy is summarized and shown in Fig. 3. As for four-wheel independent drive electric vehicle, the control effort of external yaw moment generated by

Fz3 = lf (

mg 2

Fz 4 = lf (

mg 2

2bf l may h

mg

Fz 2 = lr ( 2l

3.4. Orientated tire force allocation method

may h

2bf l

+

may h 2br l may h 2br l

)

max h 2l

)

max h 2l

)+

max h 2l

)+

max h 2l

(29)

where ax and ay represents the longitudinal acceleration and lateral 25

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accelerations of autonomous vehicle, respectively, h is the height of the center of gravity, g is the acceleration of gravity. Therefore, we can obtain the following equation 4 Fzi 4 F i = 1 zi i = 1

Fxi =

Fxi =

Fzi mg

4

Fxi

(30)

i=1

Then the matrix W is designed with thewibeing chosen as

wi =

mg 4Fzi

(31)

Therefore, matrix W can be regulated with the variation of tire vertical load. In cost function (23), the item FxT W1 F is introduced to guarantee the dynamic response ability of tire forces, and the item (1 − kf) (BxFx − ΔMz)T(BxFx − ΔMz) is used to satisfy the control requirement of external yaw moment. Accordingly, if the condition of vehicle lateral stability is severe and the grade of the estimated fault is more serious, the weight coefficient kf should be regulated to be smaller to enhance the control effort of the second item, which will promote the vehicle yaw stability and path following accuracy. Conversely, the weight coefficient kf should be regulated to be larger to increase the control effort of the first item to improve the response ability of tire force distribution. Therefore, the weight coefficient can be designed as

kf =

kf 0 ^ · g^ (t ) 1

(32)

where kf0 is a proportionality constant. 4. Simulation results

Fig. 4. The estimation results of sideslip angle and actuator fault in case study 1. (a) Sideslip angle, (b) Actuator fault.

In order to verify the effectiveness of designed fault-tolerant path following control method in this paper, two simulation cases including the lane change maneuver and the constant radius circular path maneuver are carried out in a high-fidelity CarSim-Simulink co-simulation platform. The CarSim software is applied to provide the whole vehicle dynamic model. And the designed estimation method and control strategy are achieved on the Matlab/Simulink software. The vehicle parameters are listed in Table 1. The vehicle speed in simulation is set as 20 m/s.

angle of vehicle in case study 1. As shown in Fig. 4(a), the designed sliding mode observer can track the actual vehicle sideslip angle with high estimation accuracy. Although there is a certain delay, about 0.12 s, in the estimation results, the time lag is relatively small, and the overall estimation results are in line with the requirements. In Fig. 4(b), it can be found that the proposed actuator fault estimation method can calculate the actual actuator fault in real time and has precise estimation effect, the estimation result can provide a reliable source of information for vehicle path following controller. Fig. 5 shows the path following result in case study 1, in which the ‘Reference’ represents the referenced path, the ‘FTC’ represents the path following result with fault tolerant control, and the ‘Without FTC’ represents the path following result without fault tolerant control and used to compared with FTC. As shown in Fig. 5, when actuator fault occurs, if there is no FTC, the driving path of vehicle will deviate from the ideal path seriously. This is mainly because the steering of vehicle seriously deviates from the required driving direction, resulting in the vehicle can not achieve lane-changing manoeuvre. If the FTC is applied,

4.1. Lane change maneuver with steering system fault In case study 1, the simulation of lane change maneuver is carried out. In the lane change maneuver, the vehicle runs along the road at the beginning, and then the vehicle tracks a clothoid curve path and lane changing behavior is completed during 20 m, and the width of the lane is 5 m. The vehicle continues to run straight after the lane change, during which the road adhesive coefficient is always set as 0.5. The two times gain fault of vehicle steering system is considered in case study 1 at fifth seconds. When this fault occurs, the steering angle of front wheels will be 2 times that of the original value. The simulation results in case study 1 are shown in Figs. 4–7. Fig. 4 shows the estimation results of actuator fault and sideslip Table 1 Parameters of vehicle and in-wheel motors. Symbol

Parameters

Value and units

m r lf lr bf, br Cf Cr Iz

Vehicle mass Effective radius of wheel Distances from vehicle gravity center to the front axle Distances from vehicle gravity center to the rear axle Half treads of the front(rear) wheels Equivalent cornering stiffness of front wheel Equivalent cornering stiffness of rear wheel Moment of inertia

800 kg 0.245 m 0.795 m 0.975 m 0.775 m 60,000 N/rad 40,000 N/rad 1000 kg•m2

Fig. 5. The path following result in case study 1. 26

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the effectiveness of vehicle path following control has been greatly improved and the tracking accuracy is satisfactory. Fig. 6 reveals the control performance in path following process of case study 1. As shown in Fig. 6(a), with FTC, in the process of vehicle lane change, the heading error of vehicle is bounded by −2 deg to 4 deg and finally keeps stable at 0 deg. But if the FTC method is not used, the maximum value of vehicle heading error has increased by nearly 10 times and finally keeps stable at about −3 deg. By comparing the vehicle trajectory, we can see that the vehicle of ‘Without FTC’ has deviated from the ideal path at this time. Fig. 6(b) shows the comparison result of lateral offsets between ‘FTC’ and ‘Without FTC’. When fault occurs, if there is no FTC, it can be found that the lateral offset increases quickly and arrives to −15 m at last. Contrarily, the lateral offset of FTC is kept in a small range and finally approaches to 0 when the lane change is completed. In Fig. 6(c), one can see that the yaw rate of FTC can track the referenced yaw rate with high accuracy during the whole path following period. When fault occurs, the yaw rate of ‘Without FTC’ has acutely increased and seriously deviating from the ideal value. Similarly, as we can see in Fig. 6(d), the vehicle sideslip angle is regulated within the small scope of −1 deg to 1 deg. And the sideslip angle of ‘Without FTC’ obviously increases when the actuator fault occurs. It can be found that, after lane change, the yaw rate and sideslip angle of ‘Without FTC’ also tends to stable at 0. It does not mean that the problem of actuator fault is solved by ‘Without FTC’ method. According to Fig. 5, we can find that the vehicle runs along a straight line and the steering angle of front wheel is 0 once the lane change process is achieved. That is to say, the gain fault of vehicle steering system is unable to influence the vehicle motion at this time. Actually, even without FTC, the vehicle will still run along a straight line after the lane change. That is why the yaw rate and sideslip angle of vehicle in ‘Without FTC’ simulation approach to 0 at last. But the vehicle does not run along the referenced path. Fig. 7 shows the control efforts in case study 1. In Fig. 7(a), when fault occurs, the steering angle of front wheel of ‘Without FTC’ suddenly increases, which is the primary reason of the deterioration of vehicle stability and why the actual driving path of vehicle deviates from the ideal path. In contrast, the front wheel steering angle under fault-tolerant control is well controlled and the jitter caused by sliding mode control is very small. As shown in Fig. 7(b), in the lane change process, the assigned tire force of outer-front wheel is maximal, and the assigned tire force of inner-rear wheel is submaximal, this way of distribution is conducive to generating additional yaw moment and promoting vehicle stability control. And the assigned tire force of inner-front wheel is minimal, which is good for vehicle steering and lane change. In Fig. 7(c), when fault happens, the tire forces of inner-front and outerrear wheel fluctuates sharply to produce more steering trends, which causes the instability of vehicle and the failure of path following. 4.2. Circular path maneuver with dynamic input actuator fault In case study 2, the referenced path is a curve composed of a quarter of a circular arc with the radius being 100 m and a straight line tangent to the circular arc. In simulation, the road adhesive coefficient is set as 0.8. An unknown actuator fault is considered in this simulation. We regarded the whole vehicle as an actuator, so the actuator fault is actually a dynamic input variable of vehicle model. We set the dynamic input fault as a sine wave in the simulation. The simulation results in case study 2 are shown in Figs. 8–11. As shown in Fig. 8, in case study 2, similarly, the proposed estimation method can track the vehicle sideslip angle and actuator fault in real time and the estimation accuracy are satisfactory. Thus, the estimated results can be regarded as the measurements of virtual sensors and used as the reliable inputs the fault-tolerant path following controller. According to Fig. 9, one can find that the driving path of vehicle under designed FTC can track the referenced path and is not affected by the actuator failure. However, as we can see in the results of ‘Without

Fig. 6. The control performance in case study 1. (a) Heading error, (b) Lateral offset, (c) Yaw rate, (d) Sideslip angle.

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Fig. 8. The estimation results of sideslip angle and actuator fault in case study 2. (a) Sideslip angle, (b) Actuator fault.

Fig. 7. The control efforts in case study 1. (a) Front wheel steering angle, (b) Longitudinal tire forces of ‘FTC’, (c) Longitudinal tire forces of ‘Without FTC’.

Fig. 9. The path following result in case study 2.

FTC, the yaw rate of vehicle can track the referenced yaw rate in real time and the sideslip angle is relatively small, which means that the designed controller can ensure the stability of vehicle while achieving the accurate path following control. As we can see in Fig. 11(a), compared with ‘Without FTC’, the steering angle of front wheel of FTC is well supervised by proposed controller to obtain the ideal control performance. In Fig. 11(b), the result of tire force allocation is analogous to that of in case study 1. When the vehicle tracks the arc path, it always turns right. Thus, the longitudinal tire force of front-right wheel is larger than that of frontleft wheel. Moreover, the difference between the tire forces is used to generate additional yaw moment for vehicle stability control. And as shown in Fig. 11(c), the tire force allocation result is obviously affected by actuator fault. The acute fluctuation of front wheel steering angle and tire force is also the cause of poor performance of path following and vehicle stability control if the FTC is not considered.

FTC’, under the interference of sinusoidal wave fault, the vehicle trajectory deviates severely from the set path. As shown in Fig. 10(a), if there is no FTC, the heading error of vehicle reaches to about 10 deg under the influence of actuator fault. And if the FTC is applied, the heading error of vehicle is controlled to very small by FTC. Similarly, as we can see in Fig. 10(b), the lateral offset of FTC is evidently smaller than that of ‘Without FTC’. With the FTC being used, the lateral offset in case study 2 is suppressed to a very small amount. But if without FTC, the lateral offset increases rapidly, and the maximum lateral offset reaches nearly 20 m. It indicates that the proposed fault-tolerant path following control strategy can effectively guarantee the accuracy of path following even if affected by fault. According to Fig. 10(c) and (d), the yaw rate and sideslip angle of vehicle are greatly affected and obviously increase and become unstable when fault occurs. If there is no FTC, the maximum of yaw rate and sideslip angle reaches to about 50 deg/s and 3 deg, respectively. However, with 28

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Fig. 11. The control efforts in case study 2. (a) Front wheel steering angle, (b) Longitudinal tire forces of ‘FTC’, (c) Longitudinal tire forces of ‘Without FTC’.

5. Conclusion This paper deals with the fault-tolerant path following control issue for autonomous ground electric vehicle with in-wheel motors. The vehicle dynamic model and path following model has been integrated and the problem formulation of actuator fault is provided. The sliding mode observer is used to estimate vehicle state, in which the boundary layer is introduced to decrease high-frequency buffeting phenomenon and the convergence of observer is verified. After that, the estimation method of actuator fault is designed with the above sliding mode observer being regarded as the virtual sensor. The hierarchical control method is used to achieve the overall fault-tolerant path following control. In the upper layer controller, the sliding mode control method is used to develop the path following controller, aiming to achieve the path following control and guarantee the vehicle stability simultaneously with the actuator fault. In the lower layer controller, the orientated tire force allocation method is designed to achieve the control efforts of upper layer controller, in which the tire force allocation result is adaptive to the vehicle sideslip angle and actuator fault. The simulations of lane change maneuver and circular path maneuver are carried

Fig. 10. The control performance in case study 2. (a) Heading error, (b) Lateral offset, (c) Yaw rate, (d) Sideslip angle.

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out in the CarSim-Simulink co-simulation platform. The results show that the designed estimation method is able to estimate the vehicle state and actuator fault with high accuracy and real-time tracking ability, and the proposed fault-tolerant path following control strategy has ability to handle the influences of the actuator failure, and ensure the path following accuracy and vehicle stability. The control performance and robustness of the proposed control method in this paper is proved.

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Acknowledgments

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This work was supported by the National Natural Science Foundation of China (grant no. U1564201 and U1664258), the “333 Project” of Jiangsu Province(grant no. BRA2016445), the Primary Research & Development Plan of Jiangsu Province(grant no. BE 2017129 and BE2016149), the Natural Science Foundation of Jiangsu Province(grant no. BK 20160525).

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Conflict of interests

[27]

The authors declare that there is no conflict of interests regarding the publication of this paper.

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