A Type-2 Fuzzy Approach to Driver- Shared Driving Lane Keeping Control of Semi- autonomous Vehicles under Imprecise Premise Variable

Yue Liu Beihang University Qing Xu Tsinghua University Hongyan Guo Jilin University Hui ZHANG (  [email protected] ) Beihang University https://orcid.org/0000-0002-2501-712X

Original Article

Keywords: Fuzzy control, interval type-2 fuzzy systems, Dstability , driver-automation shared control, autonomous vehicles

Posted Date: August 24th, 2021

DOI: https://doi.org/10.21203/rs.3.rs-814651/v1

License:   This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License 1

A Type-2 Fuzzy Approach to Driver-Automation Shared Driving Lane Keeping Control of Semi-autonomous Vehicles under Imprecise Premise Variable Yue Liu, Qing Xu, HongYan Guo, and Hui Zhang∗

Abstract—The driver-automation shared driving is a transi- Kd1 Driver’s proportional action about near angle tion to fully-autonomous driving, in which human driver and Kd2 Driver’s proportional action about far angle vehicular controller cooperatively share the control authority. t Preview time This paper investigates the shared steering control of semi- p autonomous vehicles with uncertainty from imprecise parameter. τa Driver’s anticipatory time By considering driver’s lane-keeping behaviour on the vehicle system, a driver-automation shared driving model is introduced for control purpose. Based on the interval type-2 (IT2) fuzzy II.INTRODUCTION theory, moreover, the driver-automation shared driving model EHICLE automation has attracted extensive attention with uncertainty from imprecise parameter is described using from academia, industry and government over the past an IT2 fuzzy model. After that, the corresponding IT2 fuzzy V controller is designed and a direct Lyapunov method is applied decades. It can reduce driver workload and road accidents to analyze the system stability. In this work, sufficient design resulted from human factors such as distraction and drowsi- conditions in terms of linear matrix inequalities are derived, ness, etc. However, there are still some barriers in the way to guarantee the closed-loop stability of the driver-automation to fully driving automation, such as the limitation of con- shared control system. Meanwhile, an H∞ performance is studied to ensure the robustness of the control system. Finally, simulation- temporary technology and anthropologic issue. Furthermore, based results are provided to demonstrate the performance of the some severe autonomous-driving accidents also indicate the proposed control method. Furthermore, an existing type-1 fuzzy human-off-the-loop driving is dangerous [1], [2]. It means that: controller is introduced as comparison, to verify the superiority although the fully driving automation is an ultimate appeal for of the proposed IT2 fuzzy controller. autonomous vehicle, the human driver still plays an important Index Terms—Fuzzy control, interval type-2 fuzzy systems, D- role in the current stage. However, inattention of human driver stability , driver-automation shared control, autonomous vehicles. does threaten the vehicle safety, and consider human driver’s physical and psychological influence on the strategic, tactical and control behavior, he/she may do misoperation in complex I.NOMENCLATURE environment. Consequently, there are many advanced vehicle assistance systems, like adaptive system (ACC), C Rear concerning stiffness r lane departure warning system (LDWS) and lane keeping as- C Front concerning stiffness f sistance system (LKAS) [3], [4], that are developed to alleviate l Distance from center of gravity to front axle f driver’s burden and improve the vehicle safety. In general, l Distance from center of gravity to rear axle r these vehicle-borne auxiliary systems could undertake driving l Distance from center of gravity to wind impact w tasks independently. For example, in a circumstance where center the road is longer and straighter and the destination is farther l Preview distance s apart, ACC could follow the preceding vehicle automatically. n Tire pneumatic trail R However, these assistance systems have their dynamic and I Vehicle yaw inertia moment z situated limitations, for instance, the bad weather condition I Inertia moment of steering system s could degrade the lead vehicle following performance of the R Steering gear ratio s ACC system [5]. Similarly, other advanced assistance systems B Steering system damping coefficient s may also outside their control bandwidth in the case where m Vehicle mass the external environment changes rapidly [5]. As a result, Hui Zhang is the corresponding author with the Email: it is necessary that human driver and vehicular assistance [email protected] system control the vehicle cooperatively. Therefore, there is Yue Liu, and Hui Zhang are with the School of Transportation Sci- ence and Engineering, Beihang University, Beijing 100091, China. e-mail: an inevitable interaction between the human driver and the ([email protected]; [email protected]) assistance system. Qing Xu is with Department of automotive engineering, Tsinghua Univer- Taking the electric power assist system(EPAS) for instance, sity, Beijing, China Hong yan Guo is with Colleague of automotive engineering, Jilin Univer- when human driver implements his/her control behavior on sity, Jilin, China the steering wheel, EPAS could assist the driver to adjust the 2 steering through an electric motor. With hands on the steering function was proposed to describe the driver activity. They wheel, the driver could perceive that the vehicle has its own in- proposed a type-1 Takagi-Sugeno fuzzy method to deal with tention. It is worthy mentioning that there may be unexpected the time-varying vehicle speed. However, although the type-1 assistance in regular driving, which can disturb the driver and fuzzy technology is useful in describing nonlinear and/or linear even be harmful to vehicle safety. Therefore, conflict between parameter-varying systems [18], it cannot describe uncertainty human driver and automation system is a challenge needed to in the membership function. Generally, the vehicle longitudinal be settled. Fortunately, shared control is a valid approach to speed in system model can be determined through the wheel- weaken conflict between them [6], [7], in which human driver speed sensor. However, the precision of measurement is lim- and automation system share the control authority of vehicle ited due to sensor performance. Usually, high precision comes collaboratively. In fact, there is no consentaneous definition for with high cost, while low-cost sensor leads to inevitable mea- shared control in academia, Abbink et al. proposed: in shared surement error. This inevitable measurement error will cause control, human and ‘robot’ are interacting congruently in a the imprecise longitudinal speed. Moreover, when the wheel perception-action cycle to perform a dynamic task that either is not free-rolling, the longitudinal speed obtained based on human or the robot could execute individually under ideal the wheel-speed sensor will also be imprecise. The imprecise circumstances [5]. Here, the ‘robot’ means a designed system longitudinal speed can cause uncertain membership function. with a degree of ‘intelligence’ and ‘autonomy’, such as the If the uncertainty is neglected when designing controller, the vehicle equipped with advanced assistance systems, which can performance of designed controller will be greatly affected. complete particular driving task independently under certain Fortunately, type-2 fuzzy technology is a valid method to circumstances. There are some other researchers that empha- further address uncertainty in the membership function. size the essence of shared control lies in the execution of The conception of type-2 fuzzy system was proposed by actions, although ‘monitor’ and ‘decision’ can also be shared Zadeh in [19]. It can be regarded as a collection of infinite [8]. This emphasis on ”low-level” execution is also central number of type-1 fuzzy systems [20], [21]. Different from to the proposed definition of shared control for automotive type-1 T-S fuzzy system, the type-2 fuzzy system has a better domain [5]. There are two different types of shared control: property in describing uncertainties which cause difficulty directly shared control and indirectly shared control [9]. The to determine exact membership function [22], [23]. There former means the ‘robot’ will interpose the driver directly are various sources of uncertainties in type-1 fuzzy system: while the latter emphasizes that the ‘robot’ will compensate (1) Measured signals activating type-1 fuzzy system may the driver’ activity. be contaminated by noise or other factors and therefore be Shared control has been extensively studied in the last uncertain; (2) Data applied to type-1 fuzzy system to modify two decades. Saleh et al. [10] designed a shared control parameters may be incomplete or fragmentary [24], [25]. All method,which was based on a closed-loop driver-vehicle-road of these can generate uncertainty in the membership function. model and the shared control law was realized through the Fortunately, it can be removed by the type-2 fuzzy technol- H2−preview control method. Huang et al. [11] presented ogy, which has been researched and demonstrated in many a shared framework of the driver and the semi-autonomous literatures. Hani et al. [26] studied the autonomous mobile vehicle, in which a data-driven shared control law using robot navigating in changing environment, in which a type-2 robust adaptive dynamics programming was studied. Li et fuzzy logic technology was developed to handle uncertainties al. [12] proposed a driver-automation shared control strategy from the changing and unstructured environments. Zhang et al. for obstacle avoidance, where a two-layer fuzzy strategy was [27] addressed the fault detection problem for continuous-time introduced to address the control authority allocation issues. fuzzy semi-Markov jump systems, in which an interval type-2 Erlien et al. [13] proposed a shared control framework for fuzzy approach was utilized to solve the parameter uncertainty obstacle avoidance using two safe driving envelopes, where of system. Kumbasar [28] proposed a self-tuning zSlices-based a model predictive control (MPC) scheme was developed to general type-2 fuzzy PI controller to handle high levels of un- determine if the driver can ensure a safe vehicle trajectory certainties and ensure the robustness of system to disturbances within proposed two envelopes and when to intervene the noise and uncertainties, where the secondary membership driver. Ji et al. [14] proposed a novel stochastic game-based functions were adjusted in an online manner. Muhuri et al. [29] shared control framework to model the steering torque in- addressed the multi-objective reliability-redundancy allocation teraction between human driver and the intelligent electric problem to ensure high system reliability in the presence of system. Li et al. [15] presented an ‘indirect optimally redundant components with uncertain parameters, shared control’ framework for steer-by-wire vehicles, in which in which an interval type-2 fuzzy technology was utilized as a ‘best-response’ driver steering model based on MPC was type-1 fuzzy technology has limitations in representing high proposed. Lu et al. [16] studied the driving performance order uncertainties. of brain-controller vehicles which are controlled by human This paper studies the driver-automation shared steering ‘mind’ through the use of a brain-computer interface. In this control problem for lane keeping performance. In order to work, they proposed a shared control method based on the attenuate the conflict between human driver and vehicular MPC strategy, to ensure the brain-control drivers have the automation system, interaction between them is necessary to maximum control authority. Nguyen et al. [17] studied the be studied. As mentioned before, the driver could perceive shared control between human driver and LKAS for lane the vehicle having its own intention with hands on the steer- keeping and obstacle avoidance, in which a fictive nonlinear ing wheel. Apart from this, the vehicular automation also 3 needs to know the driver’s behavior and provide appropri- ate assist torque based on it. Consequently, a time-varying weighting function which is related to the driver activity is introduced here to adjust the assist torque, so that an appropriate assistance level can be ensured. Considering the uncertain driver behavior, furthermore, a two-point preview driver model is used to describe the driver’s behavior. Then, a driver-automation shared driving model is obtained. As the vehicle longitudinal speed is time-varying in the driving process, the obtained model will be a linear parameter-varying model. Although type-1 T-S fuzzy theory is a valid method to address such model, it cannot remove the uncertainty in the membership function resulted from inevitable measure- ment error. Moreover, the introduced time-varying weighting function also contains time-variant variable, which can also Fig. 1. Schematic diagram of vehicle-road model. cause uncertain membership function. Therefore, a type-2 fuzzy technology is studied in this work, to address uncertainty in the membership function and the controller design issue. wind is considered. Therefore, the lateral translational motion Differing from the previous work in [30], an H∞ performance and the yaw dynamics of vehicle is described as follows: [31]: index is introduced here, to weaken the influence of external 2(Cf + Cr) 2(Crlr − Cf lf ) disturbance on the control system. It is because both the lane mv˙y = − vy + r − mvxr vx vx keeping performance and the system robustness should be + 2Cf δ + Fl, guaranteed. Besides, data from urban driving condition is also 2 2 (1) applied to verify effectiveness of the proposed method. The 2(Crlr − Cf lf ) 2(Cf lf + Crlr ) Izr˙ = vy − r main contributions of this paper are summarized as follows: vx vx 1) A new type-2 fuzzy technology is applied to remove + 2Cf lf δ + lwFl. uncertainty in the membership function resulted from mea- where vy is the lateral velocity and vx is the longitudinal surement error and the driver’s state parameter; velocity. r represents the yaw rate. m is the total vehicle mass. 2) A different control output is introduced, which considers Iz is the vehicle yaw inertia moment. lf shows the distance information of both the lane keeping performance and conflict from center of gravity to front axle and lr is the distance from between human driver and vehicular automation system; center of gravity to rear axle. lw indicates the distance from 3) A D-stability method is introduced to realize the pole center of gravity to the lateral wind impact center. Cr is the placement and then improve transient performance of the concerning stiffness of rear tires and Cf is the concerning control system. stiffness of front tires. δ represents the front wheel steering angle. In this work, in order to pursue better lane keeping perfor- III.PROBLEM STATEMENT mance, a model needs to be defined to describe the vehicle This section formulates the driver-automation shared control lateral offset. As shown in Fig.1, considering the look-ahead problem. As mentioned before, if the interaction between hu- distance, the lateral offset consists of two parts: (1) the lateral man driver and vehicle assistant system is ignored, it may lead displacement resulted from vehicle motion; (2) the component discomfort to the driver and even conflict between them. To originating from the look-ahead distance. As a result, the avoid conflict issues, the driver-automation shared control law model of vehicle lateral offset is defined as follows [32]: for a lane keeping purpose is studied. Considering uncertain y˙l = vy + lsr + vxψl, (2) driving behavior of the driver, a two-point previewed driver model is applied to the road-vehicle model and then the driver- where yl shows the lateral offset, ls is the look-ahead distance automation shared driving model can be obtained. Moreover, a and ψl is the heading error. It should be noted that the heading type-2 fuzzy technology is introduced to address time-varying error is approximatively assumed to be the yaw angle. The parameters and uncertainty in the system. assumption is valid in circumstance where the road curvature is too small to be neglected, which is realistic for the highway driving environment. Then, the following equation could be A. Driver-Automation Cooperative Driving Model obtained: ψ˙ = r. (3) In this work, a shared steering control strategy for lane l keeping performance is studied. Here, It is assumed that there In the framework of the driver-automation shared driving, are no vehicles moving into the lane on both sides of the target the human driver implement a torque on the steering wheel vehicle. Then, a lateral vehicle dynamics as shown in Fig. 1 is while the vehicular automation controller also provide an studied. In addition, in order to analyze the robustness of the assistant torque to control the vehicle. Then, a desired front- control system, effects of external disturbances like the lateral wheel steering angle is obtained through the steering system. 4

Considering the damping characteristic of steering system and where the aligning torque resulted from front-tire lateral force, the 2 2 c1 = Kd2τa a21, c2 = Kd2(τa + τa a22), c3 = Kd1, equation of front-wheel steering angle is shown as follows: Kd1 2 Ta + Td Fyf nR c4 = , c5 = Kd2τa a25Rs. I δ¨ = − − B δ,˙ vxTp s R R2 s s s For the driver-automation shared driving system, both the 2C n 2C l n 2C n T + T f R f f R f R ˙ a a driver torque and the assistant torque are system input. Mean- = 2 vy + 2 r − 2 δ − Bsδ + , Rsvx Rsvx Rs Rs (4) while, the pursuit of this work is to design an appropriate assistant torque according to the driver’s current behavior. As where I is the moment of inertia and R is the gear ratio s s the driver torque is predicted, the assistant torque could be of steering system. B is the damping coefficient of steering s designed through properly designed controller. It should be system. nR is the tire pneumatic trail. Finally, the driver- automation shared driving model for lane keeping performance emphasized that the designed assistant torque should have is obtained and the state-space model is determined by comb- ability to be adjusted according to the driver’s behavior, to avoid disturbing the human driver. ing Eqn. (1) ∼ Eqn. (4): x x˙(t)= A (t)+ B1(Ta + Td)+ B2w(t), (5) B. Driver’s adaptive need for assistance a11 a12 0 0 a15 0 0 The interaction between human driver and vehicular au- a a 0 0 a 0 0  21 22 25    tomation system plays a key role in driver-automation shared 0 1 0 0 0 0 0 control. On the one hand, the human driver must have an A = , B1 = ,  1 ls vx 0 0 0   0      adaptive control authority to the vehicle. On the other hand, the  0 0 0 0 0 1   0      assist torque from automation system should be adapt to the a a 0 0 a a  b   61 62 65 66  11 driver’s real-time activity. When the driver is in distraction, for  ⊤    B2 = b21 b22 0 0 0 0 , example, his/her currently driving performance is poor. In this case, the assistant system should provide assistance with high where   level. Moreover, the assistance level should decrease along 2(Cr +Cf ) 2(lr Cr −lf Cf ) a11 = − mv , a12 = −vx + mv , with the increase of driver’s driving performance, to ensure x − x 2Cf 2(lr Cr lf Cf ) a comfortable driving experience for the driver. In addition, a15 = m , a21 = I v , 2 2 z x 2(lr Cr +lf Cf ) 2lf Cf when the human driver is highly focused on the driving task a22 = − , a25 = , Iz vx Iz . but he/she is in overload, like the circumstance where human 2Cf nR 2Cf lf nR a61 = 2 , a62 = 2 , IsRsvx IsRsvx driver is in a dangerous situation, the need for assistance 2Cf nR −Bs a65 = − 2 , a66 = , should also be in high level. That is to say, adaptive level of IsRs Is b = 1 , b = 1 , b = lw . assistance should be provided to undertake the driving tasks 11 IsRs 21 m 22 Iz ⊤ [33]. Following these guidelines, the assistant torque from In the state-space model, x = vy r ψl yL δ δ˙ is vehicular automation system is modulated as follows: the state vector. Assistant torque Ta and driver torque Td   (9) are the system input. w(t) means external disturbance and it Ta = µ(θd(t))u(t), θd(t) ∈ [0, 1], shows the lateral wind force here. In this work, an appropriate where u(t) is the control input to be designed. The time- assistant torque, which can suit the driver’s current driving varying parameter µ(θd(t)) is a function of the driver’s ac- behavior, is desired to be designed. Considering uncertain tivity, it shows the driver’s need for assistance. Based on it, behavior of human driver, a two-point previewed driver model the assist torque Ta can be adapted according to the driver’s is introduced as follows: [17]: real-time activity. It should be noted that the driver’s activity contains two types of information: 1) the driver’s torque; 2) Td = Kd1θnear + Kd2θfar. (6) the driver’s state. Hence, the detailed description of µ(θd(t)) Here, θnear represents visual direction to the near point, which is given as follows: [17]: is related to the driver’s present activity. indicates visual θfar 2 direction to the far point, which is related to the driver’s µ(θd(t)) = ω1(θd(t) − ω2) + µmin, − σ2 σ3 expectant activity. K and K are gains in accordance with θ (t) = 1 − e (σ1TdN ) DS , d1 d2 d (10) the two angles θnear and θfar respectively, which represent T T = d . driver’s proportional actions on the two angles. The two visual dN T dmax angles θnear and θfar are defined as: Here, θd(t) represents the driver’s real-time activity. It could be yL θnear = + ψL, described by two parameters i.e. TdN , DS. TdN is the driver’s vxTp (7) 2 2 2 normalized torque and it describes the driver’s workload. DS θfar = τa a21vy +(τa + τa a22)r + τa a25Rs, (0 ≤ DS ≤ 1) shows information of driver’s state and it where Tp is the preview time and τa is the react time of human describes the driver’s attention on the driving task. Note that driver. Then, the predicted driver torque can be rewritten as it means that the driver is divorced from the driving task when follows: DS = 0, and the driver is highly focused on the driving task Td = c1 c2 c3 c4 c5 0 x, (8) when DS = 1.   5

ω1, ω2 and µmin are parameters of the parabolic equation a time-varying parameter µ(θd(t)) is introduced to adjust the µ(θd(t)). Thereinto, µmin is the minimal assistance level and assistant torque according to human driver’s activity, a global it could be obtained based on the practical demand. It is worthy optimization index aiming to reduce the discrepancy between mentioning that the minimal assistance level should guarantee human driver and automation system is also necessary. Differ- the weighting parameter µ(θd) has a large range. σ1, σ2 and ent from the work in [17], a variant control output considering σ3 are weight parameters of the driver’s torque and the driver’s difference between the driver’s torque and the assistant torque state [17]. is introduced, to attenuate conflict between them. As the uncertain driver behavior is approximately predicted by Eqn. (8) and the assistance torque is designed through Eqn. IV. DRIVER-AUTOMATION SHARED CONTROL DESIGN (9), substituting Eqn. (8) and Eqn. (9) into Eqn. (5), the driver- This section illustrates the type-2 fuzzy driver-automation automation shared driving system used for controller design shared driving model and the corresponding control strategy. purpose can be obtained as follows:

x˙(t)= Ax¯ (t)+ B¯1u(t)+ B2w(t), (11) A. Type-2 Fuzzy Driver-Automation Shared Driving Model In the driver-automation shared driving system (11), the where vehicle longitudinal velocity is time-varying, which leads to a a11 a12 0 0 a15 0 0 linear parameter-varying system. As well known, conventional a a 0 0 a 0 0  21 22 25    type-1 T-S fuzzy model is valid to describe such system 0 1 0 0 0 0 0 and then the membership function could be obtained based A¯ = , B¯1 = ,  1 ls vx 0 0 0   0      on the characteristic of the time-varying parameter. However,  0 0 0 0 0 1   0      as mentioned above, there is inevitable error in the vehicle a¯ a¯ a¯ a¯ a¯ a  ¯b   61 62 63 64 65 66  61 longitudinal speed. It will cause uncertainty in the membership a¯61 =a61 + b11c1, a¯62 = a62 + b11c2, a¯65 =a65+ b11c5, function, which cannot be addressed by the type-1 fuzzy theory. The uncertain membership function could degrade the a¯63 = b11c3, a¯64 = b11c4, ¯b61 = b11µ(θd(t)). performance of designed driver-automation shared controller. For a control system, the control output could be defined based To ensure better control performance, the uncertainty in the on the performance demand. As the lane keeping performance membership function resulted from measurement error must is studied in this work, the tracking performance should be be addressed. Differing from the existing work in which the guaranteed first. Apart from it, as the shared control strategy vehicle longitudinal speed vx is assumed to be exactly known, is proposed to reduce conflict between human driver and the type-2 fuzzy technology is introduced here, to further solve vehicular automation, the difference between them should also the uncertainty in the membership function. Apart from the be considered. Besides, when there exists assistance for human vehicle speed, the weighting function µ(θd(t)) is also time- driver, considering the road sense, the steering angle should variant within its limitation, which shows the driver’s need not change too quickly, which may cause discomfort for the for assistance. Furthermore, it is related to the driver’s state human driver. Therefore, its derivative is also an important that can be described by the parameter DS. DS is also time- factor to evaluate the proposed method. Consequently, the variant and changes in the range of its limitation, which could control output in this work is defined as: also cause uncertainty in the membership function. All of these explain the reason why a new type-2 fuzzy z(t)= ay θnear θfar δ˙ Td − Ta , model is applied here to describe the driver-automation shared = Ex(t)+ Fu(t),   driving system (11), which is expressed as follows: 0 vx 0 0 0 0 1 p 0 0 1 1 0 0 Rule i : IF f1(t) is Mi and ··· and fp(t) is Mi vxTp (12)  2 2 2  THEN ¯ ¯ E = τa a21 τa + τa a22 0 0 τa a23 0 , x˙(t)= Aix(t)+ B1iu(t)+ B2iw(t),  0 0 0 0 0 1   z(t)= Cix(t), (13)  c1 c2 c3 c4 c5 0 s   where p is the number of premise variable. Mi is the interval F = 0; 0; 0; 0; µ(θd(t)) .  type-2 fuzzy set of function fs(t) based on rule i, with Remark 1. It should be stressed that the vehicle longitudinal i ∈ {1, 2,...,r} and s ∈ {1, 2,...,p}. The local state-space ¯ n×n ¯ n×m speed vx(t) and the driver activity variable θd(t) involved in matrices Ai ∈ R , B1i ∈ R , B2i and Ci are known. system (11) are time-varying. Moreover, measurement error The interval sets in accordance with firing strength of the ith of vx(t) is inevitable in engineering and the parameter DS rule are shown as: in θd(t) is also changed. Therefore, the type-1 T-S fuzzy L U ξi(t)=[ξi (t), ξi (t)], i = 1, 2,...,r, technology depending on a precise vehicle speed in [17] may p not hold for the practical application. L ξ (t)= α 1 (f1(t)) ×···× α p (fp(t)) = α s (fs(t)), ei Mi Mi Mi Remark 2. In this work, a shared control strategy for the s=1 Yp lane keeping assistance system is studied, therefore, both of U ξi (t)= αM 1 (f1(t)) ×···× αM p (fp(t)) = αM s (fs(t)), the tracking performance and the conflict between human i i i s=1 driver and automation system should be considered. Although Y (14) 6

where α s (fs(t)) and αM s (fs(t)) represent the lower and where Gj,j ∈ {1,...,r} are the state-feedback gains to be Mi i upper grades of membership governed by the lower and upper designed. Then, the corresponding output of the interval type-2 membership functions, respectively. Note that fuzzy controller is defined as: r 0 ≤ α s (fs(t)) ≤ αM s (fs(t)) ≤ αM s (fs(t)) ≤ 1. (15) Mi i i (20) u(t)= (ξj(t)+ ξj(t))Gjx(t), j=1 Then, the interval type-2 fuzzy model of driver-automation X where shared driving system (11) is defined as L ξj (t) r ξj(t)= r L U , l=1 ξl (t)+ ξl (t) x˙(t)= ξi(t)(A¯ix(t)+ B¯1iu(t)+ B2iw(t)), ξU (t) i=1 (16) P j  Xr ξj(t)= r L U . l=1 ξl (t)+ ξl (t) z(t)= ξi(t)Cix(t), i=1 With the expressions inP (16) and (20), the closed-loop type-2 X fuzzy driver-automation shared control system can be obtained where as: r r r L U ξi(t)= ξi (t)νi(t)+ ξi (t)νi(t), ξi(t) = 1, ¯ ¯ x˙(t)= ξi(ξj + ξj) (Ai + B1iGj)x(t)+ B2iw(t) , i=1 i=1 j=1 X X X   0 ≤ νi(t) ≤ 1, 0 ≤ νi(t) ≤ 1, νi(t)+ νi(t) = 1. r z(t)= ξi [Eix(t)+ Fiu(t)] , Here, νi(t) and νi(t) are nonlinear function. A practical i=1 assumption on longitudinal speed is that vx(t) ∈ [vmin, vmax], Xr r where and . Meanwhile, the vmin = 2.5m/s vmax = 25m/s = ξi(ξj + ξj)[(Ei + FiGj)x(t)] . weighting function µ(θd(t)) is time-varying with the property i=1 j=1 X X (21) of µ(θd(t)) ∈ [µmin, µmax]. There are three premise variables 1 For simplicity, ξ (t), ξ (t) and ξ (t) are replaced with ξ , vx(t), and µ(θd(t)) involved in system. Note that the i j j i vx(t) and , respectively. Besides, they have the following number of fuzzy subsystems increases exponentially according ξj ξj to increase on the number of premise variables. Then, the characteristic: workload on computation will be increased. Here, in order r r r r to reduce the computation complexity, a new premise variable ξi = (ξj + ξj)= ξi(ξj + ξj) = 1. (22) 1 υ is introduced to decouple vx(t) and , which is shown i=1 j=1 i=1 j=1 vx(t) X X X X as follows [17]: Then, the driver-automation shared controller design is trans-

1 v0 − v1 1 formed to select matrices Gj for the closed-loop system (21). = υ + , vx(t)=(v1 − v0)υ + v0, However, when pursuing the control effect, the robustness vx(t) v1v0 v0 (17) should also be considered, which means the system itself v = v , v = v , 0 min 1 max should have ability to attenuate the influence of external where υ ∈ [0, 1]. It should be noted that the new premise disturbance on the control output z(t). An H∞ control method, variable is imprecise as the speed vx(t) is imprecise. Then, therefore, is employed to evaluate the influence mentioned the number of premise variable is reduced and the new two before, which is shown as: premise variables υ, µ(θd(t)) can be expressed as: ∥z∥2 <γ∥w∥2. (23) f1 = υ ∈ [f1min,f1max], f2 = µ(θd(t)) ∈ [f2min,f2max]. In addition, in order to obtain better transient performance, (18) a D-stability method is applied in this work. As well known, Since the number of premise variables is reduced from 3 to 2, the transient performance of control system could be improved the complexity of driver-automation shared driving system is with its eigenvalues faring away from the imaginary axis, with the cost of huge control input. Fortunately, D-stability is a significantly released. In addition, the error of vx(t) is assumed to be within ±10%, based on which the lower and upper valid method to realize it without enlarging effort of the control membership functions can be referred in Table I. system. Lemma 1. [34] D-stability: A real matrix A¯ is D-stable, i.e. all the eigenvalues of the matrix A¯ are in the LMI region D, B. Type-2 Fuzzy Controller Design ⊤ if and only if there exists a positive matrix P = P such that As the type-2 fuzzy driver-automation shared driving model ¯ ¯ ⊤ ¯⊤ is defined, the following type-2 fuzzy shared controller is MD(A, P ) = Γ P +Υ (XA)+Υ (A X) < 0. (24) applied to stabilize the system (16): O O O Specially, for a disk LMI region (q,r), the condition for the 1 p Rule j : IF f1(t) is Mj and ··· and fp(t) is Mj D-stability is −rP qP + P A¯ THEN u(t)= G x(t), (19) ≤ 0. (25) j ∗ −rP   7

TABLE I TYPE-2 FUZZYMEMBERSHIPFUNCTIONS

Lower and upper membership functions

f1 f2

f1 −0.9f1 f2 −f2 α 1 = α 1 = max α 2 = α 2 = max M M f1 −f1 M M f2 −f2 ∆= −10% 1 2 max min DS =0.5 1 3 max min 0.9f1−f1 f2−f2 α 1 = α 1 = min α 2 = α 2 = min M3 M4 f1max−f1min M2 M4 f2max−f2min

f1 −1.1f1 f2 −f2 α 1 = α 1 = max α 2 = α 2 = max M M f1 −f1 M M f2 −f2 ∆ = 10% 1 2 max min DS =1 1 3 max min 1.1f1−f1 f2−f2 α 1 = α 1 = min α 2 = α 2 = min M3 M4 f1max−f1min M2 M4 f2max−f2min

⊤ ⊤ ⊤ ⊤ ⊤ ⊤ ⊤ Lj1 + Lj1 + Ni1 + Ni1 − Vij1 − Vij1 + Qij B2i + Lj2 + Lj3 + Ni2 + Ni3 − Vij2 − Vij3 (EiX + FiNj) 2 ⊤ ⊤ ⊤ ∗ −γ I + Lj4 + Lj4 + Ni4 + Ni4 − Vij4 − Vij4 0 ≤ 0,  1  ∗ ∗ − 2 I   (26)

⊤ ⊤ ⊤ ⊤ ⊤ ⊤ ⊤ Mj1 + Mj1 + Ni1 + Ni1 − Uij1 − Uij1 + Qij B2i + Mj2 + Mj3 + Ni2 + Ni3 − Uij2 − Uij3 (EiX + FiNj) 2 ⊤ ⊤ ⊤ ∗ −γ I + Mj4 + Mj4 + Ni4 + Ni4 − Uij4 − Uij4 0 ≤ 0,  1  ∗ ∗ − 2 I   (27)

C. Stability Analysis of Type-2 Fuzzy Shared Control System and −rX qX + A¯ X + B¯ N The following theorem provides sufficient conditions for i 1i j ≤ 0. (33) ∗ −rX designing the type-2 fuzzy driver-automation shared controller   with a promised H∞ performance level. Here, Theorem 1. Given a positive scalar γ, the closed-loop type-2 ⊤ ⊤ V ··· V U ··· U fuzzy control system (21) is asymptotically and robustly stable, 11 1r 11 1r ...... the H∞ condition in (23) is satisfied and the D-stability in V =  . . .  , U =  . . .  , a disk LMI region (q,r) is ensured if there exist matrices of Vr1 ··· V1r Ur1 ··· U1r ⊤     appropriate dimensions X = X , K = G X, L , M , N , ⊤ j j j1 j1 j1 W ··· W R 0 0  , , , , , , , , , ⊤ , 11 1r 11 Lj2 Mj2 Nj2 Lj3 Mj3 Nj3 Lj4 Mj4 Nj4 Rjj1 = Rjj1 . . . . ⊤ ⊤ ⊤ W = . .. . , R = .. , Sjj1 = Sjj1, Tij1 = Tij1, Uij1, Vij1, Wij1, Rjj2 = Rjj2,  . .   0 0  ⊤ ⊤ S = S , T = T , U , V , W , U , V , W , Wr1 ··· W1r 0 0 Rrr jj2 jj2 jj2 jj2 ij4 ij4 ij4 ij2 ij2 ij2     Uij3, Vij3 and Wij3 such that S11 0 0  T11 0 0  ⊤ ⊤ S = .. , T = .. , Rjj1 0 Nj1 + Nj1 Nj2 + Nj3  0 . 0   0 . 0  ≤ ⊤ , (28) 0 Rjj ∗ Nj + N 0 0 S 0 0 T  2  4 j4  rr  rr ⊤ ⊤ Uij1 Uij2  Vij1 Vij2  Sjj1 0 Lj1 + Lj1 Lj2 + Lj3 Uij = , Vij = , ≤ ⊤ , (29) Uij3 Uij4 Vij3 Vij4 0 Sjj2 ∗ Lj4 + Lj4         Wij1 Wij2 Rjj1 0 ⊤ ⊤ Wij = , Rjj = , Wij3 Wij4 0 Rjj2 Tjj1 0 Mj1 + Mj1 Mj2 + Mj3 ≤ ⊤ , (30)     0 Tjj2 ∗ Mj4 + Mj4 Sjj1 0 Tjj1 0     Sjj = , Tjj = , 0 Sjj2 0 Tjj2 ⊤ ⊤     Wij1 + Wij1 Wij2 + Wij3 ⊤ ≤ Lj1 Lj2 Mj1 Mj2 ∗ Wij4 + W Lj = , Mj = , ij4 Lj3 Lj4 Mj3 Mj4  ⊤ ⊤  ⊤ ⊤     Mj1 + Mj1 + Li1 + Li1 Mj2 + Mj3 + Li2 + Li3 ⊤ ⊤ , Nj = Nj1 Nj2; Nj3 Nj4 . ∗ Mj4 + Mj4 + Li4 + Li4  (31) When the inequalities in (26) ∼(33) are satisfied, the state- −RV U feedback gains can be obtained as follows: ∗ −S −W < 0, (32)   −1 ∗ ∗ −T Gj = KjX , j = 1, 2,...,r. (34)   8

Proof. Considering a candidate Lyapunov function presented r , r , therefore, there j=1(ξj + ξj) = 1 i=1(ξi − ξi − ξi) = 0 as follows: is Φ = 0. By using Ψ to present r r ξ (ξ + ξ )Θ P P i=1 j=1 i j j ij ⊤ ⊤ V (t)= x(t) Px(t),P = P > 0. (35) in Eqn. (38), the following expression is obtained: r r P P Based on Eqn. (21), the time-derivative of candidate Lyapunov T T Φ+Ψ= {ξiξj(Lj + Lj + Ni + Ni +Θij) function (35) is expressed as i=1 j=1 X X ⊤ ⊤ T T V˙ (x(t))=x ˙(t) Px(t)+ x(t) P x˙(t), +ξiξj(Mj + Mj + Ni + Ni +Θij) r r T T −ξiξj(Nj + Nj ) − ξ ξ (Lj + Lj ) ⊤ ¯ ¯ i j = ξi(ξj + ξj)[ε(t) He(AiX + B1iKj)ε(t) T T T −ξ ξ (Mj + M + Li + L ) − ξ ξ (Mj + M )}. i=1 j=1 i j j i i j j X X⊤ ⊤ (40) + w(t) B2iε(t)+ ε(t) B2iw(t)]. (36) By applying Schur complement to conditions in (26) and (27) respectively, the two following equalities can be obtained: Moreover, one has L + LT + N + N T +Θ ≤ V + V T, (41) ⊤ j j i i ij ij ij r r T T T ⊤ Mj + Mj + Ni + Ni +Θij ≤ Uij + Uij . (42) z(t) z(t)= ξi(ξ + ξ )(Ei + FiGj)x(t)  j j  i=1 j=1 With conditions in (41) ∼ (42) and (28) ∼ (31), the condition X X  r r  in (40) can be transformed to r r ξi(ξm + ξm)(El + FlGm)x(t) , ⊤ ⊤ "l=1 m=1 # Φ+Ψ ≤ [ξiξj(Vij + Vij )+ ξiξj(Uij + Uij ) Xr Xr i=1 j=1 (43) ⊤ ⊤ X X ⊤ ≤2 ξi(ξ + ξj)[ε(t) (EiX + FiNj) j −ξiξjRjj − ξiξjSjj − ξiξj(Wij + Wij ) − ξiξjTjj]. i=1 j=1 X X Then, the index J can be rewritten as (EiX + FiNj)ε(t)], ⊤ ⊤ (37) J = ϵ(t) Ψϵ(t)= ϵ(t) [Ψ+Φ]ϵ(t), ⊤ −1 where ε(t) = Px(t), X = P , Kj = GjX. Besides, r(t) −RV U r(t) (44) ¯ ¯ ≤ s(t) ∗ −S −W s(t) , defining that Qij = He(AiX+B1iKj) for simplicity. It can be       seen that the type-2 fuzzy control system (21) is asymptotically q(t) ∗ ∗ −T q(t) stable if there is V˙ (x(t)) < 0. Apart from the stability, the where       robustness of control system is necessary to be studied as ⊤ r(t)= ξ1ϵ(t) ξ2ϵ(t) ··· ξrϵ(t) , mentioned before, then, an index is defined as: ⊤ s(t)= ξ ϵ(t) ξ ϵ(t) ··· ξ ϵ(t) , (45) ˙ ⊤ 2 ⊤  1 2 r  J = V (x(t)) + z(t) z(t) − γ w(t) w(t), ⊤ r r q(t)= ξ1ϵ(t) ξ2ϵ(t) ··· ξrϵ(t) . ⊤ (38) ≤ ξ (ξ + ξ )ϵ(t) Θ ϵ(t), i j j ij It can be seen from the condition (32) that J < 0. With similar i=1 j=1 X X steps in [35], negative index J indicates that the closed-loop ⊤ ⊤ ⊤ system (21) is stable and the ∞ performance is guaranteed. where ϵ(t) = [ε(t) w(t) ] , Θij = [Qij + 2(EiX + H T T 2 Moreover, it should be emphasized that all the eigenvalues of FiNj) I(EiX + FiNj) B2i; B2i − γ I]. It can be seen matrices ¯ should be within the disk LMI region to from (38) that the condition Θij < 0 can ensure J < 0. Ai (q,r) Then, the robustness of control system can be guaranteed. In ensure desired transient performance. Based on Lemma 1, the order to reduce the conservativeness, some slack matrices are following inequality is obtained: introduced here. Hence, a new function is defined as: −rP qP + P (A¯ + B¯ G ) i 1i j < 0. (46) r r ∗ −rP ⊤ ⊤   Φ= (ξi − ξi − ξi){ξj(Lj + Lj )+ ξj(Mj + Mj ) Performing the congruence transformation to (46) with i=1 j=1 diagP −1,P −1 and X = P −1, condition in (33) can be X X ⊤ −ξj(Nj + Nj )}, obtained. r r ⊤ ⊤ It is observed from Theorem 1 that H∞ performance index = {ξiξ (Lj + L + Ni + N ) j j i γ indicates the attenuation level. Through minimizing the i=1 j=1 X X ⊤ ⊤ value of γ, better disturbance attenuation performance can be +ξiξj(Mj + Mj + Ni + Ni ) obtained. Then, the following corollary is introduced: ⊤ ⊤ −ξiξj(Nj + Nj ) − ξ ξ (Lj + Lj ) ∗ i j Corollary 1. The minimum H∞ performance index γ in ⊤ ⊤ ⊤ −ξiξj(Mj + Mj + Li + Li ) − ξiξj(Mj + Mj )}, Theorem 1 can be obtained by solving the following opti- (39) mization problem: ∗ where Lj,Mj and Nj are slack matrices with appropriate γ =min γ, r dimensions. Note that ξi has the property of i=1 ξi = s.t. (26) ∼ (33). P 9

V. SIMULATION RESULTS vehicle speed and have further effect on the vehicle control. To In this section, simulation and comparison results are pro- verify the proposed controller for lane keeping performance, vided. The simulation platform structure is depicted in Fig. 2. two modes are selected in the simulation experiment. One By computing a sets of linear matrix inequalities in Theorem 1 mode introduces a longitudinal vehicle speed depending on with the disk LMI region (100, 100), the feed-back gains can the road curvature and the function between them is adopted be obtained and the optimal H∞ index γ is 0.497. It should from [37]. The other mode relies on the vehicle speed which be noted that the vehicle parameters used in the research is measured in the practically urban driving condition, which is referred from [17]. Here, the simulation test is set on shows the influence of traffic flow. the condition where humane driver and vehicular automation In the first simulation test, the provided road curvature is cooperatively undertake the driving task, with an adaptive level shown in Fig. 3 (a). According to the relationship between of assistance for the human driver. In order to demonstrate the them referred in [37], when the vehicle is running on a flat superiority of proposed type-2 fuzzy driver-automation shared street with good adhesive condition , the calculated vehicle controller, the type-1 T-S fuzzy controller from the work in speed is depicted in Fig. 3 (b). It should be noted that there [17] is introduced as the comparison. In addition, system are two types of vehicle speed i.e. desired vehicle speed constraints referred from [36], which represent the ”normal and measured vehicle speed in the simulation structure. The driving” zone for safety. vehicle speed containing error is sent to the type-2 fuzzy controller module, which could be obtained from through V = V +∆V . ∆V means the error and it is G = xmeasured xdesired x x 1 assumed that there is ±10% deviation from the sensor, which −802.5, −1551.2, −6239.1, −4802.5, −16746.4, −137.0 , is a sampled gaussian noise generated by a signal builder. G 2 =  Moreover, the lateral wind force utilized in the simulation test −806.0, −1556.5, −6252.2, −4818.0, −16786.9, −137.3 , can be referred in Fig. 3 (c). G 3 =  −802.8, −1551.1, −6239.0, −4802.1, −16746.1, −136.9 , 0.03 30 Desired Measured G 4 =  0.02 25 −803.4, −1551.9, −6241.8, −4805.7, −16756.6, −137.2 . 0.01

(47) 0 20   -0.01 15 -0.02 Road curvature (1/m) Longitudinal speed (m/s) -0.03 10 0 10 20 30 0 10 20 30 Time(s) (a) Time(s) (b)

1000

800

600

400

200 Lateral wind force (N)

0 0 10 20 30 Time(s) (a)

Fig. 3. Given road curvature and the corresponding vehicle speed.

Figs. 4 and 5 depict results of the first simulation test. Note that responses from two different shared controllers are obtained at the same driving pattern. Fig. 4 (a) indicates the vehicle lateral performance and Fig. 4(b) presents the lane keeping performance. As depicted in Figs. 4 (a) and (b), when the lateral wind functions, the proposed type-2 fuzzy controller brings lesser lateral offset and heading error. It is because the Fig. 2. Simulation schematic diagram. uncertainty resulted from error is considered, otherwise, it will degrade the performance of controller. Fig. 4 (c) shows that the Note that the proposed type-2 fuzzy controller is designed proposed type-2 fuzzy controller provides a smaller overshoot to ensure better lane keeping performance. In practice, there in the response of vehicle yaw rate, which is closely related are many factors such as the road geometry and the traffic flow to the vehicle stability. In contrast, the conventional type-1 that have influence on the driving. Both them could affect the fuzzy controller generated a larger overshoot. In addition, large 10 overshoot in steering rate should also be avoided, which will the assistance torque is bigger than that of the driver torque cause unexpected anxiety to the human driver. Similarly, it can be seen from Fig. 4 (d) that the proposed type-2 fuzzy 1 2 controller provides more smaller overshoot than the other. The 0.95 0 compared results indicate that the proposed type-2 controller 0.9 -2 outdoes the type-1 fuzzy controller. In order to explain the simulation results in detail, Table 0.85 -4 Torque (Nm) Assistant level 0.8 -6 II presents comparison results about the root mean squares Assistant Driver of vehicle states responding to two kinds of controller, and 0.75 -8 0 10 20 30 0 10 20 30 Table III shows comparison results about the infinite-norm. Time(s) (a) Time(s) (b) It can be seen from Table II and Table III that the type-2 fuzzy controller leads to little steady-state error and smaller Fig. 5. (a) Assistance level according to driver’s activity; (b) The driver torque and the assistance torque. maximum offset than the type-1 fuzzy controller during the response precess. As a result, it can be concluded that the proposed type-2 controller is superior than the traditional type- TABLE II 1 fuzzy controller. ROOT MEAN SQUARE (RMS) OF VEHICLE STATES RESPONDINGTO GIVEN ROAD CURVATURE

Vehicle state Type − 1 controller Type − 2 controller ×10-3 0.04 5 Lateral offset 8.61e − 03 4.60e − 03 Type-2 4 Type-2 0.03 Type-1 Type-1 3 Heading error 1.06e − 03 1.05e − 03

0.02 2 Yaw rate 1.24e − 03 2.60e − 04 1 0.01 Steering rate 2.64e − 03 1.29e − 03 0

Lateral offset (m) 0 Heading error (rad) -1

-0.01 -2 0 10 20 30 0 10 20 30 TABLE III Time(s) (a) Time(s) (b) INFINITE NORMOF VEHICLE STATES RESPONDINGTO GIVEN ROAD CURVATURE 0.015 0.03 Type-2 Type-2 0.01 0.02 Type-1 Type-1 Vehicle state Type − 1 controller Type − 2 controller 0.005 0.01 Lateral offset 3.13e − 02 1.30e − 02 0 0 Heading error 4.22e − 03 3.09e − 03 -0.005 -0.01 Yaw rate 1.02e − 02 2.03e − 03 Yaw rate (rad/s)

-0.01 Steering rate (rad/s) -0.02 Steering rate 2.17e − 02 1.10e − 02 -0.015 -0.03 0 10 20 30 0 10 20 30 Time(s) (c) Time(s) (d) In the second simulation test, the measured vehicle speed in urban driving condition is applied, which reflects the influence Fig. 4. Response of vehicle states in accordance with given road curvature-I. of urban traffic flow. The experiment data was employed by [38] to demonstrate the effectiveness of an adaptive cruise Fig. 5 shows information of the assistance level and the controller, in which the data were measured from an electric torque distribution, which indicates that when the lateral wind vehicle driven on a urban road (denoted as ‘Route A’ in the occurs, the human driver and the automation have same literature) in Cambridge, UK. Figs. 6 and 8 depict results of directional activity and the maximal difference between them the second simulation test, thereinto, Fig. 7 (a) depicts the is about 3.8N/m. Note that the driver torque can be predicted assistance level µ(θd(t)) according to the driver’s behavior through Eqn. (8). Looking at the information in more detail, and Fig. 7 (b) shows the corresponding torque distribution. Fig. 5 (a) depicts the assistance level in accordance with Fig. 7 (a) shows that the assistance level decreases when the current driver’s behavior. It shows that when there is no lateral wind acts, and the procedure is similar to that of the external disturbance, the automation can control the vehicle first simulation. and release human driver from the driving task. When the Fig. 8 (a) shows responses of both controllers, which are lateral wind force acts and the human driver detects it, he/she related to the lateral performance of controlled vehicle. It will act to control the steering wheel. As depicted in Fig. 5 (a), can be seen from Fig. 8 (a) that the maximal lateral offset the corresponding assistance level decreases and partial control of proposed type-2 fuzzy controller is smaller than that of authority is transferred to the human driver at the moment. the traditional type-1 controller. This shows the proposed Note that if there is a biggish external disturbance, support controller is more precise, as the new type-2 fuzzy technology of the automation may remain a high level, it is because the is applied to address the uncertainty from imprecise speed and driver could not control the vehicle completely. Therefore, the changing driver state. Fig. 8 (b) shows the lane keeping per- minimal support is still in a high level in the presence of lateral formance of both shared controllers with adaptive assistance. wind, which is about 0.76. Similarly, Fig. 5 (b) also shows that Since a smaller heading error indicates better lane keeping 11 performance, Fig. 8 (b) indicates the proposed type-2 fuzzy ×10-3 0.04 3 controller could ensure better lane keeping performance, of Type-2 Type-2 which the maximal heading error is smaller. 0.03 Type-1 2 Type-1 0.02 In addition, comparisons on responses of the yaw rate and 1 0.01 the steering rate are given in Figs. 8 (c) and (d), respectively. 0 0 It is well known that the yaw rate is an important index to -1 -0.01 Lateral offset (m) evaluate the vehicle handling stability and the value is desired -0.02 Heading error (rad) -2 to be tiny for the vehicle safety. Furthermore, steering rate -0.03 -3 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200 can also affect the driver’s driving experience, because of the Time(s) (a) Time(s) (b) feedback of road sense with his/her hands on the steering ×10-3 ×10-3 3 3 wheel. Both of them are important indicators to evaluate the Type-2 Type-2 proposed control method. Fortunately, based on comparisons 2 Type-1 2 Type-1 on responses of both controllers, it can be concluded that 1 1 the proposed type-2 fuzzy controller is much better than the 0 0 traditional type-1 fuzzy controller, as the former provides -1 -1 Yaw rate (rad/s)

smaller yaw rate and steering rate. Similarly, compared results -2 Steering rate (rad/s) -2 from Table IV and Table V also demonstrate the superiority of -3 -3 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200 proposed type-2 fuzzy controller not only in the steady-state Time(s) (c) Time(s) (d) error of control system but also in the system overshoot.

Fig. 8. Response of vehicle states in accordance with measured vehicle speed- 1000 16 I. 800 14

600 12 400 10 TABLE V 200 8 INFINITE NORMOF VEHICLE STATES RESPONDINGTO MEASURED 0 6 VEHICLE SPEED -200 4 -400 Lateral wind force (N) -600 Longitudinal speed (m/s) 2 Vehicle state Type − 1 controller Type − 2 controller -800 0 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200 Lateral offset 3.35e − 02 1.34e − 02 Time(s) (a) Time(s) (b) Heading error 2.19e − 03 2.22e − 03 Fig. 6. (a) The external disturbance; (b) The measured vehicle speed. Yaw rate 1.12e − 02 2.72e − 03 Steering rate 2.83e − 03 2.38e − 03

1.1 3

1.05 2

1 parameters in the driver-automation shared driving system 1 0 (11) are also challenges. Differing from the vehicle speed vx, 0.95 -1 Kd1 and Kd2 are related to the driver behavior. In particular, 0.9 -2 Torque (Nm)

Assistant level Kd1 represents the driver’s proportional action on near angle Assistant 0.85 -3 Driver and Kd2 describes the driver’s proportional action on far 0.8 -4 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200 angle. They indicate the driver’s contemporary behavior and Time(s) (a) Time(s) (b) anticipatory behavior, respectively. Generally, different drivers Fig. 7. (a) Assistance level according to driver’s activity; (b) The driver torque have differen parameters Kd1 and Kd2. In order to further and the assistance torque. demonstrate the superiority of type-2 fuzzy technology in dealing with high order uncertainties, an extensive simulation is tested, in which uncertain parameters Kd1 and Kd2 resulted TABLE IV from the diversity of drivers are considered. ROOT MEAN SQUARE (RMS) OF VEHICLE STATES RESPONDINGTO As mentioned above, the vehicle longitudinal velocity vx MEASURED VEHICLE SPEED and the weighting function µ(θd(t)) in the driver-automation Vehicle state Type − 1 controller Type − 2 controller shared driving system (11) are time-varying. In addition, Kd1 and K are uncertain because of the driver diversity with Lateral offset 6.20e − 03 2.23e − 03 d2 the property of K ∈ [K , K ],i = 1, 2. There 4 72 − 04 2 91 − 04 di dimin dimax Heading error . e . e 1 Kd1 Kd2 are five premise variables vx, , , and µ(θd(t)). vx vx vx Yaw rate 1.20e − 03 2.83e − 04 Similar to the previous work, the variable υ is introduced to 1 03 − 04 9 92 − 05 1 Kd1 Kd2 Steering rate . e , e decouple variables vx, , and . Then, the premise vx vx vx variables µ(θd(t)), Kd1υ and Kd2υ can be obtained. To reduce In this work, the type-2 fuzzy technology is introduced to the number of premise variables ulteriorly, a new variable address imprecise vehicle longitudinal velocity, which can lead K = Kd1/Kd2 is introduced, which has the property of to uncertainty in the membership function. In fact, apart form K ∈ [Kmin, Kmax]. It should be noted that there are K > 1 the imprecise measurement of vehicle speed, other uncertain and Kmin = Kd1min/Kd2max, Kmax = Kd1max/Kd2min. 12

Consequently, only two premise variable Kυ and µ(θd(t)) brings more modest effect on the human driver than the type- are utilized. Then, the membership function could be defined 1 fuzzy controller. In addition, results in Fig. 9 and Fig. as f1 = Kυ and f2 = µ(θd(t)). Following the same steps 10 show that the type-2 fuzzy controller has better transient mentioned above, the extensive simulation results are depicted performance, as the type-2 fuzzy controller could converge in Fig. 9 and Fig. 10. to the steady-state quickly while the type-1 fuzzy controller has brief oscillation in the response process. It is because the D-stability method is introduced in this work. As a result, it ×10-3 0.04 5 can be concluded that the type-2 fuzzy controller is superior Type-2 4 Type-2 0.03 Type-1 Type-1 to the type-1 fuzzy controller not only on the lane keeping 3 performance and vehicle stability but also on the transient 0.02 2 performance. 1 0.01

0

Lateral offset (m) 0 Heading error (rad) -1 VI.CONCLUSIONS -0.01 -2 0 10 20 30 0 10 20 30 This paper investigates a driver-automation shared control Time(s) (a) Time(s) (b) system in the presence of imprecise vehicle speed. In fact, the 0.015 0.03 vehicle longitudinal velocity is not only time-varying but also Type-2 Type-2 0.01 0.02 Type-1 Type-1 inaccurate. Meanwhile, the driver state DS in the weighting 0.005 0.01 parameter µ(θd(t)) is not constant but time-variant. Both of

0 0 them can cause uncertain membership function and if it is

-0.005 -0.01 ignored when designing controller, the controller performance

Yaw rate (rad/s) will be attenuated. In this case, the conventional type-1 T-S

-0.01 Steering rate (rad/s) -0.02

-0.015 -0.03 fuzzy technology is invalid to analyze the driver-automation 0 10 20 30 0 10 20 30 shared control system in the presence of uncertain membership Time(s) (c) Time(s) (d) function. Therefore, a type-2 fuzzy technology is developed to address it, in which the upper and lower bounds of membership Fig. 9. Response of vehicle states in accordance with given road curvature-II. function are considered. Moreover, both stability and H∞ performance are studied for the type-2 fuzzy driver-automation shared control system. The final simulation results also show

×10-3 0.04 3 advantages of the proposed type-2 fuzzy driver-automation

0.03 Type-2 2 Type-2 shared controller, compared with the existing type-1 fuzzy Type-1 Type-1 0.02 1 controller. In the future research, more complex road condition 0.01 0 will be considered to pursue better lane keeping performance. 0 -1 -0.01 In addition, a practical experiment will be done to check and Lateral offset (m) -0.02 Heading error (rad) -2 update the proposed algorithm. -0.03 -3 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200 Time(s) (a) Time(s) (b)

×10-3 ×10-3 VII.DECLARATIONS 3 3

2 Type-2 2 Type-2 Type-1 Type-1 Acknowledgements 1 1 Not applicable. 0 0

-1 -1

Yaw rate (rad/s) Author’s contributions -2 Steering rate (rad/s) -2 YL wtote the paper; QX reviesed the work; HG provided some -3 -3 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200 ideas; HZ supervised the research. Time(s) (c) Time(s) (d) Author’s information Hui Zhang is a professor with the department of Automotive Fig. 10. Response of vehicle states in accordance with measured vehicle speed curvature-II. Engineering, Beihang University, China. Funding Simulation results in Fig. 9 and Fig. 10 show that the Not applicable. type-2 fuzzy controller can promise smaller lateral offset and Availability of data and materials heading error, which means better lane keeping performance. Not applicable. Apart from it, as depicted in Fig. 9 (c) and Fig. 10 (c), the Competing interests type-2 fuzzy controller produces lesser yaw rate compared to The authors declare no competing financial interests. the type-1 fuzzy controller, indicating that it could promise Author Details better vehicle stability. Besides, a larger steering rate will Yue Liu, and Hui Zhang are with the School of Trans- influence human drivers when they put their hands on the portation Science and Engineering, Beihang University, steering wheel during the driving process. It can been seen Beijing 100091, China. e-mail: ([email protected]; from Fig. 9 (d) and Fig. 10 (d) that the type-2 fuzzy controller [email protected]) 13

Qing Xu is with Department of automotive engineering, Ts- [20] H. K. Lam and L. D. Seneviratne, “Stability analysis of interval type- inghua University, Beijing, China 2 fuzzy-model-based control systems,” IEEE Transactions on Systems Man & Cybernetics Part B Cybernetics, vol. 38, no. 3, pp. 617–28, Hong yan Guo is with Colleague of automotive engineering, 2008. Jilin University, Jilin, China [21] H. K. Lam, H. Li, C. Deters, E. L. Secco, H. A. Wurdemann, and Received: XXXX Revised: XXXXX Accepted: XXXXX K. Althoefer, “Control design for interval type-2 fuzzy systems under imperfect premise matching,” IEEE Transactions on Industrial Electron- ics, vol. 61, no. 2, pp. 956–968, 2014. [22] E. Kayacan, , and M. A. Khanesar, “Identification of nonlinear dynamic REFERENCES systems using type-2 fuzzy neural networks - A novel learning algorithm and a comparative study,” IEEE Transactions on Industrial Electronics, [1] J. C. F. D. Winter, R. Happee, M. H. Martens, and N. A. Stanton, vol. 62, no. 3, pp. 1716–1724, 2015. “Effects of and highly automated driving on [23] D. Sun, Q. Liao, and H. Ren, “Type-2 fuzzy modeling and control workload and situation awareness: A review of the empirical evidence,” for bilateral system with dynamic uncertainties and time- Transportation Research Part F Psychology & Behaviour, vol. 27, pp. varying delays,” IEEE Transactions on Industrial Electronics, vol. 65, 196–217, 2014. no. 1, pp. 447–459, 2018. [2] N. Merat, A. H. Jamson, F. C. Lai, and O. Carsten, “Highly automated [24] N. N. Karnik, J. M. Mendel, and Q. Liang, “Type-2 fuzzy logic systems,” driving, secondary task performance, and driver state.” Human Factors, IEEE Transactions on Fuzzy Systems, vol. 7, no. 6, pp. 643–658, 1999. vol. 54, no. 5, pp. 762–771, 2012. [25] O. Castillo and P. Melin, “A review on interval type-2 fuzzy logic [3] J. Wu, S. Cheng, B. Liu, and C. Liu, “A human-machine-cooperative- applications in intelligent control,” Information Sciences, vol. 279, pp. driving controller based on AFS and DYC for vehicle dynamic stability,” 615–631, 2014. Energies, vol. 10, no. 11, p. 1737, 2017. [26] H. A. Hagras, “A hierarchical type-2 fuzzy logic control architecture [4] Y. P. Kuo and T. S. Li, “GA-based fuzzy PI/PD controller for automotive for autonomous mobile robots,” IEEE Transactions on Fuzzy Systems, active suspension system,” IEEE Trans Industrial Electronic, vol. 46, vol. 12, no. 4, pp. 524–539, 2004. no. 6, pp. 1051–1056, 1999. [27] L. Zhang, H. K. Lam, Y. Sun, and H. Liang, “Fault detection for fuzzy [5] D. A. Abbink, T. Carlson, M. Mulder, J. C. de Winter, F. Aminravan, semi-Markov jump systems based on interval type-2 fuzzy approach,” T. L. Gibo, and E. R. Boer, “A topology of shared control systems IEEE Transactions on Fuzzy Systems, vol. 28, no. 10, pp. 2375–2388, finding common ground in diversity,” IEEE Transactions on Human- 2020. Machine Systems, vol. 48, no. 5, pp. 509–525, 2018. [28] T. Kumbasar and H. Hagras, “A self-tuning zslices-based general type- [6] M. Johns, B. Mok, D. Sirkin, N. Gowda, C. Smith, W. Talamonti, and 2 fuzzy PI controller,” IEEE Transactions on Fuzzy Systems, vol. 23, W. Ju, “Exploring shared control in automated driving,” in ACM/IEEE no. 4, pp. 991–1013, 2015. International Conference on Human-Robot Interaction, 2016, pp. 91–98. [29] P. K. Muhuri, Z. Ashraf, and Q. M. D. Lohani, “Multiobjective reliability [7] F. Flemisch, F. Nashashibi, N. Rauch, A. Schieben, S. Glaser, G. Temme, redundancy allocation problem with interval type-2 fuzzy uncertainty,” P. Resende, B. Vanholme, C. Loper, and G. Thomaidis, “Towards highly IEEE Transactions on Fuzzy Systems, vol. 26, no. 3, pp. 1339–1355, automated driving: Intermediate report on the HAVEit-Joint system,” in 2018. Transport Research Arena, Brussels, 2010. [30] Y. Liu and H. Zhang, “Robust driver-automation shared control for a lane keeping system using interval type 2 fuzzy method,” in 2019 IEEE [8] M. R. Endsley and D. B. Kaber, “Level of automation effects on 28th International Symposium on Industrial Electronics (ISIE). IEEE, performance, situation awareness and workload in a dynamic control 2019, pp. 1944–1949. task,” Ergonomics, vol. 42, no. 3, pp. 462–492, 1999. [31] R. Rajamani, Vehicle Dynamics and Control, 2006. [9] R. Li, S. Li, H. Gao, K. Li, B. Cheng, and D. Li, “Effects of human [32] J. Ackermann, J. Guldner, W. Sienel, R. Steinhauser, and V. I. Utkin, adaptation and trust on shared control for driver-automation cooperative “Linear and nonlinear controller design for robust automatic steering,” driving,” Tech. Rep. 2017-01-1987, 2017. IEEE Transactions on Control Systems Technology, vol. 3, no. 1, pp. [10] L. Saleh, P. Chevrel, F. Claveau, J.-F. Lafay, and F. Mars, “Shared 132–143, March 1995. steering control between a driver and an automation: Stability in the [33] D. A. Abbink, M. Mulder, and E. R. Boer, “Haptic shared control: presence of driver behavior uncertainty,” IEEE Transactions on Intelli- smoothly shifting control authority,” Cognition, Technology & Work, gent Transportation Systems, vol. 14, no. 2, pp. 974–983, 2013. vol. 14, no. 1, pp. 19–28, 2012. [11] M. Huang, W. Gao, Y. Wang, and Z. Jiang, “Data-driven shared steering [34] H. Zhang, X. Zhang, and J. Wang, “Robust gain-scheduling energy-to- control of semi-autonomous vehicles,” IEEE Transactions on Human- peak control of vehicle lateral dynamics stabilisation,” Vehicle System Machine Systems, vol. 49, no. 4, pp. 350–361, 2019. Dynamics, vol. 52, no. 3, pp. 309–340, 2014. [12] M. Li, H. Cao, G. Li, S. Zhao, X. Song, Y. Chen, and D. Cao, “A [35] J. H. Kim and B. P. Hong, “H∞ state feedback control for generalized two-layer potential-field-driven model predictive shared control towards continuous/discrete time-delay system,” Automatica, vol. 35, no. 8, pp. driver-automation cooperation,” IEEE Transactions on Intelligent Trans- 1443–1451, 1999. portation Systems, pp. 1–17, 2020. [36] N. M. Enache, M. Netto, S. Mammar, and B. Lusetti, “Driver steering [13] S. M. Erlien, S. Fujita, and J. C. Gerdes, “Shared steering control using assistance for lane departure avoidance,” Control Engineering Practice, safe envelopes for obstacle avoidance and vehicle stability,” IEEE Trans. vol. 17, no. 6, pp. 642–651, 2009. Intelligent Transportation Systems, vol. 17, no. 2, pp. 441–451, 2016. [37] M. Park, S. Lee, and W. Han, “Development of steering control system [14] X. Ji, K. Yang, X. Na, C. Lv, and Y. Liu, “Shared steering torque control for autonomous vehicle using geometry-based path tracking algorithm,” for lane change assistance: A stochastic game-theoretic approach,” IEEE Eelctronics Telecommunications Research Inst, vol. 37, no. 3, pp. 617– Transactions on Industrial Electronics, vol. 66, no. 4, pp. 3093–3105, 625, 2015. 2019. [38] A. K. Madhusudhanan and X. Na, “Effect of a traffic speed based cruise [15] R. Li, Y. Li, S. E. Li, C. Zhang, E. Burdet, and B. Cheng, “Indirect control on an electric vehicle’s performance and an energy consumption shared control for cooperative driving between driver and automation in model of an electric vehicle,” IEEE/CAA Journal of Automatica Sinica, steer-by-wire vehicles,” IEEE Transactions on Intelligent Transportation vol. 7, no. 2, pp. 386–394, 2020. Systems, pp. 1–11, 2020. [16] Y. Lu, L. Bi, and H. Li, “Model predictive-based shared control for brain- controlled driving,” IEEE Transactions on Intelligent Transportation Systems, vol. 21, no. 2, pp. 630–640, 2020. [17] A. T. Nguyen, C. Sentouh, and J. C. Popieul, “Driver-automation cooperative approach for shared steering control under multiple system constraints: Design and experiments,” IEEE Transactions on Industrial Electronics, vol. 64, no. 5, pp. 3819–3830, 2017. [18] K. Tanaka and H. O. Wang, Fuzzy control systems design and analysis: a linear matrix inequality approach. John Wiley & Sons, 2004. [19] L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning − I,” Information Sciences, vol. 8, no. 3, pp. 199–249, 1975. 14

Yue Liu received the B.Sc. degree in automotive Hui Zhang (M’15-SM’17) received the B.Sc. engineering from Henan University of Science degree in mechanical design manufacturing and and Technology, Luo Yang, China, in 2016. She automation from the Harbin Institute of Technology is currently working toward the Ph.D. degree in at Weihai,Weihai, China, in 2006, the M.Sc. degree vehicle dynamics at the School of Transportation in automotive engineering from Jilin University, Science and Engineering, Beihang University, Changchun, China, in 2008, and the Ph.D. degree Beijing, China. in mechanical engineering from the University of Her research interests include vehicle Victoria, Victoria, BC, Canada, in 2012. dynamics and advanced control, automation- He is with the School of Transportation Science driver interaction, and multivehicle cooperative and Engineering, Beihang University, Beijing, control problems. China. His research interests include diesel engine aftertreatment systems, vehicle dynamics and control, mechatronics, robust control and filtering, networked control systems, and multi-agent systems. He is an author / co-author of more than 80 peer-reviewed papers on journals and conference proceedings.

Qing Xu received his B.S. and M.S. degrees in automotive engineering from Beihang University, Beijing, China, in 2006 and 2008 respectively. During his Ph.D. research, he worked as a visiting scholar in department of mechanical science and engineering, UIUC, and got his Ph.D. degree in automotive engineering from Beihang University in 2014. From 2014 to 2016, he had his postdoctoral research in Tsinghua University. And now he is working as an assistant research professor in department of automotive engineering, Tsinghua University. Qing Xu’s main research interests include decision and control of intelligent vehicles.

Hongyan Guo (M’17) received the PH.D. degree in control theory and control engineering from the Jilin University, Changchun, China, in 2010. She joined the Jilin University, in 2011, where from 2014, she is an Associate Professor with the Department of Control Science and Engineering. In 2017, she is a Visiting Scholar with Cranfield University, Cranfield, U.K. Her current research interests include path tracking and stability control of autonomous vehicles and vehicle states estimation.