An optimal model-based trajectory following

VEHICLE SYSTEM DYNAMICS, 2017 http://dx.doi.org/10.1080/00423114.2017.1305114

An optimal model-based trajectory following architecture synthesising the lateral adaptive preview strategy and longitudinal velocity planning for highly automated vehicle Haotian Caoa,b , Xiaolin Songa , Song Zhaoc , Shan Baob and Zhi Huanga a State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, People’s Republic of China; b Human Factors Group, University of Michigan Transportation Research Institute (UMTRI), Ann Arbor, MI, USA; c College of Engineering Integrative Systems Design, University of Michigan, Ann Arbor, MI, USA

ABSTRACT

ARTICLE HISTORY

Automated driving has received a broad of attentions from the academia and industry, since it is effective to greatly reduce the severity of potential traffic accidents and achieve the ultimate automobile safety and comfort. This paper presents an optimal modelbased trajectory following architecture for highly automated vehicle in its driving tasks such as automated guidance or lane keeping, which includes a velocity-planning module, a steering controller and a velocity-tracking controller. The velocity-planning module considering the optimal time-consuming and passenger comforts simultaneously could generate a smooth velocity profile. The robust sliding mode control (SMC) steering controller with adaptive preview time strategy could not only track the target path well, but also avoid a big lateral acceleration occurred in its path-tracking progress due to a fuzzy-adaptive preview time mechanism introduced. In addition, an SMC controller with input–output linearisation method for velocity tracking is built and validated. Simulation results show this trajectory following architecture are effective and feasible for high automated driving vehicle, comparing with the Driver-in-the-Loop simulations performed by an experienced driver and novice driver, respectively. The simulation results demonstrate that the present trajectory following architecture could plan a satisfying longitudinal speed profile, track the target path well and safely when dealing with different road geometry structure, it ensures a good time efficiency and driving comfort simultaneously.

Received 10 April 2016 Revised 9 January 2017 Accepted 2 March 2017 KEYWORDS

Multi-object velocity planning; robust driver path following model; adaptive preview time; velocity tracking; sliding mode control; Driver-in-the-Loop

1. Introduction The number of the personal vehicles in China has been rapidly increasing since from last decade, according to the statistics reported by China National Statistics Bureau, the number has reached to over 240 million by the end of 2014. However, the traffic causality has become a serious public issue in China, over 58,500 people died and more than 211,000 injured due to the traffic accidents in 2014, among them, over 95% death and 93% CONTACT Xiaolin Song

[email protected]

© 2017 Informa UK Limited, trading as Taylor & Francis Group

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injured are caused by the motor cars among those traffic accidents [1]. Moreover, it also has a huge negative influence on economy costs, for example, the direct economy loss in China 2014 caused by motor vehicle crash is $160 million [1]. While in USA, statistics from Fatality Analysis Reporting System of the USA Transportation’s National Highway Traffic Safety Administration (NHTSA) also show that, 32,744 people died in motor vehicle traffic crashes in 2014 [2]. In addition, the economic cost of motor vehicle crashes that occurred in 2010 totalled $242 billion, which is equivalent to 1.6% of the U.S. Gross Domestic Product [3]. Though the number of traffic fatality is still large, as well as the economy costs, however, the overall statics for these years show that the trends are positive, since the total traffic accidents have dropped approximate 60% and the caused deaths have decreased about 48% in last decade in China. The fatal crashes have also nearly dropped around 25% from 2005 to 2014 in the USA [1,2]. Many factors could contribute to traffic safety improvements, such as improvements of drivers’ safety consciousness and habits (seat belting, avoid fatigue driving etc.), legislation establishment by the government on traffic safety, or a safer and friendly design from the vehicle manufacture, but another reason which makes cars become safer is the development of the ADASs (Advanced Driver Assistance Systems) such as EBS, LKA, and ESC, etc. The technology in driver assistance systems potentially offers additional significant reductions in the accident fatality numbers, which amount to 1.2 million people worldwide [4]. Moreover, according to the research by Kuehn et al. [5], which is based on the German Insurers Accident Research Database (IARD), shows that a car fleet equipped with Collision Mitigation Braking System (CMBS) and Lateral Guidance could avoid up to 25.1% of all car accidents in the data sample. And reports in 2014 by Insurers Accident Research [6] in German, the number has increased to 43.4% of all car accidents in the database then become preventable with ADASs equipped on cars. What’s more, reports from PricewaterhouseCoopers (PwC) also estimate that if 51% of passenger cars on the highways have adaptive cruise control and rear-end collision warning switched on, the number of critical situations is reduced by 32–82% [7]. PwC further predicts that the reduction in losses would include bodily injury (−15%), collision (−6%), property damage and protection (−14%), and personal injury protection (−10%), which is based on the annual reports on potential impacts of ADAS and autonomous car technologies on the insurance industry in 2015 [8]. Therefore, the ADASs that are designed to support the tactical and operational driving tasks are expected to improve the vehicle safety greatly. However, hazardous situations arise repeatedly in road traffic, caused by adverse conditions or human error. It is expected that future vehicles will take over more and more controls and support for better safety and mobility, from achieving partially automatic driving to full autonomous driving, as automated driving makes traffic not only safer, but also more efficient and comfortable. Thus, automated driving vehicle rise great interests from the government, industry and academia, it has been considered as the most promising approach to achieve the ultimate automobile safety and comfort. According to NHTSA, five levels of vehicle automations are defined based on the extend of drivers’ involvement, ranging from vehicles that do not have any of control systems automated (level 0) through fully automated vehicles (level 4) [9]. Obviously, the development of ADASs will also benefit from the major trend toward highly automated driving; systems found in vehicles in future will blur the boundaries among different ADASs. The ultimate ADASs of the future should be capable of automated driving in all conceivable situations with a safety level significantly

VEHICLE SYSTEM DYNAMICS

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superior to that of a human driver [10] to compensate human errors, which accounts for 90% of all accidents [4]. Therefore, in order to develop such automated vehicles, the driving tasks should be broken up into basic functional components that could be technically implemented at a certified level of maturity. Trajectory following is such a basic functional component, which is vital for automated driving to ensure that the vehicle could track a target-planned path safely. The closed-loop of the environment–driver–vehicle model is commonly considered as a local optimal problem [11]. In 1980s in last century, MacAdam presents an optimal single-point preview steering controller to minimise a cost function corresponding to lateral error [12,13]. Peng et al. present an adaptive lateral preview driver model for nonlinear vehicle by minimising a cost function including path following and yaw displacement errors [14]. Later, Sharp et al. present the concept of the multi-point preview for steering control in [15], which shows that the computation of the steering angle could be achieved by the linear combination of the preview points’ lateral offset and the vehicle’s states. Further, Sharp and Valtetsiotis[16] shows that an Linear Quadratic Regulator (LQR) state-feedback steering controller could be applied for multi-point preview steering control. Moreover, a mathematical model driver steering control including human neuromuscular dynamics using LQR method is further studied in [17]. What’s more, more control methods are applied for path tracking. For example, an LQ-Preview control law for lateral steering of a passenger vehicle is presented in [18]. Guo et al. [19] presents an adaptive fuzzy-sliding mode control (SMC) strategy used for lateral control of vision-based automated vehicles, to overcome features of nonlinearities, parametric uncertainties and external disturbances encountered in lateral control. Odhams and Cole [20], Cole et al. [21], and Timings and Cole [22] demonstrate that model predictive control (MPC) which is based on a 2DOF linear bicycle vehicle model for steering control could also be effective and feasible, due to the main advantage of MPC is the fact that it allows the current timeslot to be optimised while keeping future timeslots in account. More path-tracking problem realised by linear MPC or nonlinear MPC approach could be found in [23–26]. Though there were quite a lot of methods for vehicle path tracking, there still exist some aspects for improving. Generally, there is no velocity planning for path tracking, which means, most of them are Linear Time Invariant model, which is based on 2DOF bicycle model that demands the longitudinal speed should remain constant. Nevertheless, the vehicle velocity varies from traffic and road scenes, for example, the vehicle will decrease its speed when entering a circular lane from a straight lane. It is vital for the vehicle moves with a proper speed. Thus, besides a velocity-planning function, a longitudinal motion control should be also included. Attia et al. [27] presents paper an automatic control design for automotive driving considering both the longitudinal control (a stepping-back approach for velocity planning) and Linear Time Variant (LTV) MPC lateral control for vehicle automated guidance. However, solving an LTV model would lead to the algorithm’s time-consuming issue, especially to approach such as (N) MPC, and the velocity profile generation is based on the maximum allowed speed of the road scenario, which means, driving comfort is not considered in its velocity planning. Another problem for MPC or SMC approach in path tracking is that it will cause a very high lateral acceleration when the curvature change of the target path becomes large (e.g. a server double lane change) and brings a potential threat for vehicle stability issue, since these methods are designed for tracking the target path precisely. Therefore, we present an optimal model-based trajectory

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following architecture which synthetises the lateral adaptive preview strategy and longitudinal velocity planning and tracking for highly automated vehicle, especially when driving on a freeway or there are little interactions between other vehicles, such as automated guidance or lane keeping on highway. Contributions and their organisations in this paper are listed as follows: (1) at first, an eight-degree-of-freedom nonlinear vehicle model including a completed rear-driven powertrain is established and validated by CarSim software in Section 2. (2) A velocityplanning module with multi-object optimisation method is proposed in Section 3, which generates a smooth velocity profile that takes both the driving comfort and time-optimal purpose into consideration. (3) A robust SMC-based steering controller with an adaptive fuzzy preview-time is presented for path tracking in Section 4, which the presented controller could handle issues of the unmatched vehicle model parameters to ensure a good quality in path tracking. Furthermore, the fuzzy-adaptive preview-time strategy could lead to a smooth transition to avoiding a big lateral acceleration arising when encountering sudden change in path. (4) In order to track the planned velocity well, an SMC-based velocity-tracking controller with input–output linearisation approach is proposed in Section 5. (5) Based on the above subsystems, a trajectory following architecture synthetising the lateral adaptive preview strategy and longitudinal velocity planning is proposed eventually in Section 6. We also establish a driving performance evaluation mechanism. Two easy daily encountered scenarios, compared with Driver-in-the-Loop (DiL) simulations by an experienced driver and novice driver, validate the whole system. They are evaluated under the driving performance evaluation system to show the advantages of the automatic driving utilised by the presented trajectory following system. (6) Finally, conclusion and future work are presented in Section 7.

2. Vehicle model and its validation An eight degree-of-freedom nonlinear vehicle model, which includes a rear driven powertrain is introduced and validated in this chapter. 2.1. Vehicle chassis model An eight degree-of-freedom single-track rigid body model which includes the longitudinal position x, lateral position y, yaw ψ, roll angle φ and four wheels’ rolling would be built in this study. As shown in Figure 1, the vehicle moving with a velocity V has a lumped mass m, sprung mass ms and moment of inertia Iz , Ix corresponding to vehicle Z axis and X axis, respectively. The steering angle is denoted by δsw , through a steering gear ratio G, which results in a front wheel angle δ. According to the Newton laws, the following set of nonlinear differential equations are used to describe the motion of the vehicle. ˙ − ms hφ ψ˙ = m(˙u − v ψ)

4

Fxk − Faerox

k=1

˙ + ms hφ¨ = m(v˙ + uψ)

4 k=1

Fyk

(1)


5

Figure 1. Model illustration for 8DOF vehicle.

˙ = Mx Ix φ¨ + ms h(v˙ + uψ) Iz ψ¨ = Mz X˙ = u cos (ψ) − v sin (ψ) Y˙ = u sin (ψ) + v cos (ψ), where X and Y are the global position of the vehicle, the sub-index k = 1, 2, 3, 4 corresponds to each wheel, as shown in Figure 1. Faerox is the wind force respecting to vehicle longitudinal axis. Fxk , Fyk represents the longitudinal force and lateral force of each wheel, respectively, and donated by, 4

Fxk = Fx1 + Fx2 + Fx3 + Fx4 ,

(2)

Fyk = Fy1 + Fy2 + Fy3 + Fy4 ,

(3)

k=1 4 k=1

⎡ ⎤ ⎡ cos δ Fxi ⎢Fyi ⎥ ⎢ sin δ ⎢ ⎥=⎢ ⎣Fxj ⎦ ⎣ 0 0 Fyj

−sin δ cos δ 0 0

0 0 1 0

⎤⎡ ⎤ Fxwi 0 ⎢ ⎥ 0⎥ ⎥ ⎢Fywi ⎥ , ⎦ ⎣ 0 Fxwj ⎦ 1 Fywj

i = 1, 2, j = 3, 4,

(4)

where Fxw Fyw are the X-component of the tyre force and Y-component of the tyre force, respectively. Mxk , Mzk are the moment respecting to X axis and Z axis, which are calculated by, 1 Mz = (−Wf (Fx1 + Fx3 ) + Wr (Fx2 + Fx4 )) + a(Fy1 + Fy2 ) − b(Fy3 + Fy4 ), 2

(5)

Mx = −2(Dφ f + Dφ r )φ˙ + (ms gh − 2(Kφ f + Kφ r ))φ,

(6)

where a, b represent the distance from the CoG to the front/rear axis, Wf , Wr denote the front/rear wheelbase, Kφ f , Kφ r denote the front/rear chassis stiffness and Dφ f , Dφ r denote the front/rear chassis rolling stiffness.

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When a higher lateral acceleration rises, the vehicle demands more lateral force from the tyres, as a result, the system would become highly nonlinear. The main cause of this nonlinearity comes from the limited available lateral tyre force. Generally, the longitudinal force or lateral force of the tyre for the vehicle is obtained by a 2D table data via linear interpolation method, which is corresponding to different vertical tyre loads and tyre sideslip angle or tyre slip ratio. One common way to model the tyre force versus slip ratio or slip angle is to use the Magic Tire Formula [28]. That means the tyre force could be represented by a function of slip ratio κ, sideslip angle α, vertical loads Fz , and road adhesion constant μ, namely, Fxw = fx (κ, α, Fz , μ), Fyw = fy (κ, α, Fz , μ).

(7)

A detailed model of Magic Formula under combined slip applied in this study is interpreted in Appendix 1. The wheel speed is calculated with the driving torque distributed on the wheel, along with the tyre longitudinal forces and the brake torques if exist, the wheel dynamic equations for a rear-drive vehicle are, 1 (−FxFL reff − Tbrk − Mfr1 ) JwL 1 = (−FxFR reff − Tbrk − Mfr2 ) JwL 1 = (TwRL − FxRL reff − Tbrk − Mfr3 ) JwR 1 = (TwRR − FxRR reff − Tbrk − Mfr4 ), JwR

ω˙ wFL = ω˙ wFR ω˙ wRL ω˙ wRR

(8)

where Jwi represents the equivalent inertia for each tyre, reff is the effective radius of the tyre, Tbrk is the braking torque acting on the tyre, and Mfr represents the rolling resistance torque. Finally, the vehicle motion Equations (1), (8) could be expressed as a nonlinear state-space form as (refer Appendix 2 for the details), x˙ = f (x, u, t), y = g(x, u, t).

(9)

2.2. Rear-driven powertrain model To obtain the driven torque acting on the tyre, we also need to build a powertrain model for the vehicle, which is a rear-driven type in our case. The powertrain includes an engine, a clutch, a transmission, and a rear differential. The flow chart of the powertrain is depicted in Figure 2, and the configurations and related parameters of the vehicle powertrain are listed in Table 1. For simplicity, engine torque Te are typically characterised by a 2D torque lookup table indexed by engine angular speed ωe and throttle position γ , as shown in Figure 3. We also do not intend to consider the control of the clutch in this study, so it is always regarded as


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Figure 2. The powertrain of the rear drive vehicle. Table 1. The configuration for the presented vehicle model. Components

Values and description

Type Engine Clutch Transmission

E-class Sedan 150 kw, idle speed 750 rev/min, inertia 0.38 kg-m2 Considered always closed, inertia 0.015 kg-m2 AT, 5-speed gearbox, inertia 0.04 kg-m2 Gear ratio Efficiency

Differential Tyres

3.538, 2.06, 1.404, 1.00, 0.713 0.92, 0.92, 0.95, 0.95, 0.98

Open, Ratio 4.1, inertia: 0.02 kg-m2 225/60 R18, inertia: 0.9 kg-m2

Figure 3. Looking-up table of the engine. (a) The engine 2D-lookup for engine torque. (b) 2D-inverselookup table for throttle.

locked (the gear shifting time is usually short, thus the effects on the powertrain system is ignored), then the engine angular speed ωe are treated as ωe = ωc , where ωc denotes the output angular speed of the clutch and it is evaluated by, ωc = Rgear ωgear ,

(10)

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where Rgear is the grear ratio of the transmission gear box, and ωgear is the output angular speed of the gear box. The transmission is modelled as a 5-speed transmission (the gear ratio of each shift is shown in Table 1) and the gear ratio Rgear is automatically obtained based on the transmission speed (the shifting logic is presented in Appendix 3). Moreover, the output of the transmission gear torque Tgear could be evaluated by Tgear = Tc Rgear ηgear ,

(11)

where ηgear is the efficiency when the engine drives the wheels. Based on the assumption clarified before, the clutch is treated as locked, then ωgear is calculated by, ω˙ gear =

Tgear − Ttran . (Je + Jc )Rgear + 0.5Jtran

(12)

With transmission output shaft torque Ttran is evaluated by the torsional spring-damper, namely, Ttran = Kdr (θgear − θtran ) + Ddr (ωgear − ωtran ),

(13)

Rdiff (θwRL + θwRR ), 2 (14) Rdiff (ωwRL + ωwRR ), ωtran = 2 where Je denotes the inertia of the engine, Jc denotes the inertia of the clutch, Jtran denotes the inertia of the transmission, Kdr , Ddr are the torsional stiffness and damping coefficient of the driveline, respectively. θwRL , θwRR represent the rotation angle of the rear left wheel and rear right wheel, respectively, and finally, ωwRL , ωwRR are the angular speed of the rear left wheel and rear right wheel, respectively. What’s more, the differential system is moulded as traditional open type, TwRL TwRR are the drive torque acting on the rear wheels through the powertrain are calculated by, θtran =

1 TwRL = Tgear Rdiff_F ηdiffF − TdiffF 2 TwRR = Tgear Rdiff_F ηdiffF − TwRL .

(15)

With the torque due to torsion of the front axle TdiffF , which is calculated by, TdiffF = Kdiff (θwRL − θwRR ) + Ddiff (ωwRL − ωwRR ),

(16)

where ηdiffF is the efficiency of the differential gear, Kdiff is the stiffness of the differential shaft, and Ddiff is the damping coefficient of the differential shaft. 2.3. Vehicle model validation In order to verify the feasibility of the 8DOF nonlinear vehicle model presented in this paper, simulation tests with a step steering angle input (test-1) and sine dwell test (test2) are conducted on a high adhesion surface. The simulation results are compared with a


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Figure 4. Simulation results of model validation, comparing with CarSim model. (1) Test 1: Step-steerinput test. (a) The steering input. (b) The global trajectory. (c) Lateral acceleration of the vehicle. (d) Yaw and roll angle of the vehicle. (e) Yawrate and roll rate of the vehicle. (f) Lateral force acting on the front/rear axle. (2) Test 2: Sine with dwell steer-input test. (a) The steering input. (b) The global trajectory. (c) The lateral acceleration of the vehicle. (d) The yaw and roll angle of the vehicle. (e) The yawrate and rollrate of the vehicle. (f) The lateral forces acting on front/rear axles.

well-known commercial vehicle dynamic software CarSim, whose configurations are the same with the presented 8DOF vehicle as shown in Table 1. The data are collected when the vehicle reaches in a stable condition after 2 seconds for test-1, but the whole manoeuvre time for test-2. The vehicle parameters related to the 8DOF nonlinear vehicle model is listed in Appendix 2: Table A3. Simulation results include the steering wheel angle input, global trajectory, lateral acceleration, roll rate, yaw rate, yaw angle, roll angle and the total tyre lateral force acting on front/rear axle, which are listed in Figure 4(1): (a)–(f) and 4(2): (a)–(f), respectively. As a whole, the responses of the 8DOF nonlinear vehicle model and CarSim model are matched well, Table 2 also shows some numerical errors of the simulation results between the 8DOF nonlinear vehicle model and the CarSim model. As the errors are all in an acceptable range, and the variances of the error are relatively small, thus, it proves that the 8DOF nonlinear vehicle model could be regarded as feasible and effective, which will be adopted in this paper for a further research.

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Table 2. Error analysis of the simulation results obtained from 8DOF model and CarSim model. Err.

Yaw

Unit Test

[deg] T.1

T.2

Roll

Yaw rate

[deg] T.1

Roll rate

[deg/s] T.2

T.1

Lat. Acc.

[deg/s]

T.2

T.1

Fyf (norm.)

[g]

T.2

T.1

Fyr (norm.)

[–] T.2

T.1

[–] T.2

T.1

T.2

Me. 0.601 0.119 0.096 0.153 0.022 0.465 0.008 0.700 0.003 0.034 0.032 0.038 0.051 0.082 Var. 3e−3 6.6e−3 1.6e−5 9e−3 2.8e−5 0.083 3.0e−4 0.333 2.8e−7 5.6e−4 7.0e−7 1.1e−3 6.0e−7 1.9e−3 Abbreviations Err.: Error, T.1: Test 1, T.2: Test 2, Lat. Acc.: lateral acceleration, norm.: normalised, Me.: mean, Var.: variance Remark: the normalised value for the error of the lateral force acting on the front/rear axle is calculated by eFyf =

Fyf − F¯ yf , Fˆ yf

eFyr =

Fyr − F¯ yr , Fˆ yr

where F¯ yf , F¯ yr are the corresponding values obtained from the CarSim software, and Fˆ yf , Fˆ yr are the nominal value for normalisation. They are evaluated as the value of the tyre lateral force when the vehicle states research in stable condition after 2 s for test-1(step-steer-input test),while set as the value of the maximum tyre lateral force in the manoeuvre for test-2 (sine with dwell test).

3. Vehicle longitudinal velocity planning This section presents a velocity-planning approach based on multi-object optimisation to achieve a time-optimal, comfort, safety for vehicle driving. At first, the path definition for planning and control purpose are defined, and then followed by a detailed velocityplanning method. 3.1. Path description First, we need to define a coordinate system in order to describe the position on the road, as shown in Figure 5 (left), the origin O of the world inertial reference frame is set as a point on the road centreline. The road’s centreline could be constructed by a Bezier curve or any other appropriate parametric curve function. Then, define a coordinate system along with the road centreline, the unit vector in the road centreline normal–tangential coordinates (Frenet coordinates) system is evaluated by [29],

H (x) 1 et = ex + ey , 1 + H (x)2 1 + H (x)2 −H (x)

(17)

1

en = ex + ey , 1 + H (x)2 1 + H (x)2 where H(x) denotes the road centreline function respect to x. Assuming that the vehicle moves along the road, the path profile is composed of a series of (X, Y) data, based on the road centreline, it could be generated by,

R = Rc + br et = x −

br H (x) 1 + H (x)2

br

ex + H(x) + 1 + H (x)2

ey ,

(18)


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Figure 5. Illustration for path preview model and velocity planning. (a) The path description and driver path preview. (b) Illustration of velocity planning.

where br represents one point’s nominal distance to the centreline. Comparing with R = Xr ex + Yr ey , which yields, br H (x) Xr = x − , 1 + H (x)2 br Yr = H(x) + . 1 + H (x)2

(19)

In addition, the covered station is calculated by, si = si−1 + (Xr,i+1 − Xr,i )2 + (Yr,i+1 − Yr,i )2 .

(20)

Therefore, for a new arbitrary given station snew , the coordinate pair could be obtained by interpolation methods, namely, Xr, new = Interpolation (s, Xr , snew ), Yr,new = Interpolation (s, Xr , snew ). Further, the path curvature could be expressed as a function of the station s K(s) =

X (s)Y (s) − X (s)Y (s) 3/2

(X (s)2 + Y (s)2 )

.

(21)

3.2. Velocity profile planning Velocity planning has a significant impact on the driving safety and comfort, especially when vehicles drive in a different road geometry environment where velocity planning is required to be explicitly considered. In this paper, velocity profile generation is performed considering the velocity and acceleration constraints, such as the maximal allowed lateral and longitudinal accelerations. These constraints could considerably reduce the solution space for the velocity planning, as well as to allow the velocity planner to concentrate on the space where the optimal solution is most likely to exist. Instead of using linear velocity profiles, we specify the velocity profile as a high-order polynomial function of time, which

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is capable of generating an acceleration-continuous profile. The velocity profile u(t) is a quadruplicate function, which could be denoted by u(t) =

k=5

kak t k−1 ,

t ∈ [0, tf ],

(22)

k=1

where ak is the coefficient of the polynomial, k denotes the sequence number of the coefficient ak in the expression, tf is the final instant of one period of time. Further, the acceleration is, du(t) = k(k − 1)ak t k−2 , t ∈ [0, tf ]. dt k=5

ax (t) =

(23)

k=2

Considering the initial condition with an initial velocity u0 and initial acceleration ax0 : u(0) = a1 = u0 ,

a(0) = 2a2 = ax0 .

(24)

Namely, a1 − u0 = 0,

a2 −

ax0 = 0. 2

(25)

What’s more, the derived the jerk is, dax (t) k(k − 1)(k − 2)ak t k−3 , dt k=5

α(t) =

t ∈ [0, tf ].

(26)

k=3

And the covered station based on Equation (22) over [0, tf ] is,

s(t) =

τ =tf τ =0

u(τ ) dτ =

k=5

ak t k ,

t ∈ [0, tf ].

(27)

k=1

As described before, there exists velocity and acceleration constraints for the vehicle velocity planning, which are composed of the following terms. (1) Lateral acceleration limits, namely, ay = Ku2 ≤ ay max . Thus,

ay max (28) u≤ , K where ay max = min{μg ay_usr }, μ is the road adhesion constant and ay_usr is the maximum allowed lateral acceleration value expected by the driver. (2) In order to guarantee a pure rolling motion to avoid the wheel skidding phenomenon, it is necessary to verify that the forces transmitted to the ground is smaller than the ground friction force. For simplicity, the total tyre force is evaluated by Ft = Fx2 + Fy2 = m a2x + (Ku2 )2 ≤ μ mg.


13

Such that, a2x + K2 u4 ≤ μ2 g2 . Then,

|ax | ≤ μ2 g2 − K2 u4 .

Therefore, the acceleration constraint is, 2 2 2 4 |ax | ≤ min μ g − K u , ax_usr ,

(29)

(30)

where ax_usr represents the maximum allowed longitudinal acceleration value based on practical situations. Generally, the longitudinal acceleration of the host vehicle should be not too high to affect the comfort of passengers. As for the velocity planning, as depicted in Figure 5(b), we choose five equivalently spaced reference point over a given preview path length sp , which the reaching time of each reference point is tj , j = 1, 2 . . . , N. Therefore, we could obtain five equalities targeted to the station si , sj (ti ) −

j sp = 0, N

j = 1, 2, . . . , N.

(31)

Since it is expecting that the vehicle could move efficiently, smoothly and safely along with the target path, the proposed velocity planar aims to maximise the reference point’s velocity possibly while minimising the longitudinal jerk over the planning distance, namely, ⎡ ⎤ 1 ⎢

min ⎢ ak ,tj ∈Z ⎣

ui (ti )

τ =tN τ =0

α(τ )2 dτ

⎥ ⎥. ⎦

(32)

The control variable Z includes the coefficients of the velocity profile function and the reached time respecting to each reference point, namely, Z = [a1 a2 a3 a4 a5 t1 t2 · · · tN−1 tN ]T . Obviously, an optimisation process like (32) belongs to multi-objective programming domain, whose objective function targets to minimise the inverse of the velocity at the planning instant and the jerk for a period of time, subjecting to constraints in Equations (27)–(30). As a solution, which minimises one function often, does not minimise the others simultaneously, there is usually no unique optimal solution. Sometimes the decision maker has a goal for each objective in mind. In that case, the so-called goal programming technique can be applied. One way is to set the objective functions based on priority, and seek to minimise the deviation of the most important objective function from its goal first, before attempting to minimise the deviations of the less important objective functions from their goals. Therefore, the optimisation problem could be converted to a Min–Max problem, which means, subjecting to those velocity/acceleration constraints, the velocity planar intends to maximise the reference point’s velocity which has a minimum velocity value

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among those reference points, in the meantime, minimising the longitudinal jerk over the planning distance, such that, ⎡

⎤ 1 − Gj /wj ⎢ ⎥ u(tj ) ⎥ k = {1, 2, . . . , 5}, min max ⎢ ⎣ ⎦ τ =tN ak ,tj ∈Z 2 /w α(τ ) dτ − G N+1 N+1 τ =0

j = {1, 2, . . . , N}.

(33)

Subjected to, The equality constraints: j sp = 0, N a1 − u0 = 0, ax0 = 0. a2 − 2

sj (tj ) −

The inequality constraints: ay max 0 0, T˜ = Tˆ − T¯ is the error between T¯ and its estimation Tˆ . Since assuming T˙¯ = 0, such that, 1 ˙ 1 ˙ V˙ 2 = V˙ 1 + T˜ T˜ ≤ S(M + N δsw + R + T¯ + k˙eyp ) + T˜ Tˆ λ λ

(53)

And set the control law as, δsw =

−M − R − Tˆ − k˙eyp − sign (S) . N

Substitute Equation (54) into Equation (53), then, 1 ˙¯ ˜ ˙ V2 ≤ T −S + T − sign (S)S. λ

(54)

(55)


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Therefore, if we set the adaptive control law as

T˙¯ = Sλ(λ > 0).

(56)

Further, in order to eliminate the chattering effect, the discontinuous component sign (S) is replaced by a continuous function (S) in Equation (54), : S →

S . |S| + σ

The completed steering control law (54) rewrites as follows, δsw =

S −M − R − Tˆ − k˙eyp − |S|+σ

N

,

(57)

where Tˆ is respecting to the dynamical system (56). Finally, substitute Equations (56), (57) into Equation (53), we have 1 ˙ S2 S 1 ˙ S2 V˙ 2 ≤ S −T˜ − − T˜ Tˆ = T˜ −S + Tˆ − =− < 0, |S| + σ λ λ |S| + σ |S| + σ (58) which is held for any > 0, σ > 0, it means that control law (57) could stabilise the pathtracking system. 4.1. Lateral controller verifications This part presents verification for model with uncertainties, further, the influence of the preview time for preview SMC steering controller are also studied. 4.1.1. Simulation with model uncertainties To further investigate the robustness performance of the proposed steering controller in regular traffic scene, a routine shown in Figure 6(a) is performed under a speed as 72 km/h. The model uncertainties mainly come from the parameter variations of the vehicle, such as the tyre nonlinearities, which are arisen when the lateral acceleration becomes large, and the respecting transient cornering stiffness of the tyre will not remain unchanged but decreased. Additionally, the vehicle mass may change in a certain range in practical applications, thus, the proposed steering controller should be sufficiently robust toward these uncertainties. Then, the discussions are focused on analysing the vehicle tyre cornering stiffness and vehicle mass variations in this section. Figure 8(a) and 8(b) show that when the cornering stiffness is decreased by 50%, 25% or increased by 25% and 50%, respectively, the tracking errors has little change due to the adaptive controller designed for model uncertainty, the value of estimated T¯ would change accordingly for ensuring the proposed controller working well. As for the mass variation, similar to the variation in tyre cornering stiffness, when the mass decrease by 50%, 25% and increased by 25% and 75%, there is little change in tracking performance, as shown in Figure 8(c) and 8(d). Thus, the present steering controller has a robustness for vehicle parameter variations. What’s more, Figure 9 shows the tracking error analysis when facing with vehicle parameter variations. We can observe that the tracking error becomes large

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Figure 8. Simulation with model parameter variations. (a) The lateral error respecting to tyre cornering stiffness variation. (b) The estimated uncertainty. (c) The lateral error respecting to mass variation. (d) The estimated uncertainty.

Figure 9. Tracking error analysis for the presented controller with model parameter variations.

when the tyre cornering stiffness increased or the vehicle mass decreased. The reason is that they will lead the uncertainty terms to be larger as the coefficient term ij in Equation (40) is proportional with the tyre cornering stiffness, while universal with the mass. Therefore, there is little change in tracking error when the tyre cornering stiffness decreased or the vehicle mass increased as the corresponding uncertainty will be smaller. Additionally, from the results shown in Figure 8, we can further observe that, overall, the changes in tyre cornering stiffness could be more robust than the changes in vehicle mass, since the average tracking error of the variation in tyre cornering stiffness is smaller than the one in vehicle mass. 4.1.2. Influences of the preview-time This section presents an analysis of the influences of the preview time in this presented robust SMC controller. We conduct a double lane change manoeuvre with a moderate speed 72 km/h and high speed as 90 km/h, respectively, to further show the influences of


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Figure 10. Simulation results of a DLC manoeuvre. (a) Path following results of a DLC manoeuvre, left: 72 km/h, right: 90 km/h. (b) Lateral error of a DLC manoeuvre, left: 72 km/h, right:90 km/h. (c) Lateral acceleration results of a DLC manoeuvre, left: 72 km/h, right: 90 km/h. (d) Steering wheel angle results of a DLC manoeuvre, left: 72 km/h, right: 90 km/h.

the preview-time. The selected preview time is gradually becoming longer, from 0.1, 0.3, 0.5 to 0.8 s. The path following result, the path-tracking error, the lateral acceleration and the steering wheel angle, are plotted in Figure 10. When the speed is 90 km/h, the preview time with 0.1 s, even 0.3 s, it seems that it will lead a failure to accomplish the Double Lane Change (DLC) driving task as shown in Figure 10(a). The tracking errors in Figure 10(b) also prove that, as the preview-time becomes too short or too long, the tracking errors would become large. Therefore, it requires a longer preview time to keep the vehicle in stable for a DLC manoeuvre when the speed becomes larger. However, as shown in Figure 10(c) and 10(d), though the preview time is getting longer, the lateral acceleration and the amplitude of the steering wheel angle will be decreasing, while the tracking error will become large, indicating a decreasing in its tracking performance. Figure 11 also lists three boxplots of tracking errors, lateral acceleration and steering wheel angular speed under different speed (72 and 90 km/h) with a preview time fixed as

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H. CAO ET AL.

0.5 s, respectively. It shows a contradiction in tracking performance and handling performance, besides, the handling qualities of the vehicle will become worse with an increasing in speed. Figure 11(b) and 11(c) also show the tracking errors, lateral acceleration and steering wheel angular speed boxplots with different preview time under 72 and 90 km/h, respectively. It indicates that as the preview time is getting longer, the handing performances of the vehicle would become better, since the lateral acceleration is smaller and the steering wheel angular speed is decreasing. As for the tracking error, it seems that the preview time should be proper adopted which is neither too short nor too long in order to track the path well.

4.2. Fuzzy-adaptive preview-time for robust SMC-based steering controller Since the preview time has a significant influence on the vehicle’s tracking performance and handling performance, if the preview time could be adjusted adaptively, it could make the host vehicle track the target path well, meanwhile keeping the vehicle in stable to avoid big lateral acceleration occurred. What’s more, we should be also aware that the path curvature change is also related to the preview time, as the path curvature change becomes rapidly, it is expected that the SMC controller needs a longer preview time to achieve a smooth transition. Therefore, we present an adaptive preview-time controller based on curvature change of the preview path and vehicle’s lateral acceleration by fuzzy approach, a detailed process is shown as follows. The fuzzy control system has two inputs, which its linguistic variables are named Curv_change donating the road curvature change and Veh_Ay donating the vehicle’s lateral acceleration, both Curv_change and Veh_Ay have three values named Small, Medium, Large, respectively. Curv_change ranges from 0 to 0.05 m−1 s−1 . Generally, the curvature change calculated from the path would be noisy, so a low-pass filter will be applied to it in order to get a smooth signal. Based on simulation experiences, determination for fuzzy values respecting to curvature change is based on path information of DLC. For example, as shown in Figure 12(a), an upper bound value for an absolute ‘small curvature change’ is 0.01 m−1 s−1 , and lower bound value for an absolute ‘large curvature change’ is 0.04 m−1 s−1 . In terms of the lateral acceleration whose value varies from 0–0.8 g, if the lateral acceleration of the vehicle is less than 0.4 g, then the calculation of tyre force could be regarded as a linear form, and the tyre forces would become nonlinear when the lateral acceleration grows larger. Therefore, 0.4 g is a reasonable threshold value for accessing vehicle’s lateral acceleration. Thant means, as shown in Figure 12(b), value of lateral acceleration around 0.4 g is considered as ‘moderate/medium’, an absolute ‘large lateral acceleration’ is larger than 0.6 g and an absolute ‘small lateral acceleration’ is less than 0.1 g. There is one output named Pre_Time to represent the preview time for SMC controller. It has four values named Short, Less Medium, Medium, Long to achieve more smoothing results, which ranges from 0.2 to 0.8 s. As concluded before, the preview time should not be too small or too large. Thus, an absolute ‘short preview time’ is around 0.3 s, an absolute ‘long preview time’ is larger than 0.6 s, a ‘medium preview time’ is around 0.5 s and a ‘less medium preview time’ is around 0.4 s, as shown in Figure 12 (c). In addition, their membership function chosen as generalised bell functions that are illustrated in Figure 12.


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Figure 11. Boxplots of the DLC manoeuvre. (a) Boxplots of the track error, lateral acceleration and steering angular speed with different vehicle speed. (b) Boxplots of the track error, lateral acceleration, steering angular speed with different preview time (72 km/h). (c) Boxplots of the track error, lateral acceleration, steering angular speed with different preview time (90 km/h).

The nucleus of the fuzzy control system is made up of fuzzy rules, which in our case, the SMC driver controller intends to adopt a suitable preview time in order to keep tracking the target path, while avoiding a large lateral acceleration occurred as well, thus, the rules for SMC driver model are easy to be understood, for example:

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Figure 12. Member function and fuzzy surface of the SMC controller with adaptively preview-time. (a) The member function of the curvature change. (b) The member function of lateral acceleration. (c) The member function of the preview time. (d) fuzzy surface of the adaptive preview-time steering controller. Table 4. Robust SMC controller with adaptive preview-time fuzyy rules. Curv_change Pre_Time Veh_Ay

Small Medium Large

Small

Medium

Large

Short Less medium Medium

Less medium Medium Medium

Medium Long Long

IF Curv_change is Medium AND Veh_Ay is Medium, THEN Pre_Time is Medium; IF Curv_change is Medium AND Veh_Ay is Large, THEN Pre_Time is Long; IF Curv_change is Small AND Veh_Ay is Small, THEN Pre_Time is Short. The whole rules are listed in Table 4 and the fuzzy surface as shown in Figure 12(d) is smooth, which means the control process could be well handled. The inference engine propagates the matching of the conditions to the conclusions, generating the contribution of each rule to the control action. The well-known Mamdani’s inference method (min–max) is used to solve the fuzzy implication. Defuzzification is the transformation of the output fuzzy values that are generated by applying the inference method, which the CoA (Center of Area) method is used in this paper. The fuzzy system could be well defined, easily built and validated in Matlab. 4.3. Comparative study In order to show the effectiveness of the proposed robust SMC with adaptive preview-time (RSMC-APT) controller, several well-known driver model such as the McAdam’s optimal


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Figure 13. Simulation results of the vehicle with different steering control methods. (a) The trajectory of the vehicle with DLC manoeuvre. (b) The lateral acceleration of the vehicle (DLC). (c) The steering wheel angle of the vehicle (DLC). (d) The preview time of the RSMC-APT controller.

Figure 14. Boxplots of the tracking error, lateral acceleration and steering angular speed.

control model (The CarSim software adopt this model in its package), the MPC method, and along with the robust preview SMC (RSMC) controller presented in previous chapters are compared. The test scenario is a double lane change manoeuvre, which the vehicle performs it in a relative high speed as 90 km/h. As shown in Figure 13(a), we could observe that these entire four controllers could follow the target path well. Figure 13(b) shows the lateral acceleration of the vehicle under each controller’s influence, and Figure 13(c) shows the steering wheel angle input of each controller. What is more, the preview time of the RSMCAPT controller is plotted in Figure 13(d), which shows that the preview time could adjust itself based on the vehicle’s lateral acceleration and path curvature change as designed. Based on the plots of the lateral acceleration and the steering wheel angle, along with the boxplot drawn in Figure 14, which includes the tracking error, lateral acceleration and steering wheel angular speed, their tracking performance and handling performance vary

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each other. Overall, it seems that the tracking performance of the MPC controller and the robust SMC controller are better than the other two, however, their lateral accelerations are rather high, which nearly reach the afforded limits of the grounds. Further, the steering wheel angular speeds are higher than the other two controllers as well, which means the handling quantity of the vehicle is severer, difficult to manipulate. Actually, the tracking performance and the handling performance are conflict in critical situations such as a double lane change manoeuvre in high speed. We pursue a controller that shares a balance between the tracking performance and handling performance. Thanks to the adaptive preview time by fuzzy approach, we could observe that the handling performance of the vehicle is improved, maintaining the tracking error in an acceptable range. It seems that both the MacAdam’s model in CarSim and the RSMC-APT controller won’t cause high lateral accelerations which are approaching to the limit. When comparing with MacAdam model, based on Figures 13 and 14, overall performance of RSMC-APT controller actually is slightly worse than MacAdam model’s, but the difference is limited. What’s more, preview time in MacAdam model is generally fixed as a constant, so when dealing with different scenarios, the preview time should be properly defined, or else the tracking performance or handling performance would become worse. While the RSMC-APT model could adjust the preview time based on the road information, thus it won’t suffer this problem. Additionally, the RSMC-APT model has a robustness to vehicle parameter variations as described before, while the MacAdam model does not own this robustness feature, it is still sensitive to the parameters of the vehicle such as mass, tyre cornering stiffness. Therefore, the proposed RSMC-APT controller could track the target path well, and robustly, it also maintains a good handling performance.

5. Longitudinal velocity-tracking controller This chapter presents an SMC controller based on the input–output linearisation approach in order to track the planned velocity well, and later, a validation and comparison with some other control methods are conducted as well. Consider the longitudinal motion for one-wheel vehicle model, which is governed by, mu˙ = Fx − Froll − Fwind ,

(59)

where Fx is the tyre longitudinal force, and it is a function of the vertical tyre load, the longitudinal tyre slip and the friction coefficient. Moreover, the longitudinal tyre force could be expressed as a linear form in common driving situations with an appropriate acceleration or deceleration, namely ⎧ ⎨Cx rωuw − 1 if rωw < u, rωw − u Fx = Cx κ = Cx = max (rωw , u) ⎩Cx 1 − u if rωw ≥ u, rωw

(60)

where κ denotes the longitudinal slip, ωw is the angular speed of the tyre, Cx is the longitudinal stiffness coefficient. Generally, the rolling resistance which is much smaller than the longitudinal force is then neglected (Froll Fx ), and the wind resistance is proportional


29

with the square of the velocity, therefore, Equation (59) could be rewrite as, u˙ =

ρCrx Ar u2 Cx rωw − u − . m max( rωw , u) 2m

(61)

The wheel dynamic equation is, ω˙ w =

Tw − Fx r Tw Cx r rωw − u , = − Jw Jw Jw max( rωw , u)

(62)

where Tw represents the control torque acting on the tyre, it might be a driving torque or braking torque. Combing Equations (61) and (62), it is written as a state-space form as, ⎡ ⎤ ρCrx Ar 2 Cx rωw − u ⎡ ⎤ − u 0 ⎥ ⎢ m max (rωw , u) 2m u˙ ⎥ + ⎣ 1 ⎦ Tw =⎢ ⎣ ⎦ Cx r rωw −u ω˙ w . (63) − Jw J max(rωw , u) w

p=u Rewrite the dynamic system (63) as the form x˙ = f (x) + gTw , p = h(x). If rωw < u, in a braking condition, let, Cx r = c10 , m

Cx = c11 , m

ρCrx Ar = c12 , 2m

−

Cx r 2 = c20 , Jw

−

Cx r = c21 , Jw

1 = c22 . Jw

And such that, x=

u , ωw

⎤ ⎡ ω w c10 − c11 − c12 u2 ⎢ ⎥ u f =⎣ ⎦, ωw − c21 c20 u

g=

0 , c22

h(x) = u.

(64)

Or else with an accelerating case, Cx ρCrx Ar Cx r Cx 1 ∗ ∗ ∗ = c11 = c12 , − , = c20 , − = c21 , = c22 mr 2m Jw Jw Jw ⎡ ⎤ ∗ − c ∗ u − c u2 c10 12 11 u 0 ωw ⎢ ⎥ x= , f =⎣ , g = , h(x) = u. ⎦ u ωw c ∗ 22 c20 − c21 ωw (65)

Cx ∗ = c10 , m

Noting that the system (63) is a nonlinear one, and common control method for linear system do not work anymore, however it is easily to verify that the system processes a relative order of 2, thus the input–output linearisation approach could be applied. Let us

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take care the system (64) for braking condition first, there exists a diffeomorphism, T : D ⊂ R2 → R2 and a coordinate transformation as, u z1 h(x) ωw = = T(x) = . (66) c10 − c11 − c12 u2 Lf h(x) z2 u Equation (66) could realise the system’s input–output linearisation. Since, h(x) = u c10 c22 Lg Lf h(x) = u 2 u6 + 2c c u4 − c c ω u3 − c c u2 + c (c + c )ω u − c2 ω2 2c 11 12 12 10 w 21 10 10 11 20 w 10 w L2f h(x) = 12 . u3 (67) The system could be reformed as, z˙ 1 0 = 0 z˙ 2

1 0

L2f h(x) z1 0 L L h(x) Tw + . + 1 g f z2 Lg Lf h(x)

(68)

Let (x) := L2f h(x), (x) := Lg Lf h(x). Noting that we have the output u = z1 , thus, u¨ = (x) + (x)Tw .

(69)

Assume that the longitudinal reference speed be a continuously differentiable signal uref , and then the velocity-tracking error is, eu = u − uref

(70)

The sliding surface S of the first-order SMC is denoted as, S = e˙ u + ceu .

(71)

Still, let us definite a positive Lyapunov function as V = (1/2)S2 , the stability condition and asymptotic convergence to the surface S = 0 are achieved by the following η-attracive condition, which constrains the trajectory of the system to side along the hypersurface S, namely, V˙ = SS˙ = S(ëu + c˙eu ) = S(u¨ − u¨ ref + c˙eu ).

(72)

Let u¨ − u¨ ref + c˙eu = −η sign (S), and the control law is then chosen as, Tw =

u¨ ref − c˙eu − (x) − η sign (S) . (x)

(73)

Such that, V˙ = SS˙ = −η sign (S)S = −η|S|0.

(74)

Obviously, the control law could stabilise the system. And in order to eliminate the chattering effect, the discontinuous component sign (S) is replaced by a continuous function


31

(S), thus, Tw =

S u¨ ref − c˙eu − (x) − η |S|+σ

(x)

.

(75)

With 2 u6 + 2c c u4 − c c ω u3 − c c u2 + c (c + c )ω u − c2 ω2 2c12 11 12 12 10 w 10 21 10 11 20 w 10 w 3 u c10 c22 . (x) = u (76)

(x) =

Similarly, we can derive a control law for (65) in an accelerating case. It could be expressed as Tw∗

=

S u¨ ref − c˙eu − ∗ (x) − η |S|+σ

∗ (x)

.

(77)

With, 2 u3 ω3 + 3c∗ c u2 ω2 − c∗ c∗ u2 2c12 w w 11 12 11 21 ∗ c ω3 u + c∗ (c∗ + c∗ )ω u − c∗ c∗ ω2 −2c10 12 w 11 11 20 w 10 11 w 3 ωw u ∗ ∗ (x) = c11 c22 2 . ωw

∗ (x) =

(78)

Noting that the torque acting on the wheel might be a driving torque for acceleration (Tw > 0) or braking torque for decelerating (Tw < 0), and if Tw is the driving torque, thus, the engine torque Te and its angular speed ωe could be approximated via, Te = Tw /(Rgear Rdiff ) ωe = ωw Rgear Rdiff .

(79)

Finally, the throttle value τ as the control input for the vehicle system could be determined by looking up the 2D table, which is indexed by engine angular speed ωe and engine torqueTw as shown in Figure 3(b). Next, in order to verify the effectiveness and feasibility of the SMC longitudinal velocitytracking controller presented in this paper, it is compared with a cubic ProportionalIntegral (PI) controller as shown in Equation (80), as well as a classical PI controller (when kp3 = 0), and a fuzzy controller.

(80) ax = kp3 (u − ud )3 + kp (u − ud ) + ki (u − ud ) dt, where kp3 , kp and ki is the design parameters of the cubic-PI controller. The testing scenario is a vehicle with rather fierce velocity changes on a straight road, the vehicle decelerates from 25 to 20 m/s at the beginning, then to 15 m/s, and it accelerates to 30 m/s in the end, as shown in Figure 15(a) (the target speed). It is obvious that the host vehicle can follow the reference signal quite well with the SMC control, fuzzy control

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Figure 15. longitudinal velocity-tracking results with different control methods. (a) the velocitytracking results. (b) The throttle input (normalised) of the vehicle.

or cubic PI control (Figure 15(a)), while the performance of a normal PI method is not satisfying. Actually, the SMC controller’s tracking performance is quite good, except that it seems a little bit weaker in acceleration when comparing with the fuzzy or cubic PI, but it is stronger in decelerating phrase than the others. However, we should also be aware that the cubic PI controller is quite sensitive to its parameters’ selecting, which should be chosen carefully based on lots of simulations to achieve a good tracking performance, while the SMC controller would not suffer it. Fuzzy control is easily established with tracking error and its derivative as inputs, but it needs priori knowledge of how to dealing with a tracking mission, sometimes, it is not an easy task. Additionally, we also list the normalised throttle control input of each controller in Figure 15(b), which clearly shows that the control process is acceptable. Therefore, the SMC controller for vehicle velocity tracking is effective and feasible.

6. Simulations and verifications on vehicle trajectory following architecture We present a trajectory following architecture towards to a high-level automated vehicle for Lateral Guidance System to accomplish a driving task such as lane keeping or automatic guidance. As shown in Figure 16(a), this trajectory following system includes a velocity-planning module, longitudinal speed-tracking controller and a lateral steering controller as described in previous sections. The velocity-planning module could generate a smooth velocity profile online based on the curvature of the trajectory, so that the vehicle could travel in a maximum allowed speed while keeping the jerk in the planning period minimum in order to make the passenger feel comfortable. As the velocity planning is online, in this paper, we trig the velocity-planning module every 50 m in covered distance. In addition, the longitudinal speed-tracking controller then make the host vehicle match the planned speed and the lateral steering controller make the host vehicle follows the target path. All subsystems are built and complemented on Matlab/Simulink platform, as the system sketch is shown in Figure 16(b). We verify this optimal trajectory following system with two scenarios that are easily encountered in daily life. The vehicle is expected to move along with the target path, for velocity planning, the user-defined maximum longitudinal and lateral acceleration are 0.2 and 0.4 g, respectively. The planning distance is 50 m and the velocity-planning module will be triggered every covered distance as 50 m.


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Figure 16. Trajectory following architecture illustration and DiL simulations. (a) Illustration of the trajectory following architecture. (b) Illustration of the trajectory following Simulink model. (c) A snap of the DiL simulation by a novice driver.

6.1. Driving performance evaluations In order to show the advantage of the automatic driving which is utilised by the presented trajectory following architecture, we also conduct DiL experiments by an experienced driver and novice driver on a driving simulator to verify the feasibility and effectiveness of the presented trajectory following controller, as shown in Figure 16(c). An evaluation mechanism whose indexes include the driver steering load, lateral stability, velocity performance, driving comfort and tracking performance to access driver’s driving performance is proposed. The selected indexes for evaluating the driving performance are listed as follows. Average steering load, which is accessed by the square of the steering wheel angular speed in the driving period, namely,

1 tf J1j = (δ˙sw /δˆ˙sw )2 dt, (81) tf 0 where tf is the final time when the vehicle finish the requested course, the δˆ˙sw is a reference value, which is 180 deg/s in this paper. Average lateral stability, which is evaluated by the square of the lateral acceleration of the vehicle,

1 tf J2j = (ay /ây )2 dt, (82) tf 0

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where aˆ y is a reference value, which is 0.4 g in this paper. Apparently, a lower lateral acceleration could lead to a good value in lateral stability index. Average velocity performance, which is related to the vehicle moving speed, and it is expected to move within the velocity bounds. Thus, it is evaluated by, 1 J3j = tf

tf

(1 − ux /uˆ x )2 dt ,

0

(83)

where uˆ x is the corresponding maximum allowed velocity. Average driving comfort, which is evaluated by the longitudinal jerk of the host vehicle, obviously, a smooth accelerating or decelerating could result in better driving comforts. J4j =

1 tf

tf

t0

2

(˙ax /aˆ˙ x ) dt,

(84)

where aˆ˙ y is the reference value of the driving comfort, which the value in our paper is 0.1 g/s. Average path-tracking performance, which is evaluated by the squared lateral tracking error ey , namely,

1 tf

e2y dt. (85) J5j = tf t0 Finally, the normalised performance for different driver is calculated by, J¯ij = Jij / max{Jij }, j

(86)

where i = 1, 2, . . . , 5.j = {Experienced Driver, Novice Driver, Automatic Driving}, Then the combined performance for different users is the normalised sum of each performance’s value, such that, Jcom,j =

i=5 i=1

i=5 ! Jij / max Jij . j

(87)

i=1

6.2. Test scenario 1 As shown in Figure 17(a), the first testing scenario is one with three large over-90-degree turns where there is no inverse moving respecting to the Global-X axis, the total distance of the course is around 1000 m. The experienced driver, novice driver and the automatic driving ‘driver’ perform the scenario, respectively. The drivers are asked to finish the course to track the road centreline in a high-allowed speed as precisely as he can, but should avoid a rush accelerating/decelerating to achieve a nice driving comfort. Additionally, in order to decrease the influence caused by the human driver’s stochastic operating performance, each driver is required to perform the driving task multiple times and then a mean value of the driver’s operating performance will be adopted for comparative study. Figure 17(b) shows the lateral tracking errors of each driver, which indicates the automatic driving has its superior advantage in path tracking, much better than human drivers do. Figure 17(c)–(e)


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Figure 17. Simulation results of scenario-1. (a) The following trajectory. (b) The lateral tracking error. (c) The steering wheel angle. (d) The normalised throttle input. (e) The braking torque acting on each wheel. (f) The longitudinal acceleration. (g) The lateral acceleration. (h) The longitudinal speed. (i) The preview time of the automatic driving controller.

show the control inputs to the host vehicle, including the steering wheel angle, engine throttle and braking torques acting on each wheel, it is easily observed the control processes of the automatic driving are the smoothest among them, when compared with the novice driver and experienced driver. The corresponding longitudinal accelerations are plotted

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Figure 18. The driving performance respecting to different ‘drivers’ in scenario-1.

in Figure 17(f), which shows the maximum of the longitudinal acceleration of the vehicle driven by the experienced driver, novice driver and automatic driving is around 0.15, 0.18, 0.12 g for accelerating, respectively, and around −0.02, −0.28, −0.08 g for decelerating. All of them are within acceptable range, but the caused driving comfort varies, obviously, the automatic driving has the best driving comforts. What’s more, the automatic driving could ensure the longitudinal velocity remains within its velocity constraint, as shown in Figure 17(h). While the experienced driver sometimes will drive the vehicle in a higher speed which violates the velocity constraints as the driver determines the maximum allowed speed based on his/her driving experience in reality, and the novice driver generally tends to keep cautious to slow down when drives in the corner zone. Thus, it will lead the lateral acceleration of vehicle driven by experienced driver to be higher, but the host vehicle still moves in stable, the peak value of the lateral acceleration of the host vehicle for different drivers is around 0.5 g. A lower lateral acceleration would be obtained of the vehicle driven by novice driver since its longitudinal velocity is relatively smaller than the others. Finally, the preview time of the automatic driving controller in the simulation is plotted in Figure 17(i). As the controller designed, the preview time tends to become longer when in the turning course as its curvature change will become large, as well as its lateral acceleration. While the preview time is small when in straight course as the curvature change and lateral acceleration are both small. Figure 18 also shows each performance index value as shown in (81)–(87) and combined index value of the experienced driver, novice driver and automatic driving, respectively, by the driving performance evaluation system, which further proves the advantage of the automatic driving with the trajectory following architecture presented in our paper. Except the lateral stability which the automatic driving ranks the second, weaker than the novice driver as its longitudinal speed is smaller, the driver steering load, velocity performance, driving comfort and the path-tracking performance are the best among them. Obviously, the combined index of the automatic driving is also the best. As a whole, the automatic driving could effectively reduce the driver’s workload, improve the time efficiency and driving comforts.


37

6.3. Test scenario 2 The second one is a little different from the previous scenario, it is composed of three large twists and there exists inverse moving respecting to the Global-X axis, the target path likes an S as shown in Figure 19(a), and the length of the course is about 1500 m. Still, the experienced driver, novice driver and the automatic driving ‘driver’ perform the scenario, respectively. Again, all of the drivers are asked to finish the course to track the road centreline in a high-allowed speed as precisely as he can, but should avoid a rush accelerating/decelerating to achieve a good driving comfort. What’s more, each driver is required to perform the driving task multiple times in order to decrease the influence caused by the human driver’s stochastic operating performance, and then the mean value of the driver’s operating performance will be adopted for comparative study. Figure 19(b) shows the lateral tracking errors of each driver, which indicates the automatic driving is excellent in path tracking, very small deviations are found in its driving process, the tracking performance of the experienced driver is also good, while the tracking performance of the novice driver is not satisfying. Figure 17(c)–(e) show the control inputs to the host vehicle, including the steering wheel angle, engine throttle and braking torques acting on each wheel. Due to the rusty driving skill of the novice driver, there exists obvious oscillations in his steering wheel angle input, while there are only small corrections for the experienced driver and the steering for automatic driving is quite smooth. Indeed, it will be also related to the driver steering loads directly, obviously, the steering load of the automatic driving is the best. Overall, the control processes of the automatic driving are the smoothest among them. The longitudinal accelerations are plotted in Figure 17(f), which shows the maximum of the longitudinal acceleration of the vehicle driven by the experienced driver, novice driver and automatic driving is around 0.10, 0.08, 0.14 g for accelerating, respectively, and around −0.04, −0.04, −0.12 g for decelerating, all of them are within acceptable range. Though the peak values of the automatic driving are large, its accelerating or decelerating process is quite smooth. What’s more, the automatic driving will ensure the longitudinal velocity remains within its velocity constraint, as shown in Figure 19(h). While the experienced driver drives the vehicle in suitable high speed, which is basically within the velocity bound due to his good driving experience in DiL simulations, and the novice driver generally tends to keep cautious to slow down its speed when in this twisted road scenario. A highest lateral acceleration of the vehicle driven by novice driver, which is nearly 0.6 g, was obtained when in its turning position. The lateral acceleration of vehicle driven by the experienced driver is also higher than automatic driving due to its higher moving speed. Finally, the preview time of the automatic driving controller in the simulation is plotted in Figure 17(i), as the controller designed, the preview time tends to become longer when in the turning course when its curvature change will become large and the corresponding lateral acceleration becomes larger as well. Figure 20 shows each performance index value and combined index value of the experienced driver, novice driver and automatic driving, respectively, which are calculated by the driving performance evaluation system. It further proves the advantage of the automatic driving with the trajectory following architecture. In this scenario, except the lateral stability that is weaker when compared with the novice driver due to its lower speed, and the velocity performance that is weaker than the experienced driver ranks the second,

38

H. CAO ET AL.

Figure 19. Simulation results of scenario-2. (a) The following trajectory. (b) The lateral tracking error. (c) The steering wheel angle. (d) The normalised throttle input. (e) The braking torque acting on the wheel. (f) The longitudinal acceleration. (g) The lateral acceleration. (h) The longitudinal speed. (i) The preview time of the automatic driving controller.

the driver steering load, driving comfort and the path-tracking performance are the best among them. Still, the combined index of the automatic driving is the best. As a whole, the automatic driving could effectively reduce the driver’s steering load, ensures the time efficiency and driving comforts simultaneously. The optimal model-based trajectory following system could deal with these two scenes well. Thus, we believe it is feasible for automated driving vehicle.


39

Figure 20. Evaluation results of scenario-2 by different kinds of drivers.

7. Conclusions and future work This article presents an optimal model-based trajectory following architecture which synthetises the lateral adaptive preview strategy, longitudinal velocity planning and tracking for highly automated vehicle, when it is in a driving task such as automated guidance or lane keeping. The velocity-planning module will generate a smooth velocity profile that consider both the driving comfort and time-optimal purpose. An SMC controller with input–output linearisation technical is designed for velocity tracking, and a robust SMC steering controller with an adaptive preview-time for vehicle lateral control is proposed for path tracking. Simulation results show this trajectory following architecture are effective and feasible for high automated driving vehicle, which could plan a satisfying longitudinal speed profile, track the target path well and safely when dealing with different road geometry structure. Our future work mainly includes two aspects, first, the road curvature information in this study is assumed to known, an online predictor for road curvature could be built in our future work. Second, the host vehicle should be capable of interacting with obstacle vehicles on the road, so adaptive following system or obstacle avoidance system for automated driving could be further developed, we also try to realise the whole system implementable in real time situations.

Acknowledgements The authors gratefully appreciate constructive insightful suggestions from the reviewers and editors. The authors thank Mr Yuting Zhou and Mr Mingjun Li for their kind help in DiL simulations.

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

Funding This work is supported by the National Natural Science Foundation of China under grant numbers [51575169] and [51575167].

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References [1] China National Statistics Bureau, China Statistical Year, year of 2014 (Electronic version): 2014 traffic fatality statistic, 2015. Available from: http://www.stats.gov.cn/tjsj/ndsj/2015/indexch. htm [2] Fatality Analysis Reporting System (FARS) of the US department of transportation, Traffic Safety Facts 2014, FARS/GES Annual Report; 2016. Available from: https://crashstats.nhtsa. dot.gov/Api/Public/ViewPublication/812261 [3] Blincoe LJ, Miller TR, Zaloshnja E, et al. The economic and societal impact of motor vehicle crashes, 2010. (Revised) (Report No. DOT HS 812 013); 2015. Washington(DC): National Highway Traffic Safety Administration. [4] Volvo Trucks European Accident Research and Safety Report, Volvo, Gothenburg, Sweden; 2013. [5] Kuehn M, Hummel T, Bende J. Benefit estimation of advanced driver assistance systems for cars derived from real-life accidents. 21st International Technical Conference on the Enhanced Safety of Vehicles; June 2009, Stuttgart, Germany. [6] Kühn M, Bende J. Accident statistics and the potential of driver assistance systems, Gesamtverband der Deutschen Versicherungswirtschaft eV (GDV), Berlin; 2014, Issue Number: 47. [7] From Driver Assistance Systems to Automated Driving, VDA Magazine-Automation, Verband der Automobilindustrie e. V. (VDA), 2015. Available from: https://www.vda.de/dam/vda/ publications/2015/automation.pdf [8] PwC (PricewaterhouseCoopers), an annual report: Potential impacts of automated driver assistance systems (ADAS) and autonomous car technologies on the insurance industry, 2015, Volume 7. [9] NHTSA, Preliminary Statement of Policy Concerning Automated Vehicles.Available from: http://www.nhtsa.gov/staticfiles/rulemaking/pdf/Automated_Vehicles_Policy.pdf [10] Bengler K, Dietmayer K, Farber B, et al. Three decades of driver assistance systems: review and future perspectives. IEEE Intell Transp Syst Mag. 2014;6(4):6–22. [11] She TB. Three models of preview control. IEEE Trans Hum Factors Electron. 1966;7(2): 91–102. [12] MacAdam CC. Application of an optimal preview control for simulation of closed-loop automobile driving. IEEE Trans Syst Man Cybern. 1981;SMC-11(6):393–399. [13] MacAdam CC. An optimal preview control for linear systems. J Dyn Syst Meas Control. 1980;102(3):188–190. [14] Ungoren AY, Peng H. An adaptive lateral preview driver model. Veh Syst Dyn. 2005;43(4): 245–259. [15] Sharp RS, Casanova D, Symonds P. A mathematical model for driver steering control, with design, tuning and performance results. Veh Syst Dyn. 2000;33(5):289–326. [16] Sharp RS, Valtetsiotis V. Optimal preview car steering control. Veh Syst Dyn Suppl. 2001;35:101–117. [17] Pick AJ, Cole DJ. A mathematical model driver steering control including neuromuscular dynamics. J Dyn Syst Meas Control. 2008;130:031004-1–031004-9. [18] Saleh L, Chevrel P, Lafay JF. Chapters: optimal control with preview for lateral steering of a passenger car: design and test on a driving simulator. In: Sipahi R, et al. editors. Time delay systems: methods, applications and new trends. LNCIS 423. Berlin, Heidelberg: Springer-Verlag; 2012. p. 173–185. [19] Guo J, Li L, Li K, et al. An adaptive fuzzy sliding lateral control strategy of automated vehicles based on vision navigation. Veh Syst Dyn. 2013;51(10):1502–1517. [20] Odhams AMC, Cole DJ. Application of linear preview control to modelling human steering control. Proc Inst Mech Eng D, J Automob Eng. 2009;223(3):835–853. [21] Cole DJ, Pick AJ, Odhams AMC. Predictive and linear quadratic methods for potential application to modelling driver steering control. Veh Syst Dyn. 2006;44(3):259–284. [22] Timings JP, Cole DJ. Vehicle trajectory linearization to enable efficient optimization of the constant speed racing line. Veh Syst Dyn. 2012;50(6):883–901.


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[23] Raffo GV, Gomes GK, Normey-Rico JE. A predictive controller for autonomous vehicle path tracking. IEEE Trans Intell Transp Syst. 2009;10(1):92–102. [24] Attia R, Orjuela R, Basset M. Coupled longitudinal and lateral control strategy improving lateral stability for autonomous vehicle. In 2012 American Control Conference, Fairmont Queen Elizabeth, Montréal, Canada, June 27–June 29; 2012:6509–6514. [25] Falcone P, Borrelli F, Asgari J, et al. Predictive active steering control for autonomous vehicle systems. IEEE Trans Control Syst Technol. 2007;15(3):566–580. [26] Menhour L, Lechner D, Charara A. Design and experimental validation of linear and nonlinear vehicle steering control strategies. Veh Syst Dyn. 2012;50(6):903–938. [27] Attia R, Orjuela R, Basset M. Nonlinear cascade strategy for longitudinal control in automated vehicle guidance. Control Eng Pract. 2014;29(8):225–234. [28] Pacejka H. Tire and vehicle dynamics. Oxford: Butterworth-Heinemann; 2006. [29] Sattel T, Brandt T. From robotics to automotive: lane-keeping and collision avoidance based on elastic bands. Veh Syst Dyn 2008;46(7):597–619. [30] Brayton RK, Director SW, Hachtel GD, et al. A new algorithm for statistical circuit design based on quasi-newton methods and function splitting. IEEE Trans Circuits Syst. 1979;26:784–794. [31] Vavasis SA. Quadratic programming is in NP. Inf Process Lett. 1990;36(2):73–77. [32] Kozlov MK, Tarasov SP, Khachiyan LG. The polynomial solvability of convex quadratic programming. USSR Comput Math Math Phys. 1980;20(5):223–228. [33] Klee Victor L, Minty GJ. How good is the simplex method? In: Shisha O, editor. Inequalities III. New York: Academic Press; 1972: p. 159–175. [34] Maes CM. A regularized active-set method for sparse convex quadratic programing [PhD thesis]. Stanford University; 2011. [35] Rajamani R. Vehicle dynamics and control. Berlin: Springer Science & Business Media; 2011.

Appendix 1. Tyre combined slip theory There exists a very well-known empiric formula called ‘The Magic Formula’ (MF) [28] for those pure side slip (or longitudinal slip) and was used quite widely in vehicle engineering. Fx0j (sj ) = Dxj sin( Cxj arctan( Bxj κj − Exj (Bxj κj − arctan( Bxj κj )))),

(A1)

(A2) Fy0j (αj ) = Dyj sin( Cyj arctan( Byj αj − Eyj (Byj αj − arctan( Byj αj )))). With the table data of the lateral tyre force respect to tyre sideslip angle under the different vertical loads afforded, the pure sideslip force under an arbitrary vertical load could be obtained by linear interpolation method with an input of tyre sideslip angle. Similar approach could be applied for the condition of the pure longitudinal slip. Figure A1 shows the original data of the tyre force respecting to the slip under different loads, as well as the data obtained by MF using the parameters in Table A1. However, when dealing with the situation with a combined slip, though Pacejka et al. also presented a pure empiric equation, in order to use these tyre table data properly, the tyre force with combined slip is calculated with a basis on Equations (A1) and (A2), we will give an overall procedure in the following parts. At first, in order to simulate the tyre behaviour on different surfaces, given a friction coefficient tyre measurements μ0 (μ0 = 1 in our paper) and the friction coefficient for the simulated conditionμ, the table functions defined in Equations (A1) and (A2) are modified as follows, μ μ0 Fxj (s) = Fx0j κ , (A3) μ0 μ μ μ0 Fy0j α . (A4) Fyj (α) = μ0 μ The longitudinal and lateral slips are combined to obtain the total combined slip as, (A5) σo = σx2 + σy2 ,

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Figure A1. Tyre lateral and longitudinal characteristic curves. where σo denoting the magnitude of the theoretical slip vector, and σx =

κ , 1 + |κ|

(A6)

tan α . (A7) 1 + |κ| (Notes: the camber effect of the tyre is not considered.) Still, we will take the lateral and longitudinal components but then of the respective pure slip characteristics Fx0 and Fy0 . If we want to use the theoretical slip quantities σ0 , we must have available the pure slip characteristics with σx and σy as abscissa. This could be simply realised by the following relationships, according to: σx , (A8) Fσ xj (σx ) = Fxj 1 − |σx | σy =


43

Table A1. Parameters of the tyre magic formula. Magic formula constant Longitudinal Lateral

B

C

D

E

12.817 −0.917

1.350 1.260

μ0 Fz μ0 Fz

0.045 2.300

Fσ yj (σy ) = Fyj (arctan( σy )). Finally, the longitudinal force and the lateral force with combined slip are calculated by σxj Fσ xj (σj ), Fxwj (κi , αi ) = σj

(A9) (A10)

σyj Fσ yj (σj ). (A11) σj As for the calculating of the tyre vertical loads, assuming a fixed roll axis position and a constant roll stiffness distribution, the expression of the vertical load for each wheel could be obtained by: Fywj (κi , αi ) =

Fz1 = ε

Fdf max h mgb may bh Mφ f + , − + − 2 2L 2L LWf Wf

(A12)

Fdf max h mgb may bh Mφ f − , (A13) − + + 2 2L 2L LWf Wf max h mga may ah Mφ r Fdf + + − + Fz3 = (1 − ε) , (A14) 2 2L 2L Wr L Wr max h mga may ah Mφ r Fdf + + + − Fz4 = (1 − ε) , (A15) 2 2L 2L Wr L Wr where ε is the aero balance distribution coefficient, Fdf is the aero down force and calculated by Fdf = (1/2)Cz Ar u2 , and asy = V( β˙ + r) − hφ¨ is the centrifugal acceleration of the sprung mass, the index k ∈ {f , r} represents front or rear wheel. The longitudinal tyre slip ratio for each wheel is defined by, Fz2 = ε

κ1 =

Rw ω1 − 1, u − 0.5Wf r

(A16)

Rw ω2 − 1, (A17) u + 0.5Wf r Rw ω3 − 1, (A18) κ3 = u − 0.5Wr r Rw ω4 − 1. (A19) κ4 = u + 0.5Wr r Obviously, κk > 0 means the vehicle is accelerating, while κk < 0 means the vehicle is decelerating. Additionally, the tyre slip angle for each wheel is approximated as follows, κ2 =

α1 = arctan

v + ar − δ, u − 0.5Wf r

v + ar − δ, u + 0.5Wf r v − br α3 = arctan , u − 0.5Wr r v − br α4 = arctan . u + 0.5Wr r

α2 = arctan

(A20) (A21) (A22) (A23)

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Appendix 2. The state-space form of an 8DOF nonlinear vehicle model x˙ 1 = x2 .

(A24)

4 1 Mzk , Iz

x˙ 2 =

(A25)

k=1

4 1 1 ms h 2 Fxk − Cx ρAr x3 + x2 x4 + x2 x5 , x˙ 3 = m 2 m

(A26)

k=1

x˙ 4 =

1

J

mJ − (ms h)2

4

Fyk + ms hMx − x2 x3 ,

x˙ 5 = x6 , x˙ 6 =

(A27)

k=1

1 mJ − (ms h)2

ms h

4

(A28) Fyk + mMx ,

(A29)

k=1

x˙ 7 = x3 cos (x1 ) − x4 sin (x1 ),

(A30)

x˙ 8 = x3 sin (x1 ) + x4 cos (x1 ),

(A31)

x˙ 9 =

T1 Rw − Fx1 , Jw Jw

(A32)

x˙ 10 =

T2 Rw − Fx2 , Jw Jw

(A33)

x˙ 11 =

T3 1 − Fx3 Rw , Jw Jw

(A34)

x˙ 12 =

T4 1 − Fx4 Rw , Jw Jw

(A35)

where, J = Ix + ms h2 W W (Fx2 + Fx4 ) − (Fx1 + Fx3 ) + a(Fy1 + Fy2 ) − b(Fy3 + Fy4 ) 2 2 Mx = −(Dφf + Dφr )x6 + (ms gh − (Kφf + Kφr ))x5 ⎤⎡ ⎡ ⎤ ⎡ ⎤ Fxwi cos δ −sin δ 0 0 Fxi cos δ 0 0⎥ ⎢Fywi ⎥ ⎢Fyi ⎥ ⎢ sin δ , {i, j} = {{1, 3}, {2, 4}}. ⎣F ⎦ = ⎣ 0 0 1 0⎦ ⎣Fxwj ⎦ xj 0 0 0 1 Fxj Fywj Mz =

Reform Equations (A24) and (A35) as the state-space form, State variables x = [x1 , x2 . . . , x12 ]T , input variables u = [Fxw1 , . . . , Fxw4 , Fyw1 , . . . , Fyw4 , T1 , . . . , T4 , δ]T . The system could be expressed as a nonlinear state-space form x˙ = f (x, u, t), y = g(x, u, t).

(A36)


45

Table A2. The state variables and their descriptions. State variables x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12

Description Yaw Yaw rate Longitudinal velocity Lateral velocity Roll Roll rate Global X position Global Y position Front left wheel angular velocity Front right wheel angular velocity Rear left wheel angular velocity Rear right wheel angular velocity

Table A3. Parameters and values for the vehicle chassis model. Parameters

Value/Type

Unit

Description

m ms Ix Iz h hcg a b Wf , Wr G Kf , Kr Df , Dr ρ Cz Ar g ε

2000 1850 928 3234 0.48 0.59 1.40 1.65 1.60, 1.65 18 44,000 5000 1.225 0.18 2.8 9.8 0.5

kg kg kg · m3 kg · m3 m m m m m – N · m/rad N · m/(rad · s) kg/m3 – m2 kg · m/s2 –

The total mass of the vehicle The unsprung mass of the vehicle Vehicle roll inertia Vehicle yaw inertia The arm length of the rolling The height of the CoG Front axle distance to CoG Rear axle distance to CoG Front and rear wheel base Ratio of the SWA to wheel angle Front and rear roll stiffness Front and rear roll damping The air density Down force coefficient Cross-section area The gravitational constant Aero balance distribution coefficient

Appendix 3. The shifting logic of the gearbox The gear shifting is automatically conducted which is based on the transmission speed, upshifting occurs when the transmission speed rises above a specified level or a throttle position drops below a specified level, and similarly, downshifting occurs when the transmission speed decreases below a specified level or the throttle position rises above a specified level. A Stateflow model in Simulink is shown in Figure A3, which interprets how the gearbox accomplish its shifting. The model computes

Figure A2. The fear shifting schedule.

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H. CAO ET AL.

Figure A3. The shifting logic model in Simulink Stateflow. the upshift and downshift speed thresholds as a function of the instantaneous values of gear and throttle, as shown in Figure A2. While in a steady state, the model compares these values to the present transmission speed to determine if a shift is required. If so, it enters one of the confirm states (upshifting or downshifting), if the transmission speed no longer satisfies the shift condition, while in the confirm state, the model ignores the shift and it transitions back to steady state. This prevents extraneous shifts due to noise conditions. If the shift condition remains valid for a duration of 0.3 seconds, the model transitions through the lower junction and, depending on the current gear, it broadcasts one of the shift events. Subsequently, the model again activates steady state after a transition through one of the central junctions. The shift event, which is broadcast to the gear selection state, activates a transition to the appropriate new gear.

An optimal model-based trajectory following

An optimal model-based trajectory following

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