vehicle steer by wire

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 63, NO. 4, APRIL 2016 2251

Robust Control of a Vehicle Steer-by-WireSystem Using Adaptive Sliding Mode

Zhe Sun, Jinchuan Zheng, Member, IEEE , Zhihong Man, Member, IEEE ,and Hai Wang, Member, IEEE

Abstract—This paper presents an adaptive sliding-mode(ASM) control methodology for a vehicle steer-by-wire(SbW) system. First, the SbW system is modeled as asecond-order system from the steering motor input volt-age to the front-wheel steering angle. For simplicity, theself-aligning torque and friction arising from the tire-to-ground contact are regarded as external disturbance actingon the SbW system. Next, an ASM controller is designedfor the SbW system, which can not only cope with theparametric uncertainties in the plant model but also esti-mate the coefficient of the self-aligning torque effectively.The stability of the ASM control system is proved in thesense of Lyapunov and the guidelines for selecting thecontrol parameters are given. Finally, experiments are car-ried out for steering control to respectively follow a slalompath and a circular path under various road conditions. Itis shown that the proposed ASM controller can achievestronger robustness against various road conditions lead-ing to significantly smaller tracking errors in comparisonwith a conventional sliding-mode controller and a linearH∞ controller.

Index Terms—Adaptive sliding mode (ASM), robustcontrol, self-aligning torque, vehicle steer-by-wire (SbW)system.

I. INTRODUCTION

A S a newly developed technology, drive-by-wire technol-ogy has been playing a significant role in the automotive

industry. The main idea of drive-by-wire is to use electrome-chanical actuators and electronic control systems to replace theconventional mechanical linkages [1]. Different componentsof a vehicle constitute different parts of the by-wire technol-ogy, such as steer-by-wire (SbW) [2], [3], brake-by-wire [4],[5], shift-by-wire [6], and throttle-by-wire [7] technologies. Asone part of the drive-by-wire systems, SbW system is an inno-vative technology for automobile steering applications, whichpossesses several remarkable advantages. First, the removal ofthe steering column can eliminate the injury to a driver whenthere is a front-end collision to a large extent. Second, in a

Manuscript received February 11, 2015; revised September 23, 2015;accepted October 21, 2015. Date of publication November 10, 2015;date of current version March 8, 2016.

Z. Sun, J. Zheng, and Z. Man are with the School of Software andElectrical Engineering, Swinburne University of Technology, Melbourne,Vic., 3122, Australia (e-mail: [email protected]; [email protected];[email protected]).

H. Wang is with the School of Electrical Engineering and Automation,Hefei University of Technology, Hefei 230009, China (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIE.2015.2499246

SbW system, the traditional mechanical linkages and hydraulicsystems between the steering wheel and front wheels arereplaced by electromechanical actuators and human–machineinterfaces, which not only reduce weight, noises, vibrationsand energy consumptions but also increase the freedom andcapability for a driver to tune the steering command, such thatthe stability and maneuverability of a vehicle are improved[8]. Finally, a feedback motor is typically coupled to thesteering wheel and provides the driver with mimic haptic feed-backs of the interaction forces between front wheels and roadsurface.

As the mechanical structure of a SbW system is differentfrom traditional steering systems, large amounts of studies havebeen carried out on the mechanical modeling of SbW systemsin recent years. In [9], Gadda et al. used dc permanent mag-net motors as the steering motors and reasonably modeled theelectrical properties as the series connection of an inductor,a resistor, and a voltage source. In [10], Wang et al. utilizedtwo second-order differential equations to describe the effect oftire forces in the front wheel and the dynamics in the steeringwheel, respectively. A simple second-order model is adopted in[11] to represent the dynamics of SbW systems, where the self-aligning torque was treated as an external disturbance resistingthe steering motion away from the straight-ahead position. Inaddition, simplified second-order mathematical models wereused in [12] to describe the dynamics of the driver interfacesubsystem and the direction control subsystem, respectively.

Apart from the modeling investigation, an appropriate con-trol system is essential for the SbW system to regulate thesteering motor to generate the desired steering torques such thatthe front wheels follow the steering wheel reference commandaccurately for paths following. A variety of control method-ologies for vehicle steering systems are reported in literature,e.g., the sliding-mode-based learning control [13], robust H∞control [14], iterative learning control [15], and neural networkcontrol [16]. In particular, Yamaguchi and Murakami [17] pre-sented an adaptive control system to estimate the tire corneringstiffness, which is then used to realize the desired steering char-acteristics. In [18], Falcone et al. presented a model predictivecontrol scheme for the steering system of an autonomous vehi-cle, which computes the steering angle of front tires to follow aknown trajectory on slippery roads at possibly highest speed. In[19], a haptic feedback control strategy with adjustable controlgains is provided for a vehicle steering system to increase thedrivers’ awareness of the roadway and driving conditions and toenable the driver to avoid obstacles appearing in the road earlierthan that without feedback informations.

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2252 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 63, NO. 4, APRIL 2016

Sliding-mode control is a control methodology with strongrobustness to handle the problems of parametric uncertain-ties existing in plant models and only the upper and lowerbounds of the parameters in the plant model are needed.In [20], Wang et al. utilized a nominal feedback controllerto stabilize the nominal part of the plant model and aug-mented a sliding-mode compensator to cope with the influenceof both the unknown system dynamics and uncertain roadconditions for a vehicle steering system. Adaptive control isanother control method which plays an important role in thearea of motion control owing to its effectiveness to deal withtime-varying parameters in a plant model. In [21], Ma andNejhad proposed an independent modal space adaptive con-trol scheme and an adaptive fuzzy logic control strategy forthe manipulator vibration control and the active strut motioncontrol, respectively. In [22], an adaptive fuzzy controlleris proposed for an MEMS triaxial gyroscope, which elim-inates the effect of system nonlinearities without the needof accurate mathematical plant models. It is also quite com-mon for researchers to combine sliding-mode control withadaptive control to obtain more comprehensive advantages[23]–[28]. For instance, in order to attain accurate positioning,Sencer and Shamoto presented a robust adaptive sliding-mode(ASM) control method to reject harmonic disturbances in servosystems [29].

In this paper, we present an ASM control methodology for adeveloped SbW system, where the adaptive control is utilized toestimate the coefficient of the self-aligning torque acting on thesteering system. Then, feedforward control inputs equivalent tothe estimated self-aligning torques are generated to compen-sate for the effect of the self-aligning torques. Furthermore, afeedback controller based on sliding mode is adopted to copewith the system parametric uncertainties. The stability of theASM controller on the SbW system is proved in the sense ofLyapunov. The guidelines for selecting the control parametersare also given. Finally, experimental results are presented todemonstrate the superior performance of the proposed ASMcontroller in steering control for both a slalom path followingand a circular path following compared with a linear H∞ con-troller and a conventional sliding-mode controller for the SbWsystem.

The main contribution of this paper is that we combinean adaptation method with the sliding-mode control, which istheoretically proved to guarantee the tracking error to asymp-totically converge to zero when applied to the SbW system.The adaptation control law not only alleviates the chatter-ing problem in the sliding-mode controller but also providesexplicit estimation of the self-aligning torque that can bedirectly used by the steering wheel feedback motor to providethe driver with road feels. Furthermore, experimental resultsdemonstrate that the developed controller is simple for practi-cal implementation as well as effective to improve the trackingaccuracy.

This paper is organized as follows. Section II describes thedynamic plant model of the SbW system. In Section III, theASM controller is designed for the SbW system. Section IVpresents and compares the experimental results of the ASM

Fig. 1. Experimental setup of an SbW system.

controller, the conventional sliding-mode controller, and theH∞ controller. Section V concludes this paper.

II. PLANT MODELING

Fig. 1 shows our experimental setup for a SbW system,where the conventional mechanical linkage between the steer-ing wheel and the front wheels is removed and replaced bya steering motor (Mitsubishi HF-SP102). The steering motoris then controlled to provide appropriate torques to steer thefront wheels through a gear head, a pinion and rack gear boxand steering arms. The controller for the SbW system is imple-mented on a HP personal computer (PC) by using MATLABReal-Time Workshop and an Advantech PCI multifunction cardto collect sensor signals and to generate control input signals inreal time. A motor servo driver is used to convert the controlinput signals to current signals to drive the steering motor forsteering the front wheels. An angle sensor (MoTeC) is installedon the pinion to measure the rotation angle of the pinion.Multiplying the angle measurement by the transmission gainfrom the pinion to the front wheels yields the steering angle ofthe front wheels, which is the system output to be controlled inour SbW system. In Fig. 1(b), an angle sensor is also installedon the steering wheel to detect the angle of the steering wheelhandled by a driver. Similarly, multiplying this angle measure-ment by a scaling factor from the steering wheel to the frontwheels yields the steering wheel reference angle, which is actu-ally the reference command for the steering motor controller totrack.

SUN et al.: ROBUST CONTROL OF A VEHICLE SBW SYSTEM 2253

Based on a simplified bicycle model for vehicles [11], weshall express the plant model of the SbW system as

Jex+Bex = κu− fc − τsel

κ = κ1 · κ2 · κ3 · κ4fc = ξf sign(x)

(1)

where x represents the steering angle of the front wheels, u isthe control input of the steering motor, Je andBe are the equiv-alent moment of inertia and the viscous friction of the steeringsystem, respectively; fc is the Coulomb friction with ξf theCoulomb friction constant; τsel is the self-aligning torque actingon the steering system; κ1 is the scale factor accounting for theconversion from the steering motor input voltage to the steeringmotor output torque; κ2 is the gear ratio of the gear head; κ3 isthe gear ratio of the pinion and rack system; κ4 is the scale fac-tor to account for the transmission from the linear motion of therack to the steering angle of front wheels; and sign(·) denotesthe standard signum function.

Note that in our experimental setup, the steering motor servodriver has a much higher bandwidth than that of the mechan-ical dynamics of the SbW system. Thus, we simply use thescaling factor κ1 to approximate the model from the steeringmotor input voltage to the steering motor output torque [10].The values of the scaling factors contained in κ are given by

κ1 = 1.8

κ2 = 8.5

κ3 = 3.0

κ4 = 6.0.

(2)

Since the value of κ is fairly constant in time in our setup, itis simply regarded as a constant during the control design aswill be addressed later. However, in this paper, we consider thefollowing parametric uncertainties with the bounds given by:

|ΔJe| = |Je − Je0| ≤ ΔJe

(Je0 = 85.5 kgm2, ΔJe= 0.1Je0)

|ΔBe| = |Be −Be0| ≤ ΔBe

(Be0 = 218.8 Nms/rad, ΔBe= 0.1Be0)

|Δξf | = |ξf − ξf0| ≤ Δξf

(ξf0 = 4.2 Nm, Δξf = 0.1ξf0)

(3)

where Je0, Be0, and ξf0 denote the nominal parameters in ourplant model; ΔJe

, ΔBe, and Δξf are the parametric uncer-

tainties; ΔJe, ΔBe

, and Δξf represent the bounds of thecorresponding parameters, respectively.

As shown in Fig. 1, limited by the experiment equipment,our SbW platform cannot provide forward velocity for the frontwheels. Besides, the lack of vehicle weight distributed on thefront wheels results in the deficiency of the front-wheel cam-ber angle [8]. Hence, during the experiments, the tires are notexerted by actual self-aligning torques. However, under theassumption of small slip angles of the tire, we can utilize ahyperbolic tangent signal [10], [12] to mimic the self-aligningtorque, which can be expressed as

τsel = ρτ tanh(x) (4)

where ρτ is a time-varying coefficient with respect to variousroad conditions, and tanh(·) represents the hyperbolic tangentfunction

tanh(z) =e2z − 1

e2z + 1. (5)

Now, it is easy to obtain the following voltage signal:

usel =τsel

κ+ ns =

ρτκ

tanh(x) + ns (6)

which is artificially generated during the experiments to mimicthe actual self-aligning torque acting on the SbW system. Notethat ns denotes a white noise signal added to account forthe unmodeled dynamics and uncertainties in the self-aligningtorque. In this way, we can evaluate the robustness of thedeveloped controller to some extent through experiments.

III. CONTROL DESIGN

Our control objective is to design a robust controller such thatthe front-wheel steering angle can track a reference commandnot only fast but also accurately in the presence of modelinguncertainties and external disturbances. In order to improvetracking accuracy and robustness, an ASM controller is devel-oped to handle parametric uncertainties in the plant model andestimate the coefficient of the self-aligning torque which is thenused to design a feedforward controller to compensate for theself-aligning torque that is the primary disturbance. Finally, wealso present a conventional sliding-mode controller and an H∞controller for comparison.

A. ASM Control for SbW Systems

Define the tracking error e as

e = xr − x (7)

where xr is the steering wheel reference command assumed tobe twice differentiable. Furthermore, a linear sliding variable sis defined as

s = e+ λe (8)

where λ > 0 is to be designed.Solving the sliding-mode dynamics as given by

s = λe+ (xr − x) = 0 (9)

and neglecting the parametric uncertainties and the self-aligning torque, namely, utilizing the nominal parameters toreplace the actual ones in (1) and supposing τsel = 0, we obtainan expression of u0 which is also called the equivalent controlinput [30] as follows:

u0 =1

κ[Je0λe+ Je0xr +Be0x+ ξf0sign(x)]. (10)

In order to guarantee the robustness of the controller againstthe parametric uncertainties and the Coulomb friction fc, areaching control input u1 [31] is introduced as

u1 =1

κ[�s+Ksign(s)] (11)


where � > 0 is to be designed, and K is given by

K = ΔJeλ|e|+ ΔJe

|xr|+ ΔBe|x|+ Δξf . (12)

Lastly, a control input u2 is proposed to exclusively compensatefor the self-aligning torque acting on the SbW system

u2 =ρτκ

tanh(x) (13)

where ρτ denotes the estimation of the actual coefficient ρτ ,whose adaptation law is given by

˙ρτ = μstanh(x) (14)

with μ > 0 denotes the adaptation gain [29] to be designed ands the sliding variable as defined in (8). The design of μ will bedetailed in Section III-B.

Lemma 1: Consider the SbW system (1) with the parametricuncertainties in (3) and under the ASM control law

u = u0 + u1 + u2 (15)

with u0 defined in (10), u1 in (11), and u2 in (13), respectively.Then, the tracking error (7) of the SbW closed-loop systemcan asymptotically converge to zero for a given steering wheelreference command.

Proof: Choose the Lyapunov function as

V =1

2s2 +

1

2μJe(ρτ − ρτ )

2. (16)

Evaluating the first-order derivative of V along the systemtrajectories yields

V = ss+1

μJe(ρτ − ρτ )(ρτ − ˙ρτ ). (17)

For a specific road condition, we assume ρτ is a constantimplying ρτ = 0. Therefore, we have

V = ss+1

μJe(ρτ − ρτ )(− ˙ρτ )

= ss− 1

Je(ρτ − ρτ )stanh(x)

= s(λe+ xr − x)− 1


= s

[λe+ xr +

Be

Jex+

ξfJe

sign(x) +ρτJe

tanh(x)

− κu

Je

]− 1

Je(ρτ − ρτ )stanh(x). (18)

Substituting the control input (15) into (18) yields

V = s

{λe+ xr +

Be

Jex+

ξfJe

sign(x) +ρτJe

tanh(x)

− κ

Je

1

κ[Je0λe+ Je0xr +Be0x

+ ξf0sign(x) +�s+ ρτ tanh(x)

+ Ksign(s)]} − 1


= s

[ΔJe

Jeλe+

ΔJe

Jexr +

ΔBe

Jex

+Δξf

Jesign(x)− K

Jesign(s)− �

Jes

]

= − �

Jes2 +

Kms−K|s|Je

(19)

with

Km = ΔJeλe+ΔJe

xr +ΔBex+Δξf sign(x). (20)

According to the following inequalities:

ΔJeλes ≤ ΔJe

λ|e||s|ΔJe

xrs ≤ ΔJe|xr||s|

ΔBexs ≤ ΔBe

|x||s|Δξf sign(x)s ≤ Δξf |s|

(21)

we have

ΔJeλes+ΔJe

xrs+ΔBexs+Δξf sign(x)s

≤ ΔJeλ|e||s|+ ΔJe

|xr||s|+ ΔBe|x||s|+ Δξf |s| (22)

namely,

Kms ≤ K|s|. (23)

Based on the above analysis, we can easily conclude that

V = −�

Jes2 +

Kms−K|s|Je

< 0. (24)

The proof is thus completed. �Remark 1: In the ASM controller, we employ the adaptation

law to estimate the coefficient of the self-aligning torque ρτwithout the need of any prior information for ρτ . This is thekey benefit of adaptive estimation because the road conditionsare typically unknown to the vehicle system in reality. On theother hand, for the uncertain parameters Je, Be, and ξf , theirnominal values and bounds can be accurately identified offline.Thus, by using the sliding-mode control, the effects of theseuncertainties on the system performance can be compensatedfast and effectively.

Remark 2: The control input u1 (11) contains a discontinu-ous term Ksign(s) which may induce undesired chattering tothe control signal. To alleviate this effect, the boundary layertechnique [32], [33] can be adopted. More specific, we use thefollowing saturation function to replace the signum functionin (11):

sat(z) =

{z/ψ, if |z| < ψ

sign(z), if |z| ≥ ψ(25)

where ψ denotes the boundary layer thickness.

B. Estimation of Coefficient of Self-Aligning Torque

As aforementioned, the self-aligning torque can be estimatedby adaptively estimating its coefficient ρτ as shown in (14). To


facilitate the adjustment for either good estimation accuracy orfast estimation rate, we set the adaptation gain μ as a linear filter[29] as follows:

μ = μ1 + μ2p (26)

where μ1 > 0, μ2 > 0, and p is the Laplace operator. Notethat the derivative of the sliding variable s is given in (19).Substituting s and (26) into the adaptation law described in (14)yields

˙ρτ = μstanh(x)

= μ1stanh(x) + μ2stanh(x)

= μ1stanh(x) + μ2[−�sJe

+tanh(x)Je

ρτ

− tanh(x)Je

ρτ +Km −Ksign(s)

Je]tanh(x)

= μ1stanh(x)− μ2�stanh(x)Je

+μ2tanh2(x)

Jeρτ

− μ2tanh2(x)

Jeρτ + w (27)

with

w =μ2

Je[Km −Ksign(s)]tanh(x). (28)

Rewrite (27) by using the Laplace transform, we have[μ2tanh2(x)

Je+ p

]ρτ =

μ2tanh2(x)

Jeρτ

+(μ1Je − μ2�)stanh(x)

Je+ w.

(29)

It can be seen that if there are no parametric uncertainties exceptthe self-aligning torque acting on the SbW system, selecting thedesign parameter μ1 as

μ∗1 = μ2

�

Je(30)

leads to the term (μ1Je − μ2�)stanh(x)/Je = 0 as well asw = 0. Thus, the adaptation dynamic equation (29) reduces to

ρ∗τ =μ2tanh2(x)/Je

μ2tanh2(x)/Je + pρτ (31)

where μ∗1 and ρ∗τ denote the parameters under the ideal condi-

tion without uncertainties. However, due to the existing para-metric uncertainties in the plant model and also because theequivalent moment of inertia Je is not completely known inpractice, we simply use the nominal parameter Je0 to replaceJe in (30), i.e.,

μ1 = μ2�

Je0. (32)

Substituting (32) into (29) yields

ρτ =μ2tanh2(x)/Je

μ2tanh2(x)/Je + pρτ + da (33)

with da referred to as the adaptation perturbation, which isgiven by

da =μ2

Je[Km −Ksign(s)]tanh(x)

+ΔJe

Je0Jeμ2�stanh(x). (34)

It is noted from (34) that the adaptation perturbation da isbounded in the presence of uncertainties and actually reducesto zero without parametric uncertainties. This implies that ρτwill converge to the actual ρτ in the case of no parametricuncertainties or approach to a region close to the actual ρτwhen uncertainties arise. Moreover, we can see that the valueof μ2 affects the convergence rate of the estimation (i.e., theadaptation bandwidth). The selection of μ2 will be discussed inSection III-C.

C. Selection of Controller Parameters

To this end, we have presented the ASM control law andthe adaptation law for the self-aligning torque. It is clear thatthe stability of the overall control system can be guaranteedin the sense of Lyapunov. However, for practical implementa-tion, the controller parameters should be carefully selected totradeoff among the tracking accuracy and robustness against themeasurement noises, the system uncertainties, and unmodeledsystem dynamics.

1) Selection of λ : The parameter λ crucially determinesthe tracking bandwidth of the sliding-mode function as given by(8) and the decay rate of tracking errors on the sliding surface[29], [31]. A larger λ leads to a faster response rate and highertracking accuracy, which, however, may bring excessive high-frequency measurement noises to the system that deteriorate thetracking accuracy inversely. To account for this tradeoff, we setλ = 15 in our case.

2) Selection of � : The term with the parameter � in(11) is introduced to enforce the tracking error onto the slid-ing surface [29]. It is obvious that the control system band-width becomes higher by increasing � which leads to a fasterresponse to the reference command as well. Nevertheless, alarger � will similarly amplify the measurement noises fromthe sliding variable s and meanwhile result in a larger adap-tation perturbation da that degrades the estimation accuracy.From the actual implementations, we find that � = 45 is anacceptable value.

3) Selection of μ2 : From (33), we can see that the param-eter μ2 critically determines the adaptation bandwidth for theestimation of the self-aligning torque. Apparently, a larger μ2

leads to a higher adaptation bandwidth implying a faster con-vergence rate for the estimated coefficient ρτ to track the actualone ρτ . However, we also note that the existence of μ2 reverselyincreases the undesired adaptation perturbation da in (34).Hence, a satisfactory μ2 = 2638 is chosen for implementation.

4) Selection of ψ : It is well known that a larger value ofthe boundary layer thickness ψ (24) leads to less chattering butat the cost of reducing tracking accuracy [32]. In our paper, thedisturbance of self-aligning torque is compensated by a sep-arate control input (13), which means that a smaller value of


Fig. 2. Estimation of the coefficient of the self-aligning torque ρτ undervarious road conditions in Case 1.

K in (11) would be sufficient to overcome the remaining uncer-tainties and disturbances. This implies that the chattering effectsdue to the sign function in (11) would be much less comparedto the conventional sliding-mode control. In actual implemen-tation, we set ψ = 0.8 to obtain a satisfied balance betweencontrol smoothness and tracking accuracy.

D. Two Controllers for Comparison

To compare the benefits of the proposed ASM controller,a conventional sliding-mode controller and a linear H∞ con-troller are also designed based on the methods employed in[30] and [34], respectively. Here, we straightly give the designresults for simplicity.

1) Conventional Sliding-Mode Controller: The conven-tional sliding-mode controller [30] is given by

ucsm =1

κ(Jeλ|e|+ Je|xr|+ Be|x|+ ξf + τsel)sat(s) (35)

where sat(·) is the same saturation function defined in (25) withthe boundary layer thickness equal to 1.0; Je, Be, and ξf denotethe upper bounds of the parameters Je,Be, and ξf , respectively;and τsel denotes the upper bound of the self-aligning torqueacting on the SbW system. For a fair comparison, the upperbounds of the parameters Je, Be, and ξf of the conventionalsliding-mode controller are given by

Je = Je0 + ΔJe

Be = Be0 + ΔBe

ξf = ξf0 + Δξf .

(36)

The values of these parameters have been given in (3). We cansee that the main difference between the proposed ASM controland conventional sliding-mode control lies in that the conven-tional sliding-mode control requires a priori upper bound valueof the self-aligning torque, which may be difficult to predict inreality due to the change of road conditions; while the ASMcontroller eliminates this requirement and is able to adaptivelyestimate the self-aligning torque for various road conditions andsubsequently cancel the resistant effect of self-aligning torquethrough a feedforward path, which, on the other hand, alleviatesthe control efforts of the sliding-mode controller.

Fig. 3. Control performances of the ASM controller in Case 1.(a) Tracking profiles. (b) Tracking errors. (c) Control inputs.

2) H∞ Controller: The H∞ controller is of the followingform:

uh = 0.31xr + 20.66e+ 9.06e+ 0.79x (37)

which is designed based on a state-space approach and canguarantee an optimal bounded tracking error in the presenceof the uncertainties [34].

IV. EXPERIMENTAL RESULTS

Experiments are carried out on the actual SbW setup to verifythe designed controllers with a sampling period of 1 ms. Todemonstrate the superiority of the ASM controller, we consider


Fig. 4. Control performances of the conventional sliding-mode controllerin Case 1. (a) Tracking profiles. (b) Tracking errors. (c) Control inputs.

two different road paths for the vehicle to follow, which thusrequire different steering reference commands from the steeringwheel. Various road conditions are also considered to evaluatethe tracking performance.

A. Case 1: Steering for a Slalom Path Following

In this case, the steering wheel as shown in Fig. 1(b) ismaneuvered to generate an approximate sinusoidal waveformwhich mimics the vehicle following a slalom path. Then, thesteering wheel angle sensor collects the corresponding steeringangle. By multiplying this angle sensor signal by the corre-sponding scaling factor from the steering wheel to the frontwheels, a steering wheel reference angle is obtained and input

Fig. 5. Control performances of the H∞ controller in Case 1.(a) Tracking profiles. (b) Tracking errors. (c) Control inputs.

to the controllers under test. We also set the values of thecoefficient of the self-aligning torque ρτ in (6) as

ρτ =

⎧⎪⎨⎪⎩155, 0 < t ≤ 20 s, Snowy road

585, 20 < t ≤ 40 s, Wet asphalt road

960, 40 < t ≤ 60 s, Dry asphalt road

(38)

to represent the change of the road conditions during the exper-iments. The experimental results for this case are shown inFigs. 2–5.

From Fig. 2, we can see that the designed adaptation lawcan properly estimate various coefficients of the self-aligningtorque ρτ with respect to the change of road conditions. It isseen that there exist oscillations in the curve of the estimated


Fig. 6. Estimation of the coefficient of the self-aligning torque ρτ inCase 2.

coefficient ρτ . This is primarily caused by the high-frequencymeasurement noises from the angle sensors on the steeringwheel and the front wheel. Though we have low-pass filteredthe high-frequency noises, the cutoff frequency of the filter can-not be set too low as we also need to guarantee the fidelity ofthe signals of the measured angles. It is also noted that when thesteering angle x is zero at the beginning of the experiment, theestimated coefficient ρτ is zero as well. This is because we cansee from (33) that x = 0 implies ρτ = 0. Therefore, the adapta-tion law requires the steering angles to be varying to guaranteea good estimation of the coefficient. To solve this problem, weactually can set the initial value of ρτ as the last estimated oneduring the previous driving cycle.

The results in Figs. 3–5 obviously show that the trackingperformance of the ASM controller is superior to those of theconventional sliding-mode controller and the H∞ controller.From Fig. 3, the peak tracking errors of the ASM controllerunder the three road conditions are 0.022, 0.025, and 0.028 rad,respectively. The peak tracking errors occur at the beginning ofeach change of the road conditions, at the time of which theadaptation law starts to estimate the new value of ρτ . Oncethe estimated coefficient ρτ converges to the actual one, thepeak tracking errors consistently converge into a small regionno matter how large the magnitudes of the self-aligning torqueare. In comparison, the peak tracking errors under the conven-tional sliding-mode controller and the H∞ controller as shownin Figs. 4 and 5 are much larger than that under the ASM con-troller and also become larger with respect to the increase of theself-aligning torque.

B. Case 2: Steering for a Circular Path Following

In this case, we consider a road path being straight and fol-lowed by a circular curve which is more common in reality.This is referred to as a circular path in this paper. Similarly,we maneuver the steering wheel to generate the steering refer-ence command for such a path following as shown in Fig. 7(a).Experiments with a duration of 15 s are carried out and the coef-ficient of the self-aligning torque ρτ is set as 700 to mimic thecondition on a wet asphalt road.

Fig. 6 shows the estimation of the coefficient of the self-aligning torque under the ASM controller. We can see that the

Fig. 7. Control performances of the ASM controller in Case 2.(a) Tracking profiles. (b) Tracking errors. (c) Control inputs.

estimated ρτ deviates a bit from the actual ρτ at the steadystate, which may be primarily due to the adaptation perturba-tion da in (33) being a constant in this case. At the time of 9 s,the estimation curve contains a notched response which stemsfrom the change of the steering angle into a reverse direction.The tracking profiles of the controllers are shown in Figs. 7–9.It can be seen that the ASM controller is still superior to theconventional sliding-mode controller and the H∞ controller inthis case. The peak tracking error under the ASM controlleris 0.055 rad, which is less than both the conventional sliding-mode controller (0.059 rad) and the H∞ controller (0.07 rad).More importantly, with online estimations of the coefficientof the self-aligning torque ρτ , the tracking accuracy of the


Fig. 8. Control performances of the conventional sliding-mode controllerin Case 2. (a) Tracking profiles. (b) Tracking errors. (c) Control inputs.

ASM controller is improved to a large extent with the trackingerror almost converging to zeros eventually. Comparatively, thetracking errors under the conventional sliding-mode controllerand the H∞ controller both contain significant steady-stateerrors which are due to the insufficient capability to compensatefor the self-aligning torque.

C. Results of Steering Wheel Feedback Motor Control

In a SbW system, the feedback motor control for the steer-ing wheel is also important because it provides haptic feed-back of road conditions to the driver. For this purpose, we

Fig. 9. Control performances of the H∞ controller in Case 2.(a) Tracking profiles. (b) Tracking errors. (c) Control inputs.

employ a simple control scheme for the steering wheel feed-back motor by making its control input proportional to theestimated reaction force at the tire–road interface [35]. Assuch, the driver is provided with road feels when counteractingthe adjustable feedback forces. More specifically, the dynamicequation of the steering wheel in our setup as shown in Fig. 1 isgiven by

Jswθ +Bswθ +Dswθ = τh − τf (39)

where θ is the steering wheel angle, τh is the driver input torque,and τf is the reaction torque generated by the feedback motor.Moreover, the model parameters equal to Jsw = 0.0791 kg ·m2,Dsw = 0.2 Nm/rad, andBsw = 0.15 Nms/rad, respectively.


Fig. 10. Control performance of the steering wheel feedback motor.(a) Steering wheel angle governed by driver and feedback motor.(b) Feedback motor control torque.

By utilizing the estimated coefficient of self-aligning torqueρτ in (14), we can design the control input of the feedbackmotor as

uf =1

ζρτ tanh(x) (40)

where x is the measured front-wheel angle in (1) and ζ is apositive scaling factor to adjust the level of the feedback forceaccording to the driver’s preference. In our setup, the feedbackmotor power driver is configured to have a current feedbackservo loop. Hence, the resultant feedback motor control torqueapproximately equals

τf ≈ κfuf (41)

where κf = 1.8 is the current-torque gain.Experiments of the feedback motor control are carried out

and Fig. 10(a) shows the measured steering wheel angle outputwhich is governed by the driver input torque and the feedbackmotor torque simultaneously. This steering wheel angle outputis then used as the reference command xr denoted in (7), whichis the front wheel’s control loop aimed to follow. Fig. 10(b)presents the feedback motor control torque in response to var-ious road conditions. During the experiments, the driver canobviously feel the resistance when the steering wheel is steeredto deviate from its original position and also the variationsof feedback torques when the road condition changes. These

results demonstrate that the developed estimator for the self-aligning torques is not only effective to improve the trackingaccuracy of the front wheel’s control but also useful for thefeedback motor control to provide the driver with road feels.

V. CONCLUSION

In this paper, we developed an ASM control scheme for theSbW system which combines the sliding mode and adaptivemethod such that the resultant control system is capable ofnot only overcoming the plant parametric uncertainties but alsocompensating for the dominant self-aligning torque disturbanceeffectively. The stability of the ASM control-based SbW systemis proved in the sense of Lyapunov. Moreover, a practical adap-tation law is designed for the estimation of the coefficient ofthe self-aligning torque. The selection guidelines of the con-troller parameters are given to account for the compromisebetween the measurement noises and the desired system per-formance in practice. Finally, experiment results are presentedto demonstrate the benefits of the ASM controller with bettertracking accuracy and stronger robustness against various roadconditions in comparison with the conventional sliding-modecontroller and the linear H∞ controller.

The self-aligning torque in this paper is simply modeledby an approximate function based on the assumptions that thevehicle mass, velocity, and other related tire parameters are con-stant. In our future work, we will relax these assumptions andinvestigate a complete self-aligning torque model with theseparameters being time-varying. Under this circumstance, wewill also develop advanced adaptive estimation and controlmethod to deal with the complexity of self-aligning torque. Onthe other hand, we are currently developing a new experimen-tal SbW system on a realistic vehicle and will subsequentlyevaluate the performance of the proposed control scheme inreal driving environments. It is anticipated that the proposedcontrol scheme can enable the SbW system to track steeringcommand appropriately in practice. Based upon this perfor-mance, we may further employ an effective method for alteringthe vehicle handling dynamics [11] by adjusting the steeringcommand.

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Zhe Sun was born in Hefei, China, in 1989. Hereceived the B.E. degree in traffic and vehicleengineering from Nanjing Agricultural University,Nanjing, China, in 2011. He is currently workingtoward the Ph.D. degree in science, engineer-ing, and technology at Swinburne University ofTechnology, Melbourne, Australia. His researchinterests include vehicle dynamics and con-trol, nonlinear control, adaptive control, andmechatronics.

Jinchuan Zheng (M’13) received the B.Eng.and M.Eng. degrees in mechatronics engi-neering from Shanghai Jiao Tong University,Shanghai, China, in 1999 and 2002, respec-tively, and the Ph.D. degree in electricaland electronic engineering from NanyangTechnological University, Singapore, in 2006.

In 2005, he joined the Australian ResearchCouncil (ARC), Centre of Excellence forComplex Dynamic Systems and Control,School of Electrical and Computer Engineering,

University of Newcastle, Callaghan, Australia, as a Research Academic.From 2011 to 2012, he was a Staff Engineer with the Western DigitalHard Disk Drive R/D Center, Singapore. Currently, he is serving as aLecturer at Swinburne University of Technology, Melbourne, Australia.His research interests include mechanism design and control ofhigh-precision mechatronic systems, sensing and vibration analysis,dual-stage actuation, and vision-based control.

Zhihong Man (M’94) received the B.E. degreefrom Shanghai Jiao Tong University, Shanghai,China, the M.Sc. degree from the ChineseAcademy of Sciences, Beijing, China, and thePh.D. degree from the University of Melbourne,Melbourne, Australia, in 1982, 1987, and 1994,respectively, all in electrical engineering.

From 1994 to 1996, he was a Lecturerwith the School of Engineering, Edith CowanUniversity, Joondalup, Australia. From 1996 to2001, he was a Lecturer and then a Senior

Lecturer with the School of Engineering, University of Tasmania, Hobart,Australia. From 2002 to 2007, he was an Associate Professor ofComputer Engineering at Nanyang Technological University, Singapore.From 2007 to 2008, he was a Professor and the Head of Electrical andComputer Systems Engineering, Monash University, Sunway Campus,Bandar Sunway, Malaysia. Since 2009, he has been with the Facultyof Science, Engineering, and Technology, Swinburne University ofTechnology, Melbourne, Australia, as a Professor. His research interestsinclude nonlinear control, signal processing, robotics, neural networks,and vehicle dynamics and control.


Hai Wang (M’13) received the B.E. degreefrom Hebei Polytechnic University, Tangshan,China, in 2007, the M.E. degree from GuizhouUniversity, Guiyang, China, in 2010, and thePh.D. degree from Swinburne University ofTechnology, Melbourne, Australia, in 2013,respectively, all in electrical and electronic engi-neering.

From 2014 to 2015, he was a PostdoctoralResearch Fellow with the Faculty of Science,Engineering, and Technology, Swinburne

University of Technology. Since 2015, he has been with the School ofElectrical and Automation Engineering, Hefei University of Technology,Hefei, China, as a Professor (Huangshan Young Scholar). His researchinterests include sliding-mode control, adaptive control, robotics, neuralnetworks, nonlinear systems, and vehicle dynamics and control.

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