Virtual Prototyping of an Electric Power Steering Simulator

274 IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 14, NO. 1, MARCH 2013

Virtual Prototyping of an ElectricPower Steering SimulatorLamri Nehaoua, Mohamed Djemaï, and Philippe Pudlo

Abstract—This paper presents a simulation tool for an electricalsteering system whose aim is twofold: 1) to investigate the possibil-ity of designing a minimum clearance mechatronic platform withsensorless control methods and 2) to evaluate assistance torquecontrol feedback by considering technological specifications andhuman factor consideration. The choice has been made for adriving simulator having at least a real steering system with anelectrical power steering (EPS) device and an adequate motor toreproduce the rack load force resulting from tire/road contact,as in a real driving situation. These components are gathered toform a virtual simulator platform, which serves as a basis forfuture realization. Our main contributions concern the vehicle’sfront assembly kinematics modeling and the evaluation of the loadrack force resulting from tire/road interaction. In addition, a realapplication of the most recent virtual sensor algorithms, arisingfrom the sliding-mode observer theory for states and unknowninput estimation, is described.

Index Terms—Driving simulation, power-steering system, vir-tual sensor.

I. INTRODUCTION

THE VEHICLE has greatly evolved in less than twodecades. Nowadays, a car has complex mechatronics,

where numerous safety/comfort devices and functionalitieshave been standardized. Among others, power-assisted steeringbecame an unavoidable system. It may be described as a drivertorque amplifier used to make vehicle steering easier in givendriving maneuvers. Many actuation technologies are used forpower-assisted steering systems, which can be classified intothree main categories: 1) hydraulic type; 2) electrical type; and3) a combination of the two, with noticeable interest withinthe automotive industry toward electrical power steering (EPS).It undoubtedly constitutes an intermediate step toward thesuppression of any mechanical connection between the driverand the vehicle wheels.

Manuscript received March 17, 2012; revised July 4, 2012; accepted July 21,2012. Date of publication August 24, 2012; date of current version February 25,2013. This work was supported in part by the French National ResearchAgency (ANR) VOLHAND Project “ANR-09-VTT-14-01/06,” by the In-ternational Campus on Transportation Safety and Intermodality, by theEuropean Community, by the Nord/Pas-de-Calais Region, and by the Na-tional Center for Scientific Research. The Associate Editor for this paper wasS. S. Nedevschi.

L. Nehaoua is with the Informatics, Integrative Biology and ComplexSystems Laboratory, University of Evry Val d’Essonne, 91 025 Evry, France(e-mail: [email protected]).

M. Djemaï and P. Pudlo are with the Industrial and Human AutomationControl, Mechanical Engineering and Computer Sciences Laboratory (LAMIH,CNRS UMR 8201), University of Valenciennes and Hainaut-Cambrésis,59313 Valenciennes, France (e-mail: [email protected];[email protected]).

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

Digital Object Identifier 10.1109/TITS.2012.2211352

Nevertheless, EPS assistance control laws have no systematicfeedback design method since they are mainly based on torqueboost maps. These maps are tuned according to several de-sign criteria such as maneuverability and vibration attenuation.However, except for assistance acceptability, human factorsare omitted since torque maps are tuned in a static way. Thepresence of human-in-the-loop must be studied in depth, anddynamic assistance laws should be proposed by consideringdriver capabilities.

To evaluate the effectiveness and acceptability of futureassistance control, an experimentation tool must be designed. Infact, real race experimentations are often labored, particularlywhen involving drivers with physical disabilities. Therefore,the use of a simulation test bed seems to be an interestingalternative [1]. In this field, multiple low-cost simulators haveemerged in the last decade [2], [3]; only reduced-version EPSsimulators are designed, and most of them were built byindustrial institutions [4]–[8]. In this perspective, one of theobjectives of the ANR1 VOLHAND2 project is to design asimulation tool that includes a real EPS system that is as closeas possible to a real vehicle-steering system.

A driving simulator consists of an instrumented mechani-cal platform, ensuring permanent communication between thedriver and the simulator software. Nevertheless, some variablescannot be directly accessed or measured. Consequently, virtualsensors must be developed to measure (or estimate) them.These variables are generally divided into two different parts:1) system states (such as steering-wheel angle and its rate)and 2) unknown inputs (such as the driver torque or the loadrack force). In this area, the problem in designing virtualsensors (observers) has been addressed in several works [9].Among them, sliding-mode observers (SMOs) have been usedfor their robustness, insensitivity to the matched perturbations/uncertainties, and finite/exact time convergence [10], [11]. Inits first version, necessary and sufficient conditions must befulfilled, particularly the so-called observer-matching condi-tion. However, this assumption limits the use of these observersfor a large range of mechanical systems. This way, the useof a sliding-mode differentiator (SMD) was generalized, andSMOs have been extended to a large range of physical systems[12]–[15]. Recently, the introduction of the supertwisting al-gorithm (STA) as an absolutely continuous output injectionand the development of high-order SMD (HOSMD) have con-tributed in estimation enhancement and also in avoiding sometechnical problems, such as estimate filtration [16], [17].

1French National Research Agency.2Steering wheel for elderly and/or disabled person.

1524-9050/$31.00 © 2012 IEEE

NEHAOUA et al.: VIRTUAL PROTOTYPING OF ELECTRIC POWER STEERING SIMULATOR 275

Fig. 1. Mechatronics of the simulator and adopted mechanical architectures.

The main purpose of this paper is to describe an EPS test bedwith its overall virtual prototyping. The remainder of this paperis organized as follows: Section II exposes some propositionsto design an EPS simulator with involved components. Amodeling technique to evaluate tire kinematics and the loadrack force from tire/road contact is detailed in Section III.In Section IV, an estimation procedure is illustrated basedon SMOs to virtually measure the rack load force and thedriver torque. Finally, the last section provides simulations anddiscussions by considering different case studies.

II. OUTLINES ON THE PROPOSED TEST-BED SYSTEM

In a driving simulator, the driver is placed within severaldevices that transform his control inputs into stimuli thatcan be perceived, if correctly interpreted, by the propriocep-tive sensors. In this regard, the design of a simulator forsteering evaluation and control requires fair torque feedback.This torque is a result of the tire/road contact propagatedfrom the steering mechanical system to the driver steeringwheel.

For an EPS-like pinion type, the mechanical platform in-cludes a real EPS [steering wheel, steering column, pinionassistance motor, and rack (see Fig. 1)]. In addition, an elec-trical motor and a ball screw-nut assembly are expected forload rack force restitution. Since the overall EPS system ismechanically well defined, there is great effort in finding theminimum mechanical clearance to link the load motor to the

rack system via the ball screw-nut. The simplest way is toconvert the screw rotation into a nut translation, where the loadmotor torque is transmitted to the ball screw-nut via a pulley-belt assembly. The advantage of such a solution is the torquemultiplication ratio offered by the pulley-belt system, whichmakes it possible to choose a quite small load motor.

Aside from the mechanical platform, the simulator is builtaround various modules described here.

1) Vehicle dynamics: This is used to predict the kinematicand dynamic behavior of a vehicle according to driveractions. Its complexity relates more to the front-wheelassembly, where the main input is the rack displacement(see Section III-E). The outputs of the front-wheel assem-bly are, in one part, the tire steer/effort used to updatevehicle motion states (see Section III-B) and, in anotherpart, the reference load rack force to be restituted on thesimulator’s rack via the load motor (see Section III-D).

2) Visual/traffic environment: This aims to provide a realisticsimulation of road situations, starting from the individ-ual driver’s behavior. The objective is to immerse thedriver in realistic traffic conditions [18]. This moduleuses, in part, the vehicle motion states computed fromthe virtual vehicle module and a visual environment forprojection.

3) Low-level management: This is managed by an operat-ing system with a real-time kernel on a target personalcomputer (PC). Different control algorithms are carriedout by a standard host PC fitted with an adequate com-munication protocol. Otherwise, this module also dealswith input/output acquisition and data processing, whichgoverns different actuators.

4) Virtual sensors: Load rack force and driver torque aretwo essential quantities for EPS control feedback design.Unfortunately, it is difficult to measure these variables forcost or complexity reasons, with appropriate sensors. For-tunately, state observation and unknown input estimationare a very active research field that offers efficient meth-ods to build virtual sensors (see Section IV). This moduletakes the rack displacement from the vehicle module,the driver-steering measurement, and the EPS currentfrom the EPS torque control component. As outputs, itgives estimations of the load rack force applied by theload motor and the driver torque applied on the steeringwheel.

5) Load torque control module: Its input is the reference loadrack force, as computed by the front assembly model,and the estimated load rack force from the virtual sensorblock. This module aims to generate an adequate refer-ence current, by a given control feedback method, to drivethe load motor torque.

6) EPS torque control module: Synthesizing a control feed-back strategy intended to assist a given driver is the mainobjective for designing such test bed and constitutes thefinality of the ongoing project. More information aboutthis module can be found in [19]–[21].

In the next section, an overview of the vehicle dynamics isintroduced with special care to the front assembly modeling.


Fig. 2. Front assembly (sketched from [22]).

III. VIRTUAL VEHICLE AND SIMULATOR MODELING

A. Modeling Assumptions

In this study, a car vehicle dynamics model is considered, al-lowing the simulation of five degrees of freedom, i.e., longitudi-nal and lateral displacement, roll, pitch, and yaw rotations [22].

In normal driving situations, particularly at low speed andparking maneuver, the driver needs to be more assisted. Underthese driving conditions, the vehicle shows smooth dynamicswith moderate accelerations and low-frequency content [23].Furthermore, nonlinear terms can be neglected, and the use ofa simplified vehicle dynamics model is admissible. However, atparking maneuvers, the steer angle and the load torque appliedon the steering system rack may be important. For this, specialfocus must be brought in the description of front assemblykinematics.

For the simulator modeling, a reduced-order model of an EPSsystem is considered, which transcribes a nominal dynamicbehavior expressed by fundamental frequency below 30 Hz.Higher frequencies, which result from the vibratory behavior[24], [25], are not considered.

It is also supposed that each electric motor is driven by acurrent signal i. Electrical dynamics is not taken into accountsince it is largely faster than mechanical dynamics. Moreover,the transformation of the current signal to an equivalent voltageinput is ensured by the low-level current control loop imple-mented in the electronic hardware.

B. Geometry of the Front Wheel Assembly

The front assembly is composed of several parts, includingthe wheel rim, wheel tire, steering arm, steering rack, lowersuspension arm, and suspension shock absorber (see Fig. 2).In free motion, the wheel geometric configuration can be char-acterized by three Euler rotations with respect to the vehiclereference frame �v: rotation μ around iv , rotation η around theresulting j-axis, and steering rotation δ around vector qc. Thesuspension travel occurs along the unit vector ehc, and its effecton the wheel steering is neglected. From this assumption, weconsider that the rotation of the lower suspension arm ζ ≈ 0.Let i be the wheel mass center, at which a right-handed axisreference frame �i is attached (i for the left/right wheel). In thevehicle reference frame �v , the wheel mass center position is

given by (henceforth, all vectors are expressed in �v referenceunless otherwise stated)

rvi = rvb + rbq +Rμ,η,δriqi (1)

where rvb, rbq , and riqi are known vectors, and Rμ,η,δ =RμRηRδ is the rotation matrix transformation. Furthermore,the steering axis rqc is given by

rqc = rbc − rbq = Rμ,ηriqc

riqc = riqh + rihc. (2)

From (2), we can write the following expression:

RTμrqc = Rη

(riqh + rihc

)(3)

which represents an algebraic constraint with respect to thetwo independent variables μ and η. It remains to find thesteering angle δ from the rack displacement by consideringthe geometric constraint ‖rfw‖ = lb (the steering arm lengthis constant); then

rTfwrfw = l2b (4)

where, from Fig. 2, we have rfw = rqw −Rviriqf . Substitut-

ing this equation into (4) and after some algebraic manipula-tions, we get

−2rTqwRviriqf = l2b − riTqfr

iqf − rTqwrqw. (5)

Knowing that rotation steering Rδ transformation is given by

Rδ = eiqceiTqc +

(I3 − eiqce

iTqc

)cos δ + eqc sin δ (6)

and reporting (6) in (5), we get a trigonometric equation of theform

a1 cos δ + a2 sin δ = a3a1 = −rTqwRμη

(I3 − eiqce

iTqc

)riqf

a2 = rTqwRμηeqcriqf

a3 =12

(l2b − riTqfr

iqf − rTqwrqw

)+ rTqwRμ,ηe

iqce

iTqc r

iqf (7)

which can be solved for variable δ. Now, it is possible tocompute some wheel’s geometric variables. As shown in Fig. 3,camber angle γ and the effective tire steer δe can be derived andgiven by the following expressions:

sin γ = jTi kT , sin δe =jTi iT

jTi jT. (8)

C. Tire/Road Effort

Tire/road interaction is the most important phenomena thatcharacterize ground vehicles. Consequently, several works areundertaken, leading to a mathematical description of frictionforces. In this paper, the tire/road contact is dot shaped atpoint C, where an effort vector including the longitudinal forceFx = Fx(κ), the lateral force Fy = Fy(α, γ), and the verticalforce Fz is introduced by using an empirical tire model namedPacejka [26], with respect to the tire slip variables (κ, α) andcamber angle γ.


Fig. 3. Tire reference frame and camber (sketched from [22]).

At first, to describe the tire/road interaction, a new referenceframe �T is introduced at contact point C. kT is the normalvector to the road surface. Vector iT is obtained by iT =ji × kT , and jT completes the right-handed reference axis. Toderive the load force applied on the steering system rack, it ismore convenient to express the equivalent tire forces/momentswrench at the center of wheel i, instead of the tire/road contactpoint C; hence

F T =FxiT + FyjT + FzkT

MT =MxiT +MyjT +MzkT + F T × rCi (9)

where Mx is the tire torque about iT , My is the rollingresistance torque, and Mz = Mz(α, γ) is the alignment torque.

D. Load Rack Force

At present, all wheel kinematic variables and the equivalenttire forces/moments wrench at the wheel center are defined.Obviously, the next step concerns the calculation of load forceF r applied on the steering system rack. By using the virtualpower equilibrium [22], the power delivered by the tire/roadeffort F T /MT is equal to the power delivered by the rack forceFr; then

ΔPF r=ΔPF T /MT

→FrΔur =ΔvT

viF T +ΔωTviMT (10)

where ur is the rack displacement, and vvi and ωvi are thelinear and angular velocity vectors of the wheel center i withrespect to the vehicle reference origin v, respectively. By usingthe Jacobian vectors Δx = (∂x/∂ur)Δur, (10) becomes

Fr =∂vT

vi

∂urF T +

∂ωTvi

∂urMT . (11)

Afterward, to define the Jacobian vectors, it is necessary tocompute the linear and angular velocities of wheel center i. Bydifferentiating (1), we yield

∂vvi

∂ur=

∂ωvi

∂ur× rqi,

∂ωvi

∂ur=

∂δ

∂ureqc. (12)

Finally, to find (∂δ/∂ur), we consider the two geometricloops (rbf) and (rwf). From these loops, the position vector rbfcan be expressed as

rbf = rbq + rqf = rbw + rwf

Fig. 4. Dynamics scheme of the EPS system without the load motor.

where rbw = rbw,0 + urjv . By differentiating this equation,we get

ωvb × rbq + ωvi × rqf = urey,v + rwf . (13)

By using the geometric constraint (4), the projection of (13)on the vector rwf yields

∂δ

∂ur=

rTwfey,v

rTwf (eqc × rqf ). (14)

Nevertheless, while performing a parking maneuver, the tireside-slip angle (α) is quasi-null, whereas experimentationsshowed that the load torque is increasingly important, par-ticularly while stopping [27]. To simulate such situations, anefficient algorithm is described in [28], where the tire hysteresisbehavior is well modeled.

E. Simulator Dynamics

As stated in Section II, the simulator includes a real EPSsystem and an electrical motor for load rack force restitution.To model the whole assembly, let us consider first the steeringcolumn dynamics (see Fig. 4) given by the following equation:

Jcθc + βcθc = Td − Tcc (15)

where θc is the steering-wheel angle, Td is the steering-wheeltorque applied by the driver, and Tcc is the transmitted torquevia the EPS torque sensor modeled as

Tcc = Kc

(θc −

ur

rp

). (16)

Here, ur is the rack displacement, and rp is the pinion radius.Next, the EPS electric motor is located between the pinion andthe rack. Its mechanical equation is expressed as follows:

Jaθa + βaθa = Ta − Tcm (17)


Fig. 5. Dynamics scheme of the load motor.

where θa is the motor rotation angle, Ta = kt,aia is the motor-delivered torque, and Tcm is the transmitted torque to the rackvia the EPS gears. Finally, the rack dynamics is given as

mrur + βrur =1rp

Tcc +Na

rpTcm − Fr. (18)

The rack displacement and the EPS motor rotation are relatedby ur = (rp/Na)θa. From (17) and (18), the EPS motor/rackequivalent dynamics is

mequr + βequr =Kc

rp

(θc −

ur

rp

)+

Nakt,arp

ia − Fr. (19)

By combining (15), (16), and (19), the EPS steering systemdynamics can be written as a linear state space formulation asfollows:

x =Ax+Bia +Dζ (20)y =Cx (21)

where x = [θc, θc, ur, ur]T is the state vector, ia is the input

EPS motor-delivered current, ζ = [Td, Fr]T is a vector of ad-

ditional unknown inputs, and y = [θc, ur]T is the vector of the

measured outputs.At this stage, it remains to integrate the dynamics of the load

motor part to complete the simulator modeling (see Fig. 5).As before, the mechanical equations of the load motor and thescrew-nut are described by

Jlθl + βlθl =Tl − Tlc

Jsθs + βsθs =NlTlc +p

2πηFr (22)

where θl and θs are the load motor and the screw-nut anglepositions, respectively, and Tl = kt,lil is the motor-deliveredtorque.

On the other hand, θl, θs, and the rack displacement ur arerelated to the EPS motor angle θa by the following relations:

θs =2πpur =

1Nl

θl, θa =Na

rpur.

By reporting these relations in (22) and combining the result-ing expression with (19), we get the equivalent dynamics of theassistance motor/rack/screw-nut/load motor assembly

m′equr + β′

equr = Nlkt,lil +p

2πηrp(Tcc +Nakt,aia). (23)

Finally, with (15), the overall EPS simulator dynamics canbe expressed by the following linear state space formulation:

x = Asx+Bs[Td ia il ]T . (24)

The next section concerns the development of virtual sensorsfor unknown input estimation (for both rack load force Fr anddriver torque Td). This will be achieved by introducing theSMO theory.

IV. SLIDING-MODE OBSERVER DESIGN FOR STATES AND

UNKNOWN INPUT ESTIMATION

In this section, an application of a class of SMO developedin [17] is presented. In this class, and due to the STA, thediscontinuous output injection is replaced by a continuous out-put injection, which allows avoiding filtration of the estimatedstates and unknown inputs. In addition, unlike other SMOs(such as Walcott-Zak), the matching condition is not required,which makes the use of this type of observation techniquemore suitable, particularly for high-order systems. However,it requires the boundedness of the unknown inputs and alsotheir successive derivatives, where the Walcott-Zak observer isless restrictive. Fortunately, in mechanical system applications,the boundedness of the two first derivatives is an acceptableassumption.

A. Assumptions

Assume a linear time-invariant system (20) with n states, mmeasurable outputs, and m unknown inputs, and consider thefollowing definitions [17], [29]:

Definition 4.1: The relative degree of system (20) outputvector y with respect to unknown input vector ζ is vector[r1, . . . , rm], such that CiA

sDj = 0, and

rank(Q) = rank(D), Q =

⎡⎣CiA

r1−1D...

CiAr2−1D

⎤⎦ (25)

where i, j = 1, . . . ,m, s = 0, . . . , ri − 2, Ci is the ith line ofmatrix C, and Dj is the jth column of matrix D.

Definition 4.2: The total relative degree of system outputvector (20) with respect to the unknown inputs vector is thescalar r =

∑i ri, where i = 1, . . . ,m.

Definition 4.3: System (20) is strongly observable if andonly if the total relative degree r is equal to the rank ofobservability matrix P, i.e.,

P =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

C1

C1A...

C1An−1

...Cm

CmA...

CmAn−1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (26)

Remark 4.1: The matching condition of the Walcott-Zakobserver (rank(CD) = rank(D)) [30] is a special case of thestrong observability concept. As shown in (25), the matchingcondition implies that there must be as many independentoutputs as unknown inputs, and the relative degree vector of


Fig. 6. Virtual simulator used for simulations.

each output with respect to the unknown inputs vector is equalto or less than two.

From these definitions, the EPS output vector y = [θc, ur]T

in (20) has a relative degree vector [r1, r2] = [2, 2] with re-spect to the unknown input vector ζ = [Td, Fr]

T ; then, thetotal relative degree r = 4 is equal to the system state ordern (or equal to the rank of the observability matrix). Hence,system (20) is strongly observable. In addition, system (20)satisfies the following two assumptions [17], which makes thereconstruction of unknown input vector ζ possible:

1) Each unknown input ζi of ζ is a bounded function, where|ζi| ≤ ζi,max. (The applied driver torque and the rackload force are obviously bounded by |Td| ≤ Tmax and|Fr| ≤ Fmax.)

2) The k successive derivatives of ζi are bounded by thesame constant ζ ′i, and the (k + 1)th derivative is aLipschitzian function. As a reminder, we consider k = 1.

B. Observer Equations

Since the observability condition is fulfilled, matrix L canbe arbitrarily chosen such that A− LC is Hurwitz, and theobserver is built in the form

z =Az +Bia + L(y −Cz)x = z +P−1ω (27)

where ω = [v1,1, . . . , v1,r1 , v2,1, . . . , v2,r2 ]T , in which ω vector

components are extracted from vector v1 = [v1,1, . . . , v1,r]T

and vector v2 = [v2,1, . . . , v2,r]T . Vectors v1 and v2 are the

nonlinear parts of the observer chosen in the form of the(r + k − 1)th-order differentiator [31], as follows:

vi,1 = − 3M14i |vi,1 − yi +Ciz|

34 sign

× (vi,1 − yi +Ciz) + vi,2

vi,2 = − 2M13i |vi,2 − vi,1|

23 sign(vi,2 − vi,1) + vi,3

vi,3 = − 1.5M12i |vi,3 − vi,2|

12 sign(vi,3 − vi,2) + vi,4

vi,4 = − 1.1Misign(vi,4 − vi,3) (28)

in which i = 1, 2, Mi is chosen to be sufficiently large fromdefinition (4.1), and the two previous assumptions are chosen tobe Mi ≥ |CAri−1D|ζi,max. Finally, the unknown input vectorζ is estimated using

ζ = {PD}−1(α−P(A− LC)P−1ω

)(29)

where α = [v1,r1+1, v2,r2+1] (or α = [v1,3, v2,3]).Remark 4.2: Matrix A in (20) has one eigenvalue on the

imaginary axis ([-4.3008 ± 75.9311j, −7.8013, +28E-15]);then, system (20) needs to be stabilized and damped by intro-ducing L(y −Cz). Since (20) is strongly observable, an arbi-trary eigenvalue assignment can be made to design matrix L.Nevertheless, the new system eigenvalue should be chosen,reducing the system damping.

V. SIMULATION RESULTS

A. Simulation Configuration

Hereafter, two case studies are presented to show the ef-fectiveness of the proposed algorithms: 1) parking case ma-neuver and 2) cornering driving maneuver by using a virtualplatform (see Fig. 6), including the vehicle dynamics, thesimulator model (24), and its associated virtual sensors. In ad-dition, load torque feedback control is implemented by using aproportional-integral (PI) controller, considering reference loadrack force Fr, as computed in Section III-D, and the estimatedload rack force Fr. The PI controller output is a referencecurrent signal il, which is converted to a voltage signal toactuate the load motor. Moreover, the present simulation isundertaken for the limit case where the load rack force is atits maximum; for this, the motor assistance is disabled.

In the following, the Runge–Kutta four-integration methodis adopted. The virtual vehicle model is updated at each 10 ms.The load rack force PI control is carried out at a sampling rate of1 ms. For the observer part, the following initial conditions aresupposed to be x0 = [0, 0.5, 0, 0.005]T , and integration of theobserver equations is done at a 5-ms sampling rate. In addition,no filtration is introduced on the estimated variables.


Fig. 7. Parking maneuver case. (a) Tire steer angle δe with respect to the rack displacement ur . (b) Tire camber angle γ with respect to ur . (c) Reference drivertorque Td and simulated driver torque Tcc with respect to the steering-wheel angle θc.

Fig. 8. State and unknown input estimation for the parking maneuver case with nominal parameters. (a) Driver torque Td and load rack force Fr .(b) and (c) System states. (d)–(f) Convergence of unknown input and state estimation.

B. Case 1: Parking Maneuver

In the first scenario, the driver performs two steering-wheelturns at stop in both left and right sense. Fig. 7(a) and (b) showsthe left and right tire steer and camber angles with respect tothe rack displacement, respectively. From these figures, it isshown that the tire steer and camber angles are quite differentfor large rack displacement. This is typical for urban drivingsituations with small road curvature and low forward speed.Furthermore, for the highway driving situation, the two anglesslightly evolved in the same way.

In Fig. 7(c), the reference driver torque with respect tothe steering-wheel angle is depicted, as well as the simulateddriver torque (Tcc). It is clear that the torque sensor providedwith the EPS system gives only an approximation of the realdriver torque. This is consistent with the static way in whichthe EPS boost map is tuned. In this paper, we seek moreprecise information of the driver torque, particularly for personswho represent any disability. Otherwise, if the EPS system is

disabled, the high values of the driver torque, such as those inthis figure, are justified.

The dynamics states and the unknown inputs with theirestimations are proposed in Fig. 8, as well as all those forthe nominal case (no uncertainties on the simulator’s modelparameters). For noiseless measured signals, an exact estima-tion is achieved with a finite-time convergent, as shown inFig. 8(d)–(f).

To evaluate the robustness of the observer, a second sim-ulation is carried out by considering an uncertainty of 20%on the nominal values of the simulator’s plant parameters.Fig. 9(a)–(d) shows the state and unknown input estimation.For each variable, the estimation error compared to the nominalcase is also reported. Here, the observer ensures accurate es-timation with asymptotic convergence for almost all variables.Finally, an additive white noise is considered on the measuredsystem outputs to test the observer’s ability to deal with stochas-tic perturbations. Fig. 8(e) and (f) shows that the unknowninputs are correctly estimated with asymptotic convergence and


Fig. 9. State and unknown input estimation for the parking maneuver case with perturbations. (a) Driver torque and load rack force estimations with 20%parameter uncertainties. (b) Driver torque and load rack force estimation errors. (c) and (d) System state estimation errors. (e) and (f) Driver torque and load rackforce estimations and their convergence in the presence of measurement noises.

Fig. 10. Real vehicle experimentation conditions. (a) Road path. (b) Lane curvature. (c) Vehicle forward speed.

all those with no filtration. Moreover, it is possible to enhanceobserver performance against measurement noises by tuningparameters Mi in the HOSMD equation (28). Fortunately, theuse of a digital encoder and a controller area network bus formotor angle position measurement and acquisition considerablylimits the measured signal noises.

C. Case 2: Cornering Maneuver

The second scenario aims to test the virtual platform behaviorfor a full track road including several cornering maneuvers.This simulation is realized by employing data records includingexperimental measurements carried out by a real car vehicleon a real track road shown in Fig. 10(a). Next, a drivermodel is integrated to generate a driver torque Td according tothe measured steering-wheel angle θc, the lane curvature [seeFig. 10(b)], and the vehicle forward speed [see Fig. 10(c)] [32].In addition, an uncertainty of 20% on the nominal parametersof the used system model (24) is considered.

Once again, it can be seen in Fig. 11 that the unknown inputsare accurately reconstructed after convergence of the system

states. A closer look of the estimated variables is also depicted,where an asymptotic convergence is ensured for both states andunknown inputs (see Table I).

VI. CONCLUSION

In this paper, full prototyping of a virtual electrical power-steering test bed has been described. Some design propositionsand the overall simulator mechatronics, arising from the drivingsimulation, have been exposed.

Two main contributions are well detailed: 1) a modelingtechnique to evaluate the front assembly kinematics and theload rack force from tire/road interaction and 2) the designof virtual sensors for the dynamics states and unknown inputestimation based on the sliding-mode theory.

To prove the efficiency of each contribution, two case studieshave been illustrated in the last section for two different drivingscenarios. The first case study deals with a stopping parkingmaneuver to highlight some geometric and dynamic featuresof the front assembly. This scenario is well suitable for such


Fig. 11. Unknown input estimation. (a) Driver torque. (b) Load rack force. (c) Steering-wheel angle and rack position.

TABLE IABBREVIATIONS, NOTATION, AND NUMERICAL VALUES

validation since the driver, in that case, needs to be assistedmore. The second scenario describes full road track driving byusing experimental measurements.

APPENDIX

In (19), meq=mr+(N2a/r

2p)Ja, and βeq=βr+(N2

a/r2p)βa;

in (23) m′eq = (p/2π)meq + (2π/p)(Js +N2

l Js), and β′eq =

(p/2π)βeq + (2π/p)(βs +N2l βs). In addition, we have

A =

⎡⎢⎢⎣−βc

Jc−Kc

Jc0 Kc

rpJc

1 0 0 00 Kc

rpmeq− βeq

meq− Kc

r2pmeq

0 0 1 0

⎤⎥⎥⎦

B =

⎡⎢⎢⎣

00

− Na

rpmeq

0

⎤⎥⎥⎦

D =

⎡⎢⎢⎣

1Jc

00 00 − 1

meq

0 0

⎤⎥⎥⎦

As =

⎡⎢⎢⎣−βc

Jc−Kc

Jc0 Kc

rpJc

1 0 0 00 pKc

2πηrpm′eq

− β′eq

m′eq

− pKc

2πηr2pm′eq

0 0 1 0

⎤⎥⎥⎦

Bs =

⎡⎢⎢⎣

1Jc

0 00 0 00 pNakt,a

2πηrpm′eq

0 Nlkt,l

m′eq

0 0

⎤⎥⎥⎦ .


REFERENCES

[1] L. Nehaoua, H. Mohellebi, A. Amouri, H. Arioui, S. Espié, andA. Kheddar, “Design and control of a small-clearance driving simulator,”IEEE Trans. Veh. Technol., vol. 57, no. 2, pp. 736–746, Mar. 2008.

[2] H. Arioui, L. Nehaoua, S. Hima, N. Séguy, and S. Espié, “Mechatronics,design, and modeling of a motorcycle riding simulator,” IEEE/ASMETrans. Mechatronics, vol. 15, no. 5, pp. 805–818, Oct. 2010.

[3] H. Arioui, S. Hima, L. Nehaoua, R. J. V. Bertin, and S. Espié, “Fromdesign to experiments of a 2-DOF vehicle driving simulator,” IEEE Trans.Veh. Technol., vol. 60, no. 2, pp. 357–368, Feb. 2011.

[4] C.-J. Yeh, S.-R. Ho, M.-C. Lin, T.-H. Hu, and T.-H. Hsu, “Developmentof a test bench for tuning and validating electrical power steering controlmethod,” in Proc. IEEE Veh. Power Propulsion Conf., 2007, pp. 618–622.

[5] M. Segawa, M. Higashi, and S. Nakano, “Development of simulatorfor evaluation of steering systems,” Koyo Eng. J., vol. 168, pp. 31–37,2005.

[6] M. Segawa, M. Higashi, and S. Nakano, “Development of steeringhardware-in-the-loop (HIL) simulator,” in Proc. FISITA World Congr.,2006, vol. 10, pp. 22–27.

[7] T. Park, C. Han, and S. Lee, “Development of the electronic controlunit for the rack-actuating steer-by-wire using the hardware-in-the-loopsimulation system,” Mechatronics, vol. 15, no. 8, pp. 899–918, Oct. 2005.

[8] W. Ren, H. Chen, and J. Song, “Model-based development for an electricpower steering system,” Proc. Inst. Mech. Eng. C, J. Mech. Eng. Sci.,vol. 222, no. 7, pp. 1265–1269, Jul. 2008.

[9] R. Rajamani, D. Piyabongkarn, V. Tsourapas, and J. Y. Lew, “Parameterand state estimation in vehicle roll dynamics,” IEEE Trans. Intell. Transp.Syst., vol. 12, no. 4, pp. 1558–1567, Dec. 2011.

[10] K. Kalsi, J. Lian, S. Hui, and S. Zak, “Sliding mode observers for systemswith unknown inputs: A high-gain approach,” Automatica, vol. 46, no. 2,pp. 347–353, Feb. 2010.

[11] H. Saadaoui, N. Manamanni, M. Djemaï, J. P. Barbot, andT. Floquet, “Exact differentiation and sliding mode observers forswitched Lagrangian systems,” Nonlinear Anal. Theory, Methods Appl.,vol. 65, no. 5, pp. 1050–1069, Sep. 2006.

[12] T. Floquet and J. Barbot, “A canonical form for the design of unknowninput sliding mode observers,” in Advances in Variable Structure andSliding Mode Control. New York: Springer-Verlag, 2006.

[13] T. Floquet, C. Edwards, and S. Spurgeon, “On sliding mode observersfor systems with unknown inputs,” Int. J. Adapt. Control Signal Process.,vol. 21, no. 8/9, pp. 638–656, Oct./Nov 2007.

[14] H. Imine, L. Fridman, H. Shraim, and M. Djemaï, “Sliding modebased analysis, observer and identification of vehicle dynamics,” inLecture Notes in Control and Information Sciences, 1st ed. New York:Springer-Verlag, 2011.

[15] A. Marouf, M. Djemai, C. Sentouh, and P. Pudlo, “Driver torque androad reaction force estimation of an electric power assisted steering usingsliding mode observer with unknown inputs,” in Proc. IEEE Conf. Intell.Transp. Syst., Madeira Island, Portugal, 2010, pp. 354–359.

[16] F. J. Bejarano, L. Fridman, and A. Poznyak, “Exact state estimation forlinear systems with unknown inputs based on hierarchical super-twistingalgorithm,” Int. J. Robust Nonl. Control, vol. 17, no. 18, pp. 1734–1753,Dec. 2007.

[17] L. Fridman, A. Levant, and J. Davila, “Observation of linear systems withunknown inputs via high-order sliding-modes,” Int. J. Syst. Sci., vol. 38,no. 10, pp. 773–791, Oct. 2007.

[18] V. Punzo and B. Ciuffo, “Integration of driving and traffic simulation:Issues and first solutions,” IEEE Trans. Intell. Transp. Syst., vol. 12, no. 2,pp. 354–363, Jun. 2011.

[19] A. Marouf, C. Sentouh, M. Djemaï, and P. Pudlo, “Control of electricpower assisted steering system using sliding mode control,” in Proc. IEEEConf. Intell. Transp. Syst., Washington, DC, 2011, pp. 107–112.

[20] A. Marouf, C. Sentouh, M. Djemaï, and P. Pudlo, “Control of an electricpower assisted steering system using reference model,” in Proc. IEEEConf. Decision Control, Orlando, FL, 2011, pp. 6684–6690.

[21] J. E. Naranjo, C. Gonzalez, R. Garcia, T. de Pedro, and R. E. Haber,“Power-steering control architecture for automatic driving,” IEEE Trans.Intell. Transp. Syst., vol. 6, no. 4, pp. 406–415, Dec. 2005.

[22] G. Rill, Simulation von Kraft-Fahrzeugen. Braunschweig, Germany:Vieweg, 1994.

[23] D.-C. Liaw, H.-H. Chiang, and T.-T. Lee, “Elucidating vehicle lateraldynamics using a bifurcation analysis,” IEEE Trans. Intell. Transp. Syst.,vol. 8, no. 2, pp. 195–207, Jun. 2006.

[24] A. Badawy, J. Zuraski, F. Bolourchi, and A. Chandy, “Modeling andanalysis of an electric power steering system,” in Proc. Int. Congr. Exh.,Detroit, MI, 1999.

[25] L. Nehaoua, A. Marouf, J. J. Santin, P. Pudlo, and M. Djemai, “Towardsan electrical power-assisted steering simulator: Modelisation specifica-tions,” in Proc. IFAC Symp. Mechatron. Syst., Cambridge, MA, 2010,pp. 571–576.

[26] H. B. Pacejka, Tire and Vehicle Dynamics. Oxford, U.K.: ButterworthHeineman, 2005.

[27] R. Sharp, “On car steering torques at parking speeds,” Proc. Inst. Mech.Eng. D, J. Automobile Eng., vol. 217, no. 2, pp. 87–96, Feb. 2003.

[28] P. van der Jagt, “Prediction of steering efforts during stationary parkingmaneuvers,” in Proc. Eur. ADAMS Users’ Conf. Papers, Berlin, Germany,1999.

[29] A. Isidori, Nonlinear Control Systems. London, U.K.: Springer-Verlag,1996.

[30] S. Hui and S. H. Zak, “Observer design for systems with unknown inputs,”Int. J. App. Math. Comput. Sci., vol. 15, no. 4, pp. 431–446, Dec. 2005.

[31] A. Levant, “High-order sliding modes: Differentiation and output-feedback control,” Int. J. Control, vol. 76, no. 9/10, pp. 924–941,Jun. 2003.

[32] C. Sentouh, P. Chevrel, F. Mars, and F. Claveau, “A sensorimotor drivermodel for steering control,” in Proc. IEEE Conf. Syst., Man, Cybern.,San Antonio, TX, 2009, pp. 2462–2467.

Lamri Nehaoua received the B.S. degree in engi-neering on control and automation systems sciencefrom the University of Sétif, Sétif, Algeria, in 1999,the M.S. degree in computer vision for roboticsapplications from Clermont-Ferrand II University,Clermont-Ferrand, France, in 2002, and the Ph.D.degree from the University of Evry Val d’Essonne,Evry, France, in 2008.

He currently holds a postdoctoral position withthe Informatics, Integrative Biology and ComplexSystems Laboratory, University of Evry. His research

interests are driving simulators, driver assistance, modeling, control, and obser-vation of complex systems.

Mohamed Djemaï received the Ph.D. degree in au-tomatic control from the University Paris Sud-Orsay,France, in 1995.

From 2000 to 2008, he was an Associate Profes-sor in automatic control with the Ecole NationaleSupérieur de l’Electronique et de Ses Applications(ENSEA), Cergy, France. In 2008, he became a FullProfessor with the University of Valenciennes andHainaut-Cambrésis, Valenciennes, France, where heis also currently with the Industrial and HumanAutomation Control, Mechanical Engineering and

Computer Sciences Laboratory (CNRS UMR 8201). His research interestsare hybrid systems, sliding-mode control and observers, fault detection, andresidual generation, with application to power systems and automotive control.

Philippe Pudlo was born in France in 1970. Hereceived the Ph.D. degree in industrial and humanautomation from the University of Valenciennes andHainaut-Cambrésis, Valenciennes, France, in 1999.

He is currently a Full Professor with the Industrialand Human Automation Control, MechanicalEngineering and Computer Sciences Laboratory(LAMIH, CNRS UMR 8201), University ofValenciennes and Hainaut-Cambrésis. Since 2009,he has coordinated the ANR VOLHAND Project,whose aim is to develop personalized electric power

steering adapted for people with reduced mobility. His current researchinterests include biomechanics, modeling and simulation, ergonomics, and thehandicapped.

Documents

Virtual Prototyping of an Electric Power Steering Simulator