Modelling and control of an electromechanical steering

1

Modelling and control of an electromechanical steeringsystem in full vehicle modelsY Du1*, M A Lion1, and P Maißer2

1CAE-Methods, Volkswagen AG, Wolfsburg, Germany2Institute of Mechatronics at the Chemnitz University of Technology, Germany

The manuscript was received on 10 September 2004 and was accepted after revision for publication on 12 December 2005.

DOI: 10.1243/09596518JSCE96

Abstract: In the automotive industry, electrical and electromechanical components andsystems become more and more important. In comparison with commonly used mechanicaland hydraulic systems they offer a large number of advantages with respect to efficiency andflexibility, for example. Therefore, conventional hydraulic steering systems are being replacedmore and more with electromechanical ones. Currently, different concepts of electromechanicalsteering systems are being developed. In this work an electromechanical steering system withdouble pinions is modelled based on a uniform theory for discrete electromechanical systems.This steering system is implemented into a multibody full vehicle model and a control schemehas been developed. Subsequently, the performance of the whole electromechanical system,and especially the behaviour of the controller, has been tested with different handlingmanoeuvres.

Keywords: electromechanical steering system, electromechanical systems, multibodyvehicle model, full vehicle model, driver model, vector control, field-oriented control

1 INTRODUCTION steering wheel and gear rack. Thus, in case of a failureof the servo motor the vehicle can still be steered

Currently, an increasing importance is attached to mechanically.electromechanical components in the automotive In this research work, an asynchronous electricalindustry. An outstanding example is the electro- machine is used as the servo motor. Due to itsmechanical steering system which is characterized brushless design the asynchronous machine offersby efficiency, steering comfort, adaptability, and many advantages and is a very compact and durableenvironmental friendliness. It already fulfils a large functional unit. For the operation of this steeringnumber of requirements for steering systems of the system different input signals are required. The mostfuture [1]. In the case of electromechanical steering important signal is from the torque sensor on thedifferent concepts are known [2, 3]. In this paper, the torsion bar, which corresponds with the hand torqueconcept with double pinions as shown in Fig. 1 is of the driver applied on the steering wheel. A furtherconsidered. This kind of steering system is equipped important signal is the angular velocity of the rotor,with two pinions. While the steering pinion transfers which is necessary for a precise control of thethe steering torque applied by the driver, the torque asynchronous machine. This signal is measured bygenerated by the servo motor is introduced over means of an angular velocity sensor. An optimallyan additional servo pinion into the gear rack [4]. adapted steering support can be achieved by takingAs a fundamental property of this construction, a further information into account, e.g. the vehiclemechanical connection always survives between velocity or the steering angle. In order to generate

various steering functions, the input signals are pro-cessed by an algorithm in the control unit. Subjects* Corresponding author: 1573 CAE-Methods, Volkswagen AG,

Wolfsburg 38436, Germany. email: [email protected] of the present research work are the modelling and

JSCE96 © IMechE 2006 Proc. IMechE Vol. 220 Part I: J. Systems and Control Engineering

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The Charlesworth Group, Wakefield 01924 204830

2 Y Du, M A Lion, and P Maißer

where qm is the vector of electrical generalizedcoordinates (charges of the fundamental loops), Ais the fundamental loop matrix describing thelinear dependence between the branch chargesand the electrical generalized coordinates, and q0(t)represents the charges of the current sources in theelectrical network. Furthermore, ql is the vector of

Fig. 1 Electromechanical steering system with double mechanical generalized coordinates. The numberpinions of components of the generalized electrical and

mechanical coordinate vectors qm and ql is calledquasi degree of freedom n of the discrete EMS.control of such an electromechanical steering systemThen, the vector q=(qm ; ql) can be interpreted asin a multibody full vehicle model. The behaviour ofthe representing point of the discrete EMS in thethe whole system is tested using different handlingn-dimensional configuration space Rn. The motionmanoeuvres with the help of a driver model.equations of a discrete EMS can be written in theform

2 MODELLING OF THE ELECTROMECHANICAL M(t, ql)ql+K(t, ql, ql)+Ql(t, ql, ql)STEERING SYSTEM

+Qm(t, ql, qm, qm)=0 (1)The steering system as shown above is a typical

L(t, ql)qm+G(t, ql, ql, qm)+R(t, ql)qmelectromechanical system (EMS). It represents aphysical heterogeneous structure, which can be +C(t, ql)qm+V(t, ql)=0 (2)characterized by the interactions between electro-magnetic fields and mechanical objects [5]. The where M(t, ql) is the generalized mass matrix,dynamical behaviour and interactions within such a K(t, ql, ql) is the vector of generalized gyroscopic/system can be described by coupling the methods of Coriolis forces, Ql(t, ql, ql) is the vector of themultibody dynamics with the Kirchhoff theory [6, 7]. generalized applied forces having a pure mechanicalFor this purpose the approach of generating uniform origin (e.g. gravity, springs, frictions, and dampers),Lagrangean motion equations for the whole EMS and Qm(t, ql, qm, qm) is the vector of applied forcesworks well [7]. caused by electromagnetic fields. The matrix

L(t, ql) contains the generalized inductances and2.1 Lagrangean formalism for discrete EMS G(t, ql, ql, qm) is the vector of generalized gyroscopic/

Coriolis voltages. R(t, ql) is the matrix of generalizedTo describe the dynamical behaviour of discreteresistances, C(t, ql) represents the generalizedelectromechanical systems with the Lagrangeancapacitances, and V(t, ql) denotes the vector of theformalism, its application has been transferred firstgeneralized applied voltages.to pure electrical systems with concentrated para-

Obviously, the mechanical part (1) of the motionmeters, such as resistors, capacitors, and inductors [8].equations contains additional forces depending onThe topology of the electrical system can bethe generalized electrical coordinates. On the otherrepresented by a so-called network graph C. Thehand, the coefficient matrices appearing in theset {q:

j, x: k

s| jµC; s=1, … , 6, k=1, … , K} is called a

electrical part (2) of the motion equations can dependposition of the EMS, where q:j

are the branch chargeson the mechanical generalized coordinates. Thus, theand x: k

sthe mechanical coordinates of a given number

coupling between the mechanical and electricalof K rigid bodies. The kinematics of the EMS is deter-parts of the system is completely represented by thismined by the constraint conditions, which are giventheory. The mathematical and physical foundationsby Kirchhoff’s current law of the electrical networkand details of this uniform approach can be foundand the geometric connections between the rigidin reference [9].bodies. The vectors of the branch charges q:=(q:

j)

The theory of the Lagrangean formalism for discreteand the mechanical coordinates x:=(x: ks) which

electromechanical systems is implemented in thesatisfy the constraint conditions of the EMS at thesimulation program alaska, which was developed attime t can be written asthe Institute of Mechatronics in Chemnitz, Germany.

q:=Aqm+q0(t)The Lagrangean motion equations (1) and (2) of anEMS can be generated automatically in alaska.x:=x: (ql, t)

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3Electromechanical steering system in full vehicle models

2.2 Modelling of the induction machine electrical machine of this kind can be modelledeasily. The network graph of a three-phase induction

In general an induction machine consists of a statormachine is shown in Fig. 3.

and a rotor, which are connected by a revolute joint.From the above network graph the fundamental

An induction machine possessing a three-phase wind-loop matrix can be written as

ing system in the stator and a further three-phasewinding system in the rotor is of special interest inthis research work. In order to formulate the modelfor the induction machine the following conditionsare assumed to be fulfilled:

A=

tNNNNNNNNv

1 0 0 0 0 0 1 0 0 0 0 0

0 1 0 0 0 0 0 1 0 0 0 0

0 0 1 0 0 0 0 0 1 0 0 0

0 0 0 1 0 0 0 0 0 1 0 0

0 0 0 0 1 0 0 0 0 0 1 0

0 0 0 0 0 1 0 0 0 0 0 1

uNNNNNNNNw

T

1. The windings can be replaced by concentratedwindings.

2. The magnetization characteristic is linear andwithout saturation.

3. The magnetomechanical interactions are deter- This three-phase induction machine has seven quasimined by the field distribution in the air gap degrees of freedom. Accordingly,between the rotor and stator.

q=(qm ; ql)=(q1, q2, q3, q4, q5, q6; q7)4. Dissipative core losses and temperature depend-

ence of the resistances and inductances are =(qas , qbs , qcs , qar , qbr , qcr ; hrm)

neglected.is the vector of the generalized coordinates, where q

j,

Figure 2 shows the model of an idealized three-phase jµ{1, … , 6}, denotes the charge in the fundamentalinduction machine. In the following considerations, loop j and q

7is the mechanical rotation angle h

rmof

the indices s and r denote quantities referring to the rotor. Consequently,stator and rotor, while a, b, and c denote the three

q=(qm ; ql)=(q1, q2, q3, q4, q5, q6; q7)phases. The inductances arranged in the stator and

rotor windings are then represented by three coils =(ias , ibs , ics , iar , ibr , icr ; vrm)

staggered by 2p/3 in each case. Each of these sixis the vector of the generalized velocities, where q

j,coils forms a fundamental loop together with an

jµ{1, … , 6}, denotes the current in the fundamentalOhm’s resistance and a voltage source. If no voltageloop j and q

7is the mechanical angular velocity v

rmsource is applied in the rotor windings, a so-calledof the rotor.asynchronous machine is obtained.

If the induction machine possesses p polesWith the help of the electromechanical modelling( p

p=p/2 pole pairs), a so-called electrical angle h

rcomponents contained in alaska, e.g. capacitors (C),of the rotor can be introduced in accordance withcoils (L), resistors (R), and voltage sources (V), an

hr=pphrmConsequently, the electrical angular velocity isdefined by

vr=ppvrmThe parameters describing the physical properties ofthe electrical components are still required for thecomplete description of the induction machine. In

Fig. 2 Idealized model of a three-phase induction Fig. 3 Network graph of a three-phase inductionmachinemachine


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accordance with the fundamental assumptions, the 2.3 Electromechanical steering system in a fullOhm’s resistances r

sof the stator windings and r

rof vehicle model

the rotor windings are constant. The inductancesThe electrical machine as discussed above was

of the induction machine can be written in matriximplemented in the steering system of a multibody

form asfull vehicle model according to the double pinionconcept. For the purpose of describing the forcesbetween the road and the wheel hub, the tyre modelRMOD-K is used (see references [10] to [12]). The

L=

tNNNNNNNNv

Lls+Lss Lsm Lsm L0sr Lpsr LmsrLsm Lls+Lss Lsm Lmsr L0sr LpsrLsm Lsm Lls+Lss Lpsr Lmsr L0srL0sr Lmsr Lpsr Llr+Lrr Lrm LrmLpsr L0sr Lmsr Lrm Llr+Lrr LrmLmsr Lpsr L0sr Lrm Lrm Llr+Lrr

uNNNNNNNNw

parking torque and the steering torque at a standstillof the vehicle can also be calculated with this tyremodel. The multibody full vehicle model in com-bination with the electromechanical steering systemis shown in Fig. 4.The parameter L

ssdenotes the self-inductance of

The differential equations of this model arethe stator windings and Lsm=L

sscos(2p/3) the

generated automatically by the program alaska. Duemutual inductance between two stator windings. Lrr to the inductances the mechanical coordinate h

rmis the self-inductance of the rotor windings and

enters the electrical part (2) of the motion equations.Lrm=L

rrcos(2p/3) the mutual inductance between

On the other hand, the electromagnetic torquetwo rotor windings. Lls

is the leakage inductance ofoccurs as an additional force in the mechanicalthe stator windings and L

lrthe leakage inductance

part (1) of the motion equations. The simultaneousof the rotor windings. The relations L0sr=L

srcos h

r,

solution of the motion equations of both modelLpsr=L

srcos(h

r+2p/3), Lm

sr=L

srcos(h

r−2p/3) are con-

parts guarantees the correct description of allsidered, where Lsr

is the peak value of the mutualelectromechanical interactions occurring in theinductance between the stator and rotor windings.model.To generate the differential equations of an EMS

the program alaska requires the derivatives of theparameters describing the physical properties of 2.4 A driver model for testing thethe electrical components with respect to both the electromechanical steering systemmechanical coordinates and the time. It is recognized

In order to simulate the dynamical behaviour of thethat the parameters L0

sr, Lp

sr, and Lm

srof the inductance

electromechanical steering system with different,matrix depend on the time-dependent rotor angle h

rm.

more or less complicated driving manoeuvres, aTherefore, the partial derivative of the inductancedriver model is used. This driver model consists ofmatrix with respect to the rotor angle h

rmis not zero

both a trace controller and a velocity controller. Itand must be indicated ashas proven itself in various applications [12]. Theconcept of the trace controller, which works withpreview, is sketched in Fig. 5.

The prescribed target trace Ct(s) that the vehicle

model has to follow lies in the (Ex, E

y) plane ofqL

qhrm=

tNNNNNNNNv

0 0 0 L0∞sr Lp∞sr Lm∞sr0 0 0 Lm∞sr L0∞sr Lp∞sr0 0 0 Lp∞sr Lm∞sr L0∞sr

L0∞sr Lm∞sr Lp∞sr 0 0 0

Lp∞sr L0∞sr Lm∞sr 0 0 0

Lm∞sr Lp∞sr L0∞sr 0 0 0

uNNNNNNNNw

the inertial system and uses the arc length s asthe representation or curve parameter. The local

with L0∞sr=−p

pL

srsin h

r, Lp∞

sr=−p

pL

srsin(h

r+2p/3)

and Lm∞sr=−p

pL

srsin(h

r−2p/3). The time derivative

of the inductance matrix can then be written as

dLdt=vrm

tNNNNNNNNv

0 0 0 L0∞sr Lp∞sr Lm∞sr0 0 0 Lm∞sr L0∞sr Lp∞sr0 0 0 Lp∞sr Lm∞sr L0∞sr

L0∞sr Lm∞sr Lp∞sr 0 0 0

Lp∞sr L0∞sr Lm∞sr 0 0 0

Lm∞sr Lp∞sr L0∞sr 0 0 0

uNNNNNNNNw

Fig. 4 A multibody full vehicle model with electro-mechanical steering system


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this purpose a simple proportional controller is used

MA=kA(vsoll−vx), (6)

where kA

is a gain factor of the drive torqueregulation. A detailed description of this driver modelcan be found in reference [13].

2.5 Determination of the entire steering torqueand the desired support torque

A main characteristic of the electromechanicalsteering system is that the steering motion is givenat each time by the driver. The corresponding joint

Fig. 5 Quantities and notations for the trace control reaction forces enforcing the given motion can becalculated by means of inverse dynamics of themechanical subsystem. To this end, the generalizedtangent vector and the associated normal vector aremechanical coordinates ql are divided into so-calleddesignated with e

t(s) and e

n(s). Furthermore, v is the

intrinsic coordinates ql1, whose motion is pre-velocity vector of a vehicle-fixed reference point. The

scribed, and external coordinates ql2. Then, thetravelled distance covered on the target trace is

motion equations describing the dynamics of thedenoted by s(t), which also defines the position of amechanical subsystem can be formulated asreference point. It can be calculated by means of the

following differential equation Ml1l1

ql1+M

l1l2

ql2+Kl

1+Ql

1+QR=0 (7)

s(t)=r(s)

r(s)+d(s)v(t) ·et(s) (3) M

l2l1

ql1+M

l2l2

ql2+Kl

2+Ql

2=0 (8)

If the rheonomic constraints of the intrinsicThereby r(s) denotes the local curvature radius of thecoordinates are given astarget curve at the reference point; d(t) denotes the

lateral deviation between the vehicle and the target ql1=ql10(t)

trace and can be determined with the differentialequation the reaction forces QR can be determined from

equations (7) and (8) using inverse dynamics (see alsod(t)=v(t) ·en(s) (4)reference [14]). The calculated reaction force enforces

The direction deviation y(sV

, t) is defined by the the prescribed motion of the steering system andangle between the longitudinal axis e

xof the vehicle corresponds to the entire necessary steering torque

and a preview vector eV

, which points from the ML

. The task of the electrical machine of the electro-vehicle-fixed reference point to the preview point mechanical steering system is to supply a certainC

t(s+s

V). This preview point lies about the so-called part of the entire steering torque. In the following

preview distance sV

before the reference point on the considerations the support torque generated by thetarget trace. electrical machine is denoted as M

E. The remaining

The steering wheel angle Qs(t) required for the steering torque M

L−M

Ecorresponds with the torque

trace control is determined by means of a first-order stamped at the torsion bar and will therefore bedifferential equation linearly dependent on the lateral called the hand torque M

H. The target relationship

deviation d(t) and the direction deviation y(sV

, t) between the hand torque and the support torqueis determined or estimated by the development

Qs(t)=1

tQ

[kyy(sV , t)+k

dd(t)−Q(t)] (5) engineer in accordance with the desired steering

feeling. It is given, for example, in the form of acharacteristic field as shown in Fig. 6. The character-The behaviour of the trace controller can be adjusted

by the time constant tQ

and the two gain factors ky

istic field matches the desired value of the supporttorque M*

Eto that of the vehicle velocity and theand k

d.

The velocity controller is realized by regulating hand torque.If i denotes the gear ratio between the rotor ofthe drive torque M

Adepending on the difference

between the current velocity of the vehicle in forward the electrical machine and the steering wheel, theelectrical machine must generate a torque m*

E=iM*

E.direction v

s=v ·e

sand the desired velocity v

soll. For


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Voltage equations for the rotor

uar=

dYar

dt+i

arr

r

ubr=

dYbr

dt+i

brr

r

ucr=

dYcr

dt+i

crr

r

(10)

Constitutive equations (flux–current relationships)

Fig. 6 Characteristic field for the determination ofsupport torque

tNNNNNNNNv

Yas

Ybs

Ycs

Yar

Ybr

Ycr

uNNNNNNNNw

3 CONTROL OF THE ELECTROMECHANICALSTEERING SYSTEM

Since in this steering system the rotor of the electricalmachine is connected mechanically with the steeringwheel, the rotation of the rotor is determined by thesteering wheel motion initiated by the driver and

=

tNNNNNNNNv

Lls+L

ssL

smL

smL0

srLp

srLm

sr

Lsm

Lls+L

ssL

smLm

srL0

srLp

sr

Lsm

Lsm

Lls+L

ssLp

srLm

srL0

sr

L0sr

Lmsr

Lpsr

Llr+L

rrL

rmL

rm

Lpsr

L0sr

Lmsr

Lrm

Llr+L

rrL

rm

Lmsr

Lpsr

L0sr

Lrm

Lrm

Llr+L

rr

uNNNNNNNNw

cannot be controlled. Therefore, the control task ofthe electromechanical steering system is to producethe desired torque as accurately as possible at anygiven angular velocity of the rotor. For this highlydynamical servo drive it is necessary to employ thedynamic model of the electrical machine duringthe design of the controller. The presuppositions, tosupply the electrical machine with voltages of con- ×

tNNNNNNNNv

ias

ibs

ics

iar

ibr

icr

uNNNNNNNNw

(11)trollable amplitude, frequency, and phase, can beeasily fulfilled nowadays by efficient power switchesand fast microprocessors.

In the equations (9) and (10), ulk

with lµ{a, b, c}3.1 Model equations of an induction machine in

and kµ{s, r} denote the externally supplied voltages.the a–b–c reference frame

It should be noted that ulr=0 holds for asynchronous

electrical machines. The quantities Ylk

denote theAccording to the electrical part (2) of the motionflux linkages. The electromagnetically generated airequations the dynamic model of the inductiongap torque, which corresponds to Qm(t, ql, qm) in themachine can be expressed explicitly in the so-calledmechanical part (1) of the motion equations, can bea–b–c reference frame as follows.expressed in explicit form as

Voltage equations for the statormE=

1

2∑l∑k

qYlkqhrm

ilku

as=

dYas

dt+i

asr

s Using the matrix qL/qhrm

, this expression simplifiesto (see also reference [15])

ubs=

dYbs

dt+i

bsr

s

mE=pp√3

(Yas ibs−Ybs ias)+

pp√3

(Ybs ics−Ycs ibs)

ucs=

dYcs

dt+i

csr

s

+pp√3

(Ycs ias−Yas ics) (12)

(9)


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This system of coupled differential equations (9) The quantities to be transformed may be voltages,currents, or flux linkages. The inverse transformationto (12) shows that the induction machine is a non-

linear, multivariable system. In order to control it, a matrix from the q–d–0 to the a–b–c reference framecan be written asfield-oriented scheme [16, 17] is employed in this

research work. The decoupling of the equations canbe achieved by substituting the a–b–c referenceframe with a so-called q–d–0 reference frame. T(Q)−1=C cos Q sin Q 1

cos(Q−2p/3) sin(Q−2p/3) 1

cos(Q+2p/3) sin(Q+2p/3) 1D3.2 Model equations of an induction machine inthe q–d–0 reference frame

Applying this transformation to the voltage andA so-called q–d–0 reference frame rotating with an constitutive equations in the a–b–c reference frame,electrical angular velocity v in the same direction the corresponding equations in the q–d–0 referenceas the rotor is shown in Fig. 7. In the special case frame are the following ones.of v=0 there is a stationary reference frame. Settingv=v

e, where v

eis the so-called synchronous

Voltage equations for the statorelectrical angular velocity of the magnetic field of thestator, a synchronously rotating reference frame can

uqs=

dYqs

dt+vY

ds+r

si

qsbe obtained. The transformation of a quantity fromthe a–b–c to the q–d–0 reference frame can be carriedout with the following matrix u

ds=

dYds

dt−vY

qs+r

si

ds

u0s=

dY0s

dt+r

si

0sT(Q)=

2

3Ccos Q cos(Q−2p/3) cos(Q+2p/3)

sin Q sin(Q−2p/3) sin(Q+2p/3)

1/2 1/2 1/2 D(13)

The matrix T(Q) in fact describes a linear transform-ation in the velocity space of the planar electrical

Voltage equations for the rotormachine. Looking at Fig. 7, it is clear that Q=h=∆t

0v(t) dt+h(0) applies to the quantities belonging

uqr=

dYqr

dt+(v−v

r)Y

dr+r

ri

qrto the stator and Q=h−hr

to the quantities of therotor. Applying the matrix T(Q) to the stator or rotorquantities, the following transformation relation holds u

dr=

dYdr

dt−(v−v

r)Y

qr+r

ri

dr

u0r=

dY0r

dt+r

ri

0rC fqfdf0D=T(Q)C fafbf

cD

(14)

Constitutive equations

tNNNNNNNNv

Yqs

Yds

Y0s

Yqr

Ydr

Y0r

uNNNNNNNNw

=

tNNNNNNNNv

L∞s

0 0 Lm

0 0

0 L∞s

0 0 Lm

0

0 0 Lls

0 0 0

Lm

0 0 L∞r

0 0

0 Lm

0 0 L∞r

0

0 0 0 0 0 Llr

uNNNNNNNNw

tNNNNNNNNv

iqs

ids

i0s

iqr

idr

i0r

uNNNNNNNNw

(15)

In the equations (13), (14), and (15) the definitionsL∞

s=L

ls+3L

ss/2, L∞

r=L

lr+3L

rr/2, and L

m=3L

sr/2 and

Fig. 7 Induction machine in a q-d-0 reference frame are taken into account. In terms of q–d–0 quantities,


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the air gap torque generated by the electical machine into equation (20) the following relation holdscan be expressed in the following form (see, forexample, reference [18]) L∞r

rrLm

dYedr

dt+

1

LmYedr= ie

ds (21)mE=

32pp(Yqr idr−Ydr iqr) (16)

This relation means that the rotor flux linkage Yedr3.3 Field-oriented control of the asynchronous can be controlled, in principle, by adjusting the d

machine component of the stator current ieds

. Since itsdynamic behaviour is limited by the rotor circuit timeSelecting, for example, a synchronously rotatingconstant L∞

r/r

r, the control of Ye

dris not suitable forq–d–0 reference frame whose d axis is aligned with

fast changes in torque. Considering equation (19),the rotor field (see Fig. 8, where all quantities in thisthis task can be carried out more successfully byreference frame are characterized by the index e),controlling the q component of the stator currentthe q component of the rotor flux is always zeroie

qs; if the rotor flux linkage Ye

dris not disturbed, the

Yeqr=Lm ie

qs+L∞r ieqr=0 air gap torque can be independently adjusted by ieqs

without delay.HenceIf Ye

qr¬0, the voltage equation for the q axis of the

rotor winding without any applied voltages (ueqr=0)ie

qr=−LmL∞r

ieqs (17)

reduces to

WithYeqr=0 the expression (16) for the air gap torque

ueqr=(ve−vr)Yedr+rr ieqr=0 (22)reduces to

mE=−32ppYedr ieqr (18) Inserting equation (17) into equation (22) yields the

following control conditionSubstituting the current ieqr

using equation (17),equation (18) can be written in the desired form

ve=vr+rrLm

L∞r

ieqsYedr=vr+v2 (23)

mE=3

2pp

LmL∞rYedr ieqs (19)

Equation (23) determines the angular velocity of thewhich shows that the air gap torque can be con- q–d–0 reference frame, so that alignment of the dtrolled by adjusting either the q component of the axis with the rotor field takes place and thereforestator current or the rotor flux linkage.

Yeqr¬0 is dynamically guaranteed.

Since for asynchronous machines there are no The discussed field-oriented control scheme is anvoltages applied to the rotor windings, then ue

dr=0. effective approach for decoupling the non-linear

Taking Yeqr=0 into account the voltage equation for multivariable system of equations of an asynchronous

the d axis of the rotor reduces to machine. The aim of the control strategy is to keepthe rotor flux Ye

dras exactly as possible through

uedr=

dYedr

dt+rr iedr=0 (20) adjusting the d component of the stator current ie

dsand to control the air gap torque by adjusting the q

Inserting the flux linkage equation component of the stator current ieqs

as well as theangular velocity v

e. If a desired rotor fluxYe*

dris given,

iedr=Yedr−Lm ie

dsL∞r

the required d component of the stator current ie*ds

can be determined in accordance with equation (21)by

ie*ds=

L∞rrrLm

dYe*dr

dt+

1

LmYe*dr (24)

To generate the desired air gap torque m*E

therequired q component of the stator current ie*

qscan

be calculated according to equation (19) as follows

ie*qs=

2

3

1

pp

L∞rLm

m*EYe*dr

(25)Fig. 8 q-d-0 reference frame by rotor field orientation


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From equation (23) the following expression yields system is necessary in practical applications. In sucha field-oriented control scheme the response speedof the electrical machine is limited in principle onlyv*e=vr+

rrLmL∞r

ie*qsYe*dr=vr+v*

2(26)

by the delay of the current regulation.

The rotation angle h*e

which determines the fieldorientation will be obtained by numerical integrationof the angular velocity v*

e. This angle is used to 4 RESULTS

transform the quantities from the q–d–0 to the a–b–creference frame. In order to test and demonstrate the dynamical

behaviour of the developed model and controlIn this research work a controlled stator currentsupply is employed. The stator current components scheme for the electromechanical steering system, a

set of full vehicle simulations is carried out. Someie*as

, ie*bs

, and ie*cs

are realized by voltage supply withadditional current regulations. numerical results, obtained by the simulation pro-

gram alaska, are shown in Fig. 10. In detail, an ISOThe block diagram of the control concept for theelectromechanical steering system is shown in Fig. 9. lane change and a sinusoidal steering manoeuvre at

standstill have been simulated.The desired value of the rotor flux is matched to themechanical angular velocity of the rotor and is weak- The results demonstrate that the desired support

torque can be generated by the electrical machine inened beyond the base speed. Due to the physicallimitations of power switches this field-weakening a sufficiently exact manner. A complete and correct

Fig. 9 Control block diagram of the electromechanical steering system

Fig. 10 Simulation results of the electromechanical steering system


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8 Maißer, P. and Steigenberger, J. Zugang zur Theoriedescription of all interactions between the mechanicalelektromechanischer Systeme mittels klassischerand electrical subsystems of the steering systemMechanik, Teil 1: Elektrische Systeme in Ladungs-has been achieved through a uniform modellingformulierung. Wiss. Z. TH Ilmenau, 1974, 20(6),

technique of the whole system. 105–123.9 Maißer, P. Ein Beitrag zur Theorie diskreter elektro-

mechanischer Systeme mit Anwendungen in derManipulator-/Robotertechnik, Dissertation B, Tech-5 SUMMARYnische Hochschule Ilmenau.

10 Oertel, Ch., Eichler, M., and Fandre, A. RMOD-KIn this research work, an electromechanical steering6.0 Manual, 1999 (Gedas GmbH).system is modelled on the basis of a uniform theory

11 Eichler, E. A ride comfort tyre model for vibrationfor discrete electromechanical systems. To control anaylsis in full vehicle simulations. In Tyre modelsthe electrical machine in this steering system, a field- for vehicle dynamic analysis (Eds F. Bohm andoriented control scheme is applied. For the purpose H. P. Willumeit) Supplement to Vehicle System

Dynamics, 1997, 27, 109–122.of validating and testing the electromechanical steer-12 Eichler, M. and Lion, A. Gesamtfahrzeug-ing system, it is implemented in a multibody full

simulationen auf Prufstrecken zur Bestimmung vonvehicle model. To this end, an ISO lane change andLastkollektiven. In Berechnung und Simulation ima standstill steering manoeuvre have been simulated.Fahrzeugbau, 2000, Vol. 1559, pp. 369–398 (VDI-

The numerical results show that the electrical Berichte).machine generates the desired support torque with 13 Lion, A. Ein Kurs- und Geschwindigkeitsregler furexcellent accuracy. Based on this conceptual work EVP-Gesamtfahrzeugsimulationen mit dem Pro-the investigation of other interesting steering con- grammsystem ADAMS/CAR. Volkswagen report,

2003.cepts is possible (e.g. a variable steering gear ratio,14 Maißer, P. and Jungnickel, U. Stability of con-vehicle dynamics control with active steering, etc.).

trolled motion of a gymnast in high-speed mid airSuch an electromechanical steering system may alsomaneuvers. In Proceedings of IUTAM Symposiumserve as a basis for the implementation of steer byon Recent Developments in Non-linear Oscillations

wire functionalities. of Mechanical Systems, Hanoi, Vietnam, 1999,pp. 121–129.

15 Muller, G. Elektrische Maschinen – Betriebsverhaltenrotierender elektrischer Maschinen, 1984 (VDE-Verlag,REFERENCESBerlin/Offenbach, Germany).

16 Leonhard, W. Control of electrical drives, 2nd1 Foth, J., Gazyakan, U., Dominke, P., and Ruck, G.edition, 1996 (Springer-Verlag, Berlin).Steering systems for future requirements. In

17 Schroder, D. Elektrische Antriebe – Regelung vonEuropean Automotive Congress (EAEC7), Barcelona,Antriebssystemen, 2. Auflage, 2001 (Springer-Verlag,Spain, 1999.Berlin).2 Badawy, A., Zuraski, J., Bolourchi, F., and Chandy, A.

18 Ong, C. M. Dynamic simulation of electric machinery,Modeling and analysis of an electric power steering1998 (Prentice-Hall PTR, Upper Saddle River, Newsystem. SAE technical paper series 1999-01-0399,Jersey).1999.

3 Kozaki, Y., Hirose, G., Sekiya, S., and Miyaura, Y.Electric power steering (EPS). Motion and Control,1999, 6, 9–15.

APPENDIX4 Kwasny, O. and Manz, H. Die elektromechanischeLenkung des VW Touran. ATZ, 2005, 105(5), 464–470.

Notation5 Maißer, P., Enge, O., Freudenberg, H., and Kielau, G.Electromechanical interactions in multibody systems

A fundamental loop matrixcontaining electromechanical drives. MultibodyC matrix of the generalized capacitancesSystems Dynamics, 1997, 1(3), 281–302.e

nlocal normal vector of the target trace6 Enge, O., Kielau, G., and Maißer, P. Modelling and

simulation of discrete electromechanical systems. In et

local tangent vector of the target traceProceedings of the Third Conference on Mechatronics e

Vpreview vector of the trace controller

and Robotics: From Design Methods to Industrial ex

longitudinal axis of the vehicleApplications (Ed. J. Luckel), 1995, pp. 302–318 G vector of the generalized gyroscopic/(Teubner, Stuttgart, Germany).

Coriolis voltages7 Maißer, P. and Steigenberger, J. Lagrange-i gear ratioFormalismus fur diskrete elektromechanischei

lkcurrent in a fundamental loopSysteme. Z. Angew. Mathematik und Mechanik,

1979, 59, 717–730. k gain factor


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K vector of the generalized gyroscopic/ Ql vector of the generalized applied forces(mechanical)Coriolis forces

L matrix of the generalized inductances Qm vector of the generalized applied forces(electrically caused)L

lrleakage inductance of the rotor winding

Lls

leakage inductance of the stator winding rr

Ohm’s resistance of the rotor windingr

sOhm’s resistance of the stator windingL

rmmutual inductance between two rotorwindings R matrix of the generalized resistances

s arc lengthLrr

self-inductance of the rotor windingL

smmutual inductance between two stator s

Vpreview distance of the trace controller

ulk

supply voltage in a fundamental loopwindingsL

srpeak value of the mutual inductance v velocity vector of the vehicle-fixed reference

pointbetween stator and rotor windingsL

ssself-inductance of the stator winding V vector of the generalized applied voltages

x: vector of the mechanical coordinatesmE

air gap torqueM generalized mass matrix

d lateral deviation of the trace controllerMA

drive torque of the power trainh

eelectrical angle of the stator fieldM

Esupport torque

hr

electrical angle of the rotorMH

hand torqueh

rmmechanical angle of the rotorM

Lentire steering torque

r local curvature radius of the target tracep number of polestQ

time constant of the trace controllerpp

number of pole pairsQ

ssteering wheel angleq: vector of the branch charges

y direction deviation of the trace controllerql vector of the mechanical generalizedY

lkflux linkage in a fundamental loopcoordinates

v electrical angular velocity of the q–d–0ql1

vector of the intrinsic coordinatesreference frameql

2vector of the external coordinates

ve

electrical angular velocity of the stator fieldqm vector of the electrical generalizedv

relectrical angular velocity of the rotorcoordinates

vrm

mechanical angular velocity of the rotorq0 vector of the charges of current sourcesQR vector of the generalized reaction forces v

2slip angular velocity of the rotor


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Modelling and control of an electromechanical steering