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7/31/2019 Steer by Wire ACC06

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A New Reference Model for Steer-By-Wire Applications with Embedded

Vehicle Dynamics

Julien Coudon, Carlos Canudas-de-Wit, and Xavier Claeys

Abstract We propose a new a reference model for Steer-by-Wire applications. The model accounts for force reflectionand vehicle dynamics interaction. In particular, a Bicycle modelincluding the front axle kinematic and lateral contact forces, areexplicitly accounted during the model tuning. The model is com-pleted with a virtual feedback force, designed from an optimalfeedback control loop, that aims at imposing a particular stee-ring behavior improving the vehicle handling characteristics.This results in a model that is (vehicle) velocity dependent withgood stability properties. The model keeps such good propertieseven in presence of unmodelled tire-road dynamics, and undercertain parameter variation.

Index Terms Steer-by-Wire systems, vehicle control.I. INTRODUCTION

In recent years, important efforts have been undertaken

in drive-by-wire technologies. The Automotive industry is

highly motivated by the introduction of decoupled actuators

in order to improve driving comfort and security, reduce

vehicle cost and accelerate proving ground testing.

Steer-by-Wire is such a technology that replaces the me-

chanical interface between the driver steering wheel and the

vehicle front wheels. Two actuators are used: the steering

wheel actuator generates a force-feedback on the steering

wheel and the front wheels actuator moves the front wheels.

Expected benefits are improved safety (by the suppression

of the mechanical link between driver and wheels), better

vehicle maneuverability and handling characteristics. Steer-

by-Wire can also be further exploited to enhance vehicle

dynamic behavior.

Different Steer-by-Wire strategies have been reported in

the literature. Some of these approaches only use local feed-

back (i.e. the steering wheel angle) to produce a reaction

torque ([5], [6], [7]), computed so as to give a comfortable

driving feel. However, real road information is not trans-

mitted to the driver. As this way to compute the reaction

torque does not reflect measured road forces, the driver is

not aware of the hazards of the road, such as variations of

the road surface conditions, curbs or also ruts. This approachis similar to the unilateral tele-operation strategy.

It is thus well accepted that realistic force-feedback needs

to be transmitted to the driver in a Steer-by-Wire system. In

. This work was supported by RENAULT

. Corresponding author: C. Canudas de Wit

. J. Coudon and Xavier Claeys are with the RENAULT Research Depart-ment, Technocentre, 1 avenue du Golf, 78288 Guyancourt Cedex, FRANCE.Email: [email protected] . Tel: +33 (0)1 76 85 51 65,Fax: +33 (0)1 76 85 77 23.

. C. Canudas-de-Wit is with the Laboratoire dAutomatique de Gre-noble, ENSIEG-INPG, BP 46, 38402 St Martin dHeres, FRANCE. Email:[email protected] .

Controller

Mesures

RackFront wheel

Column

Stabilization

Reference Model(a)

Wheels AngleDriver Torque

+

- -

Cext

h r

Stabilizing

Controller

Steering system

1J s

1s

1n

vv

(, )

Reference Model

(b)

FIG . 1 . Reference model. a) Virtual mechanical structure, b) Blockdiagram of the reference model

this context, other works have developed control structures

including force-feedback strategies. [1], [2], [3], [4] show

control designs which inform the driver of the real status of

his vehicle and especially the front wheel slip. However, in

most of these applications the impact of these force-feedback

designs in the global vehicle dynamical behavior is often not

considered.

The kinematics characteristic (i.e. its geometry) of the

front axle plays a fundamental role on the chassis vehicle

stability. It also substantially impacts the steering system

comfort. Such a tradeoff is managed in todays vehicle with

conventional steering system. In the Steer-by-Wire scenario,force-feedback control design disregarding such an interac-

tion may results in poor vehicle stability properties. It is then

interesting to redesign the Steer-by-Wire strategy under the

basis of such considerations.

In this paper, we propose a new model to be used as a

reference model for Steer-by-Wire applications.

The proposed model accounts for force reflection and ve-

hicle dynamics interaction. In particular, a Bicycle model

including the front axle kinematic and lateral contact forces,

are explicitly accounted during the model tuning. The model


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A simple model of a front axle is used to link the front

tyres to the rack. piv is the steering axis inclination, dep is

the scrub radius, d is the length of the steering arm, ch is

the caster, c is the geometric trail and ct is the tyre trail. The

linear map G(s) results from substitution of (1) in (2).The force Frack applied on the rack due to external force

Fy is given by:

Frack = cos(ch)2

d(r tan(ch) + ct) Fy (3)

=(ch)

dFy

Then the resulting torque on the column due to external

forces applied on the front wheel is defined by:

Cext = Rpc Frack =(ch)

n0G(s) r (4)

where, Rpc is geometric constant (see, Fig. 1-a), and n0 =d

Rpc.

B. The reference model

1) The steering sub-system: is composed of a simple co-lumn linked to a rack by a gear. J is the total system inertia

accounting for: the steering wheel, the column, the Rack, and

the front wheels. A state-space representation of the steering

system is:

d

dt

vv

=

0 10 0

vv

01J

Cext

+

01J

u +

01J

h (5)

where r =vn

, v is the steering wheel angle, n is the desi-

red steering ratio, and u is the control input to be designed.

2) Stabilizing control loop: Equation (4) together with (5)and (1) results in the closed-loop model of the following

form:

() : x = A(V)x + Bu + Hh (6)

where x = [v,v,,]T. Note that A(V) is velocity de-

pending 2. The definitions of the A(V), B, and H matricesare given in the Appendix A. In what follows, we make

explicit the model dependence on the parameter V in view

of its consideration when scheduling controllers for different

values of V.

Under the hypothesis of constant V, model (6) defines a

linear model. This model results, without the additional feed-

back loop (u = 0), in an unstable one for certain values of V.A feedback loop needs then to be added in order to stabilizethe model while forcing this one to behave according with

certain comfort specifications.

In standard steering systems, the stabilizing effect is as-

sured by several factors among which we have: front axle

geometry, steering column friction, and external damping ef-

fects coming from hydraulic or electric power steering. Here

we propose a static feedback to introduce such stabilizing

effects.

2. For simplicity, we consider = 1 in the sequel

III. FEEDBACK LOOP DESIGN

The feedback design for u is done under the basis that

A(V) is velocity dependent. We thus wish to design a feed-back of the form:

u = K(V)x (7)

This results in a family of feedback gains parameterized as

a function of the model vehicle velocity. The closed-loop

equations are:

x = [A(V) BK(V)] x + Hh (8)

y = Cx = r (9)

A. Control specifications

Specifying good handling properties of the controlled ve-

hicle can be translated into forcing the closed-loop map

W(V) : h r, derived from Equations (8)-(9), to ex-hibit a particular transient behavior in the velocity range of

interest, i.e. , V [Vmin,Vmax].

i) Transient behavior:

settling time Ts < 1sec, overshoot D < 10%.

These specifications are typically used in commercial stee-

ring systems to characterize the vehicle handling properties

and the driver feelings; the overshoot constant D provides

a measure of the stability of the free oscillations, while the

settling time Ts influences the driver feelings with respect to

the steering system reactivity. Free oscillations refer to ve-

hicle transient behavior induced by a sudden steering wheel

drop.

It will also be suitable that the reference model, when

used in the feedback inter-connection as shown in Fig. 2,

remains stable under non-modelled effects such as tire-road

dynamics, and vehicle parameters inaccuracies. To this aim,the following specification are also in order:

ii) Stability margins:

gain margin MG > 6 dB,

phase margin M > 40 deg.

Two feedback structures will be presented next: a reduced-

order PD-feedback (IV), and a more general feedback deri-

ved from a LQ formulation (V).

IV. PD-CONTROL

The reduced-order PD-feedback is defined in (7) with

K(

V) = [

kp(

V)

,kv(

V)

,0

,0]

This control structure is interesting because its simplicity and

its direct interpretation as being the damping and the stiffness

of the virtual steering column. The complete reference model

with this additional control loop is shown by Fig. 5.

An optimization algorithm has been set in order to obtain

the best possible gain K(V) compiling with the speci-fications described previously. The proposed algorithm is

summarized in Fig. 6.

First, the velocity domain of interest, Vmin = 20 (Km/h),Vmax = 120 (Km/h) is quantified in equally Vstep-spaced n


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Fy(ch)n

Cext

G(s)

Vehicle Dynamics


+

--

h r

Reference Model

1J s

1s

1n

vv

-

Controller

kv

kp

FIG . 5 . Reference model with PD Controller.

V = [Vmin : Vstep : Vmax]

for i = 1 : VmaxVstep

Ei = {(kv, kp) |MG(kv, kp, Vi) > 6 dB

and M(kv, kp, Vi) > 40 deg.}

Criteria minimization:

J(kv, kp, Vi) =D(kv, kp, Vi)

Dmax

2+

Ts(kv, kp, Vi)

Tmaxs

2

[kv(Vi), kp(Vi)] = arg{min(kv, kp)Ei J(kv, kp, Vi)}

K(Vi) = [kp(Vi),kv(Vi),0,0]

end

FIG . 6 . PD Controller Algorithm

intervals. Then for each value Vi, i = 1,2, . . . , n ., we searchfor a controller K(Vi) = [kp(Vi),kv(Vi),0,0] satisfying thecontrol specifications (i) and (ii).

To this aim, given a velocity value Vi, we search for the

domain Ei containing all the possible values of (kv, kp)meeting the stability margins specifications, i.e., M > 40(deg) and MG > 6 (dB).

Then, among the possible values in the set Ei, the pair(kp(Vi),kv(Vi)) minimizing a cost function depending on thesettling time Ts and overshoot D is selected. This process is

repeated for each value of the vehicle speed in the domain

of interest.

The resulting gains obtained from the optimization process

are shown in Tab. I. Note that for high speeds, where stability

may be lost, the controller stabilizes the vehicle by increasing

its damping gain kv. It is thus worth to schedule this gain

with respect to V. On the other hand, the proportional gain kpcan be kept constant because its variations can be disregarded

in the considered velocity range.

The resulting stability margins and its dynamic perfor-

mances are satisfactory. The gain margin is infinite and the

phase margin remains over 40 (deg) over all the speed range

(Tab. II). The settling time of the reference model is below

TAB . I

Evolutions of PD Controller gains as a function of V.

PD Controller

V (km/h) 20 40 60 80 100 120

kv 0.9 1 1.5 4.2 6.1 7.4

kp 11.3 11.6 11.7 11.8 11.8 11.8

1 (s), and the overshoot remains under 10%. These perfor-mance indexes are robust to model parameters variations. For

instance, variations of10% on the main parameters of theRM do not induce radical decreases of the stability margins

and dynamic performances, see Tab. II.

TAB . II

Robustness of the PD Controllers

PD Controller

minMG minM maxTs maxD(dB) (deg) (s) (%)

Nominal 39 0.4 4.5

C1 10% 37 0.75 9Mtot 10% 39 0.7 5.9Bal 10% 39 0.42 5ch 10% 39 0.4 5

d 10% 37 0.7 6

V. LQ-CONTROL

Another design alternative, seeking to improve the pre-

viously obtained performance index, is to use full state-space

feedback. One possibility, explored here, is to compute K(V)from LQ-design.

Previously to state such a formulation, note that the state-

space representation (6) is controllable for all values of V,

i.e.

rank(C(V)) = rank[B,A(V) B,A2(V) B,A3(V) B] = 4

where C(V) is given in the Appendix A. Consider the nor-malized cost function

J =

0

q1

2v + q2

2v +

1

(uM)2u2

dt

with q1 =q1

(Mv )2 , q2 =

q2(Mv )

2, and Mv ,

Mv , uM, are positive

constants describing the maximal expected magnitudes of thecorresponding signals. From which we define the matrices Q

and R as follows:

Q = diag

q1

(Mv )2

,q2

(Mv )2

,0,0

, R =

1

(uM)2

Some preliminary computations have shown that possible

weights on the last two states have a negligible effect on

the final value of the controller. Therefore, only weights on

the position and velocity of the steering virtual column are

considered.

Once, the weighting matrices Q, and R are defined, the

feedback gain is computed from the associated Riccati equa-

tion, i.e.P(V)A(V) + A(V)P(V) P(V)BR1BTP(V) + Q = 0

with,

K(V) = R1BTP(V) (10)

for each V of interest. The complete reference model inclu-

ding this additional feedback loop is shown in Fig. 7

Note that each pair of values (q1, q2) defines a particularsystem behavior. Thus, we need to find among all possible

values of (q1, q2), and for each V, a pair that satisfies thecontrol specifications, in particular (i), since (ii) is already


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Fy(ch)n

Cext

G(s)

Vehicle Dynamics


+

-

h r

Reference Model

1J s

1s

1n

vv

-

LQ

Controller

(, )

FIG . 7 . Reference model with LQ Controller.

embedded in the problem formulation. Nevertheless, phase

margins need to be checked if other dynamic effects, not

captured by model (6), are to be considered. This issue is

discussed next.

The control is designed using the model (6). This model

does not include the tire-road friction dynamic phenomenon

known as Tyre relaxation length. In order to take it into

account, the lateral force static model (2) can be replaced

by the following model:

Fy =C1

BalV

s + 11 (11)

We call () the extended system defined by () togetherwith the tire-road dynamics (11).

This dynamics has a negative effect on the system sta-

bility since it reduces the phase margin. However, it is not

interesting to explicitly include such a phenomenon from the

beginning, because this dynamics will increases the dimen-

sion of the model forcing the construction of an observer on

the basis of an approximated model. Instead, the stability ofthe global system, with the additional parameter Bal and the

LQ controller, has to be checked a posteriori. This is done

in the algorithm presented hereafter.

The optimization algorithm is quite different from the one

used to compute PD controller. It basically searches a pair

(q1, q2) that conforms with the requested specifications. Anadditional test is also added to account for the effect of

the tire dynamic: the stability margins are calculated on the

complete system (). The values of (q1, q2) which give anon-stabilizing controller for this extended model are thus

eliminated, those that give stabilizing controllers are kept.

Finally, the pair (q1, q2) preserving the stability for the

extended system (), and yielding the best tradeoff betweena small settling time Ts and a small overshoot D is retained.

As before, this process is repeated for each value of the

vehicle speed, Vi. The algorithm is summarized in Fig. 8.

Tab. III shows the gains values resulting from this algo-

rithm.

A brief look to dynamic performances shows that this

controller provides very good performances; the settling time

never exceeds 0.45 (s), and the overshoot is kept below

0.33%. The obtained stability margins (including tire dyna-mics) are also very satisfactory providing very high degree

V = [Vmin : Vstep : Vmax]

for i = 1 : VmaxVstep

Compute Ei

Ei = {(q1, q2) |MG((q1, q2, Vi)) > 6 dB

and M((q1, q2, Vi)) > 40 deg.}

Optimize (q1, q2)

[q1(Vi), q2(Vi)] = arg{min(q1, q

2)Ei

2

J(q1, q2, V)}

J(q1, q2, Vi) =D(q1, q2, Vi)

Dmax

2+

Ts(q1, q2, Vi)

Tmaxs

2

Compute K(Vi) from Eq.(10)

end

FIG . 8 . LQ Controller Algorithm

TAB . III

Evolutions of LQ Controller gains as a function of V.

LQ Controller

V (km/h) 20 40 60 80 100 120

k1 21.5 20.9 21.5 21.3 21.9 11.9

k2 3.2 2.1 3.2 4.8 5.3 5.2

k3 36.9 17.5 13.3 11.9 10.9 11.2

k4 202 198 203 205 202 199

of robustness: MG is infinite and the phase margin is grea-

ter than 76 (deg). Consequently, variations in some of the

physical model parameters do not impact much the phase

margin. The settling time and the overshoot do not increase

significantly neither. The precise values are given in Tab. IV.

TAB . IV

Robustness of the LQ Controller

LQ ControllerminMG minM maxTs maxD

(dB) (deg) (s) (%)

Nominal 77 0.37 0.33

C1 10% 76 0.45 2Mtot 10% 77 0.37 0.4Bal 10% 75 0.37 1.5ch 10% 77 0.37 0.45

d 10% 77 0.42 1.5

V I . SIMULATION RESULTS

This section compares in simulations the two models pro-

posed in the previous sections. The Model has been tuned

by designing feedback loops satisfying the desired properties:

the phase margin M is over than 40 (deg) and gain margin

MG is infinite. The settling time remains under 1 second and

the overshoot never exceeds 5%. Exact values are given inTab. II and Tab. IV. Note that the LQ-design leads to a lower

overshoot.

A. Free oscillations

Fig. 9 shows the yaw rate transients under a sudden stee-

ring wheel drop by the driver. Under this scenario, the self

aligning torque makes the yaw rate return to zero. Conven-

tional steering systems exhibit, in this case, a certain level of

overshooting as the figure shows. The two reference model

responses, also shown in the figure, demonstrate the model

ability to reduce such phenomena.


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4.5 5 5.5 6 6.5 7

0.04

0.02

0

0.02

0.04

0.06

Temps (s)

Yawr

ate(rad/s)

Conventional Steering

PD Controller

LQ Controller

FIG . 9 . Yaw rate transients under a sudden steering wheel drop.

1 2 3 4 5 6 7 810

9

8

7

6

5

4

3

2

Temps (s)

Torque(Nm)

LQ Controller

PD Controller

FIG . 10 . Front sliding scenario (level differences are du to the differentcharacteristics of each controller).

As a consequence of this oscillation attenuation, the total

torque, T,

T = Kx + Cext

which is feeded back to the driver, is also well damped

(simulation are not shown here for lack of space) improving

the driver feelings.

B. Changes in the road adhesion coefficient

One of the objectives of the proposed model is to be

sensitive to on-line variations of the vehicle interaction with

the road. This goal is obtained by the presence of the inputs

(measured signals) Cext,, and in the model, such that the

driver is aware of the real status of his vehicle. For instance,

in case of a sudden front sliding (i.e. an abrupt change in

), the driver has to feel a brutal drop of the total torque-

feedback T on the steering wheel.The simulation scenario, to evaluate such property, repro-

duces a driver intention of preserving the wheels orientation

under a sudden front sliding. For such a purpose, a simple

outer PD feedback loop is used to regulate the desired vehicle

orientation. It has the steering wheel angle position as its

input, and h as its output.Fig. 10 shows the time-evolution of T under a sudden

drop of from 1 to 0.2. It can be observed that in both

cases, T reflects such a variation, with enough magnitude

to allow the driver to be informed of the real vehicle status.

VII. CONCLUSIONS

We had introduced a new reference model for Steer-by-

Wire applications. The model is designed to reproduce a

virtual steering structure combining mechanical components

(composed by the steering wheel, the rack, and the front tires)

with a feedback loop (named here controller). The model has

the driver torque, the auto-alignment torque, the yaw-rate,

and the slip angle as the inputs. The model output is the frontwheel angle. These extra inputs, allows for better assessment

of the vehicle dynamic evolution as well as variations on the

tire/road interaction.

We have presented two different ways of designing this

loop; a reduced-order PD structure, and a optimal LQ feed-

back. This last, as expected, performs over the PD feedback.

Because the inclusion of the vehicle dynamics, the obtained

model turns out to be velocity dependent. For all the conside-

red velocity range, the model has been shown to have good

stability properties, to fulfill the transient specifications, and

to keep such good properties even in presence of unmodelled

tire-road dynamics, and under certain parameters variations.

APPENDIX

A. matrix definitions

A(V) =

0 1 0 0

C1 K0

J d0

l1 C1 K0J V

C1 K0J

C1 l1d Iz

0 C1 l

21+C2 l

22

V Iz

C2 l2C1 l1Iz

C1Mtot V d

0 1 +C2 l2C1 l1

Mtot V2

C1+C2Mtot V

BT

=

0 1J

0 0

, H =

0 1J

0 0

where K0 =(ch)

d.

C =

0 1J

0 C1 K0

J2 d

1J

0 C1 K0

J2 d

K0 C21 l

21

J2 V d Iz+

K0 C21

J2 Mtot V d

0 0

C1 l1

d Iz J

(C1 l21+C2 l

22) C1 l1

J V I2z d +

(C2 l2C1 l1) C1

J I z Mtot V d

0 0C1

Mtot V d J(1 +

C2 l2C1 l1Mtot V

2 )C1 l1J d Iz

(C1+C2) C1J M 2

totV2 d

REFERENCES

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