STEER-BY-WIRE SUSPENSION ANDSTEERING DESIGN FOR CONTROLLABILITY

AND OBSERVABILITY

S. Laws ∗ C. Gadda ∗ S. Kohn ∗ P. Yih ∗

J.C. Gerdes ∗ J.C. Milroy ∗

∗ Department of Mechanical EngineeringStanford University

Abstract: With steer-by-wire, suspension and steering systems are no longergoverned by purely mechanical criteria but must instead be designed accordingto mechatronic considerations. This paper demonstrates an approach to steer-by-wire suspension design in which geometric design variables are chosen to render thetire reaction torque about the steer axis highly predictable. This predictability, inturn, clarifies the connection between the vehicle dynamics and steering dynamics,enabling full state observer and controller design. Copyright c©2005 IFAC

Keywords: Steer-by-wire, Mechatronics, Observability, Controllability, VehicleDynamics

1. INTRODUCTION

Steer-by-wire technology promises to deliver sig-nificant benefits, ranging from relaxed packag-ing constraints to advanced safety systems suchas lanekeeping assistance. While by-wire systemspose new engineering challenges, they also openup new design opportunities for suspension andsteering systems. Currently, suspension and steer-ing are designed to provide good vehicle handlingcharacteristics and driver feel. With steer-by-wiretechnology, handling and feel become softwarefeatures customizable on a per-driver basis. Assuspension and steering make this transition tofully mechatronic systems, traditional design cri-teria must be re-examined. This paper presentsa look at design from the perspective of inherentcontrollability and observability.

The key idea of the controller and observer struc-ture assumed in this paper is that the steering mo-tor dynamics and the vehicle dynamics are linkedthrough the tire forces and their reactions aboutthe steer axis. In Yih and Gerdes (2004), this rela-

tionship is used to develop an effective estimatorfor sideslip angle. Combined with a yaw rate mea-surement, the full vehicle state is then available forcontrol, providing a range of possibilities for con-trol algorithms (Yih and Gerdes, 2004; Vilaplanaet al., 2004). Gadda et al. (2004) showed thatthis link between vehicle and steering dynamics,if known, could also be exploited for developingsteer-by-wire diagnostics. Thus the predictabilityof the connection between the steering system andthe tire forces becomes central to observability,controllability, and diagnostics.

The geometry of the steering and suspension sys-tems strongly influences the nature of the steeraxis reaction torque. Thus the need for a pre-dictable torque can be translated into specificconstraints on system kinematics. The core of thepaper focuses on describing these constraints anddiscussing how they are influenced by geometricdesign parameters. These ideas are further de-veloped through an analysis and redesign of anexisting steer-by-wire system for improved con-trollability and observability.

Fig. 1. Stanford’s “P1” Steer-by-wire vehicle2. STEER-BY-WIRE CONTROL

PHILOSOPHY

In previous research at Stanford, the authors havemodified a Corvette to steer-by-wire configura-tion (Yih et al., 2003) and, more recently, de-veloped P1, a by-wire vehicle with independentfront-wheel steering (Figure 1). For both vehicles,the control scheme must account for both a lackof state information about the vehicle motionand disturbances on the steering system due tothe front tire forces. By incorporating a physicalmodel of the steering, these two problems can belinked and simultaneously solved. While vehicledynamics are inherently nonlinear, the followingderivation simplifies the problem to a linear frame-work to establish basic intuition behind the ap-proach.

bar

Uy

Ux

Uβ

lrα

rfα

lfα

rrαrδ

lδ

Fig. 2. Vehicle schematic and nomenclature

The planar dynamics of the vehicle can be mod-eled using the bicycle model, where the width ofthe vehicle is considered negligible. In this case, aslight extension to the bicycle model is used whichconsiders left and right steer angles (δl and δr)separately (Figure 2). Small angle approximationsare used and lateral tire force is assumed to beproportional to the tire slip angle, so that a linearmodel of the planar vehicle dynamics is developed,given by the following:

xv = Avxv + Bvδ (1)

where

xv =[β r

]T

δ =[δl δr

]T

Av =

−C0

mUx

C1

mU2x

− 1

C1

Iz− C2

IzUx

Bv =

Cαf

2mUx

Cαf

2mUx

Cαfa

2Iz

Cαfa

2Iz

C0 = Cαf + Cαr

C1 = Cαrb− Cαfa

C2 = Cαfa2 + Cαrb2

where δl and δr are the left and right steer angles,β is the sideslip angle, r is the yaw rate, Iz is thepolar moment of inertia of the vehicle, Cαf andCαr are the front and rear cornering stiffnesses, aand b are the distances from the center of gravityto the front and rear axles, m is the mass of thevehicle, and Ux is the forward velocity.

The steer-by-wire system at each wheel is a DCmotor and gearbox connected via a parallelogramfour-bar linkage to the wheel, enabling the follow-ing linear model of the steering:

˙xm = Amxm + Bm

[i τa

]T (2)

where

xm =[δ δ

]T

Am =

0 1

0 − bm

Jm

Bm =

0 0rgηkm

Jm− 1

Jm

where δ is the steering angle, Jm is the effectivemoment of inertia of the steering system, bm isthe effective damping of the steering system, rg

is the gearbox ratio, η is the combined efficiencyof the motor and gearbox, and km is the motorconstant relating torque to current. The inputs tothis model are the current to the motor, i, and thealigning torque, τa.

Tire forces generate a reaction torque about thesteer axis. During most driving, the major con-tribution to this reaction torque is the aligningtorque τa, which is the reaction torque due tolateral tire forces. It is related to the vehicle stateby the following equation:

τa = −Cαf (tp + tm)(β +a

Uxr − δ) (3)

where tp and tm are the pneumatic and mechani-cal trails of the tire (Figure 3). Pneumatic trail isthe distance from the center of the tire contactpatch to the centroid of the force distributionon the tire contact patch. This value is typi-cally about 20-25mm for passenger cars. Although

constant for most handling regions, as the tireapproaches its friction limit, the pneumatic traildecreases toward zero. In order to arrive at a linearmodel, tm and tp are assumed to be constant (asare Cαf and Cαr), and the steer axis reactiontorque is assumed to be equal to the aligningtorque.

Combining (1), (2), and (3) yields a linear state-space model of the vehicle that is controllableand observable using measurements of only thesteering angles and yaw rate. The accuracy ofthis model is limited by the assumptions thatthe aligning torque is a linear function of thevehicle states and that it is the only contributorto steer axis reaction torque. The next sectionexamines how the steering system geometry canbe designed to minimize the contributions to steeraxis reaction torque not captured in this linearmodel.

3. CURRENT STEERING GEOMETRY ANDDESIGN CONSIDERATIONS

The steering and suspension systems of the P1 by-wire vehicle are shown in Figure 4. The suspensionis a double wishbone design with a four-bar steer-ing linkage.

The mechanism by which a motor steers an in-dividual wheel is comprised of two components:the steering knuckle (or upright), which definesthe relative positions of the wheel and steer axis,and the steering linkage, which defines the relativepositions of the steer axis and the motor axis.The design of each of these components can beanalyzed separately.

3.1 Steering Knuckle Parameter Definitions

There are five main geometric parameters to con-sider in steering knuckle designs: the wheel radius(Rl), caster angle (θc), kingpin inclination angle(θk), scrub radius (d), and mechanical trail (tm).These parameters are illustrated in Figure 3 withthe car at its nominal suspension position and zerosteer angle. Caster and kingpin inclination anglesare defined relative to the car, while mechanicaltrail and scrub radius are defined relative to thewheel. Caster and kingpin inclination angles donot change with steer angle, but mechanical trailand scrub radius may. Most suspensions are de-signed such that the effect of vertical suspensiontravel on these measurements is small.

The values of these five parameters for the cur-rent design are within the typical ranges of mostpassenger cars. These values are given in Table 1.

Ste

er A

xis

Left Side View

Stee

r Axi

s

θcθk

Rear View

tm dtp

k

w

ss

k

Fig. 3. Left and rear views of the left-hand wheel

Motor &Gearbox

Shock & Spring

Lower Suspension Arm

Upper Suspension Arm

Steering Knuckle

Tie rod

Steering KnuckleBall Joints

Fig. 4. View of current suspension and steeringsystems

Table 1. Steering Knuckle Parameters

Parameters at Typical Current P1

Zero Steer Values Design

Wheel Radius (Rl) 300 to 350 mm 320 mm

Caster Angle (θc) 4 to 7◦ 5.5◦

Kingpin Inclination 7 to 15◦ 13.2◦

Angle (θk)

Scrub Radius (d) -75 to 75 mm 50.5 mmMechanical Trail (tm) 15 to 50 mm 28 mm

3.2 Effects of Knuckle Parameter Selection

Most important to steer-by-wire design is howthese parameters effect the steer axis reactiontorque. This is important from a mechanism de-sign standpoint since it is what must be canceledby the steering actuator to maintain wheel posi-tion. It is also important from a state estimationstandpoint since it is how tire forces (and, by amodel, vehicle states) are observed by the steeringtorque.

There are three main contributions to the steeraxis reaction torque:

• The cross-product of lateral tire forces withtotal trail

• The effect of suspension jacking

• The cross-product of longitudinal tire forceswith scrub radius

The relative importance of these three parts de-pends on vehicle speed.

Effects on Steer Axis Reaction Torque from TotalTrail At higher speeds (>5 m/s) when highlateral forces can be generated with relativelysmall steer angles, the cross-product of lateraltire forces with total trail dominates the steeraxis reaction torque. This is the only contributionconsidered in the simple model given in (3). In atraditional steering system, this torque providesthe return-to-center feel at higher speeds. In asteer-by-wire system, larger total trails not onlywill increase the gain used to sense lateral tireforces but will also increase the actuator effortneeded to maintain a non-zero steer angle.

The following equations provide an analytic ex-pression for mechanical trail (tm) for the left wheelas a function of steer angle (δ):

k =1√

tan2 θc + tan2 θk + 1

− tan θc

− tan θk

1

(4)

P (δ) = (1− cos δ)kkT +

+

cos δ k3 sin δ −k2 sin δ−k3 sin δ cos δ k1 sin δk2 sin δ −k1 sin δ cos δ

(5)

s = [x y z]P (δ) [−Rl tan θc d Rl]T (6)

w = [x y z]P (δ) [0 1 0]T (7)

k = [x y z] k (8)

l = s−Rlz − z · ww

‖z − z · ww‖(9)

tm = (l− z · lz · k

k) · w − w · zz

‖w − w · zz‖(10)

In (5), P (δ) is a rotation matrix which rotatesabout the steering axis, given by k in (8). Theorientation of the wheel is represented by w in(7). The location of the center of wheel and thelocation of the contact patch, both relative to thenominal intersection of the steering axis with theground, are given by s and l in (6) and (9). Vectorsk, w, and s are illustrated in Figure 3.

A plot of mechanical trail for the current steer-by-wire design is given in Figure 5. Note thatmechanical trail changes very significantly withsteer angle, even passing through zero and goingnegative. When mechanical trail passes throughzero, the vehicle states are unobservable, and itbecomes impossible to estimate vehicle sideslip.

Effects on Steer Axis Reaction Torque from Sus-pension Jacking At lower speeds (<5 m/s) when

−30 −20 −10 0 10 20 30−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Steer Angle (deg)

Mec

hani

cal T

rail

(m)

Current Design

Proposed Design

Fig. 5. Mechanical trail change on the left wheelas a function of steer angle

the contribution from total trail and lateral forcesis small, the effect of suspension jacking can dom-inate the steer axis reaction torque. As the wheelsturn in and out, they lower and raise the car,respectively. Suspension jacking is the torque feltabout the steer axis as a result of this motion.Jacking torque is simply the reaction torque aboutthe steer axis from tire normal forces.

Jacking torque (τj) as a function of steer angle (δ)and normal force (Fz) is given by (4, 5, 6, 7, 8, 9)and the equation below:

τj = k · (l× Fzz) (11)

Figure 6 illustrates the jacking torque effect onP1. The jacking torque tends to be large onlywhen turning out at high steer angles. When thewheels are linked together in a traditional steeringsystem, the jacking torques of each wheel nearlycancel at low steer angles. At high steer angles,the jacking torque of the wheel that is turningout dominates, resulting in a return-to-center feel.This is less important at high speeds since, fora given amount of steer angle, the torque dueto lateral tire forces becomes much larger thanjacking torque.

Effects on Steer Axis Reaction Torque from ScrubRadius The steer axis reaction torque contribu-tions from scrub radius are generally undesirableand often minimized by design. The primary mo-tivation for having a non-zero scrub radius is thatat near-zero vehicle speed, it allows the tire to rollslightly while turning.

Turning with too little scrub radius at near-zerospeeds would require very high actuator effortand result in excessive tire wear. However, a largescrub radius is also not desirable as it allows longi-tudinal tire forces (i.e. braking) to have a notableeffect on steer axis reaction torque. When both

−30 −20 −10 0 10 20 30−80

−70

−60

−50

−40

−30

−20

−10

0

10

20

Steer Angle (deg)

Jack

ing

Tor

que

(N−

m)

Current DesignProposed Design

Fig. 6. Jacking torque on the left wheel as afunction of steer angle

wheels are linked together in a traditional steeringsystem, the contributions from the left and rightsides generally cancel out. When the wheels aredecoupled as in the P1 steer-by-wire system, theeffects of longitudinal tire forces are undesirablefor vehicle state estimation and control.

3.3 Steering Linkage Design

The P1 steer-by-wire vehicle uses a parallelogramfour-bar steering linkage (Figure 4). This avoidsall prismatic joints, maintaining high efficiencyand backdrivability. It also maintains a 1:1 rela-tionship between motor and steer angles so thateffective inertias are constant.

The steering linkage should be designed to min-imize roll steer. Because steering linkages tie to-gether parts that do and do not move with verticalsuspension travel, it is possible that the wheelsmay steer when the vehicle rolls. In traditionalsteering systems, this effect may be desirable togive a proper feel to the driver. In steer-by-wiresystems, this creates an effect that is difficultto measure, causing a deterioration in state es-timation ability. The roll steer characteristics aredetermined by the position of the inboard tie rodend, which can be selected with the same processused in traditional steering system design.

4. EXPERIMENTAL RESULTS

One way to validate the steer axis reaction torquemodel is to use experimental data from P1 to com-pare estimates of steer axis reaction torque to itsmeasured value. The maneuvers were performedat a slow speed of 4 m/s to allow high steer angleswhich better illuminate steering nonlinearities.Longitudinal force contributions to the reactiontorque are small enough to be neglected since themaneuvers were performed without braking.

21 22 23 24 25 26 27 28 29 30

−20

0

20

δ (d

eg)

21 22 23 24 25 26 27 28 29 30

−200

−150

−100

−50

0

50

100

Ste

er a

xis

reac

tion

torq

ue (

Nm

)

Time (s)

MeasuredLinear modelNonlinear model

Fig. 7. Steer axis reaction torque estimation onleft wheel

The measured value of steer axis reaction torqueis computed using a measurement of steer motorcurrent with a gearbox and motor model. Thismodel includes friction, efficiency, damping, andinertia contributions. The estimates of steer axisreaction torque are computed using measurementsof vehicle states and steer angles with two differentmodels. The first is the linear aligning momentmodel given in (3); the second is the nonlinearmodel developed in the previous section thatincludes jacking torque and a varying mechanicaltrail.

The results for the left wheel are given in Fig-ure 7. The largely constant offset between the twoestimates is due primarily to the contribution ofjacking torque. Near the end of the test, when theleft wheel is turned far inward, this offset growsnotably. This is due to the increasing mechanicaltrail. By including both of these, the nonlinearmodel tracks the measured value much better.Clearly, the effects of these two contributions can-not be neglected with P1.

5. AN IMPROVED STEERING GEOMETRYDESIGN

Based on this discussion, the suspension and steer-ing systems on P1 can be improved by:

• Reducing mechanical trail changes with steerangle

• Reducing suspension jacking torque• Reducing effects of longitudinal forces on

steer axis reaction torque

A significant reduction in kingpin inclination an-gle will reduce mechanical trail changes and the

effects of suspension jacking. A reduction in scrubradius will reduce the impact of longitudinal forceson steer axis reaction torque. These are altered bychanging the position of the two ball joints on thesteering knuckle (illustrated in Figure 4).

Changes to ball joint positions cannot be madearbitrarily. One reason is that the distance be-tween the ball joints should not decrease much.This would increase the force-loading on suspen-sion members, increasing unwanted complianceeffects that reduce the accuracy of steer anglemeasurement. Another is packaging. The posi-tions of wheels and brakes place constraints onthe available positions for ball joints. There arethree ways to position the ball joints:

• Both ball joints inside the rim of the wheel.Although this allows for low kingpin angles,it requires that they must be close together,giving undesirable force-loading characteris-tics.

• Lower ball joint inside the rim and upper balljoint inboard of the rim and tire. This allowsthe upper ball joint to be positioned higherand further away from the lower ball joint,but makes it difficult to attain both a smallscrub radius and kingpin inclination angle.

• Lower ball joint inside the rim and upper balljoint above the rim and tire. This providesadequate separation distance between theball joints and allows both a small scrubradius and kingpin inclination angle at theexpense of consuming packaging space abovethe tire. This is known as a tall knuckledesign.

For the P1 steer-by-wire vehicle, the last optionabove is the best choice. Due to packaging con-straints, a zero kingpin inclination angle is un-desirable. The proposed knuckle design is sum-marized by the parameters in table 2. The newparameters give mechanical trail and suspensionjacking characteristics given in Figures 5 and 6.

Table 2. Steering Knuckle Parameters

Parameters at Current P1 Proposed

Zero Steer Design Design

Wheel Radius (Rl) 320 mm 320 mm

Caster Angle (θc) 5.5◦ 6.3◦

Kingpin Inclination 13.2◦ 1.9◦

Angle (θk)

Scrub Radius (d) 50.5 mm 28 mmMechanical Trail (tm) 28 mm 33 mm

The basic form of the proposed design itself is notnew; tall knuckle designs exist in production carstoday. However, the motivations behind and theadvantages of selecting this design are quite dif-ferent. They are based not on traditional steeringfeel but rather their implications on controllabilityand observability.

6. CONCLUSIONS

The design considerations for individual wheelsteer-by-wire are quite different than those for tra-ditional steering systems. More suitable steeringgeometries can be developed by examining howthe geometric parameters of the steering systeminfluence observability and controllability.

The design criteria established in the paper com-pliment existing suspension and steering systemdesign strategies by providing a framework withwhich to analyze the performance of steer-by-wiresuspension designs. They provide the necessarylink between traditional suspension and steeringdesign and new steer-by-wire technology.

7. ACKNOWLEDGMENTS

The authors would like to acknowledge NissanMotor Corporation for sponsoring this research.Special thanks to Toshimi Abo, Kazutaka Adachi,Tomoko Inoue, Takeshi Mitamura, Dr. KimioKanai, and Masaharu Asano for their support ofthis project.

REFERENCES

Feick, S. and M. Pandit (2000). Steer-by-wire asa mechatronic implementation. SAE WorldCongress. 2000-01-0823.

Gadda, C. D., P. Yih and J. C. Gerdes (2004).Incorporating a model of vehicle dynamics ina diagnostic system for steer-by-wire vehicles.Proceedings of AVEC pp. 779–784.

Vilaplana, Miguel A., Douglas J. Leith andWilliam E. Leithead (2004). Control ofsideslip and yaw rate in cars equipped with4-wheel steer-by-wire. SAE World Congress.2004-01-2076.

Yih, P. and J.C. Gerdes (2004). Steer-by-wire forvehicle state estimation and control. Proceed-ings of AVEC pp. 785–790.

Yih, P., J. Ryu and J.C. Gerdes (2003). Mod-ification of vehicle handling characteristicsvia steer-by-wire. Proceedings of the Ameri-can Control Conference.