dynamic-deﬂection tire modeling for low-speed vehicle...

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Shiang-Lung KooDepartment of Mechanical Engineering,

University of California at Berkeley,Berkeley, CA 94720-1720e-mail: [email protected]

Han-Shue TanCalifornia PATH,

University of California at Berkeley,Institute of Transportation Studies,

Richmond, CA 94804-4698e-mail: [email protected]

Dynamic-Deflection TireModeling for Low-Speed VehicleLateral DynamicsVehicle lateral dynamics depends heavily on the tire characteristics. Accordingly, a num-ber of tire models were developed to capture the tire behaviors. Among them, the empiri-cal tire models, generally obtained through lab tests, are commonly used in vehicledynamics and control analyses. However, the empirical models often do not reflect theactual dynamic interactions between tire and vehicle under real operational environ-ments, especially at low vehicle speeds. This paper proposes a dynamic-deflection tiremodel, which can be incorporated with any conventional vehicle model to accuratelypredict the resonant mode in the vehicle yaw motion as well as steering lag behavior atlow speeds. A snowblower was tested as an example and the data gathered verified thepredictions from the improved vehicle lateral model. The simulation results show thatthese often-ignored characteristics can significantly impact the steering control designsfor vehicle lane-keeping maneuvers at low speeds. �DOI: 10.1115/1.2745847�

IntroductionVehicle lateral characteristics are crucial in steering control

unctions of the advanced vehicle control systems �e.g., automatedighway systems �AHS� �1� and advanced chassis control system2��. In particular, the low-speed characteristics are often over-ooked and are very important to vehicle control applications,uch as automatic steering systems for bus precision docking �3�,he automated rotary plow �4�, and the automated rubber tire gan-ry �RTG� �5�. An accurate tire model is often required to facilitatehe design or implementation of these advanced control systems.

Literature review shows that a number of tire models were de-eloped for various purposes. These models generally describedhe relations between tire deflections and tire forces. However,hey all have certain limitations at low vehicle speeds. Tire modelsan be divided into three categories: finite element models, ana-ytical models, and empirical models. �I� The finite element mod-ls are constructed to underscore specific tire characteristics suchs reinforcement, inflation pressure, rubber elastic behavior, rimontacts, and inertia forces. These models have the ability to de-cribe tire behavior in detail �6�. They are usually employed in acientific study to perform simulations to predict tire elastic de-ormation and tire force. Complex numerical approaches are com-only used to solve the equations in these models but are often

oo cumbersome for vehicle dynamics analysis and control. �II�he analytical model calculates the tire forces and predicts thessential tire elastic characteristics by focusing on the physicalroperties of a tire. Well-known models include the beam-on-lastic-foundation model and the stretched-string model �7�. Theyrovide a basic understanding of tire lateral behavior. However,he solutions at a nonsteady-state involve solving partial differen-ial equations. Therefore, it is not easy to incorporate the PDE

odels into vehicle control formulation. �III� The empirical models based on approximations of the relationship between the tireateral forces and the slip angles. These relationships can be ob-ained through steady-state lab tests �8� or dynamic lab tests9,10�. The linear tire model and the magic formula tire model arewo well-known examples that describe steady-state tire behav-ors. The relaxation length tire model, essentially a tire model with

Contributed by the Dynamic Systems, Measurement, and Control Division ofSME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CON-

ROL. Manuscript received June 23, 2005; final manuscript received January 10,
007. Review conducted by Hemant M. Sardar.
ournal of Dynamic Systems, Measurement, and ControlCopyright © 20

ded 21 Apr 2009 to 219.239.98.208. Redistribution subject to ASM

an additional first-order lag filter, was proposed to describe thedynamical relation between the slip angle and the lateral force forsmall slip angles based on the dynamic lab test results �11�. Forlarge slip angles, an ad-hoc method that combines the magic for-mula tire model with the first-order lag was employed �12�. Manystudies on the tire dynamic behavior indicate that this lag propertyis evident at very low speeds.

The empirical tire models are widely used in vehicle dynamicsanalysis, control designs, and driving simulators. However, theseempirical models have two major limitations. �I� They do notrepresent comprehensive tire lateral dynamics. For example, it isdifficult for them to explain the tire shimmy problem �13�, whichdoes impact vehicle lateral dynamics. �II� They do not considerthe actual tire-vehicle interactions in a real vehicle configuration.These models are generally simplified too far to capture both theimportant secondary tire behaviors as well as the actual interac-tions between tire and vehicle in a real vehicle configuration.

The main goals included improving the empirical tire modeland preserving the ability of the easy applications to control anddynamics analysis. This paper describes the development of asimple dynamic-deflection tire model as well as the resulting im-provements on the tire-vehicle interface. Low-speed experimentswith a snowblower were used to validate the model improve-ments. The H-infinity theory and �-synthesis are used to designdifferent steering controllers that show the impact of these specificlow-speed dynamics to the vehicle controls. Low-speed vehicledynamics turn out to be a key factor in designing high-gain auto-matic snowblower steering control. This important field of dynam-ics is rarely discussed in literature on vehicle control.

This paper is organized as follows: Sec. 2 formulates the prob-lem; Sec. 3 focuses on the development of the improved tire andvehicle models; the predicted results of the improved models arediscussed in Sec. 4; Sec. 5 describes the experimental setup andparameter identification; Secs. 6 and 7 investigate, respectively,the impact on vehicle handling dynamics and vehicle lateral con-trol; Sec. 8 summarizes the paper.

2 Problem DescriptionThe vehicle consists of four subsystems: actuation devices,

tires, unsprung inertias, and the vehicle body. The actuation de-vices execute the driver commands to drive the unsprung inertia.Based on the ground conditions and the states of the unsprung
inertias, the tires generate forces through the unsprung inertia to
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ontrol the vehicle body. The common vehicle lateral models,uch as the bicycle model �14�, the 3-DOF vehicle lateral model15�, and the 6-DOF vehicle model �16�, are all within the frame-ork. Each of these models is composed of subsystems with vary-

ng degrees of complexity.The main interests of this paper lie in the areas of tire lateral

ehavior and tire-vehicle interface. Most tire models employed inredicting vehicle lateral controls describe the linear �or nonlin-ar� static tire behavior with one exception: the relaxation lengthire model inserts a speed-dependent first order lag. A large num-er of vehicle lateral models neglect the dynamics of unsprungnertia. The vehicle body, the unsprung inertia, and the actuationevices are usually treated as a whole. The tire forces are thenirectly applied to this lumped-together variation of the vehicleody, and the steering mechanism simply imposes a geometriconstraint between the wheels and the vehicle body. However, thisimplification may not be applicable under low-speed conditions,hen the wheels and the vehicle body have noticeable relativeotions. For example, a vehicle yaw motion can be observed

learly when the driver swivels the steering wheel in a vehiclehich is nearly standing still. This observation indicates that the

tire yaw suspension” between the vehicle and the ground, and theteering “dynamics” of the front wheels cannot be neglected atery low speeds.

To study the impact of these often-ignored tire and interfaceynamics, an improved tire model with an augmented tire-vehiclenterface is needed. This tire model should have the followingroperties: �I� it is relatively simple and linearizable for most con-rol system synthesis and analysis; �II� it includes the commonlynown tire characteristics, such as the side slip angle and the lagehavior of the tire force; �III� when incorporated into a vehicleateral model, this model describes the essential modes of theehicle body; �IV� the resulting tire-vehicle interface captures thenternal dynamics between the tires and the vehicle body thatannot be represented by a typical vehicle suspension model.

Improved Vehicle Lateral DynamicsThis section describes the improved vehicle lateral model,

hich includes the proposed dynamic-deflection tire model andhe resulting tire-vehicle interface mechanism.

3.1 Dynamic-Deflection Tire Model. In principle, vehicleody dynamics have six degrees of freedom �6 DOF�. A typicalehicle suspension mechanism allows large relative motions in theoll, pitch, and vertical directions between the vehicle body andhe unsprung inertia. The suspension mechanism is generally verytiff in longitudinal, lateral, and yaw directions since it directlyransmits the tire forces without internal vibrations. Therefore, theuspension mechanism significantly impacts the vehicle dynamicsn the roll, pitch, and vertical axes. As a result, the tire “compli-nce” characteristics become the most dominant sort of dynamics

Fig. 1 Top view of „a… contact patch for tire latera

n the remaining axes: longitudinal, lateral, and yaw. The pro-

94 / Vol. 129, JULY 2007


posed tire model was developed along these three principal axes.This 3-DOF tire model should capture the elastic properties of

the tires in order to describe tire suspension behaviors that mayimpact vehicle dynamics. Three independent sets of nonlinearsprings and dampers are utilized to represent the tire modes. Thesprings and dampers are typically characterized by the dynamicrelation between tire deflections and forces. The tire deflections�x ,�y ,�yaw are defined as the wheel displacements with respect tothe tire contact patch in the three principal axes of the wheel. Thetire tread within the contact patch is assumed to be in contact withthe ground without sliding. The expressions for these nonlinearsprings and dampers are shown in Eqs. �1�–�3�. These constitutethe 3-DOF dynamic-deflection tire model, denoted by the 3-DOFDDT model,

Flong = fFx��x,�x� �1�

Flat = fFy��y,�y� �2�

Myaw = fMz��yaw,�yaw� �3�

where �x and �y represent the longitudinal and lateral tire deflec-tions, respectively, and �yaw is the yaw slip angle of the tire.Under the small-deflection assumption, Eqs. �4�–�6� give the lin-ear representation of the 3-DOT DDT model. The tire lateral forceis concentrated behind the center of the contact patch by a dis-tance dtrail �pneumatic trail�, which helps generate the self-aligning moment,

Flong = Dlong�x + Clong�x �4�

Flat = Dlat�y + Clat�y �5�

Myaw = Dyaw�yaw + Cyaw�yaw �6�

where Dlong , Dlat, and Dyaw, are the tire longitudinal, lateral, andyaw damping coefficients, respectively; Clongitudinal , Clateral, andCyaw are the tire longitudinal, lateral, and yaw spring constants,respectively.

Among these tire deflections, the yaw and lateral deflectionschange the direction the wheel is traveling. This is a result fromthe facts: �I� the deflection of tire tread is continuous; �II� the tiretread within the contact patch is in contact with the ground with-out sliding. The kinematics of the wheels and contact patch underthe yaw and lateral tire deflections are described as follows.

Equation �7� defines the slip angle in two different ways: thespeed ratio between the wheel rolling velocity vr and the contactpatch side velocity vs and the displacement ratio between the tirelateral deflection �y and the lateral relaxation length �lat,

�lat � �y/�lat = vs/vr �7�

This equation indicates that the traveling direction changes as

eflection, „b… contact patch for tire yaw deflection
l d
soon as the tire lateral deflection occurs. Figure 1�a� shows the

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iagrams of the deformed tire contact patch without traction. Theateral relaxation length �lat is defined as the distance between theheel center and the hypothetical point O. The point O is located

t the intersection of the tangential line of the deformed footprintnd the wheel plane. The lateral relaxation length is generallyonsidered to remain unchanged under various lateral loads �17�.

In the inertia coordinates, a wheel yaw angle, and a contact-atch yaw angle are defined as �w and �cp, respectively. By defi-ition,

�yaw = �w − �cp �8�

s shown in Fig. 1�b�, point A represents the center of the contactatch and point B is the location ahead of the contact patch by aistance �yaw �yaw relaxation length� along the tire equatorialine. Given the wheel speed vr, the speed of each material pointetween A and B with respect to the wheel center is also vr.

1

ithout tire lateral forces, the intersection angles between theelocity directions of the points and the wheel center plane varyrom �cp�t� to �w�t� �from A to B� when the yaw slip angle occurs.he transport of these material points into the contact patch con-

inues to alter �cp�t�. The small angle between �cp�t� and �w�t�etermines the rate of �cp�t�, as shown in Eq. �9�,

�cp�t� = vr��w�t� − �cp�t��/�yaw �9�

aking the Laplace transform of Eq. �9�, the transfer functionrom �w to �cp is given by

�cp =vr/�yaw

s + vr/�yaw�w �10�

quation �10� shows that �cp�t� follows �w�t� with the lag of

r /�yaw. By defining actual and effective steering angles as ��w−�s−�d and �eff��cp−�s−�d, respectively, where �s repre-

ents the vehicle body yaw angle with respect to the road refer-nce frame and �d is the yaw angle of the road reference frameith respect to the inertia frame, the relation between � and �eff

an then be expressed as

�eff =vr/�yaw

s + vr/�yaw� �11�

n the above equation, the vehicle yaw angle ��s+�d� does notppear because changing this angle does not generate tire yaw slipngles directly but slightly alters the rolling speeds of each wheel.

The tire dynamics described by the 3-DOF DDT model shouldomply with the empirical tire characteristic: the relationship be-ween the lateral force Fy and the slip angle � as found in the labests. Since the slip angle � is defined as the angle between theheel center plane and the traveling direction of the wheel center,

t can be rewritten as

� = �lat + �yaw �12�In addition, Eq. �13� shows the governing equation of the Re-

axation Length Tire model �denoted by the RLT model�,

Fy = C�

vr/�

s + vr/�� RLT model� �13�

here � is the relaxation length and C� is the tire corneringtiffness.

By combining Eq. �4� and Eq. �7�, lateral force in the 3-DOFDT model can be rewritten in terms of the lateral slip angle �lat,

s shown in Eq. �14�. By using Eqs. �10� and �12�, Eq. �15� giveshe tire lateral force in terms of the side slip angle �,

Flat = �Dlats + Clat��lat�lat �14�

1Since the velocity of the material point at A with respect to the ground is zero¯
wheel center=vA.
ournal of Dynamic Systems, Measurement, and Control


=�Dlat�lats + Clat�lat�vr/�yaw

s + vr/�yaw� �15�

Note that the wheel yaw angle is the same as the steering angle inthe lab test scenario �i.e., �w=��. The corner frequency of�Dlat ,�lats+Clat ,�lat� is generally on the order of 5 Hz. To keepthe low-frequency characteristic in Eq. �15� the same as that inEq. �13�, it is easy to see that

Clat = C�/�lat �16�

�yaw = � �17�

3.2 Improved Tire-Vehicle Interface. The most evident rela-tive motions in the tire-vehicle interface are the suspension mo-tions between the vehicle body and the unsprung mass, the deflec-tions between the wheel and the tire contact patch, and the frontwheel steering motion. Conventional vehicle lateral models deter-mine the tire forces based upon the vehicle geometry and veloci-ties, but not the tire deflections. The resultant tire forces are usu-ally described along lateral and longitudinal axes and applieddirectly to the unsprung mass. Because the relative lateral andlongitudinal motions between the unsprung inertia and the vehiclebody are very small, the tire forces along these two axes essen-tially determine the yaw moment of the vehicle. Since the momentof inertia of the front wheels is far less than that of the vehiclebody, the yaw dynamics of the front wheels are often ignored. Asa result, the relative motions that are sometimes included in thevehicle models are simply the roll, pitch, and heave suspensionmotions.

It is easy to see that the other two relative motions cannot becaptured by the conventional suspension models. The 3-DOFDDT model uses the tire deflections to calculate the tire forces andmoments and thus, creates a natural tire suspension to describe therelative motions between the wheel and the contact patch. Thefollowing two examples demonstrate the “tire suspension behav-iors” along vehicle yaw and lateral axes.

Example 1—Tire Yaw Suspension Behavior: At a vehiclespeed of 0 mph, turning the front wheel assembly requires mo-ment, generated by the steering mechanism. The reaction momentforces the vehicle body to turn in the opposite direction. Such avehicle yaw motion can be formulated using the 3-DOF DDTmodel. The moments of inertia of the steering components thatturn with the vehicle body are lumped together into I1; the mo-ment of inertia of the front wheel assembly is denoted by I2. Theequations of motion for the two moments of inertia are given by

I1�s = � Mi + � �18�

I2��s + �� = Mf − � �19�

where � is the internal moment between I1 and I2.In principle, the 3-DOF DDT model requires an independent set

of longitudinal, lateral, and yaw displacements for the contactpatch of each tire to calculate tire forces and moments in a planarmotion. To avoid unnecessary complexity and to focus on vehiclelateral behavior, yu

f and yur correspond to the lateral positions of

the contact patch at the front and rear axles in the road referenceframe; �eff represent the yaw angle of the contact patch at the frontaxle. For convenience, �yu ,�u� are defined as

yu = �l2yuf + l1yu

r�/�l1 + l2� �20�

�u = �yuf − yu

r�/�l1 + l2� �21�

By applying the 3-DOF DDT tire model, the moments exerted on
I1 and I2 are shown in Eqs. �22� and �23�, respectively,
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� Mi = − 2�Dlatf l1 + Dlat

r l2��s − �u� − 2�Clatf l1 + Clat

r l2��s − �u�

− �Dlongf + Dlong

r ��s − �u�d − �Clongf + Clong

r ��s − �u�d�22�

Mf = − �2Dyawf �� − �eff� + 2Cyaw

f �� − �eff�� 23�

here l1 and l2 are the distances from the front axle to vehicle.G. and from the rear axle to vehicle C.G., respectively; d is the

rack width of the vehicle; �u=0 in this example. By combiningqs. �18�–�23�, a second-order dynamic relation between the ve-icle yaw angle and the steering angle is shown in Eq. �24�, de-cribing the vehicle yaw characteristic with respect to the “tireaw suspension,”

�s

�= − �I2s2 + 2Dyaw

f s + 2Cyawf �/��I1 + I2�s2 + �2�Dlat

f l1 + Dlatr l2�

+ �Dlongf + Dlong

r �d�s + �2�Clatf l1 + Clat

r l2� + �Clongf + Clong

r �d��24�

Example 2—Tire Lateral Suspension Behavior: A vehicle israveling at a constant speed vr on a straight road. A lateral forceext is applied to the vehicle C.G. and the vehicle exhibits mainlylateral motion. For convenience, the vehicle yaw dynamics can

e ignored. The kinematical equation for the contact patch and thequation of motion for the vehicle are shown as

yu = vr�ys − yu�/�lat �25�

without loss of generality, �lat=�latf =�lat

r is assumed�

Mys = Flatf + Flat

r + Fext �26�

here M is the vehicle mass; ys is the lateral displacement of theehicle C.G. in the road reference coordinates. The tire lateralorces on the front and rear axles are expressed as

Flatf = − 2Dlat

f �ys − yu� − 2Clatf �ys − yu� �27�

Flatr = − 2Dlat

r �ys − yu� − 2Clatr �ys − yu� �28�

ombining Eqs. �25�–�28�, Eq. �29� shows a third-order relation-hip between the input force Fext and the lateral displacement ys

ys

Fext=

s + vr/�lat

�Ms2 + �2Dlatf + 2Dlat

r + Mvr/�lat�s + �2Clatf + 2Clat

r ��s�29�

quation �30� shows the governing equation of a typical bicycleodel under the same situation,

ys

Fext=

vr

�Mvrs + 2�C�f + C�

r ��s=

vr

�Mvrs + 2�Clatf + Clat

r ��lat�s�30�

hen vr is very small or equal to zero, Eq. �29� describes the “tireateral suspension behavior.” The typical bicycle model cannotxplain this vehicle lateral motion, as shown in Eq. �30�. When vrs very large, the two models exhibit similar characteristics �ex-ept at high frequency�.

3.3 Improved Vehicle Lateral Model. The yaw and lateralotions are the two most dominant portions of vehicle dynamics

n lateral control. As discussed in Sec. 3.2, the often-ignored tireynamics affect both vehicle yaw and lateral characteristics. Aimple bicycle model incorporated with the 3-DOF DDT modelan be used to investigate the impact of these neglected dynamicso vehicle yaw and lateral behaviors.

For a vehicle traveling at a constant velocity vr, the transla-ional and angular velocities of the contact patch centers of theour tires are expressed in Eqs. �31� and �32�. These equations are
erived from the vehicle geometry and the tire deflections,
96 / Vol. 129, JULY 2007


yu = vr�u + vr�effl2/�l1 + l2� + vr�ys − yu�/�lat �31�

�u = vr�eff/�l1 + l2� + vr��s − �u�/�lat − �d �32�The dynamic equations of the improved bicycle model are givenby

Mys = Flatf + Flat

r − Mvr�d �33�

I1�s = � Mi + � �34�

I2��s + �� = Mf − � �35�Based on the 3-DOF DDT model, the tire forces and moments areexpressed in Eqs. �36�–�39�,

Flatf = − 2Dlat

f �ys − yu + l1��s − �u�� − 2Clatf �ys − yu + l1��s − �u��

�36�

Flatr = − 2Dlat

r �ys − yu + l2��s − �u�� − 2Clatf �ys − yu + l2��s − �u��

�37�

� Mi = − c1��s − �u� − k1��s − �u� + Flatf l1 − Flat

f l2 �38�

Mf = − c2�� − �eff� − k2��eff� �39�where

k1 =�Clong

f + Clongr ��l1 + l2�2

k2 = 2Cyawf

c1 =�Dlong

f + Dlongf ��l1 + l2�2

c1 = 2Dyawf

In general, the damping force and the inertia force on I2 are rela-tively small compared with its spring force at low frequencies.

The two respective terms I2� and c2��− �eff� in Eqs. �35� and �39�can be ignored. This sacrifices the plant response accuracy at highfrequencies but facilitates control designs by lowering the plantorder. By combining Eq. �11� and Eqs. �31�–�39�, a seventh orderstate-space representation can be found in Appendix A. The statevariables are �yu ys ys �u �s �s �eff �T and the input is thesteering angle �.

4 Analysis of the Improved ModelsSection 4.1 compares the DDT model and the RLT model in a

simulated test scenario. Section 4.2 shows the improvements inpredictions of vehicle dynamics from the frequency-domain per-spective.

4.1 Comparison Between RLT Model and DDT Model in aSimulated Test Scenario. The RLT and DDT models utilize asimilar first-order relationship. Section 3.1 has shown that the twomodels typically exhibit the same low-frequency characteristicswhen testing tires. This section compares the two models under anatypical simulated test scenario and the results show again that thetwo models can have very similar behavior at low frequencies.

This simulated test was designed to investigate the response ofa single tire in lateral velocities from external lateral forces. In thisscenario, the lateral force Fext is applied to the center of the wheelsuch that the wheel yaw motion can be ignored. The lateral veloc-ity of the wheel is vs. The governing equations of the two models
in this simulated test are shown below.


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4.1.1 RLT Model. The equation of motion for the RLT models given by

mwsvs = − C�

vr/�

s + vr/�

vs

vr+ Fext �40�

here mw is the total mass of the tire, the wheel, and the verticaload.

By rearranging Eq. �40�, the transfer function from Fext to vsan be calculated as

Table 1 Identified parameters of the snowblower

20,500 kg C�r 350,000 N/rad Dlat

f 9,000 N s/m

1 168,250 kg m2 �yaw 0.45 m Dlatr 9,000 N s/m

1 1.3 m �lat 1.0 m k1 500,000 N/m

2 2.2 m Clatf 350,000 N/m k2 80,000 N/m

�f 350,000 N/rad Clat

r 350,000 N/m c1 10,000 Ns/m

Fig. 2 Frequency response from steering ang

Fig. 3 Frequency response from steering angle



vs

Fext=

s + �vr/��mws2 + �mwvr/��s + �C�/��

�41�

4.1.2 3-DOF DDT Model. The governing equations for the3-DOF DDT model are expressed as

ycp = vr�y/�lat �42�

mwvs = mw�ycp + �y� = − Dlat�y − Clat�y + Fext �43�

where ycp is lateral displacement of tire contact patch with respectto the test road. By combining Eqs. �42� and �43�, Eq. �44� showsthe transfer function from Fext to vs,

vs

Fext=

ycp + �y

Fext=

s + �vr/�lat�mws2 + �mwvr/�lat + Dlat�s + Clat

�44�

By using Eq. �16�, it is clear that the differences between Eq.�41� and Eq. �44� lie on the following two terms: �lat and Dlat. If� is equal to �lat, the frequency responses of the two transfer

o yaw rate at „a… V=0.5 m/s and „b… V=20 m/s

ateral acceleration at „a… 0.5 m/s and „b… 20 m/s
to l
functions are almost the same at high speeds. At low speeds, the

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amping constant Dlat dominates the damping term �mwvr /�latDlat� for the DDT model. This simulated test shows the follow-

ng facts:

• The two tire models have the similar characteristics at highspeeds in this simulated test scenario.

• Raising the vehicle speed increases the dynamic dampingterm mwvr /�lat in Eq. �41� and Eq. �44�.

• The RLT model exhibits the undamped motion at nearlyzero speed; the 3-DOF DDT model keeps the damping co-efficient, Dlat. Therefore, the 3-DOF DDT model matchesmore closely to true tire behavior at low speeds.

4.2 Analysis of Improved Bicycle Model With the DDTodel. This section compares four vehicle lateral models, the

eometric model, bicycle model, bicycle model with the RLTodel, and bicycle model with the DDT model. These compari-

ons show the model improvements made in the vehicle yaw andateral characteristics from the DDT model under dynamic steer-ng inputs. The governing equations of the above four models can

ig. 4 „a… Trajectories of the front and rear wheels and the ateering angle to yaw rate. „b… Under various yaw relaxation0.4 m/s. „d… Under different tire damping constants, V=0.4 m

e found in Appendices B, C, D and A, respectively. Inserting the

98 / Vol. 129, JULY 2007


identified parameters of the snowblower from Table 12 into thefour models, Figs. 2�a� and 2�b� show their respective frequencyresponses from the steering angle to the yawrate at low and highspeeds; Figs. 3�a� and 3�b� illustrate the corresponding responsesfrom steering angle to lateral acceleration at the vehicle C.G.

At high speeds, there are only small differences present amongthe three “bicycle-based” models. The geometric model does notaccurately match any of the bicycle-based models due to its ne-glected tire dynamics. At low speeds and low frequencies, thegeometric model and the bicycle model are equivalent. The bi-cycle model with the RLT model matches the original bicyclemodel up to 0.5 Hz. Only the bicycle model with the DDT modeland the bicycle model with the RLT model exhibit resonantmodes. The following list compares the discrepancies of the re-sulting low-speed vehicle dynamics between the DDT and RLTmodels.

2The identification procedure of the snowblower parameter will be discussed in

ociated slip angles at steady state; frequency response fromgth, V=0.4 m/s. „c… Under various lateral relaxation length, V

sslen/s.

Sec. 5.



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• Using the DDT model, the vehicle phase starts to drop sig-nificantly above 0.1 Hz. This decrease shows that the ve-hicle exhibits a steering lag behavior at low frequencies.None of the other models predicts this characteristic.

• At very low speeds, the bicycle model with RLT has anearly undamped system response; the bicycle model withDDT keeps the tire damping force.

• At zero vehicle speed, only the bicycle model has the sin-gularity problem �divided by vr=0�. Both the geometricmodel and the bicycle model with RLT have null gains forthe transfer functions. The bicycle model with DDT is theonly model which exhibits the resonant yaw motion at zerospeed.

• The above observations make it clear that the vehicle modelwith DDT is the only tire model that matches real-worldbehavior of vehicles and tires at low speeds.

Experimental Setup and Parameter IdentificationThe improved bicycle model was validated using test data fromsnowblower. The snowblower is a form of massive snow re-oval equipment with very stiff suspension, which makes it

niquely convenient in testing the dynamic validity of the 3-DOFDT model. The sensors installed on the snowblower were the

ig. 5 Comparison among various models-frequency response.6 m/s, „c… at 0.45 m/s and 1.6 m/s

from steering angle to yaw rate „a… at zero speed, „b… at 0.45 and

teering encoder, the yawrate sensor, and the lateral position sen-



Fig. 6 Lateral displacement versus time for „a… V=0.9 m/s, fre-quency varying from 0.3 to 0.6 Hz; „b… V=1.2 m/s, frequency�0.2 Hz; „c… V=1.5 m/s, frequency�0.6 Hz; „d… V=1.6 m/s,
frequency�0.7 Hz; „e… V=2.5 m/s, frequency�0.5 Hz
JULY 2007, Vol. 129 / 399


sqe

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ors. The frequency sweep technique was used to obtain the fre-uency responses from steering angles to sensor outputs at differ-nt velocities.

The identified parameters of the snowblower are shown inable 1. The inertias and the vehicle dimensions are the knownarameters; the other parameters can be estimated by using theollowing procedure:

�1� When the vehicle is cornering at steady state, its trajectoryis a circle. As shown in Fig. 4�a�, the triangle formed by theradii of the front and rear wheels and the wheelbase can beused to determine the front and rear slip angle � f and �r.

From the equation in Appendix C, the force balanceequation of the bicycle model at steady state is

Mvr�s = C�f � −

ys + l1�s

vr� + C�

r 0 −ys + l2�s

vr� �45�

This equation originally results from

C�f � f + C�

r �r = Mvr�s �46�The moment balance equation of the bicycle model atsteady state is

C�f l1� −

ys + l1�s

vr� − C�

r l20 −ys − l2�s

vr� = 0 �47�

From Eqs. �45� and �47�, the relation between the steeringangle and the yaw rate is given by

�s

�=

vr

�l1 + l2� +Mvr

l1 + l2 l1

C�f −

l2

C�r � �48�

From Eqs. �46� and �48�, the vehicle parameters C�f and C�

r

can be obtained.�2� Figure 4�b� shows the frequency response to various yaw

relaxation length �yaw. When �yaw increases, the steeringlag behavior at low frequency is more evident. �yaw can bechosen by minimizing the squared error, which is defined as

esq = �i

�n

�Gi�j�n� − Gexp,i�j�n��22

where Gi�j�n� is the complex value of the transfer functionof �s /� at the speed vi and the frequency �n and Gexp,i�j�n�is the experimental result of �s /�. In the example of thispaper, two speeds �0.45 m/s and 1.6 m/s� and six differentfrequencies �0.1–0.6 Hz� were utilized when identifyingthe parameters of the snowblower.

�3� The snowblower uses the same type of tires for the frontand the rear axles with very similar cornering stiffness. Forconvenience, the lateral relaxation length of the front andrear tires is assumed to be the same. Figure 4�c� shows the

˙

Fig. 7 „a… The controlled plant with weighting fu�-synthesis framework; „c… the weighting functio

response of �s /� to various �lat. This lateral relaxation

00 / Vol. 129, JULY 2007


length determines the natural frequency of the yaw mode.When �lat is close to 1 m, the natural frequency of the yawmode is approximately 0.8 Hz. Based on Eqs. �16� and�38�, Clat

f , Clatr , and k1 can be easily calculated. k2 is used to

adjust the low-frequency gain of �s /� at zero vehicle speed.�4� Similarly, assume that Dlat

f =Dlatr . These damping constants

determine the damping ratio of the yaw mode, whichchanges the magnitude of the resonant peak. Figure 4�d�shows the frequency response to various tire damping co-efficients. When Dlat

f is around 9000 N s/m, the improvedbicycle model matches the resonant peaks.

6 Vehicle Lateral Dynamics ValidationThis section validates the DDT model using the experimental

data. Figure 5�a� uses both the original bicycle model and thebicycle model with the DDT model �denoted by the improvedbicycle model later� to estimate the “best” matching frequencyresponse from steering input to yawrate at zero vehicle speed. Thesolid line illustrates data from the improved bicycle model; theasterisks represent the experimental data. The original bicyclemodel does not appear in this figure due to the singularityproblem.3 This clearly indicates that the improved bicycle modelis the only model that has the appropriate degrees of freedom tomatch the test data between zero to almost zero speed, includingthe resonant peak at 0.8 Hz.

Figure 5�b� plots the responses from steering input to yaw rateat 0.4 m/s and 1.6 m/s. The solid lines illustrate the matchedresults of the improved bicycle model and the dash lines representthe fitted results of the original bicycle model. It is easy to see thatthe improved bicycle model matches the experimental data verywell �especially the phase characteristic at low frequency and theresonant peak at 0.8 Hz�. It is impossible to adjust the parametersin the original bicycle model to match these frequency responsesat low speeds. The original bicycle model can fit the experimentalresults up to 0.1 Hz at very low frequency. The discrepanciesbetween the bicycle models with and without DDT in phase andgain plots are up to 200 deg and 10 dB, respectively. Such dis-crepancies have the potential to impact any closed-loop controllerdesign.

Figure 5�c� shows the response from steering input to yaw rateat the same speed. The solid lines illustrate the matched results ofthe improved bicycle model and the dashed lines represent thepredicted results of the bicycle model with RLT. As expected inSec. 4.2, at the speed of 0.45 m/s the bicycle model with RLTshows an excessive resonant peak; 10 dB more than that from theimproved bicycle model, and it also does not exhibit the steeringlag behavior as the improved bicycle model predicts.

The model validation on the vehicle lateral characteristics isillustrated by comparing the measured lateral displacements with

3

tions; „b… the closed-loop interconnection usingfor controller design

ncns

The bicycle model has almost null gains when the speed is extremely small.



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hose obtained by the improved bicycle model. The experimentsere conducted at various vehicle speeds on a straight road. The

teering angle was swiveled at different frequencies; the outputas the lateral displacement from the front displacement sensor.he same steering input, vehicle speed, and initial condition were

ed into the improved bicycle model. Figures 6�a�–6�e� show theesults. These predicted results match the experimental data veryell.

Impact on Vehicle Lateral ControlIn this section, the snowblower was used as an example to

emonstrate the closed-loop impact of often-ignored tire behaviorn vehicle lateral control. The goal of the controller design is toeep the plow head at the road center line. To meet the perfor-ance and robustness requirements, two steering controllers �C1

nd C2� are individually developed, based on the original andmproved bicycle models �P1 and P2�, respectively. The � synthe-is and the H infinity theory are applied to the control design.igure 7�a� and 7�b� show the setup of the robust performanceroblem. The objective of the � synthesis is to find a controller Kuch that sup

��R

��T�→��j��, where is a prescribed value

ig. 8 Controller bode plot from head position error to steer-ng angle

Fig. 9 Step response of the nominal closed-loop



�e.g., =0.95�, w= �w d�T, and z= �z e�T �18,19�. Figure 7�c� plotsthe frequency responses of the performance and uncertainty func-tions, Wp and Wu. Wp is selected to limit the steady-state trackingerror within ±25%. Based on the parameter variation of the origi-nal bicycle model, Wu is assumed, up to 10% of the nominalmodel at low frequencies and 100% at high frequencies.

D-K iteration is used to approximate the solution to this robustperformance problem �20�. Figure 8 shows the frequency re-sponses of the steering controllers C1 and C2 at two differentspeeds, 1 m/s and 2 m/s. Below 0.2 Hz, there is a gain discrep-ancy of 8 dB between C1 and C2. At high frequencies, C2 includesa “notch” filter naturally to address the resonant mode at 0.8 Hz.

Figures 9�a� and 9�b� show the step responses of the two nomi-nal closed-loop systems, P1C1 / �1+ P1C1� and P2C2 / �1+ P2C2� atthe velocities of 1 m/s �solid line�, 2 m/s �dotted line�, and 3 m/s�dashed line�. The step responses all meet the performance re-quirements. When comparing the two plots, the simulation resultsof the controlled bicycle model are slightly better than those fromthe bicycle model with DDT, the reason being that the originalbicycle model creates larger phase margins at the crossover fre-quency than does the improved bicycle model.

Fig. 10 Step response of the closed-loop system C1P2 / „1+C1P2…

systems „a… C1P1 / „1+P1C1…; „b… C2P2 / „1+P2C2…

JULY 2007, Vol. 129 / 401


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A

w

A

4

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In Fig. 10, the step responses of the “mismatched” closed-loopystem P2C1 / �1+ P2C1� show that stability problems arise for ve-icle speeds below 2 m/s. This indicates that low-speed tire dy-amics introduces excessive uncertainties in the bicycle modelhen the controller is designed when certain tire dynamics which

re often ignored are not considered. Although, without thenowledge of such tire dynamics, C1 can be redesigned by in-reasing uncertainty weighting for the original bicycle model, it isery likely to overestimate the uncertainty weighting and end upith an inferior controller.It has been shown that the often-ignored tire behaviors are very

mportant to low-speed control applications on heavy vehicles.hese heavy vehicles generally have larger tires �possibly longer

elaxation length� and lower resonant frequency than the passen-er vehicles. The steering lag and the resonant mode on the heavyehicles are more evident at lower frequency. On the passengerars, these tire behaviors dominate the vehicle lateral characteris-
ics at a relatively higher frequency. When the required bandwidth
��s − �d

� �0 0

��s −

02 / Vol. 129, JULY 2007


of a controlled passenger car is higher, these behaviors start toplay an important role in the control design. The steering lagbehavior �nonminimum phase phenomenon� yields additionalphase lag around the crossover frequency and the resonant modemay cause the stability problem. Both issues increase the diffi-culty of the control design.

8 ConclusionA linearizable dynamic-deflection tire model is proposed for

low-speed vehicle lateral dynamics and control. This tire modelnot only describes the empirical tire behaviors but also capturesthe often-ignored tire suspension modes. This model is easilyimplemented with the existing vehicle lateral models. When inte-grated into a vehicle lateral model, the new tire model providessufficient degrees of freedom to match the low-speed vehicle testdata. The simulation of the vehicle lane-keeping control demon-strates that the low-speed vehicle lateral dynamics contributed by
the tires is crucial for steering control.
ppendix A: State Space Form of the Improved Bicycle Model

1 0 0 0 0 0 0

0 1 0 0 0 0 0

− 2�Dlatf + Dlat

r � 0 M − 2�Dlatf l1 − Dlat

r l2� 0 0 0

0 0 0 1 0 0 0

0 0 0 0 1 0 0

− �2Dlatf l1 − 2Dlat

r l2� 0 0 − �2Dlatf l2 + 2Dlat

r l22 + c1� 0 �I1 + I2� 0

0 0 0 0 0 0 1

� yu

ys

ys

�u

�s

�s

�eff

� = A11 A12 0 A14 0 0 A17

0 0 1 0 0 0 0

A31 A32 A33 A34 A35 A36 0

0 0 0 A44 A45 0 A47

0 0 0 0 0 1 0

A61 A62 A63 A64 A65 A66 A67

0 0 0 0 0 0 A77

� yu

ys

ys

�u

�s

�s

�eff

�+

0

0

0

0

0

− k2

vr/�yaw

�� + 0

0

− Mvr�d

− �d

0

0

0

�here

A11 = vr/�lat A12 = vr/�lat A14 = vr

A17 = l2vr/�l1 + l2� A31 = 2�Clatf + Clat

r � A32 = − 2�Clatf + Clat

r �

A33 = − 2�Dlatf + Dlat

r � A34 = 2�Clatf l1 − Clat

r l2� A35 = − 2�Clatf l1 − Clat

r l2�

A36 = − 2�Dlatf l1 − Dlat

r l2� A44 = − vr/�lat A45 = vr/�lat

A47 = vr/�l1 + l2� A61 = 2�Clatf l1 − Clat

r l2� A62 = − 2�Clatf l1 − Clat

r l2�

A63 = − 2�Dlatf l1 − Dlat

r l2� A64 = 2�Clatf l1

2 + Clatr l2

2� + k1 A65 = − 2�Clatf l1

2 + Clatr l2

2� − k1

A66 = − �Dlatf l1

2 + Dlatr l2

2� − c1 A76 = vr/�yaw A77 = − vr/�yaw

ppendix B: State Space Form of the Geometric Model

ys=

0 vr ys+

l2vr/�l1 + l2��

�d� �

vr/�l1 + l2� �Transactions of the ASME


A

w

A

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ppendix C: State Space Form of the Original Bicycle Model

ys

ys

�s

�s

� = 0 1 0 0

0− 2�C�

f + C�r �

Mvr

2�C�f + C�

r �M

− 2�C�f l1 − C�

r l2�Mvr

0 0 0 1

0− 2�C�

f l1 − C�r l2�

Izzvr

2�C�f l1 − C�

r l2�Izz

− 2�C�f l1

2 + C�r l2

2�Izzvr

� ys

ys

�s

�s

� + 0

2C�f /M

0

2C�f l1/Izz

�� + 0

2�l1C�f − l2C�

r ��d/�Mvr� − vr�d

0

− 2�l12C�

f + l22C�

r ��d/�Izzvr��

here Izz the inertia moment of the vehicle.

ppendix D: State Space Form of the Bicycle Model With RLT Model

ys

ys

�s

�s

Flatf

Flatr

� = 0 1 0 0 0 0

0 0 0 0 1/M 1/M

0 0 0 1 0 0

0 0 0 0 l1/Izz − l2/Izz

0 − 2C�f /� 2vrC�

f /� − 2C�f l1/� − vr/� 0

0 − 2C�r /� 2vrC�

r /� 2C�r l2/� 0 − vr/�

� ys

ys

�s

�s

Flatf

Flatr

� + 0

0

0

0

2C�f yr/�

0

�� + 0

− vr�d

0

0

2C�f l1/�

− 2C�r l1/�

�
eferences�1� Shladover, S. E., 1991, “Program on Advanced Technology for the Highway
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