Simulation-Based Engineering LabUniversity of Wisconsin-Madison

Technical Report TR-2019-04

A MATLABR© Implementation of a Set of Three Vehicle Dynamics

Models

Shuping Chen and Dan Negrut

October 14, 2019

Abstract

Three commonly used vehicle models; i.e., a bicycle model [1], an 8-DOF vehiclemodel [2], and a 14-DOF vehicle model [2], are discussed as implemented in MATLABR©

and made available as open source in the public domain. Three different steering inputcommands, including sine wave steering, step steering and ramp steering, are appliedin vehicle steering simulation and results are discussed insofar as their correctnessis concerned. These models are envisioned to be used in conjunction with modelpredictive control (MPC) as applied in the context of multi-body dynamics, see [3, 4].These vehicle models are used in several other projects that investigate the use of MPCfor autonomous vehicle driving, see [5–7].

Keywords: Vehicle Modeling, Tire Modeling, Steering Simulation

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Contents

1 Introduction 3

2 Vehicle Dynamics Modeling 32.1 Tire Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Bicycle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 14-DOF Vehicle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 8-DOF vehicle model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Simulation and Analysis 153.1 Vehicle Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.1 Bicycle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2.2 8-DOF Vehicle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2.3 14-DOF Vehicle Model . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Conclusions and Future Work 26

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1 Introduction

Vehicle dynamics modeling is a crucial basis for Autonomous Ground Vehicles, which is con-cerned with a mobile vehicle’s ability to drive safely and generate optimal control commands.The fidelity of a dynamic model is defined as presenting the movement and forces on a ve-hicle accurately with given conditions. Three models, the bicycle model, the 8-DOF vehiclemodel and the 14-DOF vehicle model, are generated in order to implement the model predic-tive control for path tracking in MATLABR©. The current work presents, through numericalsimulation, the robustness and validity of three dynamic models with different inputs. Inthis report, these vehicle models are developed and implemented respectively and simulationresults of the vehicle response are presented under the conditions of sine wave steering input,step steering input and ramp steering input.

The paper is structured as follows. Section 2 describes three vehicle dynamics models indetail. The simulation results of three different steering angle inputs are presented in Section3. This is then followed by concluding remarks in Section 4 which discusses future researchdirection.

2 Vehicle Dynamics Modeling

This section describes the tire and vehicle model used for simulations. The nomenclatureused in the bicycle model depicted in Figures 1. We denote by Fl and Fc the longitudinal andlateral tire forces respectively, Iz is the car inertia along z axis, X and Y are the absolute carposition inertial coordinates, a and b are the car geometry (distance of front and rear wheelfrom center of mass), m is the car mass, s is the slip ratio, y and x are the local longitudinaland lateral coordinates in car body frame, ẋ and ẏ are the vehicle longitudinal and lateralspeeds, respectively. α is the tire slip angle, δ is the wheel steering angle, ψ is the headingangle(yaw angle), ψ̇ is yaw rate and β is vehicle sideslip angle. C.M. is the abbreviation ofCenter of Mass. The lower subscripts (·)f and (·)r specialize a variable at the front wheeland the rear wheel, respectively [8].

2.1 Tire Modeling

Except for aerodynamics forces and gravity, all of the forces which affect vehicle behaviorare provided by the tires. Because tire forces produce primary external influence and theyhave highly nonlinear performance, it is essential to use a realistic tire model, especiallywhen investigating large control inputs that results in response near the limits of the linearcharacter scale of the tire. The tire lateral and longitudinal forces are assumed to dependon normal force, slip angle, surface friction, and slip ratio. However, when the slip ratioand slip angle are limited within small values, the tire model can be simplified and generatelinearied lateral force and longitudinal force [9].

Under this assumption, the tire lateral force can be given as Fc = Cα(µ, Fz)α, where Cαis tire cornering stiffness related to tire normal force and tire-road friction coefficient; and

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tire longitudinal force can be given as Fl = Cx(µ, Fz)sf,r, where sf,r is the slip ratio of fronttire or rear tire, Cx is the tire longitudinal stiffness which also related to the tire normalforce and tire-road friction coefficient.

2.2 Bicycle Model

The single-track vehicle model [10] or ‘bicycle model’ [9] [11] is often used in the controllerdesign, the schematic of this model is depicted in Figure 1. Several simplifications andassumptions are employed as listed below:

1. Neglecting the effect of steering system, combining the left and right tires of front andrear axles respectively, taking the front wheel steering angle as the input directly, i.e.,δf = δ, δr = 0;

2. Assuming that the vehicle only moves in yaw plane, neglecting the effect of suspension,so the vertical displacement, the pitch angle and the roll angle are considered zero;

3. The full vehicle are treated as rigid body and the lateral acceleration coefficient (i.e.,the ratio between lateral acceleration and gravity acceleration) is limited within 0.4;

4. Assuming that the left and right tire slip angle are equal, the front wheel steer angle issmall and the lateral forces are proportional to tire slip angles; Then, the lateral forcecan be given as Fc = Cαα, where Cα denotes the tire cornering stiffness and α denotesthe tire slip angle;

5. Assuming that the effect of aerodynamics and the load transfer between the left andright tires are neglected.

Figure 1: Bicycle Model [9] [10] [11]

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With the assumption 1 to 5, the bicycle model can be described as [8] [9]:

m(ẍ− ẏψ̇) = Flfcosδ − Fcfsinδ + Flr (1)m(ÿ + ẋψ̇) = Flfsinδ + Fcfcosδ + Fcr (2)

Izψ̈ = a(Fcfcosδ + Flfsinδ)− bFcr. (3)

Assuming that the vehicle side slip angle and the tire slip angle are small, the approximationof linearized side slip angle and tire slip angle can be simplified as below [10]:

β = arctan(ẏ

ẋ) ≈ ẏ

ẋ, αf = arctan(β+

aψ̇

ẋ)− δ ≈ β+ aψ̇

ẋ− δ, αr = arctan(β−

bψ̇

ẋ) ≈ β− bψ̇

ẋ.

The position of the vehicle expressed in global coordinates can be calculated as follows [8] [9]:

Ẏ =ẋsinψ + ẏcosψ (4)

Ẋ =ẋcosψ − ẏsinψ. (5)

Choosing[ẏ, ẋ, ψ, ψ̇, Y,X

]as the state variable, with Eq. (1) through Eq. (5), small angle

assumption and linear tire model, then the nonlinear vehicle model equations are given as

ÿ =− ẋψ̇ +Cαf

( ẏ + aψ̇ẋ

− δ)

m+Cαr( ẏ − bψ̇

ẋ

)m

(6)

ẍ =ẏψ̇ +sfCxf + srCxr

m−Cαf

( ẏ + aψ̇ẋ

− δ)δ

m(7)

ψ̈ =1

Iz

(aCαf

( ẏ + aψ̇ẋ

− δ)− bCαr

( ẏ − bψ̇ẋ

))(8)

Ẏ =ẋsinψ + ẏcosψ (9)

Ẋ =ẋcosψ − ẏsinψ. (10)

2.3 14-DOF Vehicle Model

In order to better represent the vehicle lateral and yaw dynamics as well as coupling ofyaw-roll motion due to the transient lateral load transfer during extreme maneuvers, higherorder model such as 8-DOF and 14-DOF are also used in rollover studies. A 14-DOF vehiclemodel, which considers the suspension at each corner, has the same benefits of an 8-DOFvehicle model, with the additional capabilities of predicting vehicle pitch and heave motions.It also offers the flexibility of modeling nonlinear springs and dampers and can simulatethe vehicle responses to normal force inputs in the case of an active suspension system.Moreover, the 14-DOF model, unlike the 8-DOF model, can predict vehicle behavior even

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after wheel lift-off and thus can be used in developing and testing the validity of rolloverprediction/prevention strategies [2].

The Figure 2 shows the schematic of the two axle vehicle model used to investigate vehicleroll response to steering and torque inputs. This schematic includes 6 DOF at the vehiclelumped mass C.M. and 2 DOF at each of the four wheels, including vertical suspension traveland wheel spin. The body is modeled as being rigid, with body-fixed coordinates attached atthe C.M. and aligned in principal directions (coordinate frame 1). u, v, w indicate forward,lateral, and vertical velocities, respectively, of the sprung mass [2].

Figure 2: Schematic of 14-DOF vehicle model with one-dimensional suspension and coordi-nate frames [2]

The force and velocity components in the right front corner of a vehicle is depicted inFigure 3. The velocity usrf , vsrf , and wsrf are the velocities of the right front strut mountingpoint in the longitudinal, lateral, and vertical directions, respectively, in the body-fixedcoordinate frame, which is attached to the sprung mass C.M. (coordinate frame 1). Thesevelocities can be obtained by transforming the C.M. velocities as [2]usrfvsrf

wsrf

=

0 0cf2

0 0 a−cf

2−a 0

ωxωyωz

+uvw

. (11)For the left front corner:uslfvslf

wslf

= 0 0

−cf2

0 0 acf2−a 0

ωxωyωz

+uvw

. (12)6

Figure 3: Description of forces and velocities at the right front corner of a vehicle [2]

For the left rear corner: uslrvslrwslr

= 0 0

−cr2

0 0 −bcr2

b 0

ωxωyωz

+uvw

. (13)For the right rear corner:

usrrvsrrwsrr

=

0 0cr2

0 0 −b−cr

2b 0

ωxωyωz

+uvw

. (14)The forces Fxsrf , Fysrf and Fzsrf are the forces transmitted to the sprung mass along the

longitudinal, lateral, and vertical directions, respectively, of coordinate frame 1. The forcesFxgsrf , Fygsrf , and Fzgsrf are the forces acting at the tire ground contact patch in the samecoordinate frame 1. These forces can be written in terms of the tire forces Fxgrf , Fygrf , andFzgrf by projecting its components along coordinate frame 2 as [2]FxgsrfFygsrf

Fzgsrf

=1 0 00 cosφ sinφ

0 −sinφ cosφ

cosφ 0 −sinφ0 1 0sinθ 0 cosθ

FxgrfFygrfFzgrf

. (15)The forces Fxgrf and Fygrf are obtained by resolving the longitudinal(Fxtrf ) and cornering(Fytrf )

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forces at the tire contact patch as [2]

Fxgrf =Fxtrfcosδ − Fytrfsinδ; (16)Fygrf =Fytrfcosδ + Fxtrfsinδ; (17)

where the δ is the steering angle at the road wheels.The linear tire model has been used in the development of the tire forces Fxtrf and Fytrf .

The longitudinal and lateral slips used in the tire model were calculated as follows [2]:

srf =

(rrfωrf −

(ugrfcosδ + vgrfsinδ

))|(ugrfcosδ + vgrfsinδ

)|

(18)

αrf =tan−1(vgrfugrf

)− δ. (19)

The ugrf and vgrf , longitudinal and lateral velocities at the tire contact patch of the rightfront wheel, can be determined as [2]

ugrf =cosθ(uurf − ωy · rrf

)+ sinθ

(wurfcosφ+ sinφ

(ωx · rrf + vurf

))(20)

vgrf =cosφ(vurf + ωx · rrf

)− wurfsinφ. (21)

The longitudinal (uurf ) and lateral (vurf ) velocities of the unsprung mass in the bodycoordinate frame used in the earlier equations can simply be written as [2]

uurf =usrf − lsrfωy (22)vurf =vsrf + lsrfωx, (23)

where lsrf is the instantaneous length of the strut. The derivative of uurf and vurf is givenas

u̇urf =u̇srf −((ωsrf − ωurf )ωy + lsrf ω̇y

)(24)

v̇urf =v̇srf +((ωsrf − ωurf )ωy + lsrf ω̇y

). (25)

The unsprung mass vertical velocity wurf represents the degree of freedom correspondingto the suspension deflection and can be expressed by applying Newton’s law for the verticalmotion of the unsprung mass as [2]

muẇurf =cosφ(cosθ

(Fzgrf −mug

)+ sinθFxgrf

)− sinφFygrf − Fdzrf

− xsrf · ksf − ẋsrf · bsf −mu(vurf · ωx − uurf · ωy

), (26)

where ksf is the suspension stiffness, bsf the suspension damping coefficient, and xsrf theinstantaneous compression of the right front suspension spring. The force, Fdzrf , representsthe additional load transfer that occurs at the wheels through the suspension links becauseof the reaction force to the force transmitted to the sprung mass through the roll center [2].

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The instantaneous suspension spring deflection xsrf is given as [2]

ẋsrf = −wsrf + wurf . (27)

The vertical force Fzgrf acting at the tire ground contact patch in coordinate frame 2can be written in terms of the tire stiffness (ktf ) and the instantaneous tire deflection (xtrf )as [2]

Fzgrf = Fztrf = xtrfktf . (28)

The instantaneous tire deflection xtrf in equation (28) is given as [2]

ẋtrf = wgrf − wuirf = wgrf −(cosθ

(wurfcosφ+ vurfsinφ

)− uurfsinθ

), (29)

where wuirf is the vertical velocity of the wheel center in the inertial coordinate frame. Forthe simulations in this article, it is assumed that the vertical velocity wgrf at the tire contactpatch is zero (smooth road). It should be noted that even though the tire is assumedto remain at a fixed angle with the strut, the vertical stiffness of the tire, ktf , is alwaysconsidered to be normal to the ground between the ground and the wheel center [2].

The instantaneous tire radius is then determined as [2]

rrf =r0 − xtrfcosθcosφ

. (30)

To account for the wheel lift-off, when the tire radial compression becomes less than zero,the tire normal force Fzgrf is set equal to zero. In addition, the instantaneous tire radius isconsidered equal to the nominal tire radius until it returns to the road surface [2].

If xtrf < 0 then Fzgrf = 0 and rrf = r0. (31)

As the tire normal force becomes zero, no lateral (Fygrf ) and longitudinal (Fxgrf ) tireforces are developed at that contact patch. Thus, the only forces acting at that suspensioncorner are the weight and inertia forces of the unsprung mass, which are very small inmagnitude. As the cardan angles and appropriate coordinate transformations between thebody-fixed coordinate frame 1 and the coordinate frame 2 at the tire-ground contact patchare considered for all the forces and velocities in the system, the model is able to simulatevehicle behavior after wheel lift-off and during the rollover event with just the modificationmentioned in (31) [2].

The instantaneous length of the strut lsrf is given as [2]

lsrf = lsif −(xsrf − xsif

), (32)

where lsif is the initial length of strut and xsif is the initial suspension spring deflection.The initial length of the strut lsif is taken such that [2]

lsif = h− (r0 − xtif ), (33)

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where xtif is the initial tire compression.The initial spring compression xsif and the initial tire compression xtif are determined

from the static conditions as [2]

xsif =mb

2(a+ b)ksf(34)

xtif =(mb/2(a+ b) +muf )

ktf. (35)

For the left front corner, the left rear corner and the right rear coner, the instantaneouslength of the strut, the initial length of strut and the initial suspension spring deflection, theinitial length of the strut and the initial spring compression and the initial tire compressionare the same as those of the right front corner.

The forces Fxsrf and Fysrf transmitted to the sprung mass along the u− and v−axesof the body-fixed coordinate frame are obtained after subtracting the components of theunsprung mass weight and inertia forces from the corresponding forces Fxgsrf and Fygsrfacting at the tire contact patch as [2]

Fxsrf =Fxgsrf +mugsinθ −muu̇urf +muωzvurf −muωywurf (36)Fysrf =Fygsrf −mugsinφcosθ −muv̇urf +muωxwurf −muωzuurf . (37)

The vertical force Fzsrf transmitted to the sprung mass through the strut is given as [2]

Fzsrf = xsrfksrf + ẋsrfbsrf . (38)

Figure 4 shows the forces and velocities in the roll plane of, for example, the frontsuspension. Generally, the roll center height is defined with reference to the ground. However,for the development of this model, the front and rear roll centers are assumed to be fixed atdistances hrcf and hrcr, respectively, below the sprung mass C.M. along the negative w-axisof the body-fixed coordinate frame 1. Moreover, the roll center is simply considered to be apoint of application of the forces transmitted to the sprung mass through the suspension linksand not as a kinematic constraint. In the figure 4, Fzslf and Fzsrf are the forces transmittedto the sprung mass through the struts. Fyslf and Fysrf represet the lateral forces transmittedto the sprung mass through the suspension links. In the absence of a roll center, i.e., whenthe roll center is assumed to be in the ground plane, the total roll moment transmitted tothe sprung mass at, for example, the right front corner along the ωx direction is given as [2]

Mxrf =Fygsrf(lsrf + rrf

)−(mugsinφcosθ +muu̇urf −muωxwurf +muωzuurf

)· lsrf

=Fygsrf · rrf + Fysrf · lsrf , (39)

for the left front corner:

Mxlf =Fygslf(lslf + rlf

)−(mugsinφcosθ +muu̇ulf −muωxwulf +muωzuulf

)· lslf

=Fygslf · rlf + Fyslf · lslf , (40)

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Figure 4: Forces and velocities in the front suspension roll plane [2]

When a roll center is modeled as shown in Fig. 4, the roll moment Mxrf transmitted tothe sprung mass by the right front corner suspension is given as [2]

Mxrf = Fysrfhrcf . (41)

Thus, the inclusion of a roll center reduces the total roll moment transferred to the sprungmass by the front suspension. The difference between the roll moments in the absence of theroll center and when the roll center is considered acts directly on the unsprung mass and isresponsible for the link load transfer forces (jacking forces), Fdzlf and Fdzrf . These forcescan be estimated as [2]

Fdzrf = −Fdzlf =Fygsrfrrf + Fysrf lsrf + Fygslfrlf + Fyslf lslf − (Fysrf + Fyslf )hrcf

cf. (42)

The moments Myrf and Mzrf transmitted to the sprung mass at, for example, the rightfront corner by the suspension along the ωy and ωz directions can be given as [2]

Myrf =−(Fxsgrf (lsrf + rrf )− (−mugsinθ +muu̇urf −muωzvurf +muωywurf ) · lsrf

)=−

(Fxsgrf · rrf + Fxsrf · lsrf

)(43)

Mzrf =0. (44)

For the rear suspension, the total roll moment transmitted to the sprung mass at the leftrear and right rear corner has the same form as the front suspension.

Thus, the forces of Fdzlr and Fdzrr can be estimated as [2]

Fdzrr = −Fdzlr =Fygslrrlr + Fyslrlslr + Fygsrrrrr + Fysrrlsrr − (Fyslr + Fysrr)hrcr

cr. (45)

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The equation of motion for the 6 DOF of the sprung mass model can now be derivedfrom the direct application of Newton’s laws for the system as [2]

m(u̇+ ωyw − ωzv) =∑

(Fxsij) +mgsinθ (46)

m(v̇ + ωzu− ωxw) =∑

(Fysij)−mgsinφcosθ (47)

m(ẇ + ωxv − ωyu) =∑

(Fzsij + Fdzij)−mgcosφcosθ (48)

Jxω̇x =∑

(Mxij) +(Fzslf − Fzsrf )cf + (Fzslr − Fzsrr)cr

2(49)

Jyω̇y =∑

(Myij) + (Fzslr + Fzsrr)b− (Fzslf + Fzsrf )a (50)

Jzω̇z =∑

(Mzij) + (Fyslf + Fysrf )a− (Fyslr + Fysrr)b

+(−Fxslf + Fxsrf )cf + (−Fxslr + Fxsrr)cr

2. (51)

where m is the sprung mass and the subscript ‘ij ’ denotes left front (lf ), right front (rf ),left rear (lr), and right rear (rr).

The cardan angles θ, ψ, φ needed in the aforementioned equations are obtained by per-forming the integration of the following equations [2],

θ̇ =ωycosφ− ωzsinφ (52)

ψ̇ =ωysinφ

cosθ+ωzcosφ

cosθ(53)

φ̇ =ωx + ωysinφtanθ + ωzcosφtanθ. (54)

The rotational dynamics for each wheels can be given as [12]

Jwω̇rf = Tdrf − Tbrf − rrfFxtrf , (55)

where Tdrf is the driving torque transmitted to the wheel, Tbrf is the wheel braking torque.

2.4 8-DOF vehicle model

An 8-DOF full vehicle model is often used as a simplified lower order model for studyingvehicle handling in scenarios which do not involve significant longitudinal accelerations. Inthis section, a formulation for the 8-DOF model, adapted from various references [2] [12],that can match the 14-DOF model reasonably accurately is presented [2].

The schematic of the 8-DOF full vehicle model is given in Figure 5. The model has fourdegrees of freedom for the chassis velocities and one degree of freedom at each of the fourwheels representing the wheel spin dynamics. The chassis velocities include the longitudinalvelocity, u, the lateral velocity, v, the roll angular velocity, ωx, and the yaw angular velocity,ωz. The pitch and heave motions are not modeled and the front and rear suspensions are

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represented simply by their respective equivalent roll stiffness (kψf/kψr) and roll dampingcoefficients (bψf/bψr) [2].

Figure 5: Schematic of 8-DOF full vehicle model [2]

The equations of motion for the chassis velocities are obtained as given subsequently [2]:

mt(u̇− ωzv

)=∑

Fxgij + (mufa−murb)ω2z − 2hrcmωzωx (56)

mt(v̇ + ωzu

)=∑

Fygij + (murb−mufa)ω̇z + hrcmω̇x (57)

Jzω̇z + Jxzω̇x = (Fyglf + Fygrf ) a− (Fyglr + Fygrr) b+(Fxgrf − Fxglf

)cf

2

+

(Fxgrr − Fxglr

)cr

2+(murb−mufa

)(v̇ + ωzu

)(58)(

Jx +mh2rc

)ω̇x + Jxzω̇z =mghrcφ−

(kφf + kφr

)φ−

(bφf + bφr

)φ̇+ hrcm

(v̇ + ωzu

), (59)

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where

hrc =hrcfb+ hrcra

a+ b. (60)

In these equations, the forces Fxgij and Fygij are the longitudinal and lateral forces at thefour tire contact patches and the subscript ‘ij ’ denotes lf, rf, lr, and rr. As before, hrcf andhrcr are the vertical distances of the front and rear roll centers below the sprung mass C.M.,and thus hrc is the vertical distance from the sprung mass C.M. to the vehicle roll center.It should be noted that as equation for the roll degree of freedom is written by consideringmoments acting about the vehicle roll center. It should be noted that as equation (59) for theroll degree of freedom is written by considering moments acting about the vehicle roll centerrather than the sprung mass C.M., the roll inertia of the sprung mass about the vehicle rollcenter

(Jx +mh

2rc

)is considered in equation (59) [2].

The equations for the wheel dynamics and the longitudinal and lateral tire forces arethe same as those used in the 14-DOF model. The longitudinal and lateral velocities at, forexample, the right front tire contact patch required in these equations are given as [2]

ugrf =u+ωzcf

2(61)

vgrf =v + ωza. (62)

The normal forces at the four tires are determined as [2]

Fzglf =mgb

2(a+ b)+mufg

2−

(mufhufcf

+mb(hcg − hrcf

)cf (a+ b)

)(v̇ + ωzu

)−(kφf + bφf

)cf

−(mhcg +mufhuf +murhur

)(u̇− ωzv

)2(a+ b)

(63)

Fzgrf =mgb

2(a+ b)+mufg

2+

(mufhufcf

+mb(hcg − hrcf

)cf (a+ b)

)(v̇ + ωzu

)+

(kφf + bφf

)cf

−(mhcg +mufhuf +murhur

)(u̇− ωzv

)2(a+ b)

(64)

Fzglr =mga

2(a+ b)+murg

2−

(murhurcr

+ma(hcg − hrcr

)cr(a+ b)

)(v̇ + ωzu

)−(kφrφ+ bφrφ̇

)cr

+

(mhcg +mufhuf +murhur

)(u̇− ωzv

)2(a+ b)

(65)

Fzgrr =mga

2(a+ b)+murg

2+

(murhurcr

+ma(hcg − hrcr

)cr(a+ b)

)(v̇ + ωzu

)+

(kφrφ+ bφrφ̇

)cr

+

(mhcg +mufhuf +murhur

)(u̇− ωzv

)2(a+ b)

. (66)

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These equations are fairly simple and linearized. It is possible to include several additionalterms in the equations for the chassis velocities as well as the tire forces. However, in ourexperience, these terms have very little effect on the vehicle responses and can be ignored [2].

3 Simulation and Analysis

3.1 Vehicle Model Parameters

Parameters used for vehicle model simulations are listed as below:

Table 1: Parameters for vehicle dynamics modeling

Symbol Parameter description Value

m Sprung mass 1400(kg)Jx Sprung mass roll inertia 900(kgm

2)Jy Sprung mass yaw inertia 2000(kgm

2)Jz Sprung mass pitch inertia 2420(kgm

2)Jxz Sprung mass inertia 90(kgm

2)a Distance of sprung mass C.M. from front axle 1.14(m)b Distance of sprung mass C.M. from rear axle 1.4(m)h Sprung mass C.M. height 0.75(m)cf Front track width 1.5 (m)cr Rear track width 1.5 (m)ksf Front suspension stiffness 35000 (N/m)ksr Rear suspension stiffness 30000 (N/m)bsf Front suspension damping coefficient 2500 (Ns/m)bsr Rear suspension damping coefficient 2000 (Ns/m)muf Front unsprung mass 80 (kg)mur Rear unsprung mass 80 (kg)ktf Front tire stiffness 200000 (N/m)ktr Rear tire stiffness 200000 (N/m)Cαf Front right tire cornering stiffness 44000 (N/rad)Cαr Rear right tire cornering stiffness 47000 (N/rad)Cxf Front right tire longitudinal stiffness 5000 (N)Cxr Rear right tire longitudinal stiffness 5000 (N)r0 Nominal tire radius 0.285 (m)Jw Tire/wheel roll inertia 1 (kgm

2)hrcf Front roll center distance below sprung mass C.M. 0.65 (m)hrcr Rear roll center distance below sprung mass C.M. 0.6 (m)

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3.2 Simulation Results

Assuming the longitudinal velocity is constant in the simulation, three different steering inputsignals are implemented to three vehicle models and the responses are studied to understandthe effect of simplifications.

3.2.1 Bicycle Model

1. Take sinusoidal steering wheel angle with amplitude at 0.0087(rad), i.e. 0.5(deg) asinput, and fix the vehicle longitudinal velocity at 33.73(m/s), the output of the vehiclemodel includes lateral velocity, yaw angle and yaw rate, which are listed as follows,

Figure 6: Bicycle model response of road wheel sine wave steer input at a speed of 33.73m/s

2. Take step steering wheel angle as input with amplitude 0.0087(rad) and the samevehicle longitudinal velocity as before, the output of the vehicle model includes lateralvelocity, yaw angle and yaw rate, which are listed as follows,

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Figure 7: Bicycle model response of road wheel step steer input at a speed of 33.73m/s

3. Take ramp steering wheel angle as input with gradient at 3(deg/s). The vehicle longi-tudinal velocity is fixed at 50(km/h), the output of the vehicle model includes lateralvelocity, yaw angle and yaw rate, which are listed as follows,

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Figure 8: Bicycle model response of road wheel ramp steer 3deg/s input at a speed of 50km/h

3.2.2 8-DOF Vehicle Model

1. Take sinusoidal steering wheel angle with amplitude at 0.0087(rad) as input, and main-tain the vehicle longitudinal velocity at 33.73(m/s), the output of the vehicle modelincludes roll angle, yaw angle, yaw rate, lateral velocity, lateral acceleration, which arelisted as follows,

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Figure 9: 8-DOF Vehicle response of road wheel sine wave steer input at a speed of 33.73m/s

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2. Take step steering wheel angle as input with amplitude at 0.0087(rad) and the vehiclelongitudinal velocity at 33.73(m/s), the outputs of the vehicle model include roll angle,yaw angle, yaw rate, lateral velocity, lateral acceleration, which are listed as follows,

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Figure 10: 8-DOF Vehicle response of road wheel step steer input at a speed of 33.73m/s

3. Take ramp steering wheel angle as input with gradient at 3(deg/s) and the vehiclelongitudinal velocity fixed at 50(km/h), the outputs of the vehicle model include rollangle, yaw angle, yaw rate, lateral velocity, lateral acceleration, which list as follows,

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Figure 11: 8-DOF Vehicle response of road wheel ramp steer 3deg/s input at a speed of50km/h

3.2.3 14-DOF Vehicle Model

1. Take sinusoidal steering wheel angle as input with amplitude 0.0087(rad) and thevehicle longitudinal velocity is 33.73(m/s), the output of the vehicle model includes rollangle, yaw angle, yaw rate, lateral velocity, lateral acceleration, which list as follows,

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Figure 12: 14-DOF Vehicle response of road wheel sine wave steer input at a speed of33.73m/s

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2. Take step steering wheel angle as input with amplitude 0.0087(rad) and the vehiclelongitudinal velocity is 33.73(m/s), the outputs of the vehicle model include roll angle,yaw angle, yaw rate, lateral velocity, lateral acceleration, which list as follows,

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Figure 13: 14-DOF Vehicle response of road wheel step steer input at a speed of 33.73m/s

3. Take ramp steering wheel angle as input with gradient at 3(deg/s) and the vehiclelongitudinal velocity at 50(km/h), the outputs of the vehicle model include roll angle,yaw angle, yaw rate, lateral velocity, lateral acceleration, which list as follows,

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Figure 14: 14-DOF Vehicle response of road wheel ramp steer 3deg/s input at a speed of50km/h

4 Conclusions and Future Work

A bicycle model, an 8-DOF vehicle model and a 14-DOF vehicle model were developed andimplemented in MATLABR©. The bicycle model, assuming the vehicle only moves in yawplane and neglecting the load transfer between the left and right tires, is often used in thevehicle controller design due to its simplicity. The 8-DOF vehicle considered the longitudinal,the lateral, the roll and the yaw motion of the chassis, and four wheel spin dynamics. The14-DOF vehicle model, which considered the suspension at each corner, has the same benefitsof an 8-DOF vehicle model, with the additional capabilities of predicting vehicle pitch andheave motions. Different front road wheel steering angles are taken as input signals and theiroutput responses are studied and analyzed.

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In the future work, model predictive control for path following will be implemented byusing lower fidelity vehicle model as the prediction model in the controller and using higherfidelity vehicle model as the plant.

Acknowledgments

The authors would like to thank Heran Shen of Columbia University for his comments onan early version of this document.

References

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IntroductionVehicle Dynamics ModelingTire ModelingBicycle Model14-DOF Vehicle Model8-DOF vehicle model

Simulation and AnalysisVehicle Model ParametersSimulation ResultsBicycle Model8-DOF Vehicle Model14-DOF Vehicle Model

Conclusions and Future Work