on the handling of automated vehicles: modeling
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European Journal of Mechanics / A Solids 90 (2021) 104372
A0
Contents lists available at ScienceDirect
European Journal of Mechanics / A Solids
journal homepage: www.elsevier.com/locate/ejmsol
On the handling of automated vehicles: Modeling, bifurcation analysis, andexperimentsSanghoon Oh a,โ, Sergei S. Avedisov a,b, Gรกbor Orosz a,c
a Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USAb Toyota InfoTechnology Center, Mountain View, CA 94043, USAc Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor, MI 48109, USA
A R T I C L E I N F O
Keywords:Automated vehiclesBicycle modelNonholonomic dynamicsBifurcation analysisSteady state cornering
A B S T R A C T
The dynamics of automated vehicles are studied and the existence and stability of steady state corneringmaneuvers are investigated. Utilizing tools from analytical mechanics, namely the Appellian approach, vehiclemodels are constructed which incorporate geometrical nonlinearities and differentiate between front wheeldrive (FWD) and rear wheel drive (RWD) automobiles. The dynamics of these models are studied usingbifurcation analysis. Stable and unstable steady states are mapped out as a function of speed and steeringangle. It is demonstrated that RWD and FWD vehicles exhibit different behavior, especially when they operateclose to their handling limits. The theoretical results are verified experimentally using an automated vehicleperforming safety critical maneuvers.
1. Introduction
Automated vehicles are likely to enter our roads soon and they needto satisfy more strict safety requirements compared to their human-driven counterparts (Ersal et al., 2020). For motion planning andcontrol of automated vehicles choosing an appropriate model is impor-tant. The model must have high enough fidelity to be able to predictthe motion of the vehicle accurately, while in the meantime it mustbe simple enough so it can be run in realtime. When focusing on thelateral and yaw motion, the so-called single track or bicycle model isoften used to plan the motion of the vehicle and to track the plannedtrajectory by steering the vehicle (Popp and Schiehlen, 2010; Schrammet al., 2014; Rajamani, 2012; Ulsoy et al., 2012). In this paper weconsider this modeling approach and show that by adding importantgeometric nonlinearities one can improve the predictive power ofbicycle models without significantly increasing their complexity.
While in many scenarios assuming rigid wheels can be used tosimplify the model (Paden et al., 2016; Vรกrszegi et al., 2019), takinginto account the deformation of the tires is important when tryingto predict and control the motion of the vehicle; especially, close toits handling limits (Liu et al., 2016). One of the earliest work aboutthe bicycle model with tires can be found in Riekert and Schunck(1940), and since then there has been a large number of publicationsfollowing this approach. These models mainly differ in how the tire-road interaction is modeled. Since the lateral deformation of a tire leadsto the formation of a slip angle between the direction of the wheel
โ Corresponding author.E-mail address: [email protected] (S. Oh).
plane and the direction of velocity of the wheel center, the lateral forcearising due to this deformation is often expressed as a function of thetire slip angle (Pacejka, 2005). Nevertheless, more precise tire modelsthat capture the effects of the tire deformation along the contact patchhave been recently proposed and analyzed (Beregi and Takรกcs, 2019;Mi et al., 2020).
In general, the lateral force is a nonlinear function of the slip angleand this nonlinearity was taken into account in prior studies usingthe bicycle model. In earlier works, bifurcations of a bicycle modelwith nonlinear tire characteristics were investigated (Ono et al., 1998;Della Rossa et al., 2012; Farroni et al., 2013) and steering controllerswere designed in order to maintain lateral stability (Voser et al., 2010).More recent studies about lateral stabilization and operating vehiclesat their handling limits relied on boundaries in state space related tothe peak lateral tire forces (Hwan Jeon et al., 2013; Erlien et al., 2013).However, the actual region of attraction of steady states may differ fromthese boundaries depending on the steering angle and the drive type.Moreover, mapping out the detailed state space picture may allow oneto design high-performance controllers. Indeed, many recent researchefforts in automated vehicle control is related to driving vehicles attheir handling limits (Li et al., 2020; Berntorp et al., 2020; Wurts et al.,2020; Lu et al., 2021), including the control of unstable motions likedrifting (Velenis et al., 2011; Goh et al., 2020), in order to improve thevehicleโs agility. However, existing models do not differentiate between
vailable online 27 July 2021997-7538/ยฉ 2021 Elsevier Masson SAS. All rights reserved.
https://doi.org/10.1016/j.euromechsol.2021.104372Received 27 January 2021; Received in revised form 29 June 2021; Accepted 18 J
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European Journal of Mechanics / A Solids 90 (2021) 104372S. Oh et al.
Table 1Parameters of a 2016 KIA Soul automobile used in this paper.
Parameter name Value Unit
๐ Vehicle mass 1110 kg๐ฝG Vehicle mass moment of inertia 1343 kg m2
๐ Vehicle center of mass position 1.03 m๐ Vehicle center of mass position 1.54 m๐ = ๐ + ๐ Wheelbase 2.57 m๐ Distributed tire stiffness 4 ร 106 N/m2
๐ Half contact length 0.1 m๐ถ = 2๐๐2 Cornering stiffness 8 ร 104 N/rad๐ Sliding friction coefficient 0.6๐0 Static friction coefficient 0.9
drive types, which provide a fundamental limitation for the predictivepower of the models as well as for the performance of the designedcontrollers. This limitation is particularly important to overcome if onewishes to exploit the opportunities brought by the integration of vehicleautomation and electrification (Zhang et al., 2018).
To answer this challenge, in this study we utilize the Appellianapproach (Gantmacher, 1975; De Sapio, 2017) and construct nonlinearbicycle models which, in addition to the tire nonlinearities, incorporategeometric nonlinearities. We demonstrate that these models are able tobring out the qualitative differences between rear wheel drive (RWD)and front wheel drive (FWD) automobiles. We use bifurcation analysisto investigate the steady state cornering while varying the steeringangle and the speed of the vehicle. The bifurcation diagrams and statespace pictures reveal stable and unstable turning motions and theregions of attractions for the former ones. These diagrams allow us toexplore the similarities and differences between the behavior of RWDand FWD vehicles. The theoretical results are validated with a help ofan automated vehicle driven up to its handling limits. We also pointout specific unstable motions which, if stabilized by appropriate controldesign, can significantly enhance the maneuverability of automatedvehicles.
The paper is organized as follows. In Section 2 we describe thegoverning equations and the cornering characteristics of a so-calledtraditional model with tire nonlinearity. In Section 3 we provide thederivation of the equations of motion of the proposed nonlinear modelsfor RWD and FWD vehicles using the Appellian approach and showthat these models simplify to the traditional model for small steeringangles. Section 4 contains the analysis of steady state cornering fordifferent models using numerical continuation. In Section 5 we discussthe experimental results that validate the FWD model. In Section 6 weconclude our work and propose future research directions.
2. Traditional bicycle model
In this section we describe a version of the bicycle model that hasbeen used in the literature. We refer to this as the traditional bicyclemodel in the remaining of the paper. While discussing this model wealso introduce some basic notations and terminology related to thesteady state cornering. Finally, we point out some of the shortcomingsof this traditional model.
Fig. 1(a) shows the schematic diagram of a bicycle model for anautomobile with front wheel steering. In this model the front wheelsare united as one wheel and the rear wheels are also united as onewheel. The mass of the vehicle is ๐, its mass moment of inertia aboutthe center of mass G is denoted by ๐ฝG, the wheel base in given by ๐, thedistance between the center of the front axle F and the center of massG is given by ๐ while the distance between the center of the rear axleR and the center of mass G is given by ๐. Indeed we have ๐ = ๐+๐. Theparameters used in this paper are listed in Table 1.
The vehicle can be described by three configuration coordinatesand these are typically chosen to be the position of the center of mass(๐ฅ , ๐ฆ ) and the yaw angle ๐ . The velocity state is typically given
2
G G
Fig. 1. (a) Bicycle model with geometric, kinematics and dynamics quantities indi-cated. (b) Nonlinear tire characteristics for brush model (blue) and magic formula(green). The location of peak force ๐ผpk , the beginning of the sliding region ๐ผsl,the cornering stiffness ๐ถ = d๐น
d๐ผ(0) and the saturation force ๐๐น๐ง are indicated. (For
interpretation of the references to color in this figure legend, the reader is referred tothe web version of this article.)
by the angular velocity ๐ called the yaw rate, and by the velocitycomponents of the center of mass G resolved in the frame fixed to thevehicleโs body, referred to as longitudinal velocity ๐ฃ and lateral velocity๐; see Fig. 1(a). The kinematic relationships between the derivativesof the configuration coordinates and the velocity components can beexpressed by
๏ฟฝ๏ฟฝG = ๐ฃ cos๐ โ ๐ sin๐,
๏ฟฝ๏ฟฝG = ๐ฃ sin๐ + ๐ cos๐,
๏ฟฝ๏ฟฝ = ๐.
(1)
Also, we assume that the steering angle ๐พ can be assigned by thecontroller of the automated vehicle which is the input in the dynamicalsystems constructed below.
The rest of the modeling effort shall go into deriving differentialequations for the longitudinal velocity ๐ฃ, the lateral velocity ๐, and theyaw rate ๐. Due to the elasticity of the rear tire, the velocity ๐ฏR of thewheel center point R is not aligned with the rear wheel plane but itforms the slip angle ๐ผR. Similarly, for the front wheel, the velocity ๐ฏFof the wheel center point F is not aligned with the front wheel planebut it forms the slip angle ๐ผF. The lateral tire deformations lead to thelateral forces ๐นR and ๐นF that are indicated in Fig. 1(a).
By approximating the deformation along the contact patch with astraight line, the lateral force can be expressed as a function of the slipangle as given in the Appendix and visualized by the blue curve inFig. 1(b). This is often referred to as the brush model. The slip angle๐ผpk corresponds to the peak value of the lateral force while ๐ผsl marks thelimit of complete sliding. When the slip angle exceeds the latter value,the lateral force saturates at ๐๐น๐ง, as indicated in the figure, where ๐น๐ง isthe normal force and ๐ is the sliding friction (as opposed to the staticfriction that is denoted by ๐0). Finally, the derivative ๐ถ = d๐น
d๐ผ (0) is alsohighlighted. This is referred to as the cornering stiffness and for the
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European Journal of Mechanics / A Solids 90 (2021) 104372S. Oh et al.
inwmo
l
octtl
T๐ผitb
wTc
๐
mstti
oi
s
iwade
brush model it can be calculated as ๐ถ = 2๐๐2 where ๐ denotes thedistributed tire stiffness and ๐ is the half contact length; see Table 1for the numerical values used in the paper.
For comparison, an empirical model, called the magic formula(Pacejka, 2005), is also shown in Fig. 1(b) by the green curve; seecorresponding formula in the Appendix. The four independent param-eters appearing in the nonlinear tire characteristics are matched forthe brush model and the magic formula by matching the location andthe value of the peak, the saturating value of the lateral force, and thecornering stiffness. In this paper we consider the brush model but theresults shall look very similar for the magic formula. We remark thatthe tire deformations also lead to aligning moments but those havetypically small effects in scenarios investigated in our paper, and thus,they are neglected. Finally, we mention that the longitudinal slip isneglected in this paper in order keep the complexity of the model low.We found that the longitudinal forces required to keep the longitudinalvelocity constant are typically much smaller than the peak lateral force.They may reach around 10%โ15% of the peak lateral force in a smallvelocity range, where one may use higher degree of freedom modelsincorporating both lateral and longitudinal tire deformations.
2.1. Equations of motion
When focusing on handling it is often considered that the longi-tudinal velocity is constant, i.e., ๐ฃ(๐ก) โก ๐ฃโ, which can be ensured byapplying the appropriate driving force. Moreover, it is often assumedthat the steering angle and the slip angles are small and the geometricalnonlinearities can be neglected, which lead to
๐ผR = โ๐ โ ๐ ๐๐ฃโ
,
๐ผF = ๐พ โ ๐ + ๐ ๐๐ฃโ
,(2)
cf. (29) and (37) for the nonlinear versions developed further be-low. Similarly, one may neglect the geometrical nonlinearities whencalculating the force components normal to the body and obtain
๏ฟฝ๏ฟฝ =๐นR(๐ผR) + ๐นF(๐ผF)
๐โ ๐ฃโ๐,
๏ฟฝ๏ฟฝ =โ๐ ๐นR(๐ผR) + ๐ ๐นF(๐ผF)
๐ฝG,
(3)
that are the NewtonโEuler equations for the lateral and rotationalmotions, respectively; cf. (28) and (36) for the nonlinear versionsdeveloped further below. Here ๐นR and ๐นF denote lateral tire forces andthese depend on the rear and front slip angles ๐ผR and ๐ผF as shownin Fig. 1, and in turn depend on the state variables ๐ and ๐ and thenput ๐พ according to (2). In summary, (1), (2), (3) provide us with fiveonlinear differential equations for the state variables ๐ฅG, ๐ฆG, ๐, ๐, ๐ith input ๐พ. Notice that these equations are not fully coupled: oneay first solve (2), (3) to obtain velocities ๐, ๐, and then solve (1) to
btain the configuration coordinates ๐ฅG, ๐ฆG, ๐ .In order to further simplify the matter, Eqs. (2), (3) are often
inearized around the rectilinear motion yielding
[
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
]
=
โก
โข
โข
โข
โฃ
โ๐ถR + ๐ถF๐๐ฃโ
๐ ๐ถR โ ๐ ๐ถF๐๐ฃโ
โ ๐ฃโ
๐ ๐ถR โ ๐ ๐ถF๐ฝG๐ฃโ
โ๐2๐ถR + ๐2๐ถF
๐ฝG๐ฃโ
โค
โฅ
โฅ
โฅ
โฆ
[
๐๐
]
+
โก
โข
โข
โข
โฃ
๐ถF๐๐ ๐ถF๐ฝG
โค
โฅ
โฅ
โฅ
โฆ
๐พ (4)
where ๐ถR =๐๐นR๐๐ผR
(0) and ๐ถF =๐๐นF๐๐ผF
(0) denote the cornering stiffnessf the rear and the front wheels, respectively. For simplicity, the so-alled traditional linear model (1), (4) is often used instead of theraditional nonlinear model (1), (2), (3). However, due to the linearire characteristics the former model is only valid far from the handlingimits, i.e., in the domain ๐ผ โช ๐ผ in Fig. 1(b).
3
pk d
2.2. Steady state cornering
Steady state cornering corresponds to negotiating a curve of con-stant radius by keeping the steering angle constant. Such a test canbe conducted with an automated vehicle as explained further below inorder to validate dynamical models and design controllers.
When substituting ๐(๐ก) โก ๐โ, ๐(๐ก) โก ๐โ, ๐พ(๐ก) โก ๐พโ into (2), (3) one mayobtain
๐ผโR = โ๐โ โ ๐ ๐โ
๐ฃโ,
๐ผโF = ๐พโ โ ๐โ + ๐ ๐โ
๐ฃโ,
0 = ๐นR(๐ผโR) + ๐นF(๐ผโF) โ ๐๐ฃโ๐โ,
0 = โ๐ ๐นR(๐ผโR) + ๐ ๐นF(๐ผโF).
(5)
hese can be solved for the state variables ๐โ, ๐โ and the slip anglesโR, ๐ผ
โF while taking into account the nonlinear tire force character-
stics (41), (42) (or alternatively (46)). Due to these nonlinearitieshe solutions can typically be found numerically (as detailed furtherelow).
According to (1) the corresponding configuration coordinates read
๐ฅโG(๐ก) =๐ฃโ
๐โ sin(๐โ๐ก + ๐0) +๐โ
๐โ cos(๐โ๐ก + ๐0) + ๐ฅ0,
๐ฆโG(๐ก) = โ ๐ฃโ
๐โ cos(๐โ๐ก + ๐0) +๐โ
๐โ sin(๐โ๐ก + ๐0) + ๐ฆ0,
๐โ(๐ก) = ๐โ๐ก + ๐0,
(6)
here the constants ๐ฅ0, ๐ฆ0, ๐0 are obtained from the initial conditions.he radius of the circular arc ran by the center of mass G can bealculated as
G =โ
(๐ฅโG โ ๐ฅ0)2 + (๐ฆโG โ ๐ฆ0)2 =
โ
(๐ฃโ)2 + (๐โ)2
๐โ . (7)
Similarly all points of the automobile move along circular paths. Forexample, the corresponding radius of the rear axle center point R isgiven by
๐ R =
โ
(๐ฃโ)2 + (๐โ โ ๐ ๐โ)2
๐โ . (8)
Of particular importance is the difference ๐ผโF โ ๐ผโR whose sign deter-ines whether the vehicles displays understeer (๐ผโF โ ๐ผโR > 0) or over-
teer (๐ผโF โ ๐ผโR < 0) behavior. Most production vehicles are designedo have understeer characteristics, but, as we will demonstrate belowhey may exhibit oversteer behavior for specific sets of parameters andnitial conditions.
For small steering angles (and correspondingly small slip angles)ne may utilize the linearized Eqs. (4) instead of (2), (3), which resultn[
๐โ
๐โ
]
=[
๐ถR + ๐ถF โ(๐ ๐ถR โ ๐ ๐ถF) + ๐(๐ฃโ)2
โ(๐ ๐ถR โ ๐ ๐ถF) ๐2๐ถR + ๐2๐ถF
]โ1 [ ๐ถF๐ฃโ
๐ ๐ถF๐ฃโ
]
๐พโ. (9)
Substituting these into (2) one may derive
๐ผโF โ ๐ผโR = ๐พus๐ฃโ๐โ
๐, (10)
where the so-called understeer coefficient
๐พus =๐๐๐
(
๐๐ถF
โ ๐๐ถR
)
, (11)
determines the understeer (๐พus > 0) or oversteer (๐พus < 0) behavior formall slip angles.
For most production automobiles understeer characteristic is typ-cally ensured by having the engine at the front (that results in theeight distribution ๐น๐ง,R < ๐น๐ง,F โ ๐ > ๐) and using the same type of tirest the rear and the front (that yields ๐ถR โ ๐ถF); see, for example, theata of a 2016 KIA Soul in Table 1. For some high-end automobiles thengine is more toward the middle/rear (that results in the even weightistribution ๐น โ ๐น โ ๐ โ ๐).
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European Journal of Mechanics / A Solids 90 (2021) 104372S. Oh et al.
Fig. 2. The critical longitudinal velocities (13) and (15) as a function of the steeringangle ๐พโ in case of steady state cornering.
In some special scenarios, analytical solutions of the nonlinearEqs. (5) exist and these solutions can lead to further insight. Let usconsider the case when both tires have the maximal tire force; see๐ผ = ๐ผpk in Fig. 1(b). Using the static weight distribution ๐น๐ง,R = ๐๐๐โ๐and ๐น๐ง,F = ๐๐๐โ๐ in (44) we obtain
๐ผโR = arctan
(
๐0๐๐๐2๐๐2๐
(
1 โ2๐3๐0
)โ1)
,
๐ผโF = arctan
(
๐0๐๐๐2๐๐2๐
(
1 โ2๐3๐0
)โ1)
,
๐นR(๐ผโR) =๐0๐๐๐3๐
( 43โ๐๐0
)(
1 โ2๐3๐0
)โ2,
๐นF(๐ผโF) =๐0๐๐๐
3๐
( 43โ๐๐0
)(
1 โ2๐3๐0
)โ2.
(12)
Substituting the last two equations of (12) into the third equation of(5) and substituting the first two equations of (12) into the first twoequations of (5) while using the approximation arctan ๐ผ โ ๐ผ, we cancalculate the critical velocity:
๐ฃโcr,1 โ
โ
โ
โ
โ
โ
โ
โ
โ
๐0๐๐3
( 43โ๐๐0
)(
1 โ2๐3๐0
)โ2
๐พโ โ๐0๐๐(๐ โ ๐)
2๐๐2๐
(
1 โ2๐3๐0
)โ1, (13)
which only exist for ๐พโ > ๐0๐๐(๐โ๐)2๐๐2๐
(
1 โ 2๐3๐0
)โ1. When the vehicle
reaches this speed both tires provide the maximum available lateralforce, that is, increasing the speed further one may expect the handlingto deteriorate.
Another special scenario of interest is when the both wheel are atthe sliding limit (๐ผ = ๐ผsl in Fig. 1(b)) which yields
๐ผโR = arctan( 3๐0๐๐๐
2๐๐2๐
)
, ๐นR(๐ผโR) =๐๐๐๐๐
,
๐ผโF = arctan( 3๐0๐๐๐
2๐๐2๐
)
, ๐นF(๐ผโF) =๐๐๐๐๐
,(14)
according to (41) and (43) with static weight distribution. Now, sub-stituting the last two equations of (14) into the third equation of(5) and substituting the first two equations of (14) into the first twoequations of (5), the approximation arctan ๐ผ โ ๐ผ leads to the secondcritical velocity:
๐ฃโcr,2 โโ
โ
โ
โ
โ
๐๐๐
๐พโ โ3๐0๐๐(๐ โ ๐)
2๐๐2๐
, (15)
which only exists for ๐พโ > 3๐0๐๐(๐โ๐)2๐๐2๐ . When this speed is reached both
tires enters full sliding and the vehicle becomes uncontrollable.
4
The critical velocities (13) and (15) are shown as a function of thesteering angle ๐พโ in Fig. 2. Indeed, these velocities only exist abovecertain steering angles as indicated by the dashed vertical asymptotes.Also, observe that for large steering angles the two critical velocitiesare close to each other. We remark that there exist no analytical proofthat the above special states (where both tires have maximal tire forceand where both tires are at the limit of full sliding) actually exist.However, in Section 4 we will demonstrate by numerical continuationthat they do in fact show up for the traditional nonlinear model (1),(2), (3) where that the analytical formulae (13) and (15) provide goodapproximations of the critical speeds.
Finally, we remark that apart from finding the steady states bysolving (5) one may also evaluate the stability of steady state corneringby linearizing (2), (3) about the steady state. In particular, defining theperturbations ๏ฟฝ๏ฟฝ = ๐ โ ๐โ, ๏ฟฝ๏ฟฝ = ๐ โ ๐โ, ๏ฟฝ๏ฟฝ = ๐พ โ ๐พโ one may obtain thelinearized system
[ ๐๐
]
=
โก
โข
โข
โข
โข
โฃ
โ๐ถโ
R + ๐ถโF
๐๐ฃโ๐ ๐ถโ
R โ ๐ ๐ถโF
๐๐ฃโโ ๐ฃโ
๐ ๐ถโR โ ๐ ๐ถโ
F๐ฝG๐ฃโ
โ๐2๐ถโ
R + ๐2๐ถโF
๐ฝG๐ฃโ
โค
โฅ
โฅ
โฅ
โฅ
โฆ
[
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
]
+
โก
โข
โข
โข
โข
โฃ
๐ถโF๐๐ ๐ถโ
F๐ฝG
โค
โฅ
โฅ
โฅ
โฅ
โฆ
๏ฟฝ๏ฟฝ (16)
where ๐ถโR =
๐๐นR๐๐ผR
(๐ผโR) and ๐ถโF =
๐๐นF๐๐ผF
(๐ผโF) are the generalized corner-ing stiffnesses. Note that (4) is special version of (16) where thelinearization happens around the rectilinear motion.
In order to derive stability conditions for the generalized corneringstiffnesses we assume the trial solution ๏ฟฝ๏ฟฝ = ๐ e๐ ๐ก, ๏ฟฝ๏ฟฝ = ๐ e๐ ๐ก, ๐, ๐, ๐ โ C,which results in the characteristic equation
๐ 2 + ๐๐ + ๐ = 0, (17)
where
๐ =๐ถโ
R + ๐ถโF
๐๐ฃโ+๐2๐ถโ
R + ๐2๐ถโF
๐ฝG๐ฃโ,
๐ =๐2๐ถโ
R๐ถโF
๐๐ฝG(๐ฃโ)2+๐ ๐ถโ
R โ ๐ ๐ถโF
๐ฝG.
(18)
The RouthโHurwitz stability criteria ๐ > 0 and ๐ > 0 can be visualizedin the (๐ถโ
R, ๐ถโF )-plane as depicted in Fig. 3 by the green shaded domain.
The boundary given by ๐ = 0 (blue line) denote Hopf bifurcation: acomplex conjugate pair of characteristic roots crosses the imaginaryaxis and oscillations arise. This is highlighted by the two panels inthe right hand side that show the characteristic roots for two differentparameter choices. On the other hand, the boundary given by ๐ = 0 (redcurve) denote fold bifurcation: a real eigenvalue crosses the imaginaryaxis and stability is lost in a non-oscillatory way. The two panels in theleft hand side illustrate this case by showing the characteristic rootsfor two different parameter choices. At the intersection of the stabilityboundaries (๐ถโ
R = ๐ถโF = 0) there exits a double zero characteristic root.
The special states discussed above (with both tires having peak force orboth tires are being at the sliding limit) are in fact correspond to thiscase.
While the traditional nonlinear model discussed in this section canprovide insight into the dynamics of automobiles, it only incorporatesthe tire nonlinearities and omits the geometric nonlinearities. As willbe shown below geometric nonlinearities can have significant effects onthe dynamics, especially when the vehicle operates close to its handlinglimits. Moreover, these effects differ for front wheel drive and rearwheel drive automobiles which demand the modeling efforts discussedin the next section.
3. Bicycle models for different drive types
In this section we derive models for rear wheel drive (RWD) andfront wheel drive (FWD) automobiles while taking into account thegeometric nonlinearities as well as the nonlinear tire characteristics.
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๐ฅ
Fig. 3. Stability chart for the traditional nonlinear model on the plane of the generalized cornering stiffnesses ๐ถโR and ๐ถโ
F . The blue and red stability boundaries correspond toHopf and fold bifurcations, respectively, while the stable region is shaded green. The characteristic roots are depicted in the panels on the left and the right for four differentparameter sets. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
๐
We distinguish the drive types by considering different kinematic con-straints and the equations of motion are derived by using the Appellianframework. We show that both models simplify to the traditionalnonlinear model, presented in the previous section, when assumingsmall steering and slip angles.3.1. Rear wheel drive model
Recall Fig. 1. Again we use the three generalized coordinates(๐ฅG, ๐ฆG, ๐) describing the position of the center of mass G and theorientation of the vehicle in the plane. We assume that the rear wheelis rolling with constant speed, that is, the longitudinal velocity of thecenter of the rear wheel is constant and it is denoted by ๐ฃโ. This isequal to the longitudinal velocity of the center of mass G, leading tothe kinematic constraint
G cos๐ + ๏ฟฝ๏ฟฝG sin๐ = ๐ฃโ, (19)
which is ensured by applying the appropriate driving force at the rear.Since we have 3 configuration coordinates and 1 kinematic constraintwe need 3 โ 1 = 2 pseudo velocities when applying the Appellianformalism (Gantmacher, 1975; De Sapio, 2017). These are chosen tobe the lateral velocity of center of mass G and the yaw rate:
๐ = โ๏ฟฝ๏ฟฝG sin๐ + ๏ฟฝ๏ฟฝG cos๐,๐ = ๏ฟฝ๏ฟฝ .
(20)
Eqs. (19), (20) can be solved for the generalized velocities:
๏ฟฝ๏ฟฝG = ๐ฃโ cos๐ โ ๐ sin๐,
๏ฟฝ๏ฟฝG = ๐ฃโ sin๐ + ๐ cos๐,
๏ฟฝ๏ฟฝ = ๐,
(21)
where the first two rows give the coordinate transformation for thevelocity of the center of mass G between the Earth fixed frame (๏ฟฝ๏ฟฝG, ๏ฟฝ๏ฟฝG)and the body fixed frame (๐ฃโ, ๐).
The acceleration energy of the system can be calculated as
๐ = 12๐(
๏ฟฝ๏ฟฝ2G + ๏ฟฝ๏ฟฝ2G)
+ 12๐ฝG๏ฟฝ๏ฟฝ
2
= 1๐(
๏ฟฝ๏ฟฝ2 + 2๐ฃโ๐๏ฟฝ๏ฟฝ)
+ 1๐ฝ ๏ฟฝ๏ฟฝ2 +โฏ(22)
5
2 2 G
where โฆ represent terms that do not contain the pseudo acceleration and we used the derivative of (21) when deriving the last row.
The virtual power generated by the lateral tire forces ๐ R and ๐ F canbe calculated as
๐ฟ๐ = ๐ R โ ๐ฟ๐ฏR + ๐ F โ ๐ฟ๐ฏF, (23)
where ๐ฟ๐ฏR and ๐ฟ๐ฏF are the virtual velocities of the wheel center pointsR and F, respectively. In order to calculate the dot products in (23) itis convenient to resolve the velocities and forces in a frame attached tothe vehicle body:
๐ฏR =[
๐ฃโ
๐ โ ๐ ๐
]
body, ๐ R =
[
0๐นR
]
body,
๐ฏF =[
๐ฃโ
๐ + ๐ ๐
]
body, ๐ F =
[
โ๐นF sin ๐พ๐นF cos ๐พ
]
body.
(24)
Substituting these into (23) and using ๐ฟ๐ฃโ = 0 we obtain
๐ฟ๐ = (๐นR + ๐นF cos ๐พ)โโโโโโโโโโโโโโโโโ
๐ฑ๐
๐ฟ๐ + (โ๐ ๐นR + ๐ ๐นF cos ๐พ)โโโโโโโโโโโโโโโโโโโโโโโโโโโ
๐ฑ๐
๐ฟ๐, (25)
which results in the pseudo forces
๐ฑ๐ = ๐นR + ๐นF cos ๐พ,
๐ฑ๐ = โ๐ ๐นR + ๐ ๐นF cos ๐พ.(26)
Notice that ๐ฑ๐ is the lateral component of the resultant tire force while๐ฑ๐ is the resultant moment of these forces about the center of mass G.
The Appell equations are given as๐๐๐๏ฟฝ๏ฟฝ
= ๐ฑ๐ ,
๐๐๐๏ฟฝ๏ฟฝ
= ๐ฑ๐,(27)
which lead to
๏ฟฝ๏ฟฝ =๐นR(๐ผR) + ๐นF(๐ผF) cos ๐พ
๐โ ๐ฃโ๐,
๏ฟฝ๏ฟฝ =โ๐ ๐นR(๐ผR) + ๐ ๐นF(๐ผF) cos ๐พ ,
(28)
๐ฝG
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European Journal of Mechanics / A Solids 90 (2021) 104372S. Oh et al.
w
t
tooatm
aCo(
3
vsac
๐ฅ
ucpg
tf
w
ac
where we spelled out the that tire forces depend on the slip angles.Using the component of the velocities in (24) we obtain
tan ๐ผR = โ๐ โ ๐ ๐๐ฃโ
,
tan(๐ผF โ ๐พ) = โ๐ + ๐ ๐๐ฃโ
,(29)
here the second equation can also be rewritten as
tan ๐ผF =๐ฃโ tan ๐พ โ (๐ + ๐ ๐)๐ฃโ + (๐ + ๐ ๐) tan ๐พ
, (30)
hat is a more practical form when substituting into (41).The equations of motion of the rear wheel drive automobile are
hen described by (21), (28), (29). These provide us with 5 firstrder differential equations which are often referred to as 2.5 degreesf freedom (DOF). Also, notice that when considering small steeringngles and small slip angles, (28) simplifies to (3) and (29) simplifieso (2). That is, the rear wheel drive model simplifies to the traditionalodel under the small angle assumption.
We also remark that the kinematic equations for the RWD modelre the same as those for the traditional model (cf. (21) and (1)).onsequently, in case of steady state cornering the radii of the centerf mass ๐ G and the rear wheel center point ๐ R can be calculated by7) and (8), respectively.
.2. Front wheel drive model
A similar derivation can be performed for the front wheel driveehicle where we assume that the front wheel is rolling with a constantpeed. That is, the velocity component of the center of the front wheelligned with the wheel plane is constant and it is denoted by ๏ฟฝ๏ฟฝโ. Theorresponding kinematic constraint can be written as
G cos(๐ + ๐พ) + ๏ฟฝ๏ฟฝG sin(๐ + ๐พ) + ๐ ๏ฟฝ๏ฟฝ sin ๐พ = ๏ฟฝ๏ฟฝโ, (31)
sing the velocity components of the center of mass G. Again wehoose the lateral velocity of center of mass G and the yaw rate asseudo velocities; cf. (20). The system (20), (31) can be solved for theeneralized velocities:
๏ฟฝ๏ฟฝG =( ๏ฟฝ๏ฟฝโ
cos ๐พโ (๐ + ๐ ๐) tan ๐พ
)
cos๐ โ ๐ sin๐,
๏ฟฝ๏ฟฝG =( ๏ฟฝ๏ฟฝโ
sin ๐พโ (๐ + ๐ ๐) tan ๐พ
)
sin๐ + ๐ cos๐,
๏ฟฝ๏ฟฝ = ๐,
(32)
when ๐พ โ ๐2 , where the first two rows give the coordinate transforma-
ion for the velocity of the center of mass G between the Earth fixedrame (๏ฟฝ๏ฟฝG, ๏ฟฝ๏ฟฝG) and the body fixed frame (๏ฟฝ๏ฟฝโโ cos ๐พ โ (๐ + ๐ ๐) tan ๐พ, ๐);
cf. (21).Using the time derivative of (32) we can calculate the acceleration
energy:
๐ =12๐(
๏ฟฝ๏ฟฝ2G + ๏ฟฝ๏ฟฝ2G)
+ 12๐ฝG๏ฟฝ๏ฟฝ
2
=12
๐cos2 ๐พ
๏ฟฝ๏ฟฝ2 + 12(
๐ฝG + ๐๐2 tan2 ๐พ)
๏ฟฝ๏ฟฝ2 + ๐๐ tan2 ๐พ ๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝ
+ ๐(
๏ฟฝ๏ฟฝโ
cos ๐พโ ๐ ๐ tan ๐พ
)
๐ ๏ฟฝ๏ฟฝ
+ ๐tan ๐พcos2 ๐พ
(
๐ + ๐ ๐ โ ๏ฟฝ๏ฟฝโ sin ๐พ)
๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝ
+ ๐๐tan ๐พcos2 ๐พ
(
๐ + ๐ ๐ โ ๏ฟฝ๏ฟฝโ sin ๐พ)
๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝ
+ ๐๐ ๐ ๐ tan ๐พ ๏ฟฝ๏ฟฝ +โฏ ,
(33)
here again โฆ refer to the terms without the pseudo acceleration ๏ฟฝ๏ฟฝ.
6
The velocities of the wheel center points can be expressed as
๐ฏR =โก
โข
โข
โฃ
๏ฟฝ๏ฟฝโ
cos ๐พโ (๐ + ๐ ๐) tan ๐พ
๐ โ ๐ ๐
โค
โฅ
โฅ
โฆbody
,
๐ฏF =โก
โข
โข
โฃ
๏ฟฝ๏ฟฝโ
cos ๐พโ (๐ + ๐ ๐) tan ๐พ
๐ + ๐ ๐
โค
โฅ
โฅ
โฆbody
,
(34)
in the body fixed frame while the lateral forces are still given by (24).These are used when calculating the virtual power (23) and yield thepseudo forces
๐ฑ๐ = ๐นR +๐นFcos ๐พ
,
๐ฑ๐ = โ๐ ๐นR +๐ ๐นFcos ๐พ
,(35)
cf. (25), (26).Then the Appell Eqs. (27) become
[ ๐cos2 ๐พ
๐ ๐ tan2 ๐พ
๐ ๐ tan2 ๐พ ๐ฝG + ๐๐2 tan2 ๐พ
]
[
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
]
+
โก
โข
โข
โข
โฃ
๐tan ๐พcos2 ๐พ
(
๐+๐ ๐โ๏ฟฝ๏ฟฝโ sin ๐พ)
๏ฟฝ๏ฟฝ + ๐(
๏ฟฝ๏ฟฝโ
cos ๐พโ๐ ๐ tan ๐พ
)
๐
๐ ๐tan ๐พcos2 ๐พ
(
๐+๐ ๐โ๏ฟฝ๏ฟฝโ sin ๐พ)
๏ฟฝ๏ฟฝ + ๐๐ ๐ ๐ tan ๐พ
โค
โฅ
โฅ
โฅ
โฆ
=
โก
โข
โข
โข
โฃ
๐นR(๐ผR) +๐นF(๐ผF)cos ๐พ
โ๐ ๐นR(๐ผR) +๐ ๐นF(๐ผF)cos ๐พ
โค
โฅ
โฅ
โฅ
โฆ
(36)
where the dependency of the tire forces on the slip angles ๐ผR and ๐ผFre spelled out again, cf. (28). These can be determined by using theomponents of the velocities in (34):
tan ๐ผR = โ(๐ โ ๐ ๐) cos ๐พ
๏ฟฝ๏ฟฝโ โ (๐ + ๐ ๐) sin ๐พ,
tan(๐ผF โ ๐พ) = โ(๐ + ๐ ๐) cos ๐พ
๏ฟฝ๏ฟฝโ โ (๐ + ๐ ๐) sin ๐พ,
(37)
where the second formula can be simplified to the more practicalform
tan ๐ผF = tan ๐พ โ ๐ + ๐ ๐๏ฟฝ๏ฟฝโ cos ๐พ
, (38)
that can be used in (41).The equations of motion of the front wheel drive automobile are
given by the 5 first order Eqs. (32), (36), (37). Again, considering smallsteering angles and small slip angles, which also yields ๏ฟฝ๏ฟฝโ โ ๐ฃโ, (36)simplifies to (3) and (37) simplifies to (2). That is, the front wheel drivemodel also simplifies to the traditional model under the small angleassumption. Note however that the kinematics of the FWD model isgiven by (32) (and not by (21)) which result in the radii
๐ G =
โ
( ๏ฟฝ๏ฟฝโ
cos ๐พโโ (๐โ + ๐ ๐โ) tan ๐พโ
)2+ (๐โ)2
๐โ , (39)
and
๐ R =
โ
( ๏ฟฝ๏ฟฝโ
cos ๐พโโ (๐โ + ๐ ๐โ) tan ๐พโ
)2+ (๐โ โ ๐ ๐โ)2
๐โ , (40)
in case of steady state cornering (cf. (7) and (8)).
4. Nonlinear analysis
In this section we analyze the nonlinear dynamics of the modelsdeveloped above by utilizing numerical continuation. Namely, we studythe traditional nonlinear model (2), (3) described in Section 2, the rearwheel drive model (28), (29), (30) and the front wheel drive model
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Fig. 4. Bifurcation diagrams (a,b,d,e,g,h) and phase portraits (c,f,i) for three models with steering angle ๐พโ = 2 [deg]. Stable and unstable steady states are denoted by green andred colors, respectively. Gray shading indicates the domains where the slip angles are smaller than ๐ผpk . In the bifurcation diagrams the blue stars indicate Hopf bifurcations,magenta crosses denote fold bifurcations, and black diamonds mark singularities. (For interpretation of the references to color in this figure legend, the reader is referred to theweb version of this article.)
(36), (37), (38) developed in Section 3. Note that for the analysis belowthe kinematic Eqs. (1), (21), and (32) can be dropped.
We utilize the numerical continuation software DDE-Biftool (Sieberet al., 2014) in order to find steady states while fixing the steering angle๐พโ and changing the velocity ๐ฃโ (for the traditional and RWD models)or the velocity ๏ฟฝ๏ฟฝโ (for the FWD model). The steady state equationswere shown in (5) for the traditional model but we do not spell outthe formulae for the RWD and FWD models for simplicity. Apart fromfinding the steady state lateral velocity ๐โ, steady state yaw rate ๐โ, andsteady state slip angles ๐ผโR, ๐ผ
โF, DDE-biftool also evaluates stability by
linearizing the equations around the steady state. These were calculatedanalytically in (16) for the traditional model but are not spelled out forthe RWD and FWD models due to algebraic complexity.
Fig. 4 shows the numerical continuation results when setting thesteering angle to ๐พโ = 2 [deg]. The top, middle, and bottom rows corre-spond to the traditional, RWD, and FWD models, respectively. Panels(a) and (b) display the bifurcation diagrams for the traditional model.In particular, the steady state slip angles ๐ผโR and ๐ผโF are plotted asfunction of the longitudinal velocity ๐ฃโ. Stable and unstable states areshown by green and red curves, respectively. Gray shading highlightsthe domain where the slip angles are below the limit ๐ผpk , i.e., |๐ผR| โค ๐ผpkand |๐ผF| โค ๐ผpk ; cf. (2) and Fig. 1(b).
The stable steady state stays within this region and slip anglesincrease with the speed. This state correspond to negotiating a curvewithout saturating the tires and we refer to this as regular turning in therest of the paper. There are also two unstable states which move closerto the stable state as the speed increases. The upper unstable steadystate corresponds to the motion where the rear tire is saturated and thevehicle is taking a sharper turn compared to the regular turning motion,so we refer to this as sharp turning in the rest of the paper. On the other
7
hand, the lower unstable steady state corresponds to drifting, i.e., thevehicle is turning (with saturated rear tire) to the opposite directionthan the steering wheelโs direction would suggest.
The phase portrait in Fig. 4(c) is drawn for ๐ฃโ = 20 [mโs]; cf. solidvertical line in panels (a) and (b). Stable and unstable steady states areindicated by green and red points, respectively, while trajectories fordifferent initial conditions are shown by blue curves. The gray domainindicates where |๐ผR| โค ๐ผpk and |๐ผF| โค ๐ผpk and it is obtained via (2). Thisprovides an estimate for the region of attraction for stable steady state,which is given accurately by the stable manifolds of the unstable steadystates. When choosing initial condition outside of these manifolds thevehicle spins away from the stable motion. This is often utilized by lawenforcement officers to stop a fleeing vehicle: the pursuing vehicle shallhit the rear of the fleeing vehicle hard enough to make it spin.
Fig. 4(d,e) show the bifurcation diagrams for the RWD model. Thecorresponding phase portrait for ๐ฃโ = 20 [mโs] is shown in Fig. 4(f)where the gray regions are obtained from |๐ผR| โค ๐ผpk and |๐ผF| โค ๐ผpkusing (29), (30). When comparing the results with the traditionalmodel, a high level of similarity can be noticed. That is, for smallsteering angles the traditional and the RWD models are essentiallyequivalent in the speed regime investigated.
For the FWD model the bifurcation diagrams and the phase portraitare displayed in Fig. 4(g,h) and (i), respectively, and (37), (38) is usedto obtain the gray shaded domain in panel (i) from |๐ผR| โค ๐ผpk and|๐ผF| โค ๐ผpk . While the stable branch behaves similar as in the traditionaland RWD cases, the unstable branches show qualitative differences. Inparticular, the upper branch, corresponding to sharp turning, displaysa singularity at small speed (indicated by black diamond) while thedrifting branch at the bottom folds back (marked by magenta cross)and also gains stability via a Hopf bifurcation (denoted by blue star)
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Fig. 5. Bifurcation diagrams (a,b,d,e,g,h) and phase portraits (c,f,i) for three models with steering angle ๐พโ = 8 [deg]. Stable and unstable steady states are denoted by green andred colors, respectively. Gray shading indicates the domains where the slip angles are smaller than ๐ผpk . In the bifurcation diagrams the blue stars indicate Hopf bifurcations,magenta crosses denote fold bifurcations, and black diamonds mark singularities. (For interpretation of the references to color in this figure legend, the reader is referred to theweb version of this article.)
for larger speed. Due to the fold, there exist two drifting solutions, oneof these is outside of the window in Fig. 4(i). The Hopf bifurcation issubcritical (Guckenheimer and Holmes, 1983; Kuznetsov, 2004) andresults in an unstable periodic orbit around the stable steady state.The amplitude of the periodic orbit is very small and this orbit givesthe region of attraction of the stable steady state. These qualitativedifferences, however, only influence the behavior for large slip anglesor for small speed. The rest of the diagram still looks akin to that ofthe traditional model.
We remark that from Fig. 4 one may also infer how the bifurcationdiagrams and phase portraits look like in the limit ๐พโ = 0. In this case,the stable steady state remains at zero as the speed increases. Thiscorresponds to the rectilinear motion. The unstable steady states appearsymmetrically around the stable one and these correspond to specialmotions where the vehicle turns despite zero steering angle.
Fig. 5 depicts the bifurcation diagrams and phase portraits forsteering angle ๐พโ = 8 [deg]. For the traditional model the bifurcationdiagrams (panels (a,b)) show qualitative differences compared to the๐พโ = 2 [deg] case in Fig. 4(a,b). As the speed is increased the stablebranch reaches the peak slip angle where it meets with the upperunstable branch of sharp turning via a degenerate fold bifurcation(indicated by magenta cross). The corresponding speed ๐ฃโcr,1, given by(13), is highlighted by the vertical dashed line. Such critical point onlyexist due to the โโsymmetryโโ of the traditional model which causesboth wheels to reach the handling limit at the same speed. However,reminiscences of this bifurcation will show up for the RWD and FWDmodels as explained below.
As the speed is increased further a stable branch arises for whichthe rear slip angle stays smaller than the peak value ๐ผpk while thefront slip angle becomes larger and the front tire saturates. An unstable
8
branch with both slip angles being larger than the peak value alsoarises. This ends in a singularity (marked by black diamond) when thespeed reaches the critical value ๐ฃโcr,2, given by (15), that is marked bythe vertical dashedโdotted line. For higher speeds only two solutionsexists: the stable steady state with saturated front tire and the unstabledrifting state with saturated rear tire. These are shown in Fig. 5(c)that highlights that the stable state has a small region of attractionwhich is not anymore approximated by the gray shaded domain givenby |๐ผR| โค ๐ผpk and |๐ผF| โค ๐ผpk ; cf. (2).
The bifurcation diagrams of the RWD model, displayed inFig. 5(d,e), show some similarities to the those of the traditional model.The biggest change is that the degenerate fold bifurcation is broken,resulting in two separate branches. The stable branch behaves similarto the traditional model while the unstable branch folds back (asmarked by the magenta cross). The phase portrait in Fig. 5(f) revealsthat there also exist multiple unstable drifting states, one of which isoutside of the window in the bifurcation diagram as the slip angles are(unrealistically) large. Again, the shaded region given by |๐ผR| โค ๐ผpk and|๐ผF| โค ๐ผpk (cf. (29), (30)) is not related to the region of attraction of thestable steady state.
For the FWD model, as depicted in Fig. 5(g,h), the degenerate foldbifurcation is broken in another way, resulting in two folding branches(see magenta crosses). For lower speed we obtain steady states of stableregular turning and unstable sharp turning (with saturated rear tire).For higher speed we have sharp turning motions with saturated tiresthat are mostly unstable (see top branches). Even when stability isgained via subcritical Hopf bifurcation (see blue stars) the resultingstable motion has very small region of attraction (given by a smallperiodic orbit). Similar behavior can be observed for the branchescorresponding to drifting (see bottom branches). The phase portrait in
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Fig. 6. Experimental results for steady state cornering when the steering wheel angle is set to 180 degrees and the speed is set to 10.0 m/s (black), 12.5 m/s (blue), 15.0 m/s(orange), 17.5 m/s (gray). (a) Trajectories of the rear axle center point R. (b,c,d) Speed of the point R, steering wheel angle, turning radius of the point R as function of time.(e) Turning radius as a function of speed. The colored curves correspond to those on the left side of Fig. 5(g,h). (f) Trajectories of R for regular turning and sharp turning for setspeed 12.5 m/s obtained by simulations. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 5(i) suggests that no stable motion with practically-sized regionof attraction exists for large speed, as opposed to what suggested bythe gray shaded domain given by |๐ผR| โค ๐ผpk and |๐ผF| โค ๐ผpk ; cf. (37),(38). The most notable phenomenon observed for the FWD model isthat there exist a range of speed where neither regular nor fast turningmotion happens. This behavior will be confirmed experimentally inSection 5 using an automated vehicle.
To summarize, the bifurcation analysis performed above revealsthe similarities and differences between different models. For smallsteering angles the traditional, RWD, and FWD models show similarbehavior (except for small speed) but such statement does not holdfor larger steering angles. In the latter case, the traditional model canstill provide some guidance, in particular regarding the critical speed,but differentiating the drive type seems critical in understanding thedynamics at large steering angles and speeds. In the next section weshow experimental results obtained for a front wheel drive vehicle andcompare those to the theoretical results discussed above.
9
Table 2Experimental results for steady state cornering with steering wheel angle set to 180degrees.
Set speed Measured speed |๐ฃR| SWA Turning radius ๐ R
10.0 [m/s] 9.75 ยฑ 0.17 [m/s] 180.02 ยฑ 0.34 [deg] 16.72 ยฑ 0.41 [m]12.5 [m/s] 11.92 ยฑ 0.19 [m/s] 180.11 ยฑ 0.95 [deg] 19.53 ยฑ 0.45 [m]15.0 [m/s] 14.02 ยฑ 0.26 [m/s] 178.27 ยฑ 2.37 [deg] 24.91 ยฑ 0.80 [m]17.5 [m/s] 15.56 ยฑ 0.73 [m/s] 176.13 ยฑ 4.14 [deg] 30.99 ยฑ 2.65 [m]
5. Experimental validation
In this section, we present the results of the steady state corneringexperiments and compare to those obtained by bifurcation analysis inthe previous section. In particular, we demonstrate that the experimen-tal results match with the theoretical results for a front wheel driveautomobile when using the FWD model developed in Section 3.2 thatincludes the appropriate geometric nonlinearities. We emphasize that
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European Journal of Mechanics / A Solids 90 (2021) 104372S. Oh et al.
S
w
a
๐ผ
the RWD model developed in Section 3.1 and the traditional modelanalyzed in Section 2 would give qualitatively different behavior.
A front wheel drive automated KIA Soul with specifications givenin Table 1 were used for the experiment. We tune the parameters๐ = 1600 [kg], ๐ฝG = 2000 [kgm2], ๐ = 2 ร 106 [Nโm2], ๐ = ๐0 = 1.2 in ac-cordance with our experimental setup, but these do not lead to qual-itative changes in the bifurcation diagrams. The driving tests wereperformed on dry asphalt at the University of Michigan test track calledMcity. The steering angle was held constant by the human operatorwhile the speed was held constant by the controller. GPS data wascollected together with vehicle data including the steering angle andthe wheel based velocity. The GPS antenna was mounted above thecenter of the rear axle (see point R in Fig. 1) and below we display theposition and speed of this point.
The experimental results are shown in Fig. 6 where panel (a) depictsthe traces of the rear center point R for four different set speed values:10.0 [m/s] (black), 12.5 [m/s] (blue), 15.0 [m/s] (orange), 17.5 [m/s](gray). The steering wheel angle is set to 180 degrees which corre-sponds to ๐พโ โ 11 degrees at the wheel (given the steering ratio 15.7).Corresponding to the constant steering angle the vehicle is moving on acircular path and the size of this circle increases with the speed. Steadystate cornering cannot be maintained for a full circle for the largest setspeed as shown by the gray trajectory. We remark that in reality thepaths observed during the experiments for different set speed valueswere not concentric but we shifted them to increase the readability ofthe figure.
Fig. 6(b) displays the corresponding speed as a function of time; seeTable 2 for the corresponding mean and standard deviation values. Onemay notice that the higher the set speed is the larger the speed errorsare. This is due to the fact that the longitudinal control was achievedby a proportional controller that did not compensate for the resistancearising from steering, i.e., the force component ๐นF sin ๐พโ โ 0.2๐นF; seeFig. 1(a). The steering wheel angle SWA and the turning radius ๐ Rare shown in Fig. 6(c) and (d), respectively, as function of time, whilethe corresponding mean and standard deviation values are given inTable 2. Again, observe that the errors of the steering wheel angle andthe turning radius increase with the set speed and for set speed 17.5[m/s] the steady state cannot be sustained. We remark that the turningradius we obtained by locally fitting arcs to the trajectories using awindow size of 19 points.
Fig. 6(e) summarizes the experimental results by displaying theturning radius as function of the measured speed. The dots mark themean values while the error bars indicate the standard deviations foreach experiment. These results are compared with those obtained viabifurcation analysis; cf. Fig. 5(g,h). In particular, the radius ๐ R iscalculated using (40) with ๐พโ = 11 [deg], while we also use |๐ฃR| =๐โ๐ R; cf. (34), (40). There is good agreement between the theoreticallyobtained stable branch (green) and the experimental measurements(black, blue, orange, gray squares). In particular, we emphasize thequalitative agreement that no steady state exists above a critical speed;cf. the magenta cross denoting the fold bifurcation and the data pointfor the highest speed (where no steady state was found). We remarkthat such qualitative agreement holds for an extended set of vehicleparameters, that is, this behavior is a robust feature of the FWD modeldeveloped in this paper.
While the measurement data fit the stable branch in Fig. 6(e), theunstable branch also reveals some interesting dynamics. In particular,observe that the unstable steady state indeed corresponds to sharpturning (smaller radius than the stable steady state cornering) withsaturated rear tire; cf. Fig. 5(g,h). The corresponding trajectories ofthe rear axle center point R, obtained via numerical simulations, aredepicted in Fig. 6(f) for set speed 12.5 [m/s]. When linearizing the FWDmodel (36), (37), (38) around these steady states one obtains a modelsimilar to (16). This way one can show that the sharp turning motionis controllable (at least at the linear level) by modulating the steer-ing angle. This suggests that, with the appropriate controllers, FWDautomated automobiles may execute sharp turns, and thus, achieve
10
enhanced maneuverability.
6. Conclusion
High fidelity single track models were developed to describe thelateral and yaw motion of automated vehicles. The models differentiatebetween front wheel drive and rear wheel drive automobiles by in-corporating the geometric nonlinearities beside the tire nonlinearities.The steady state cornering behavior was analyzed with the help ofnumerical bifurcation analysis that allowed us to compare the newlydeveloped models to a traditional nonlinear model which omits geo-metric nonlinearities. It was demonstrated that for small steering anglesthe three models were essentially equivalent and they all exhibitedthree qualitatively different motions: regular turning, sharp turning,and drifting. In the latter two cases the rear tires were saturated.For larger steering angles similar motions appeared but the modelsexhibited significant differences about how the motions changed as thespeed of the vehicle was increased. The front wheel drive model wasvalidated experimentally by using an automated vehicle on a test track.Experiments confirmed the existence of a region where neither regularnor sharp turning motions existed. The potential of exploiting sharpturning and drifting motions to enhance maneuverability of automatedvehicles has been highlighted but the corresponding control design wasleft for future research. Including the effects of combined slip in themodels is another important future direction.
CRediT authorship contribution statement
Sanghoon Oh: Modeling, Simulation, Analysis, Paper writing.ergei S. Avedisov: Modeling, Experiment, Analysis, Paper writing.Gรกbor Orosz: Modeling, Experiment, Analysis, Paper writing.
Declaration of competing interest
The authors declare that they have no known competing financialinterests or personal relationships that could have appeared toinfluence the work reported in this paper.
Acknowledgments
This research was partially supported by the Mobility Transforma-tion Center at the University of Michigan, USA. The authors thankChaozhe He for his help in collecting the experimental data and ac-knowledge the insightful discussions with Dรฉnes Takรกcs on nonholo-nomic vehicle models.
Appendix
Here we provide the algebraic formulae for the nonlinear tire char-acteristics shown in Fig. 1(b). For the brush model (blue curve), assum-ing parabolic pressure distribution along the tire-ground contact regionone may obtain the form
๐น (๐ผ) =
{
๐1 tan ๐ผ + ๐2 tan2 ๐ผ sgn ๐ผ + ๐3 tan3 ๐ผ, if 0 โค |๐ผ| < ๐ผsl,๐๐น๐ง sgn๐ผ, if ๐ผsl < |๐ผ|,
(41)
ith coefficients๐1 = 2๐๐2,
๐2 = โ(2๐๐2)2
3๐0๐น๐ง
(
2 โ๐๐0
)
,
๐3 =(2๐๐2)3
(3๐0๐น๐ง)2(
1 โ2๐3๐0
)
,
(42)
nd sliding limit
= arctan( 3๐0๐น๐ง ). (43)
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European Journal of Mechanics / A Solids 90 (2021) 104372S. Oh et al.
a
(
Here ๐ is the contact patch half-length, ๐ is the distributed lateralstiffness of the tires, ๐น๐ง is the vertical load on the axles, ๐ and ๐0 arethe friction coefficients between the tires and the road for sliding andsticking, respectively. Notice that ๐ and ๐ only show up as 2๐๐2 in theabove formulae. A simplified version of this model with ๐ = ๐0 can befound in Fiala (1954)
Using (41), (42) one may derive the location and the value of thepeak as
๐ผpk = arctan
(
๐0๐น๐ง2๐๐2
(
1 โ2๐3๐0
)โ1)
,
๐น (๐ผpk ) = ๐0๐น๐ง( 43โ๐๐0
)(
1 โ2๐3๐0
)โ2,
(44)
nd the cornering stiffnessd๐นd๐ผ
(0) = 2๐๐2. (45)
The green curve in Fig. 1(b) is given by the magic formula
๐น (๐ผ) = ๐ท sin(
๏ฟฝ๏ฟฝ arctan(
๐ต(1โ๐ธ) tan ๐ผ + ๐ธ arctan(
๐ต tan ๐ผ)
)
)
, (46)
where the four independent parameters ๐ด,๐ต, ๏ฟฝ๏ฟฝ, ๐ท are typically identi-fied experimentally. One may calculate the lateral force correspondingto complete sliding
๐น (๐โ2) = ๐ท sin(๏ฟฝ๏ฟฝ๐โ2), (47)
the location and the value of the peak
๐ผpk = arctan( tan
(
๐ (๏ฟฝ๏ฟฝ, ๐ธ))
๐ต
)
,
๐น (๐ผpk ) = ๐ท,(48)
where ๐ (๏ฟฝ๏ฟฝ, ๐ธ) is the solution of the equation
1 โ ๐ธ) tan ๐ง + ๐ธ๐ง + tan ๐2๏ฟฝ๏ฟฝ
= 0, (49)
for ๐ง, and the cornering stiffness is given byd๐นd๐ผ
(0) = ๐ต๏ฟฝ๏ฟฝ๐ท. (50)
Then comparing (41), (44), (45) with (47), (48), (50) the parametersof the brush model and the magic formula can be matched.
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