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Effect of vertical and lateral couplingbetween tyre and road on vehiclerolloverYinong Li a , Wei Sun a , Jingying Huang a , Ling Zheng a & YanyangWang aa The State Key Laboratory of Mechanical Transmission , ChongqingUniversity , No. 174 Shazhengjie, Shapingba, Chongqing , 400044 ,People's Republic of ChinaPublished online: 07 May 2013.

To cite this article: Yinong Li , Wei Sun , Jingying Huang , Ling Zheng & Yanyang Wang (2013)Effect of vertical and lateral coupling between tyre and road on vehicle rollover, Vehicle SystemDynamics: International Journal of Vehicle Mechanics and Mobility, 51:8, 1216-1241, DOI:10.1080/00423114.2013.791395

To link to this article: http://dx.doi.org/10.1080/00423114.2013.791395

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Vehicle System Dynamics, 2013Vol. 51, No. 8, 1216–1241, http://dx.doi.org/10.1080/00423114.2013.791395

Effect of vertical and lateral coupling between tyre and roadon vehicle rollover

Yinong Li*, Wei Sun, Jingying Huang, Ling Zheng and Yanyang Wang

The State Key Laboratory of Mechanical Transmission, Chongqing University No. 174 Shazhengjie,Shapingba, Chongqing 400044, People’s Republic of China

(Received 20 February 2013; final version received 25 March 2013 )

The vehicle stability involves many aspects, such as the anti-rollover stability in extreme steeringoperations and the vehicle lateral stability in normal steering operations. The relationships betweenvehicle stabilities in extreme and normal circumstances obtain less attention according to the presentresearch works. In this paper, the coupling interactions between vehicle anti-rollover and lateral stabil-ity, as well as the effect of road excitation, are taken into account on the vehicle rollover analysis. Theresults in this paper indicate that some parameters influence the different vehicle stabilities diverselyor even contradictorily. And it has been found that there are contradictions between the vehicle rollovermitigation performance and the lateral stability. The direct cause for the contradiction is the lateralcoupling between tyres and road. Tyres with high adhesion capacity imply that the vehicle possesses ahigh performance ability to keep driving direction, whereas the rollover risk of this vehicle increasesdue to the greater lateral force that tyres can provide. Furthermore, these contradictions are inten-sified indirectly by the vertical coupling between tyres and road. The excitation from road not onlydeteriorates the tyres’ adhesive condition, but also has a considerable effect on the rollover in somecases.

Keywords: untripped vehicle rollover; anti-rollover stability; lateral stability; road excitation;coupling effect

1. Introduction

Vehicle rollovers are dangerous incidents and have a higher fatality rate than other kindsof crashes. Of the nearly 9.1 million passenger car, SUV, pickup and van crashes in 2010,only 2.1% involved a rollover. However, rollovers accounted for nearly 35% of all deathsfrom passenger vehicle crashes. In 2010, more than 7600 people died in vehicle rollovercrashes [1].

According to the NHTSA-2003-14622 document [2], the vehicle rollover could be classifiedas off-road rollover, on-road tripped rollover and on-road untripped rollover.

The tripped rollovers (on- or off-road) are going to occur when a vehicle leaves the roadwayand slides sideways, digging its tyres into soft soil or striking an object such as a curb orguardrail. The high tripping force applied to the tyres in these situations can cause the vehicleto roll over. To prevent this kind of rollover, rollover warning systems or collision avoidancealarm systems are utilised.

*Corresponding author. Email: ynli@cqu.edu.cn

© 2013 Taylor & Francis

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Vehicle System Dynamics 1217

Table 1. Examples of vehicle dynamic excitations.

Excitation type Steering wheel input Road surface input

shock Fishhook manoeuvres Road bumpsSteady-state vibration Slalom test Washboarding roadStochastic vibration – Stochastic road surface

The vehicle on-road untripped rollover is mainly caused by the lateral acceleration duringcornering operations, e.g. high-speed collision avoidance manoeuvres. And the excessive rollangle of vehicle body induced by lateral acceleration is the direct reason for rollover.

Rollovers, more so than other types of crashes, are complex and particularly violent innature, reflect the interaction of the driver, road, vehicle, and environmental factors. So whilevehicle type does play a significant role in rollover incidents, other factors such as driverbehaviour and road as well as environmental conditions can also cause vehicle rollover.

Based on the view of vibration and shock, the excitation to vehicle dynamics could beclassified as shock, steady-state vibration and stochastic vibration. Also, the excitation couldcome from steering wheel or road according to different source. Table 1 lists the examples foreach kind of excitation. Generally, uneven road is used for testing the ride comfort of vehicles,and the steering manoeuvres are designed mainly for testing the stability of vehicles. The con-ventional analyses usually treat these aspects separately, which are insufficient considerationsin some extreme situation.

In order to test the vehicle stability repetitively, many testing operations and standards arespecified to regulate the driver behaviours, e.g. Passenger car/trailer combinations – lateralstability test of ISO 9815, ‘Elk’ or ‘Moose’ test of ISO 3888-2:2011. Among these standardsand tests, fishhook specified by NHTSA-2003-14622 is most commonly used for vehiclerollover tests.

To eliminate the possibility of on-road untripped rollovers, the common solution is toincrease vehicle stability by tuning vehicle parameters (passive method) [3,4] or equippingwith active systems such as ESP or Active Suspensions (active method) [5–7]. These existingresearches mainly focus on the vehicle’s anti-rollover stability, which leave other factors, suchas the general lateral stability and road conditions, out of consideration.

Besides the anti-rollover performance, the vehicle stability involves many other aspects,e.g. the lateral stability such as oversteering tendency and sideslip, yet the current researchworks [8–11] generally treated these features independently.

Sadri and Wu [12,13] utilised the Lyapunov function to analyse the lateral stability andinvestigated the rollover stability region within the phase diagram of lateral velocity and yawrate. Nam et al. [14] constructed an observer to estimate the roll angle through lateral tyre forceand thus proposed a direct roll moment controller for four- wheel steering vehicle. Similarly,Du et al. [15] developed a yaw moment controller, whereas the controller parameters aredependent on the vehicle forward velocity and the controller is designed by solving finitenumbers of linear matrix inequalities (LMIs). Both Nam and Du’s methods could achieveadequate responses of sideslip angle and yaw rate. Lu et al. [16] discussed control authorityand the effective working region of different chassis key subsystems for rollover prevention,and further utilised MR semi-active suspension as key actuator to implement integrated controlwith braking and steering [17,18]. These existing researches can improve the steering or anti-rollover stability, however the couplings between lateral stability and roll dynamics are seldomconsidered integrally. The role and effect of lateral stability such as sideslip in rollover analysisneed to be considered further.

Moreover, though road condition is not mandatory in the definition of on-road untrippedrollover, the excitation from uneven road is inevitable, especially that the uneven road profile

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1218 Y. Li et al.

is an important reason for the vehicle body roll motion. The contributions of road excitationto rollover have not been studied thoroughly yet.

In light of the above opinions, the lateral coupling of tyre and road, namely the tyre lateraladhesive capacity is discussed comprehensively in this paper for the conjoint analysis ofvehicle anti-rollover and other stabilities. Meanwhile, the vertical coupling of tyre and road,i.e. the attachment of tyres to the road is integrated into the analysis of vehicle rollovers bythe combination of the fishhook manoeuvres and uneven road excitation.

This paper is organised as follows: First of all, the vehicle models as well as nonlinear tyremodel are derived, followed by a preliminary analysis on the model characteristics. Subse-quently, based on the fishhook manoeuvre method and the comparison of different rollovercritical factors (RCFs), the relationships between vehicle lateral and anti-rollover stability areinvestigated. Then the effect of uneven road on vehicle rollover is investigated. And finally,the conclusions drawn from this research are presented.

2. Vehicle models

The vehicle model consists of three sub-models: the vibration and roll model that emphasiseson sprung mass roll motion, which is the direct cause of the vehicle rollover; the vehiclesteering model that is used for the calculation of the lateral acceleration, which is the mainreason for the vehicle body roll motion; the tyre model that provides the lateral force forsteering model to calculate the lateral acceleration.

2.1. Tyre model

The primary purpose of the tyre model is to work out the lateral force that tyre can provide.When the vehicle is cornering, the roll motion of vehicle body will arouse lateral load transferon tyres, and this transfer leads to the variation of tyre force. To take account of this effect,tyre load is considered with the slip angle. The Magic-Formula (MF) tyre models [19] areadopted here because they are easy to implement, and have been widely proved accurate inmany vehicle dynamics simulation. Though they may not fit for low-speed conditions, in thisrollover study, the vehicle speed is usually high enough. For pure slip conditions, the nonlinearMF equations can be described as

Fy = D sin{C arctan{Bα − E[Bα − arctan(Bα)]}},

B = b3 sin(2 arctan(Fz/b4))

CD, C = b0,

D = b1F2z + b2Fz

1000, E = b5Fz + b6,

(1)

where the unit of lateral force Fy and tyre load Fz is kilo Newton, the unit of slip angle α isradian. B, C, D and E are the empirical parameters, respectively.

Two types of tyres are chosen to discuss the relationships between the vehicle anti-rolloverand lateral stability performance. Tyre A stands for high adhesive capacity tyres, and tyre Bstands for low adhesive capacity tyres, as shown in Table 2. Subjected to the same tyre load,the lateral force responses of the two types of tyres are shown in Figure 1(a). It can be seenthat the extreme slip angle of tyre A is smaller than tyre B when the maximum lateral forceis achieved, whereas the lateral force tyre A can provide is larger than tyre B. The visualisednonlinear force responses of the MF tyres models to tyre load and slip angle are shown inFigure 1(b).

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Vehicle System Dynamics 1219

Table 2. Characteristics of MF tyre models.

Extreme slip Maximum adhesiveTyre type angle (◦) coefficient Note

Tyre A 8 0.9064 High lateral adhesive capacityTyre B 12 0.6564 Low lateral adhesive capacity

–32 –24 –16 –8 0 8 16 24 32–8000

–4000

0

4000

8000

slip angle (º)

Tyr

e la

tera

l for

ce (

N)

tyre Atyre B

010

20

04000

80000

4000

8000

slip angle (º)Tyre load (N)

(a) (b)

Tyr

e la

tera

l for

ce (

N) tyre A

tyre B

Figure 1. Comparison of the two MF tyre models: (a) lateral force of the two tyres and (b) force response of thetwo tyres.

a b

γ

vx

vy

1δ1α

2α

Figure 2. Two-DOF vehicle manoeuvre model.

2.2. Vehicle steering model

The steering model employed in this paper is a typical two-DOF system as shown in Figure 2.It expresses the lateral and yaw motions of vehicle. The linear bicycle model would resultin significant discrepancies with the real vehicle response especially during extreme driveoperations such as the fishhook manoeuvre used in this study. The discrepancies come mainlyfrom the nonlinear response of tyres. Though the bicycle model is employed in the research,only the geometric relationships of this model are utilised, and the discrepancies of nonlinearforce response are complemented by using four MF tyre models instead of linear corneringstiffness model for tyres. A similar method is already validated in reference [20].

When the vehicle is cornering, the lateral force is applied on the centre of gravity (CG)of vehicle body, and thus induces the rollover. The equations of the steering model can beexpressed as follows:

mv(vy + vxγ ) = (Fy1 + Fy3) cos(δ1) + (Fy2 + Fy4) cos(δ2), (2)

Izγ =∑

Mz = a(Fy1 + Fy3) cos(δ1) − b(Fy2 + Fy4) cos(δ2). (3)

The wheel steering angle is usually small enough, and only considering the front wheelsteering, the geometry relations are

α1 = arctan

(vy + aγ

vx

)− δ1 ≈ vy + aγ

vx− δ1, α2 = arctan

(vy − bγ

vx

)≈ vy − bγ

vx.

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The above Equations (2) and (3) could be rewritten in a differential equation form:

vy = [(Fy1 + Fy3) cos(δ1) + (Fy2 + Fy4) cos(δ2)]mv

− vxγ , (4)

γ = [a(Fy1 + Fy3) cos(δ1) − b(Fy2 + Fy4) cos(δ2)]Iz

. (5)

And the lateral acceleration is

ay = vy + vxγ . (6)

2.3. Vibration and roll model

The vehicle vibration and roll model is a commonly used seven-DOF model, as shown inFigure 3. This model is widely used and proved reasonably accurate in many studies such asvehicle stability [18,20,21] or ride comfort analysis [22]. What is important in this study isthat the lateral acceleration is brought into the roll motion of vehicle body [23], and the rollmotion can be expressed as follows:

Ixϕ =∑

Mx = (Fs1 − Fs3)m

2+ (Fs2 − Fs4)

n

2+ msayd cos ϕ + msgd sin ϕ. (7)

Assuming that the angle is small enough, and then the above equation could be

Ixϕ == (Fs1 − Fs3)m

2+ (Fs2 − Fs4)

n

2+ msayd + msg dϕ, (8)

where the suspension forces are

Fsi = csi(zui − zsi) + ksi(zui − zsi). (9)

According to geometry relations:

zs1 = z − aθ + mϕ

2, zs2 = z + bθ + nϕ

2,

zs3 = z − aθ − mϕ

2, zs4 = z + bθ − nϕ

2.

Figure 3. Seven-DOF vehicle vibration model.

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Vehicle System Dynamics 1221

Then the vertical vibration and pitch motion of sprung mass are

msz = Fs1 + Fs2 + Fs3 + Fs4, (10)

Iyθ =∑

My = (Fs2 + Fs4)b − (Fs1 + Fs3)a. (11)

And the vertical vibration of unsprung

mui zui = kti(zgi − zui) + ksi(zsi − zui) + csi(zsi − zui). (12)

As an available solution [20,24], this model is chosen to discuss the effect of the vertical andlateral coupling between tyre and road on vehicle rollover and stabilities.

The tyre load can be calculated by summing the static and dynamic load, where the staticload for each tyre is

Fsz1 = Fs

z3 = bmvg

2(a + b), Fs

z2 = Fsz4 = amvg

2(a + b).

Then the total load for each tyre is

Fzi = Fszi + kti(zgi − zui). (13)

The trigger condition of termination for the test is when the inner tyres lift off the ground:

Fz1 ≤ 0, Fz2 ≤ 0 or Fz3 ≤ 0, Fz4 ≤ 0.

In order to obtain the matrix form of Equations (7)–(12), define

[zs] = [zs1 zs2 zs3 zs4]T, [z] = [z θ ϕ]T,

[zu] = [zu1 zu2 zu3 zu4]T, [zg] = [zg1 zg2 zg3 zg4]T

Kt = diag(kt1, kt2, kt3, kt4), Ks = diag(ks1, ks2, ks3, ks4), Cs = diag(cs1, cs2, cs3, cs4)

Ms = diag(ms, Iy, Ix), Mu = diag(mu1, mu2, mu3, mu4)

H =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 −am

2

1 bn

2

1 −a−m

2

1 b−n

2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

, L =⎡⎣0 0 0

0 0 00 0 1

⎤⎦ , u1 = [0 0 ay]T, u2 = [zg].

Then the above equations can be written in a matrix form, separated into sprung and unsprungmass part:

[z] = M−1s (ms dgL − HTKsH)[z] − M−1HTCsH[z]

+ M−1s HTKs[zu] + M−1

s HTCs[zu] + ms dM−1s u1, (14)

[zu] = M−1u KsH[z] + M−1

u CsH[z]− (M−1

u Ks + M−1u Kt)[zu] − M−1

u Cs[zu] + M−1u Ktu2. (15)

As can be seen, there are two inputs for this model: the vehicle lateral acceleration andvertical road excitation. The lateral acceleration influences the lateral coupling of tyres androad, while the vertical road excitation has an influence on the vertical coupling of tyres androad. The excitation from road mainly deteriorates the ride comfort, impacts the attachmentsof tyres to road, disturbs the vehicle body roll motion and reduces the manoeuvring stability.

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Table 3. Energy distribution of vehicle vibration modes.

Energy distribution on different DOF (%)Order of Naturalmodal frequency (Hz) Z θ φ zu1 zu2 zu3 zu4

1 1.11 78.62 21.29 0.00 0.04 0.00 0.04 0.002 1.27 21.29 78.60 0.00 0.00 0.05 0.00 0.053 1.37 0.00 0.00 99.87 0.03 0.03 0.03 0.034 11.96 0.00 0.00 0.00 24.95 25.05 24.95 25.055 11.96 0.07 0.02 0.00 46.91 3.05 46.91 3.056 11.96 0.02 0.09 0.00 3.05 46.90 3.05 46.907 11.96 0.00 0.00 0.13 25.02 24.92 25.02 24.92

2.4. Preliminary analysis on model characteristics

A compact front wheel drive sedan is utilised in this study. The vehicle is designed well inhandling and dynamics, the parameters are acquired from the NHTSA official data [25], andmore details of this vehicle are listed in the appendix. A preliminary analysis is conducted toobtain the characteristics of this vehicle.

In the vehicle suspension system, damping is a complicated physical quantity, which mayinclude the effect of friction and absorber as well as other uncertainties. So the actual dampingeffect is usually represented by an equivalent coefficient.

In the commonly used seven-DOF vibration model of vehicle, the damping is usually treatedas viscous damping, as defined in equation (9). It is common knowledge that the viscousdamping mainly has effect on response amplitude as well as nature frequency. The greater thedamping is, the smaller the response amplitude and nature frequency are.

To conduct a nature frequency analysis, Equations (14) and (15) must be written as aundamped free vibration form:

MZ + KZ = 0, (16)

where

M =[

Ms 03×4

04×3 Mu

], K =

[HTKsH −HTKs

−KsH (Ks + Kt)

], Z =

[ [z][zu]

].

The modal coupling results are given in Table 3. The bold values indicate the principal modeof the vehicle vibration system. As can be seen, the vehicle body roll motion is decoupledwith other degree of freedom, it composes 99.87% of the vibration energy in the third modal,and the natural frequency is 1.37 Hz. The resonance of vehicle body roll motion may inducerollover when the vehicle is excited at 1.37 Hz. The excitation may be from road disturbancesas well as steering manoeuvres. Moreover, the bounce of wheels converges at 11.96 Hz. Thebounce of wheels may lead to the detachment of tyres from the road, which will deterioratethe adhesion conditions of tyres and finally make the vehicle lose stability.

The transfer responses of the vehicle body roll motion and wheel bounce to different exci-tations are shown in Figure 4 and Table 4. As can be seen, except the response of wheelbounce to road excitation, the resonance frequencies of other motions are at 1.15 Hz, whichis approximately equal to the vehicle body roll motion frequency of 1.37 Hz. And the littledifference is due to damping effect as mentioned earlier.

Both the lateral acceleration and road excitation can arouse the vehicle body roll motion aswell as the wheel bounce motion. The vehicle is likely to roll over due to the excessive rollangle, or is easy to lose stability due to the uneven tyre load distribution induced by wheel

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Vehicle System Dynamics 1223

0.2 0.5 1 2 5 10 200

0.01

0.02(a)

(b)

(c)

(d)

Frequency (Hz)

mag

(ra

ds/

m)

0.2 0.5 1 2 5 10 20–200

–100

0

Frequency (Hz)

phas

e (º

)

0.2 0.5 1 2 5 10 200

0.5

1

1.5 × 10–3

Frequency (Hz)

mag

(s2 )

0.2 0.5 1 2 5 10 20–300

–200

–100

0

Frequency (Hz)ph

ase

(º)

0.2 0.5 1 2 5 10 200

0.2

0.4

0.6

0.8

Frequency (Hz)

mag

(ra

d/m

)

0.2 0.5 1 2 5 10 20–300

–200

–100

0

Frequency (Hz)

phas

e (º

)

0.2 0.5 1 2 5 10 200

1

2

3

Frequency (Hz)

mag

0.2 0.5 1 2 5 10 20–200

–100

0

Frequency (Hz)

phas

e (º

)

Figure 4. FRs of vehicle model: (a) (1) magnitude FR of roll angle to lateral acceleration, (a) (2) phase FR of rollangle to lateral acceleration, (b) (1) magnitude FR of wheel bounce to lateral acceleration, (b) (2) phase FR of wheelbounce to lateral acceleration, (c) (1) magnitude FR of roll angle to road excitation, (c) (2) phase FR of roll angle toroad excitation, (d) (1) magnitude FR of wheel, and (d) (2) phase FR of wheel bounce to road excitation.

Table 4. FRs of vehicle model.

Response

Excitation Roll angle Wheel bounce

Lateral acceleration Figure 4(a) (1) Figure 4(a) (2) Figure 4(b) (1) Figure 4(b) (2)Road excitation Figure 4(c) (1) Figure 4(c) (2) Figure 4(d) (1) Figure 4(d) (2)

bounce. As a result of these consequences, the lateral and anti-rollover stability should beconsidered as a whole response to the action of both steering and road excitations. The impactof these coupling effects on the vehicle stabilities will be further discussed in the followingsection.

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1224 Y. Li et al.

Figure 5. Steering wheel signal of fishhook manoeuvre 1a.

3. Effect of lateral coupling between tyre and road on vehicle stabilities

3.1. Fishhook manoeuvre and RCF

The fishhook is a test that stresses the roll-over tendency of vehicles by inducing it with atypical emergency manoeuvre. The name of this test is taken for the shape of vehicle runningtrajectory. There are two types of fishhook manoeuvre, the fixed time fishhook (fishhook 1a)and the roll rate feedback fishhook (fishhook 1b). Due to the simplicity of implementation,the fishhook 1a is chosen to investigate the effect of lateral coupling between tyre and road.The steering wheel signal is shown in Figure 5 [2]. The rollovers often occur at the time of T1

and T2, and the second stage, T2, is the most dangerous time.The static stability factor (SSF) is a basic indicator to evaluate the vehicle’s resistance

capability of rollover. It is defined as SSF = (the half-track width of the vehicle)/(the heightof the CG of the vehicle). The smaller the SSF, the more likely the vehicle is to roll over.

However, the SSF only represent the static characteristics, and is not suitable for some activerollover avoidance or warning system. As a consequence, modifications to the SSF [26,27] ordifferent dynamic stability indexes [28] are proposed, these indicators include the effects oflateral movement of vehicle CG during vehicle body roll motion, suspension jacking forces,the dynamic overshoot in the roll angle or other factors. In this paper, several RCFs are chosento make a comparison for the further study. The RCF1 [29] is

RCF1 = g(m + n)

4− |ϕ|dg − |ay|(h − zs) − |ϕ|Ix

ms. (17)

The RCF1 involves the decrease in vehicle half-track width due to roll angle, the increasein vehicle CG height due to the CG vertical vibration, and the influence of roll moment ofthe sprung mass. A smaller value of RCF1 indicates a lower rollover resistance capability ofa vehicle, and the rollover occurs when RCF becomes negative.

Another commonly used RCF [30] is

RCF2 =∣∣∣∣ 4hay

(m + n)g

∣∣∣∣ . (18)

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Vehicle System Dynamics 1225

And the lateral load transfer rate (LTR) is also chosen as RCF [31] in some studies:

RCF3 = LTR =∣∣∣∣Fz1 + Fz2 − Fz3 − Fz4

Fz1 + Fz2 + Fz3 + Fz4

∣∣∣∣ . (19)

The larger RCF2 or RCF3 is, the more likely that the vehicle rolls over, and if the RCF2

or RCF3 is greater than 1, it is regarded that rollover has already occurred. To normalise thenumerical range of the three RCFs, a transformation of RCF1 is needed:

RCF′1 = 1

0.5RCF1/ max(RCF1) + 0.5− 1, (20)

where obviously

max(RCF1) = g(m + n)

4

then

RCF′1 = 1

0.5RCF1/[g(m + n)/4] + 0.5− 1. (21)

A comparison of the three RCFs subject to the same fishhook manoeuvre is shown in Figure 6.In the simulations of this research, in order to make a comparison between different RCFs,and investigate the effect of road fluctuation on rollover, a RCF value less than 1 is beneficialfor the purpose of continuity in simulations. So, by trial and error, the simulation speed isset as 54 km/h, which is enough to arouse a significant RCF value whereas the terminationconditions will not be triggered. As can be seen, the RCF1 is more sensitive to the rollover, andhave a higher safety threshold at the beginning of T2 in fishhook. The RCF2 is less sensitivebut can give a higher safety threshold in the entire manoeuvring. The RCF3 mainly focuseson the asymmetric load on different tyres, and its rollover safety threshold is much lower thanthat of RCF1.

Besides the sensitivity to vehicle stability, the cost of the three RCFs should also be noticed.RCF2 requires the fewest parameters, which means fewer sensors are needed. The RCF1 needsmore parameters and computation, but is more sensitive to the rollover detection.

According to these comparisons, the RCF1 is more sensitive to the rollover and have a highersafety threshold, besides, it can express more effect of different factors, such as the oscillationof roll acceleration, the movement of both roll inertia and vehicle mass, so it is more superiorin the rollover analysis [20].

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

Time (s)

RC

F

RCF1

RCF2

RCF3

Figure 6. Comparison of different RCFs.

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3.2. Tyre and vehicle response to fishhook 1a manoeuvre

As mentioned previously, the existing researches treat the lateral stability [8,9] and anti-rolloverstability [4,20] independently or partially. In this paper, the different stabilities are intendedto be analysed correlatively.

Some indicators are chosen to evaluate these stabilities. For the anti-rollover stability, itis featured by the RCF. For the over or under steering tendency, it is easy to be evaluatedaccording to the vehicle trajectory. As for the lateral stability, which means whether there issideslip of vehicle, many researches [32,33] utilise the tyre friction ellipse to carry out theanalysis. However, a single parameter, the tyre slip angle, is used to estimate whether thevehicle is stable [34] in this paper. When the slip angle of tyre exceeds the extreme value,the sideslip is then regarded as occurring. The benefit of this method is that only one simpleparameter is needed.

The responses of the four tyres subjected to a fishhook 1a manoeuvre are shown in Figure 7,where the vehicle speed is 54 km/h. The 3D curve of tyre response includes tyre lateral force,the corresponding slip angle and tyre load. It shows that the curves of tyres 1 and 3 covera larger range than their counterparts. And the load of tyre 4 is near zero, which means thattyre 4 is about to lift off the ground.

The slip angles are extracted and shown in Figure 8, the slip angle of the rear wheels, i.e.α2, is approximately smaller than 8◦ (extreme slip angle mentioned in Table 2) during thewhole manoeuvre period. However, the front tyre slip angle, α1, reaches approximately 15◦in the first stage of fishhook manoeuvre, which means the sideslip occurs on the front tyres.Moreover, the α1 rises even to 17◦ in the second stage, i.e. the sideslip is more serious.

–20–10

010

05000

–7500

–2500

2500

7500

tyre load (N)slip angle (º)

tyre

late

ral f

orce

(N

)

tyre 1 tyre 2 tyre 3 tyre 4 MF tyre

Figure 7. Force responses of tyre to fishhook manoeuvre.

0 2 4 6 8 10–18

–9

0

9

18

time (s)

slip

ang

le (

º)

1

2

Figure 8. Slip angle responses of tyre to fishhook manoeuvre.

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Vehicle System Dynamics 1227

-10 -5 0 5 10–40

–20

0

20

Roll angle (º)

Rol

l rat

e (d

eg/s

) pole 1

pole 2

pole 3

Figure 9. Roll phase respect to fishhook manoeuvre.

0 20 40 60 80–100

–80

–60

–40

–20

0

20

Forward direction (m)

Late

ral d

irect

ion

(m)

Figure 10. Vehicle trajectory in fishhook manoeuvre.

The phase diagram of vehicle body roll response is shown in Figure 9. There are three poles.Pole 1 represents the vehicle entrance speed into fishhook manoeuvre, pole 2 and pole 3 standsfor the first and second steering stage in fishhook manoeuvre, separately. The closure of theroll phase diagram implies that the vehicle can return to steady state eventually. The vehicletrajectory is also shown in Figure 10.

3.3. Evaluation of vehicle stabilities

Quite a number of vehicle parameters could be adjusted to improve vehicle dynamics, e.g.vehicle mass, location of the CG, the stiffness, characteristics of suspension and tyre, etc. Theinfluence of these parameters on the single stability performance of vehicle has already beenillustrated by many researches [3,4,35], and will not be repeated here. In this paper, only thecross effect on different stabilities is discussed.

The evaluation between two types of tyre is made, with high and low adhesive capacityrespectively, as listed in Table 2. The original reference vehicle is equipped with high adhesivetyres (tyre A), and the contrast vehicle is equipped with low adhesive tyres (tyre B). Thecomparison results shown in Figure 11 indicate that the sideslip tendency of contrast vehicleincreases significantly, especially in the second stage of the manoeuvre, from less than 15◦ tomore than 45◦. So tyres with high adhesive capacity possess reliable lateral stability.

However, according to the roll phase diagram shown in Figure 12, the roll angle and rollrate decrease slightly. This means that the roll motion of the vehicle body is more stabilised.As a result, the rollover risk of contrast vehicle is greatly reduced as shown in Figure 13, theRCF drops from 0.76 to 0.48. We can say the anti-rollover stability is increased by 36.8%.So tyres with high adhesive capacity possess a high risk to rollover. However, there is also a

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0 2 4 6 8 10

45

30

15

0

–15

–30

–45

Time (s)

Slip

ang

le (

º)

(tyre A)1

(tyre A) 2

(tyre B) 1 (tyre B) 2

Figure 11. Slip angle responses of different types of tyre.

–10 –5 0 5 10

45

–45

–30

–15

0

15

30

Roll angle (º)

Rol

l rat

e (d

eg/s

)

tyre A

tyre B

pole 1

pole 2

pole 3

Figure 12. Roll phase responses of different types of tyre.

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

Time (s)

RC

F

(tyreA)RCF1

(tyreB)RCF1

Figure 13. RCF1 responses of different types of tyre.

new problem. The phase diagram stays in the third pole and cannot get closed. This meansthe vehicle is keeping side slipping in the second stage of fishhook manoeuvre and thus thelateral stability of this vehicle is decreasing.

These two phenomena reveal that there is contradiction between different vehicle stabilityperformances, and tyre adhesive capacity is one key reason for this contradiction. Tyres withhigh adhesive capacity possess reliable lateral stability, but a high risk to rollover. It can beexplained with the smaller force that low adhesive tyres can provide. The lateral force that lowadhesive tyres can provide is too small to afford a large lateral acceleration. And the smallerlateral acceleration can only induce a smaller roll angle on vehicle body, and thus reduce therollover risks ultimately. On the other hand, tyres with high adhesive capacity guarantee largerlateral force to achieve a higher steering capability, but the larger lateral force is easy to inducevehicle’s rollover.

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Vehicle System Dynamics 1229

0 25 50 75 100–100

–80

–60

–40

–20

0

20

Forward direction (m)

Late

ral d

irect

ion

(m)

tyre A tyre B

Figure 14. Trajectories of different types of tyre.

The vehicle trajectory also changes very differently as shown in Figure 14. It becomes quitesharper than the original trajectory due to sideslip of tyres both in the first and second stageof fishhook manoeuvre.

4. Effect of vertical coupling between tyre and road on rollover analysis

As analysed previously, not all the extreme steering manoeuvre will end in rollover, andsometimes the side slip should be the major concern. However, the excessive roll angle ofsprung mass do play a role of direct reason for vehicle’s rollover, and the roll motion of sprungmass is mainly caused by the lateral acceleration induced by cornering on vehicle body. Inaddition to these factors, another noticeable and common reason for vehicle body roll motionis the excitation from uneven road, and its effect is inevitable and should not be ignored. Next,we will look into this factor.

4.1. Effect of the road phase difference on vehicle body roll motion

A harmonic and periodical excitation from road is uncommon; however, its effect on vehiclebody roll motion is concise and intuitive. So it is necessary to look at the effect of this kindof excitation. The periodical road is a kind of steady-state sine excitation. The magnitude-frequency response (FR) of this excitation has already been explained in Section 2.4. Thissection focuses on the effect of the road phase difference between the left and right tyres.

Supposing that the vehicle wheel base is much smaller than the wavelength of the roadfluctuation, which means that the front tyres are in the same phase as that of the rear tyres,then the vertical excitation from road applied on left and right tyres are

zg1 = G0 sin(2π ft − σ), (22)

zg3 = G0 sin(2π ft), (23)

where f is the frequency of the road fluctuation, σ the phase difference between left and rightside tyres and G0 stands for the degree of roughness.

Then the road excitation on the left tyres relative to the right is

zg1 − zg3 = G0 sin(ft − σ) − G0 sin(ft)

= G′0 sin(ft − σ ′), (24)

where

tan(σ ′) = sin(σ )

cos(σ ) − 1, G′

0 = 2G0

∣∣∣sin(σ

2

)∣∣∣ . (25)

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0.2 0.5 1 2 5 10 20

0180

360540

7200

0.5

1

1.5

Frequency (Hz)phase

difference (º)

mag

(ra

d/m

)

Figure 15. Rolling response subject to road excitation.

0 2 4 6 8 100

0.5

1

1.5

Time (s)

RC

F

RCF on periodical road RCF without road

Figure 16. RCF1 response to the periodical road.

Drawn from the Equations (24) and (25), the conclusion is that the magnitude of the relativeroad excitation is determined only by the phase difference, and obviously the magnitude ismaximal when the phase difference is 180◦. Combining Equation (24) with Figure 4(c), it iseasy to obtain the results in Figure 15, which shows the synthesis response when vehicle issubjected to different frequency and phase excitation. As can be concluded, the extreme case iswhen the excitation amplitude is maximal and the excitation frequency is inducing resonanceof vehicle body, i.e. the phase difference is 180◦ and the frequency is approximately 1.15 Hz.

The comparison of fishhook manoeuvres between with and without road excitation is shownin Figure 16. The extreme case is chosen to see how severe the road effect is: the phasedifference between the two sides of tyres is set to be 180◦; the fluctuating magnitude of roadis 0.02 m (approximately equivalent to the B class road). And the road excitation frequency isabout 1.15 Hz with the condition that the wavelength of road is 13 m and the vehicle speed is54 km/h (54 km/h divided by 13 m is 1.15 Hz).

As can be seen, when excited by road in the extreme condition, the RCF1 reaches 1.309,which exceeds 1 significantly, i.e. the vehicle rolls over. However, the RCF1 is only 0.7227without the road excitation. The effect of this kind of road makes the risk of rollover increaseby 81.13%. According to this result, the uneven road is an important factor that cannot beneglected in the vehicle rollover analysis.

4.2. Effect of the stochastic road on vehicle stabilities

The stochastic road is a more common excitation than the periodical road. Its effect on vehiclerollover is investigated as follows. First a model of stochastic road excitation for each tyreis established. Then the coherence analysis of this road model is carried out, which is animportant factor for the vehicle body roll motion.

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Vehicle System Dynamics 1231

Table 5. Value of G0 (unit: 10−6 m3).

Road class Lower limit Geometric mean Upper limit

A – 16 32B 32 64 128C 128 256 512D 512 1024 2048E 2048 4096 8192F 8192 16,384 32,768G 32,768 65,536 131,072H 131,072 262,144 524,288

4.2.1. Transfer function model of the stochastic road

The excitation on tyre 1 is selected as reference of the other tyres. The road roughness at tyre 1is expressed by the filtered white noise model:

zg1 = −2π f0zg1 + 2π√

G0vxw, (26)

where w stands for the white noise disturbance of the road.According to ISO 8608, the value of the degree of roughness, G0, is determined by road

class, as shown in Table 5.Assuming the transfer function of the road excitation from tyre 1 to tyre 3, i.e. the transfer

characteristic of road excitation from left tyres to right is

G31(s) = Zg3(s)

Zg1(s)= a0 + a1s + a2s2

b0 + b1s + b2s2. (27)

And the transfer function of the road excitation from tyre 1 to tyre 2, i.e. the road excitationfrom front tyres to rear tyres, is a pure time delay:

zg2(t) = zg1(t − td). (28)

Then the transfer function from front to rear could be written in 2-order Pade approximationform:

G21(s) = Zg2(s)

Zg1(s)= 1 − (td/2)s + (t2

d/12)s2

1 + (td/2)s + (t2d/12)s2

, (29)

where td = (a + b)/vx, and a + b is the wheel base.The road excitation on tyre 4 needs to be worked out through the transfer path. As shown

in Figure 17. The transfer path could be either path 1 or path 2. However, the final transferfunction from tyre 2 to tyre 4 needs to include the time delay between tyre 2 and tyre 1, whereasthis situation does not happen in the transfer path 1. For this reason, the simpler path 1 is usedto express road excitation at tyre 4:

G41(s) = Zg4(s)

Zg1(s)= Zg4(s)

Zg3(s)

Zg3(s)

Zg1(s)= G43(s)G31(s) = G21(s)G31(s). (30)

With Equations (26)–(30), the uneven road excitation model at four tyres has been accom-plished.

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Tyre 3 Tyre 4

Tyre 1 Tyre 2

Transfer Path 1

Transfer Path 2

Figure 17. Road excitation model.

4.2.2. Coherence analysis of the road excitation

The difference between the road excitations on left and right tyres arouses a significant portionof the vehicle body roll motion, and this difference is determined by the coherency betweenthe left and right road excitation. As a result, the coherence analysis of road excitation is anessential part in this rollover analysis.

According to the control theory, for the single input single output system, the FR functionfrom input to output is defined as follows:

H(f ) = Sxy(f )

Sxx(f ), (31)

where Sxy(f ) is the cross-spectrum between input and output, and Sxx(f ) is the self-spectrumof input.

And the coherence function between input and output is

coh2xy(f ) = |Sxy(f )|2

S2xx(f )Syy(f ). (32)

For the stochastic road, the statistic characteristics of left and right side of road should bethe same, i.e. Sxx(f ) = Syy(f ). Then from the above Equations (31) and (32), there is

cohxy(f ) = |H(f )|. (33)

Moreover, the FR function is a special form of transfer function, i.e.

H(2π f ) = G(j2π f ). (34)

As a result, the coherence function, cohxy(f ), is the magnitude-frequency characteristics ofthe model expressed by the transfer function G(s).

Therefore, coherence serves as the description of stochastic road. On the basis of thecoherence analysis, the vehicle rollover subjects to different stochastic road can be investigated.

4.2.3. Effect of stochastic road on vehicle rollover

Four typical types of road are exhibited in Figure 18. The coh1 shows low coherency in thebroadband frequency domain. The coh2 shows low coherency in the low-frequency domain.The coh3 show slow coherency in the high-frequency domain. And the coh4 shows highcoherency in the broadband frequency domain.

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Vehicle System Dynamics 1233

0 5 10 15 200

0.5

1

1.5

Frequency (Hz)

coh

coh1 coh2 coh3 coh4

Figure 18. Different coherency of roads.

0.2 0.5 1 2 5 2010

10–3

10–5

10–7

10–9

Frequency (Hz)

PS

D (

m2 /H

z)

coh1 coh2 coh3 coh4

Figure 19. PSD of roads with different coherency.

Low coherency means that the coherency between the road excitation at left and right tyresis small, normally is smaller than 0.2 as can be seen from Figure 18; high coherency meansthe aforementioned coherency is large, normally is great than 0.8 as shown in the Figure 18.Thus, the coherency can be a presentation of road quality. The higher the coherency is, theflatter the road is in lateral section.

Low-frequency domain represents the frequency domain that covers the natural frequencyof vehicle body roll motion, which is of a relatively low frequency. And in this case, it is about1.37 Hz in Table 3. Whereas high-frequency domain represents the frequency domain thatcovers the natural frequency of wheel bounce motion, which is of a relatively high frequency.And in this case it is about 11.96 Hz in Table 3. And broadband frequency domain representsthe frequency domain that covers both of the two aforementioned natural frequencies. Thefrequency here can be a presentation of the road scale in the longitudinal section. The highfrequency signifies micro scale, and low frequency signifies the macro scale.

These four different types of stochastic roads exhibit different coherency at the naturalfrequencies of vehicle body roll motion as well as wheel bounce; they also reflect the excitationof the road with different qualities in a different scale. It is efficient to see the effect of roadexcitation on the response of sprung and unsprung mass according to these four types of road.

Though the coherencies and the time histories of these roads are different, the power spectraldensity (PSDs) of these roads are the same, as shown in Figures 19–23, which is a class Croad.

The time histories of the coh1 and coh2 roads are almost the same, and so do the coh3 andcoh4. The conclusion is that the time history of the road is mainly determined by coherencyin the low-frequency domain. So the following discussion will focus on the roads of coh2 andcoh3 because they are more representative.

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1234 Y. Li et al.

0 5 10 15 20–0.04

–0.02

0

0.02

0.04

0.06

Time (s)

Roa

d (m

)

tyre1 tyre2 tyre3 tyre4

Figure 20. Time history of road with coh1.

0 5 10 15 20–0.04

–0.02

0

0.02

0.04

0.06

Time (s)

Roa

d (m

)

tyre1 tyre2 tyre3 tyre4

Figure 21. Time history of road with coh2.

0 5 10 15 20–0.04

–0.02

0

0.02

0.04

0.06

Time (s)

Roa

d (m

)

tyre1 tyre2 tyre3 tyre4

Figure 22. Time history of road with coh3.

0 5 10 15 20–0.04

–0.02

0

0.02

0.04

0.06

Time (s)

Roa

d (m

)

tyre1 tyre2 tyre3 tyre4

Figure 23. Time history of road with coh4.

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Vehicle System Dynamics 1235

0 2 4 6 8 100

0.5

1

Time (s)

RC

F 1

coh3 coh2 without road

Figure 24. RCF1 responses to different road types.

0 2 4 6 8 100

0.5

1

Time (s)

RC

F 1

D class road C class road B class road without road

Figure 25. RCF1 responses to different road classes.

The fishhook manoeuvre is conducted on the class C road, with coherency of coh2 and coh3,respectively. The results are shown in Figure 24. It seems that there is no apparent distinctionbetween the two conditions, only that the coh2 road is more likely to arouse the rollover riskat time about 3 or 6 s, while the most dangerous time should be at about 2.5 s when there isno road excitation.

Among these roads, coh3 is more common in reality. The RCF results of this coherency ondifferent road classes are shown in Figure 25. As can be seen, the risk of rollover increasesproportionally to road class. And the road excitation contributes to the RCF more at non-extreme time, e.g. at time about 3 or 6 s, and especially in the first stage of fishhook manoeuvre,e.g. at time about 1.8 s, whereas the extreme time should be at about 2.5 s. In extreme cases,a class D road is enough to induce vehicle rollover. More comparison details are summarisedin Table 6. We can see that class C road already arouses a noticeable increase in rollover risk,more than 12%.

Table 6. Comparison of RCF value with different road excitation.

Road type RCF Relative change

Without road 0.7227 –coh2 C Class 0.8144 +12.69%coh3 B Class 0.7658 +5.96%coh3 C Class 0.8105 +12.15%coh3 D class (peak 1) 1.068 +47.78%coh3 D class (peak 2) 0.8957 +23.94%

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–20–100

100

5000

–7500

–2500

2500

7500

Tyre load (N)

slip angle (º)

Tyr

e la

tera

l for

ce (

N) without road

with road MF tyre

Figure 26. Force response of tyre 4 respect to fishhook manoeuvre on the stochastic road.

Table 7. Tyre force response.

Max. tyre load Min. tyre load Max. lateral force Min. lateral force

Without road 5040 N 1653 N 4664 N −2439 NWith road 5523 N 1075 N 5241 N −3327 NRelative change +9.58% −34.97% +12.37% −34.41%

4.2.4. Coupling effect of road on lateral and anti-rollover stability

As mentioned previously, the vertical coupling state of tyres and road determines the conditionof how tyres are attached to the road, i.e. the road hold performance of tyres. If the verticalcoupling state between tyres and road is bad, the tyres easily jump off the road. This phe-nomenon is not good for the vehicle stability because no adhesive force (both lateral andlongitudinal) could be provided by tyres. In some high-speed conditions, vehicle is likely tolose control if the tyres separate from the road. So a symmetrical, continuous contact betweenthe four tyres and road is essential for vehicle stability. In some extreme case, the tyre load maydrop to zero, which is an even worse condition. And according to the example of Figure 26,this is a possibility. Figure 26 shows an example of deteriorative vertical coupling state of tyresand road, the force response of tyre 4 due to excitation of class C road with coh3 and fishhookmanoeuvre. More details are listed in Table 7, as can be drawn from Figure 26 and Table 7,the minimum tyre load drops 34.97%, and the minimum lateral force that tyre can provideddecreased by 36.41%, this imply that the vehicle lateral stability decreases when consideringthe road excitation.

To comprehensively evaluate the effect of stochastic road excitation on vehicle anti-rolloverand lateral stability, RCF1 is still a convenient indicator for vehicle rollover, and the RCF3

proposed in the previous section is just suitable to monitor the variation of tyre load and thusvehicle lateral stability.

Because the combined condition of road excitation and fishhook manoeuvre is a non-stationary state, the time–frequency analysis such as short-time Fourier transform or wavelettransform is more suitable for a thorough investigation. However, for simplicity, here theroad excitation is considered independently, and the final result of RCF is assumed to be thelinear superposition of the two excitations. Then the focus of RCF can be limited only tothe proportion aroused by road excitation and the conventional fast fourier transform (FFT)method could be applied.

The RCF1 and RCF3 induced by the class C road with different coherencies are shown inFigures 27 and 28, and the FFT results are shown in Figures 29 and 30. From the time domainresults, it can be seen that the coh2 road can arouse a higher RCF1, while a relative lower RCF3

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Vehicle System Dynamics 1237

0 5 10 15 200

0.02

0.04

0.06

Time (s)

RC

F 1

coh2 coh3

Figure 27. RCF1 time history to different road types.

0 5 10 15 200

0.05

0.1

0.15

0.2

Time (s)

RC

F 3

coh3

coh2

Figure 28. RCF3 time history to different road types.

0 5 10 15 200

1

2

3x 10-3

Frequency (Hz)

RC

F 1

coh2

coh3

Figure 29. FFT results of RCF1 to different road types.

than the coh3. The frequency domain results also illustrate the same phenomenon, though notvery significant.

These results indicate that road excitation with low coherency in the low-frequency domainhas a greater effect on vehicle rollover than lateral stability. Conversely, the road of highcoherency in the low-frequency domain has more effect on vehicle lateral stability thanrollover. Therefore, on the same section of road, the vehicle anti-rollover stability and lateralstability are also in contradiction.

Table 8 summarises the increased percentage of RCF due to the excitation of different classcoh3 road. It reveals that the increase in RCF3 is greater than RCF1, and the class C road isalready enough to have an obvious effect on vehicle stabilities, where the risk of rollover mayincrease by 6% and the lateral stability decreases by 12%.

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1238 Y. Li et al.

0 5 10 15 200

2

4

6x 10–3

Frequency (Hz)

RC

F 3

coh3

coh2

Figure 30. FFT results of RCF3 to different road types.

Table 8. Increased percentage of RCF under road excitation.

Road class

RCF B C D E

RCF1 2.5% 6% 10% 25%RCF3 6% 12% 26% 180%

5. Conclusions

The vehicle stability involves many different performances, of which the anti-rollover stabilityis an important one. The direct reason for rollover is the excessive roll angle, which is deter-mined by both the road excitation and lateral acceleration. Moreover, the lateral accelerationis determined exactly by the vehicle lateral stability, which refers to the ability to preventover steering tendency or sideslip. On the other hand, the vehicle body roll motion in turn hascrucial influence on the tyre load, which will affect the vehicle lateral stability significantly.

As a result, the relationships and interactions among these different stability performancesare considered integrally. The role and effect of lateral stability such as sideslip in rolloveranalysis are investigated in this paper, and the important findings are as follows:

(1) There are contradictions between the vehicle lateral and anti-rollover stability. Whencornering, the vehicle with higher lateral stability possesses a higher risk of rollover.Meanwhile, on the same section of the road, the effects of road excitation on lateral or anti-rollover stability are different. Road excitation with low coherency in the low-frequencydomain has a greater effect on vehicle rollovers than lateral stability.

(2) The coupling between tyres and road is a key reason for the above contradiction. Theforce that tyres can provide, i.e. the lateral coupling between tyres and road, is the directcause of this contradiction. High adhesive capacity tyres mean that the vehicle has a higherperformance to maintain driving trajectory, while the greater lateral force that tyres provideraises the risk of rollover. On the other hand, the interaction force and attachment betweentyres and road, i.e. the vertical coupling between tyres and road is an indirect factor thatintensifies this contradiction. The vertical coupling between tyres and road disturbs the tyrelateral force and vehicle body roll motion, and thus the lateral and anti-rollover stability.So the lateral coupling is regarded as a direct factor and vertical coupling is an indirectfactor. These results suggest that a coordinated control system is needed for the differentobjectives of vehicle stability performances, and the key lies in the distribution of the tyreload.

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Vehicle System Dynamics 1239

(3) The effect of road excitation on vehicle rollover is considerable in some extreme situation.Periodical road, though uncommon, will result in the vehicle resonant vibration, andincreases the rollover risk easily and greatly. Stochastic road has some similar effects, andmay further induce the rollover at the moment which was supposed to be safe if there isno road excitation.

In this paper, the focus is limited to the tyre adhesive capacity and the road excitation, both ofwhich have explicit impact on the conclusions. However, the wheel alignment parameters, theeffect of vehicle acceleration or braking, and the yaw moment induced by tyre longitudinalforce distribution are not considered, since the action mechanism of these factors is implicitand intricate. A more comprehensive research is needed to make a thorough investigation onthe contradictions of vehicle stabilities, and these factors will be taken into account in the nextwork, which is going to focus on the proposition of the multi-objective coordinated controlsystem for the different objectives of vehicle stability performances.

Acknowledgements

This paper is supported by National Natural Science Foundation of China (Grant No. 51275541 and 51005256),National Natural Science Foundation of Chongqing (Grant No. cstc2013jjB70001) and Foundation of state keylaboratory of Mechanical Transmission (Grant No. 0301002109165).

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Appendix. Nomenclature and value

Name Value Unit Expression

a 1.178 m Longitudinal distance of from CG to front axleb 1.464 m Longitudinal distance of from CG to rear axlem 1.458 m Front track widthn 1.455 m Rear track widthh 0.506 m Height of the sprung mass CG above the groundd 0.256 m Distance between CG and roll axis

ms 1403.3 kg Sprung mass/vehicle body massmv 1559.2 kg Vehicle total massmui kg Unsprung massIz 2340 kg m2 Moment of inertia about the z-axis of the vehicle total

massIy 2131 kg m2 Moment of inertia about the y-axis of the sprung massIx 522 kg m2 Moment of inertia about the x-axis of the sprung mass

Fyi N Lateral force provide by each tyreFs

zi N Static load for each tyreFsi N Force of each suspensionksi 20,000 N/m Stiffness for each suspensionkti 200,000 N/m Vertical stiffness for each tyrecsi 1500 N s/m Damping coefficient for each suspensionα1 radian Slip angle of front tyresα2 radian Slip angle of rear tyresδ1 radian Front wheel steering angleγ radian Vehicle yaw angleφ radian Vehicle roll angleθ radian Vehicle pitch anglez m Vertical displacement of sprung mass at CGzui m Vertical displacement of unsprung masszsi m Vertical deformation of suspensionzgi m Vertical fluctuation of road profileay m/s2 Vehicle lateral accelerationvy m/s Vehicle lateral speedvx 54 km/h Vehicle speedf0 0.01 Hz Cutoff frequency for road irregularitiesg 10 m/s2 Acceleration of gravityT1 0.25 s 250 ms dwell time in the first stage of fishhook manoeuvreT2 3 s 3s pause in the second stage of fishhook manoeuvreCG Centre of gravityi 1, 2, 3, 4 Tyre subscript stands for left front, left rear, right front

and right rear separately

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