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This article was downloaded by: [Monash University Library]On: 05 December 2014, At: 10:07Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
Vehicle System Dynamics: InternationalJournal of Vehicle Mechanics andMobilityPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/nvsd20
Minimum time cornering: the effectof road surface and car transmissionlayoutDavide Tavernini a , Matteo Massaro a , Efstathios Velenis b ,Diomidis I. Katzourakis c & Roberto Lot aa Department of Industrial Engineering , University of Padova , ViaVenezia 1, Padova , 35131 , Italyb Department of Automotive Engineering , Cranfield University ,Cranfield , Bedfordshire , MK43 0AL , UKc Research and Development , Volvo Cars Corporation , Goteborg ,SE 40531 , SwedenPublished online: 26 Jun 2013.
To cite this article: Davide Tavernini , Matteo Massaro , Efstathios Velenis , Diomidis I. Katzourakis& Roberto Lot (2013) Minimum time cornering: the effect of road surface and car transmissionlayout, Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility, 51:10,1533-1547, DOI: 10.1080/00423114.2013.813557
To link to this article: http://dx.doi.org/10.1080/00423114.2013.813557
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Vehicle System Dynamics, 2013Vol. 51, No. 10, 1533–1547, http://dx.doi.org/10.1080/00423114.2013.813557
Minimum time cornering: the effect of road surface and cartransmission layout
Davide Taverninia, Matteo Massaroa*, Efstathios Velenisb, Diomidis I. Katzourakisc
and Roberto Lota
aDepartment of Industrial Engineering, University of Padova, Via Venezia 1, Padova 35131, Italy,bDepartment of Automotive Engineering, Cranfield University, Cranfield, Bedfordshire MK43 0AL,
UK, cResearch and Development, Volvo Cars Corporation, Goteborg SE 40531, Sweden
(Received 29 March 2013; final version received 3 June 2013 )
This paper investigates the minimum time/limit handling car manoeuvring through nonlinear optimalcontrol techniques. The resulting ‘optimal driver’ controls the car at its physical limits. The focus is oncornering: different road surfaces (dry and wet paved road, dirt and gravel off-road) and transmissionlayouts (rear-wheel-drive, front-wheel-drive and all-wheel-drive) are considered. Low-drift pavedcircuit-like manoeuvres and aggressive/high-drift even counter-steering rally like manoeuvres arefound depending on terrain/layout combinations. The results shed a light on the optimality of limithandling techniques.
Keywords: car; limit handling; optimal control; minimum time; drifting; tyre; off-road; rally
1. Introduction
In recent years, there is a growing interest in the investigation of the so-called aggressivemanoeuvring (see, e.g. [1–3]), following the idea that high-drift and even counter-steeringmanoeuvres may be more efficient than typical low-drift manoeuvres under certain road–tyrecharacteristics and vehicle layout. In particular, experimental evidence shows that rally driversare used to such manoeuvres, thus suggesting that under low friction conditions this drivingstrategy could be even optimal from the minimum time point of view. In addition, a betterunderstanding of such limit handling conditions could lead to advanced electronic stabilitycontrol systems with an extended operating envelope.
As an experimental example of aggressive manoeuvres by expert drivers, Figure 1 showsa rear-wheel-drive (RWD) car performing a 180◦ turn on off-road surface. The recordedmanoeuvre starts with the car travelling at a speed of V = 67 kph in straight motion. After0.25 s (between the first two snapshots of the trajectory plot) the driver starts braking (stillin straight motion) and after 1.25 s (between the third and fourth snapshots) he steers inwardthe curve (negative δ) while still braking. At 3 s (seventh snapshot) the driver throttles on andsteers outward the curve (positive δ, counter-steering). The minimum speed of 26 kph andmaximum vehicle drift/slip angle of β = 25◦ are reached at 4 s (ninth snapshot). From here
*Corresponding author. Email: [email protected]
© 2013 Taylor & Francis
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0 2 4 6 8 100
50
100V
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/h)
0 2 4 6 8 10
−20
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d (º
)b
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50
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Thr
ottle
(%
)
0 2 4 6 8 100
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Time (s)
Bra
ke (
%)
rearfront
Figure 1. Experimental data of a RWD car performing a 180◦ curve on a off-road surface. The trajectory snapshotsare with 0.5 s time increment.
on the driver accelerates while exiting the curve. The steering angle is rather vibrating and canbe noted that full throttle phases are associated with counter-steering phases.
On the other side, an experimental example of low vehicle drift manoeuver is reported inFigure 2, where a front-wheel-drive (FWD) car performs a 180◦ turn on a paved track. Similar
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Figure 2. Experimental data of a FWD car performing a 180◦ curve on paved track. The trajectory snapshots arewith 0.5 s time increment.
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Vehicle System Dynamics 1535
to the previous case, the recorded manoeuvre starts with the car in straight motion at a speedof 63 kph. The driver starts braking and at 1 s (second snapshot) he steers inward the curve.The steer is kept inward the curve (negative δ) all along curve, with a peak value of 22◦ at 4 s(eighth snapshot). The vehicle drift β remains low (<5◦). Nevertheless the lateral accelerationreaches 0.9g, i.e. the manoeuvre is quite demanding (it is a C-segment medium car driven byan expert driver).
The present work addresses the optimality of cornering manoeuvring through the use ofnonlinear optimal control techniques. The control problem is defined as follows: find theminimum time ‘optimal’ manoeuvre (in terms of vehicle states, trajectory and control inputs)given the car characteristics, tyre–road characteristics, road geometry and driver limitations(control bandwidth and magnitude). The resulting nonlinear optimal control problem is solvedusing the indirect method detailed in [4]. Driver’s input is limited in frequency and magnitudeto reproduce real drivers’ limitations.[5]
The method has been used successfully in the past for minimum time manoeuvring ofmotorcycles.[6,7] Similar approaches on minimum time manoeuvring are reported in [8,9](based on direct methods) and [10] (limited to constant speed).
The car model employed herein is based on the well-known single-track model pioneeredin [11–13] and included in most vehicle dynamics textbooks.[14,15] The model has beenenriched with nonlinear tyres, coupling between longitudinal and lateral tyre forces [16] aswell as longitudinal load transfer. In [2], the capability of this simple model to reproducecomplex and aggressive manoeuvres consistent with experimental data is shown.
The work is organised as follows. In Section 2 the mathematical formulation of the problemis presented together with the car model and optimisation method, in Section 3 the effect ofroad condition on road/tyre characteristics is discussed, and finally in Section 4 the minimumtime simulations with different road–tyre characteristics and vehicle layout are presented.
2. Mathematical formulation
The goal is to find the minimum time manoeuvre (and related input history) of a given vehi-cle running on a given road while accounting for the environment constraints. Nonlinearoptimal control techniques will be employed since the problem is especially nonlinear whenconsidering limit handling manoeuvres.
The vehicle, tyre and road models will be described first, then optimal control problem willbe formulated.
2.1. Vehicle model
A single-track model including nonlinear tyres and front/rear load transfer is employed andshown in Figure 3. For each axle the wheel includes the contribution of the left and the right
f
f
s
Figure 3. Car model with variables.
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wheel of the real vehicle. Pitch and roll rotations are neglected. The equations of motionconsist of Newton–Euler equations in the longitudinal, lateral and yaw directions
V = g(sF cos(δ − β) − fF sin(δ − β) + sR cos β + fR sin β) − kDV 2(cos β)3
M, (1)
β = −� + g
V(sF sin(δ − β) + fF cos(δ − β) − sR sin β + fF cos β) + kDV sin β(cos β)2
M,
(2)
� = Mg
IG(a(fF cos δ + sF sin δ) − bfR), (3)
where V is the absolute vehicle velocity, β the vehicle drift angle, � the yaw rate, fR and fF thenormalised lateral forces, sR and sF the normalised longitudinal forces and δ the steer angle.Note that the lateral and longitudinal forces are normalised with the weight Mg. Note also thatthe absolute velocity V and drift angle β are used in place of the longitudinal velocity u andlateral velocity v which are often used instead. The reason is that in limit handling conditionthe two differ substantially (e.g. on a 90◦ drift u is null) and the absolute V is the one the driveraims to control.
The axles’ equilibria are expressed using the following equations of motion:
ωR = (γR − r sR) M g
IM, (4)
ωF = (γF − r sF) M g
IM, (5)
where γR and γF are the rear and front axles propulsive/braking torques, ωR and ωF are wheelspin velocities.
Normalised vertical loads on the front and rear axles are computed from the vertical andpitch equations of motion which are solved to give
nR0 = a
a + b+ h(sF cos δ − fF sin δ + sR)
a + b, (6)
nF0 = b
a + b− h(sF cos δ − fF sin δ + sR)
a + b, (7)
where h is the height of the centre of mass (CoM) from the ground. To easily account for thesuspension pitch dynamics, the current vertical loads are computed from relaxation equations
τNnR + nR = nR0, (8)
τNnF + nF = nF0, (9)
where τN approximates the non-instantaneous load transfer related to suspension dynamics.Tyre forces depend on longitudinal slip κ , lateral slip λ and tyre vertical load n.Rear and front (practical) longitudinal slips κR and κF as well as (practical) rear and front
lateral slips λR and λF are defined as follows:
κR = ωR r − V cos β
V cos β, (10)
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Vehicle System Dynamics 1537
κF = ωF r − V cos(β − δ) − � a sin δ
V cos(β − δ) + � a sin δ, (11)
λR = − arctan
(−� b + V sin β
V cos β
), (12)
λF = − arctan
(V sin(−δ + β) + cos δ � a
V cos(−δ + β) + sin δ � a
). (13)
The coupling between longitudinal and lateral forces is accounted using the similarity methoddescribed in [15, Chapter 3]. In practice, the theoretical longitudinal slip σx, lateral slip σy andequivalent slip σ are computed from the practical slips κ and λ
σx = κ
1 + κ, σy = tan λ
1 + κ, σ =
√σx
2 + σy2, (14)
and used to compute the lateral μf and longitudinal μs tyre friction coefficient
μf = σy
σDλ sin[Cλ arctan{σBλ − Eλ(σBλ − arctan σBλ)}], (15)
μs = σx
σDκ sin[Cκ arctan{σBκ − Eκ(σBκ − arctan σBκ)}], (16)
necessary to compute the steady-state tyre longitudinal sR,F and lateral fR,F forces
sR0 = nR μsR(λR, κR), sF0 = nF μsF(λF, κF), (17)
fR0 = nR μfR(λR, κR), fF0 = nF μfF(λF, κF), (18)
B, C, D and E being the well-known Pacejka coefficients.Current longitudinal and lateral forces are computed from steady-state values using the
relaxation equations
σr fRV cos β
+ fR = fR0, (19)
σr fFV cos β
+ fF = fF0, (20)
σr sR
V cos β+ sR = sR0, (21)
σr sF
V cos β+ sF = sF0, (22)
where σr is the relaxation length both for lateral and longitudinal forces.Finally, it is fundamental to track the vehicle position on the road during the manoeuvre.
This is achieved using three curvilinear coordinates: the position of the vehicle along the roadss, the lateral position with respect to the road centre line sn and the angle α of the vehiclewith respect to the road centre line (Figure 4). The related equations are
ss = V cos(α + β)
1 − snK, (23)
sn = V sin(α + β), (24)
α = � − KV cos(α + β)
1 − snK, (25)
where K is the local curvature of the road.
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x
y
K
ss
sn
v
u a
Figure 4. Curvilinear coordinates.
Summarising, the vehicle model consists of 14 differential equations (1–5, 8, 9, 19–22,23–25) and as many state variables
x = {V , β, �, ωR, ωF, fR, fF, sR, sF, nR, nF, ss, sn, α}T. (26)
The control vector consists of two elements
u = {δ, γt}T, (27)
where δ is the steer angle and γt is the total driving/braking torque which is split between therear and front axles according to the torque distribution factor kt
γR = ktγt and γF = (1 − kt)γt . (28)
2.2. Optimal control
Optimal control aims at minimising a certain cost function subject to a certain number ofequality and/or inequality constraints.
In this case, the cost function is the total manoeuvre time T , while the vehicle equations ofmotion are included in the optimisation as equality constraints
f(x, x, u) = 0, (29)
where f consists of Equations (1–5, 8, 9, 19–22, 23–25), x is the state vector reported inEquation (26) and u are the driver controls of Equation (27).
Since the maximum steering angle on vehicles is limited to a certain δmax, a correspondinginequality constraint is added as follows:
δ ≤ |δmax|. (30)
To include the limited bandwidth of real drivers, also the maximum steer angle rate is limitedto δmax
δ ≤ |δmax|. (31)
A similar bandwidth limitation is added to the whole propulsive/braking torque, i.e. to thedriver throttling/braking actions
γt ≤ |γtmax |. (32)
To account for the limited engine power, another inequality constraint is added, stating thatthe whole propulsive torque (rear axle plus front axle) is limited
(γtktωR + γt(1 − kt)ωF)Mg ≤ Pmax. (33)
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Vehicle System Dynamics 1539
In order to constrain the vehicle trajectory within the road boundaries the following inequalityconstraints are added:
−Lw ≤ sn − b sin α ≤ Rw, (34)
−Lw ≤ sn + a sin α ≤ Rw, (35)
where Rw and Lw are right and left road widths, respectively.Initial and final conditions, e.g. vehicle velocities and positions on the road at the start and
finish line, are given by
b1(x(0), u(0)) = 0 and b2(x(T), u(T)) = 0. (36)
The problem is solved using an indirect method approach. In practice, the constrained problemdefined above is transformed into an unconstrained problem, by means of Lagrange multipliersand penalty functions. From the first variation of the unconstrained problem a boundary valueproblem is obtained, and the finite difference discretisation yields to a large nonlinear system.Full details on the solver are reported in [4].
The result of the optimal control problem completely describes both the dynamics of thevehicle system during the manoeuvre and the driver inputs that produce it. One of the mainadvantages of this method is that no driving rules have to be predefined.
3. Road–tyre interaction
Nowadays, measuring the tyre properties in terms of forces/torques as a function of slipsis a consolidated procedure among car and tyre manufacturers. Nevertheless, it is worth toremember that what is measured is not the absolute tyre response, rather the response of thetyre on the specific tested surface is measured. In other words, the same tyre behaves differentlyon different road surfaces. However, experience highlighted some general patterns (see, e.g.[14,17]). In particular, when using the same tyre on different road asphalts, the corneringstiffness keeps almost unchanged while there may be changes on the friction peak coefficient.Even when moving to wet asphalt (in non-hydroplaning condition) the cornering stiffnessdoes not change significantly while the friction peak is greatly reduced. In practice, as longas the ground surface can be considered rigid with respect to the tyre carcass, the tyre hasbasically the same cornering stiffness (as predicted by theoretical brush and string modelsin [15, Chapters 3 and 5]). When off-road surfaces are considered (e.g. gravel) the groundcannot be considered rigid with respect to the tyre, and a reduction on the cornering stiffnessis therefore expected.
In [18,19], it is highlighted that the changes in the tyre force vs. slip curve are in the directionof a reduction of the cornering stiffness, while in [20] it is shown that the peak of the lateralforce moves towards high slips and even disappears (in very soft surfaces a bulldozing effectrelated to the plowing of the soil may even take place). The same adjustments characterise thecurve of longitudinal forces vs. longitudinal slip.[19]
In the current paper, four types of tyre–road interaction curves named tyre 1, tyre 2, tyre 3and tyre 4 are considered and simulated to evaluate their effect on optimal driving strategies.They are obtained by varying the Pacejka coefficients B, C, D and E, see Table 1 (same valuesare used for the lateral and longitudinal forces). In other words, we consider the same tyre ondifferent road surfaces.
The first tyre–road characteristic (tyre 1, blue curve in Figure 5) has a typical high-adherencecurve (e.g. dry paved road): the cornering stiffness and longitudinal stiffness are 10 rad−1, thefriction peak value is 1 and is reached at a low slip value of 0.15.
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Table 1. Pacejka tyre parameters.
Tyre 1B 6.8488 D 1.0C 1.4601 E −3.6121
Tyre 2B 11.415 D 0.6C 1.4601 E −0.20939
Tyre 3B 15.289 D 0.6C 1.0901 E 0.86215
Tyre 4B 1.5289 D 0.6C 1.0901 E −0.95084
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
Tyre 1Tyre 2 Tyre 4
Tyre 3
s
m
Figure 5. Tyre–road friction coefficient μ as a function of the slip σ .
The second tyre (tyre 2, red curve in Figure 5) differs from the previous one only by thefriction peak value, which is reduced to 0.6, and can be considered typical of a wet paved road(no hydroplaning) or a reduced friction paved road.
The third (tyre 3, green curve in Figure 5) corresponds to a tyre on a dirt off-road surface:again the cornering/longitudinal stiffness is 10 rad−1 (like tyre 1 and tyre 2), the maximumfriction is 0.6 (like tyre 2) but it is reached at very high slip values, in other words there is nopeak in the operating range of slips and the curve is monotonic.
The last curve, tyre 4 (orange curve in Figure 5) is an extreme off-road curve (e.g.gravel), where the tyre is rolling on a soft surface that contributes to the reduction of cor-nering/longitudinal stiffness, reduction in the maximum friction and absence of a visible peakin the force vs. slip curve. The cornering/longitudinal stiffness is 1 rad−1, the maximumfriction is 0.6 (like tyre 2 and tyre 3) and is reached at very high slips (monotonic behaviour).
It is expected that typical off-road surfaces would be in between tyre 3 and tyre 4.
4. Simulations
The aim is to investigate how road surface (i.e. different tyre–road curves) and transmissionlayout affect minimum time manoeuvre strategies. The simulations are performed on a classicmanoeuvre for vehicle dynamics: the U-turn. In particular, a 180◦ turn with a curvature radiusof 10 m (on the centre line) connects two straight sections of 30 m. The road width is 10 m.
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Vehicle System Dynamics 1541
Table 2. Parameters of the vehicle model.
Parameter Symbol Value Unit
Gravity g 9.81 m/s2
Mass M 1300 kgTotal yaw inertia IG 2000 kg m2
Height of CoM h 0.50 mDistance of CoM from rear b 1.53 mDistance of CoM from front a 0.96 mAxle inertia IM 1.8 kg m2
Wheel radius r 0.28 mDrag coefficient (1/2ρCxA) KD 0.2107 kg/mTyre relaxation length σr 0.50 mSuspensions time-lag τN 0.25 sPropulsive torque distribution (FWD/RWD/AWD) kt 0/1/0.5Braking torque distribution kt 0.4Engine power Pmax 110 kW
Table 3. Boundary conditions (BC).
Variable Initial BC Final BC
V 55 kph Freeβ 0 0� 0 0ωR Trim FreeωF Trim FreefR 0 FreefF 0 FreesR 0 FreesF 0 FreenR Trim FreenF Trim Freess 0 10π + 60 msn 0 Freeα 0 0
The same vehicle is simulated with the four road–tyre adhesion curves discussed in theprevious section and with different transmission layout, namely FWD, RWD and all-wheel-drive (AWD). The transmission layout is simply changed by varying the parameter kt ofEquations (28) and (33) in propulsive condition. In particular, it is kt = 0 in case of FWD,kt = 1 in case of RWD and 0 < kt < 1 in case of AWD. In braking condition kt is the samefor all cases. The car parameters are reported in Table 2.
Boundary conditions are the same for all the simulations: at the starting line the vehicleis going straight on the centre of the road with a speed of 55 kph, while on the finish linethe vehicle is only requested to go straight and parallel to the centre line, see Table 3 for asummary of initial and final conditions. Finally, Table 4 summarises manoeuvre time and exitspeed for all simulated conditions.
Minimum time optimal manoeuvres in case of tyre 1 (paved dry) and different transmissionlayout are reported in Figure 6. All the three manoeuvres are characterised by low vehicle drift(β < 6◦), low tyre slips (indeed maximum friction coefficient is achieved at small slips) andthe trajectories are those typical of racing cars on paved circuits, where the driver uses mostof the available width of the road reaching the apex of the manoeuvre in the middle of the turnand exiting the curve close to the outer border. The RWD vehicle travels on a wider path whencomparing with the FWD, both when entering and exiting of the turn. The AWD trajectory
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Table 4. Summary of exit speed and manoeuvre time for the simulations inFigures 6–9.
Tyre Layout Manoeuvre time (s) Exit speed (kph)
Tyre 1 AWD 6140 73FWD 6223 70RWD 6510 67
Tyre 2 AWD 7375 67FWD 7601 57RWD 8094 51
Tyre 3 AWD 7400 66FWD 7860 55RWD 8369 49
Tyre 4 AWD 7850 59FWD 8572 50RWD 9215 44
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V(K
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Figure 6. Simulations with tyre 1, high-adherence (paved dry).
is in between the two. Under this road condition all the vehicles perform a sort of pendulummanoeuvre while entering the turn, to exploit the road width and reduce the vehicle trajectorycurvature. After the pendulum, the steer is always inward the curve (no counter-steering) andthe tyre slips remain on low values, close to the friction peak (no skidding). The AWD hasthe minimum time of travel (6.140 s), followed by the FWD (6.223 s) and RWD (6.510 s). Itis expected that the AWD is the fastest (indeed the traction effort is distributed over the twoaxles), while the reason the FWD is faster than the RWD is mainly related to the fact thatin the current vehicle the CoM is shifted toward the front axle (static load distribution front
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Vehicle System Dynamics 1543
to rear is 61/39), thus giving a traction advantage to the FWD layout over the RWD. Alsowhen comparing exit speeds, AWD ranks first (73 kph), followed by FWD (70 kph) and RWD(67 kph).
Minimum time manoeuvres for tyre 2 (paved wet) are depicted in Figure 7. Again they arecharacterised by low vehicle drift β, low tyre slips κ , λ and resemble again the trajectoriestypical of racing cars on paved track. There are no relevant differences among different tractionlayout until the exit of the turn, where the RWD vehicle travels again on a wider path. Theinitial pendulum which characterised tyre 1 is significantly less pronounced. The steer is onceagain always inward the curve. The quickest manoeuvre is performed by the AWD (7.375 s).Higher travel times are recorded for the FWD (7.601 s) and RWD (8.094 s). The same rankingholds for final speeds (AWD: 67 kph, FWD: 57 kph, RWD: 51 kph). As expected in all thecases with tyre 2 the travelling times are higher than those obtained with tyre 1 and the finalspeeds lower, due to the reduced friction peak coefficient.
When considering tyre 3 (loose off-road) some novelties appear. The minimum timemanoeuvres depicted in Figure 8 can be considered aggressive, with vehicle drift β up to45◦ for the AWD, 30◦ for the RWD and 15◦ for the FWD. The sideslip λR of the rear axlefollows the same trend of the vehicle drift β. On the other hand, the highest front axle slip λF
is reached by the AWD, followed by the FWD and RWD. In particular, the front axle sideslipof the RWD is almost null during the whole curve (ss > 30 m). It can be observed that in allthree cases the trigger to the drift is the driver’s steering action inward the curve while braking.This is consistent with the observations reported in [2] on the trail-braking technique used byrally drivers. In trail-braking, the driver takes advantage of the forward weight shift duringbraking, which increases the oversteering behaviour of the vehicle. In all the cases, the steer
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Figure 8. Simulations with tyre 3, off-road (dirt).
angle reaches a peak value close to 15◦ inward the curve at ss = 25 m. Then, depending ontransmission layout different strategies are adopted. In the case of RWD and AWD, the steerangle is reduced to zero (at ss = 30 m and ss = 35 m, respectively) while the vehicle is stillbraking. When passing through the null steer angle condition the driver throttles on to enter acounter-steering power-slide drifting. In the case of FWD instead, the driver slightly increasesthe steer angle inward the turn and throttles on shortly after (ss = 38 m). No pendulum ispresent on the initial straight. Similarly to the previous cases, RWD exploits the whole roadwidth while exiting the turn, although with significant drift values. On the contrary, AWDand FWD still exit the turn on similar trajectories, but this time they keep close to the roadcentre line, even though with very different vehicle drift β (AWD is much more drifting thanthe FWD) throughout most of the manoeuvre (25 m < ss < 75 m). The inspection of the steerangle is even more interesting. While the FWD is consistently inward the curve (almost steady17◦ along the turn), the AWD is slightly outward the curve (35 < ss < 65 m) and the RWD isconsistently outward the turn (counter-steering).
The considerations on time of travel and finish speeds remain those of tyre 1 and tyre 2: theAWD is the fastest with a travel time of 7.400 s and speed of 66 kph, followed by the FWD(7.860 s and 55 kph) and RWD (8.369 s and 49 kph).
The last case corresponding to tyre 4 (extreme off-road) is reported in Figure 9. The vehicledrift angles β denote aggressive manoeuvres, and reaches values higher than 60◦ for theAWD, above 50◦ for the RWD and 40◦ for the FWD, slightly before the middle of the turn(ss = 40 m). The manoeuvres are somehow close to those reported for the loose off-roadtyre with the differences among different layouts amplified. The inspection of the steer anglereveals three markedly different behaviours: steer inward the curve for the FWD (almost steady18◦ for most of the turn), counter-steering for RWD (almost steady 16◦ for most of the turn)
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Figure 9. Simulations with tyre 4, extreme off-road (gravel).
and almost null/slightly counter-steering angle for the AWD (mean value 3◦). In the case ofRWD and AWD the drifting is again triggered by a steering action inward the curve whilebraking, and then reduced (null close to ss = 30 m) while throttling to a power-slide condition(markedly counter-steering for RWD and slightly counter-steering/null steer for AWD). FWDuses a slightly different strategy: it initially steers inward the turn, then temporarily reducesthe steer at ss = 20 m, then steers again inward the curve and accelerates like RWD and AWDfrom ss = 30 m to reach an almost steady turning condition. Again the minimum manoeuvretime and maximum exit speed are achieved with the AWD (7.850 s and 59 kph), followed bythe FWD (8.572 s and 50 kph) and RWD (9.215 s and 44 kph). This time, all three vehicleshave exit trajectories far from the outer limit of the road and a kind of late apex corneringstrategy. The late apex line corresponds to exiting the corner near the inner limit of the road.Rally driving involves limited knowledge of the road geometry and optimal braking pointscompared with the highly rehearsed closed circuit driving. The late apex trajectory is favouredby rally drivers [21] as a safer approach to corners, to account for cases where they have miss-judged the braking point before the corner. If the driver is late in braking before the cornera wider trajectory through the corner is necessary to accommodate the higher entry speed. Adriver aiming for a late apex line can afford a wider trajectory (still within the limits of theroad) when carrying higher speed in the corner entry. It turns out that this approach is not onlysafer, but corresponds also to the minimum travel time strategy when the road is low frictionand soft (with respect to the tyre).
Summarising, the minimum time optimal cornering manoeuvre is characterised by smallvehicle drift and tyre slip angles (paved track-like strategy), when the tyre–road curve is char-acterised by a marked peak of the friction coefficient at low slip values, while is characterised
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by high-drift and tyre slip angles (rally like strategy) when the tyre–road curve is charac-terised by a monotonic friction vs. slips curve (e.g. friction peak at very high slip values).The maximum friction coefficient itself is not sufficient to change the minimum time optimalmanoeuvre from the low vehicle drift pattern to the high vehicle drift one. In other words,aggressive manoeuvres with high vehicle drift angles are minimum time optimal on off-roadconditions, where the friction curve is monotonic or with the friction peak placed at very hightyre slips, even outside the range of slips actually engaged. The effect of cornering stiffness isnot as important as the position of the friction peak. In other words, a reduction in the corner-ing stiffness amplifies the drifting behaviour introduced by tyre–road monotonic interactiontypical of off-road condition.
5. Conclusions
A simple yet effective vehicle model has been described and used to investigate minimumtime cornering strategies on different road surfaces with different transmission layout. Theminimum time manoeuvre on asphalt surfaces is characterised by low vehicle drift angles andtrajectories that resemble those of racing cars on paved track circuits, where the whole roadwidth is exploited. When moving to off-road conditions, the minimum time manoeuvre ischaracterised by aggressive, high-drift, even counter-steering (RWD and AWD) manoeuvresand rally like trajectories which keep the vehicle far from the outer border of the road.
Acknowledgement
This work was partially supported by an EPSRC First Grant Award – award number EP/I037792/01.
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