Design and Simulation of an ABS Control Scheme

for a Formula Student Prototype

Joao Pedro Ferrojoaoferro@tecnico.ulisboa.pt

Instituto Superior Tecnico, Lisboa, Portugal

May 2014

Abstract

This thesis addresses the design of an anti-lock braking system (ABS) seeking the implementationon a Formula Student racing prototype. The proposed scheme utilizes a cascade control architecture:a PID-type fuzzy controller with a Takagi-Sugeno-Kang fuzzy inference system is designed for wheelslip control in the outer loop, whilst a brake pressure PD controller is adopted in the inner loop. Awheel slip estimation solution, resorted to a complementary filter, is also developed. The performanceof the ABS is assessed with a full vehicle model, sustained by vehicle dynamic principles. The modelis integrated with brakeline dynamics and a tire friction model based on reliable experimental data.Straight line brake simulations are performed and results are evaluated in terms of braking efficiency,under different and variable conditions. A complete lap within a typical Formula Student circuit is alsosimulated with and without the use of ABS. Keywords: anti-lock braking system (ABS), FormulaStudent, fuzzy controller, PID controller, wheel slip estimation, vehicle model, tyre model

1. Introduction

The ABS is an active safety system present in mostof passenger cars and trucks. It prevents wheel lock-up and minimizes braking distance, while ensuringthe steering stability of the vehicle under hard brak-ing. From a typical tire friction curve, depicted inFig. 1, this is accomplished by controlling longitu-dinal wheel slip SX (Eq. 1) within a desired rangewhere peak longitudinal tire force is attained whilemaintaining adequate lateral force.

SX =ωWRL − vX

vX(1)

where ωW is the wheel-spin velocity, vX is the lon-gitudinal vehicle velocity, and RL is the tire loadedradius.

As the optimum value of SX do not coincide forboth tire longitudinal and lateral forces, respec-tively FXT and FY T , a compromise is frequentlynecessary near the absolute value of 0.2. Hence, theABS design is inherently related with the problemof wheel slip control. In practical terms, since ve-hicle speed vX cannot be directly controlled, wheelslip control is executed by controlling wheel angularvelocity ωW alone, which in turn is accomplishedby manipulating the brake pressure on the wheelcaliper pcal.

1.1. Background

Beyond the inestimable contribution to road safety,ABS may also be used in racecars as a driving-aidsystem for enhanced braking performance and re-duced tire wear. Although forbidden in most high-end racing disciplines like Formula 1, driving-aidslike ABS are allowed in Formula Student (FS) andteams may benefit from its implementation. FS isa motosport and engineering competition betweenuniversity teams across the globe, where studentsare challenged to design and build a single-seat rac-ing car. At Instituto Superior Tecnico, ProjectoFST team has recently built its fifth and most inno-vative prototype FST 05e. The availability of newdevelopment tools (sensors on the car, computercar model) allied with the constant seek for bestperformance possible has cleared the path for theimplementation of an ABS for the first time. Thepurpose of this work is thus to design and model aneffective and reliable solution for the ABS controlproblem that can also be simple and inexpensive toimplement on a FS prototype.

1.2. Literature Review

Several approaches to the ABS control problemhave been developed and documented in the liter-ature. Conventional threshold is the simplest andprimary technique used for wheel slip [1, 2]. How-

1

0− 0.04− 0.08− 0.12− 0.16− 0.20− 0.24− 0.28− 4,000

− 3,500

− 3,000

− 2,500

− 2,000

− 1,500

− 1,000

− 500

0

SX

FT(N)

FXT(µ = 0.65, Pacejka)FXT(µ = 0.65, TTC)FXT(µ = 0.2, Pacejka)FYT(α = − 6o, TTC)

UnstableRegion

StableRegion

SXref

Region

Figure 1: ABS action zones.

ever, the pronounced nonlinearity and uncertaintyof the system to be controlled, mainly due to thecomplex friction between tire and road interaction,has demanded for other adaptive and robust con-trol methods. Some of the most relevant and ef-ficient approaches were reviewed, namely the hy-bridization of conventional PID [3, 4], sliding-modecontrol [5, 6, 7, 8], fuzzy control [9, 10] and neuro-fuzzy control [11, 12, 13].

2. Vehicle and Tire Model

For the ABS design process and further simulation,a vehicle model with 14 DOFs is built upon themechanical, physical and aerodynamic principles ofvehicle dynamics [14, 15]. For an ABS applications,dynamics under braking and tire-road interactionare the most important aspects to consider. Thedeveloped full vehicle model is composed by fiveinterdependent sub-models, as represented in Fig. 2.

Horizontal dynamics is a 3-degrees of freedom(DOFs) model of vehicle position (xE and yE)and orientation (yaw angle ψ) in the inertialreference plane. It resembles the kinematics of

BrakelineDyamics

Tb WheelDynamics

SX TireModel

FXT

HorizontalDynamics a

VerticalDynamics

v

FZT

Fbp

Figure 2: Vehicle dynamics overview.

the vehicle’s CG as a result of the applied tireforces and moments from the Tire Model.

Vertical dynamics are modelled in 7-DOFs vi-brational model of the vehicle subjected toload transfer (proportional to acceleration) andaerodynamic forces (proportional to velocity).This model allows the transient computation ofthe individual tire vertical loads, and the ver-tical displacements of the sprung mass zs, un-sprung masses zu,FL, zu,FR, zu,RL and zu,RR,

2

roll φ, and pitch θ.

Brakeline Dynamics model the hydraulic brak-ing system, which for an ABS-equipped vehi-cle also include a hydraulic modulator. Givenan external input of pedal brake force, it out-puts the brake torque that is sent to the WheelDynamics.

Wheel dynamics applies the brake torque fromBrakeline Dynamics and the current tire lon-gitudinal force from the Tire Model and calcu-lates the resulting wheel angular acceleration.The real-time calculation of longitudinal slipratio SX that is returned to the Tire Model isalso part of the Wheel Dynamics.

Tire model is an empirical model of the com-plex friction dynamics between the tire and thepavement. It maps the tire forces and momentsas a function of a set of input variables, namelythe longitudinal slip ratio SX and tire verticalload.

2.1. Tire Model

Tires are the primary source of the tractive, brak-ing, and cornering forces/torques that provide thehandling and control of a car. An accurate tire fric-tion model is, thus, of absolute importance for ABSdesign. Three types of tire models were designedand compared:

Pacejka Model The most widely used semi-empirical tire friction model. It is given bythe so-called parametric Magic Formula (MF)(Eq. 2) [16], whose coefficients support a setparametric functions dependent on the rollingconditions (e.g., tire vertical load).

y = D sin[C arctan{Bx− E(Bx

− arctanBx)}] (2)

Y (X) = y(x) + SV (3)

x = X + SH (4)

Burkhardt Model Is a simpler parametric modelgiven by the following velocity-dependent para-metric expression [6, 17]:

F (x) =[A(1− e−Bx

)− Cx

]e−Dxv (5)

Neural Network Model Is a versatile self-developed model that takes advantage ofthe effectiveness of artificial neural network(ANN) for nonlinear curve fitting.

The aforementioned models are used to fit reli-able experimental data retrieved from the tests con-ducted at Calspan Tire Research Facility (TIRF).

For the parametric Pacejka and Burkhardt models,the least squares method approach(Eq. 6) is usedto find the set of parameters p = pi that minimizeserror to the experimental data. Neural Networkmodel fitting is achieved by iterative network train-ing.

minp

∑

i

r2i =

∑

i

(F (p, xi)− yi)2(6)

Fitting results, exhibited in Fig. 3 for the Pacejkamodel, indicate that this provided accurate longitu-dinal tire force FXT mapping for a wider range ofpossible SX , including the wheel lock-up case whereSX reaches -1. For the other two models, though agood fit is achieved within the limits of experimen-tal data availability, the accurateness of the modelis compromised for extrapolation beyond this limit.

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−4,000

−3,000

−2,000

−1,000

0

1,000

2,000

3,000

4,000

SX

FX

T(N

)

FZT = 350 lbsFZT = 150 lbsFZT = 250 lbsFZT = 50 lbs

Figure 1.7: Pacejka Model vs. TTC Data for FZT = {50, 150, 250, 350} lbs.

9

Figure 3: Pacejka Model vs. TTC Data for FZT ={150, 250, 350} lbs.

3. ABS Design

3.1. Proposed Approach

The cascade control architecture with two feedbackloops shown in Fig. 4 is proposed for the ABS prob-lem. On the outer loop, a PID-fuzzy wheel slipcontroller compares the actual wheel slip SX witha given reference and outputs a reference pressureprefb accordingly. This reference pressure is thensent to a conventional PD brake pressure controller,in the inner loop, which in turn sends a commandsignal u to the hydraulic modulator (actuator) sothe reference pressure is matched.

This cascade arrangement means that the pres-sure controller deals only with the fast brakelinedynamics, allowing the wheel slip controller to ex-clusively deal with the slower remaining vehicle dy-namics, enhancing the combined performance of thecontrollers. Moreover, the combination of the well-known PID principles with the intuitive theory and

3

Wheel Slip Controller

PressureController

pbref

BrakeSystem

WheelModel

Pressure Sensor

Wheel Velocity Sensor

AccelerometerVehicle and Tire Model

u Tb

ωW

pb

SX ref

aWheel Slip Estimator

SX

- -

Figure 4: Proposed control structure.

computational efficiency of fuzzy inference systems(FISs) provides a simple design and implementationsolution.

3.2. Brake Pressure Controller

Since the hydraulic modulator (actuator) only re-ceives binary signals for the inlet and outlet valves,the analog control signal u outputted by the pres-sure controller has to be previously converted in twoon-off signals uin and uout (Fig. 5). The pulse-widthmodulation (PWM) technique is used for the pur-pose. Two sawtooth wave signals with frequencyfc are generated, one ranging between [0 1] for theinlet valve, and [−1 0[ for the outlet signal. Fig-ure 6 depicts the conversion method for an examplecontrol signal.

0 5 · 10−2 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−0.5

0

0.5

1

1.5

Time [s]

uin

0 5 · 10−2 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−1.5

−1

−0.5

0

0.5

1

1.5

Time [s]

u

Sawtooth [01]Sawtooth [− 10]

0 5 · 10−2 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−0.5

0

0.5

1

1.5

Time [s]

uout

Figure 1.3: PWM implementation with fc = 50 Hz.

4

Figure 6: PWM implementation with fc = 50 Hz.

Due to the resulting dynamic characteristics ofthe inner loop plant, i.e., brakeline and hydraulicmodulator system, the regular ZieglerNichols meth-

ods for open and closed loop cannot be applied.Therefore, the model of the PWM conversion andthe plant is approximated by the linearisable func-tion (assuming pmc =∞):

dp

dt= a · tanh(b · u+ h) + v (7)

PD gains are then tuned, using Matlab’s PIDTuning tool with the approximate system, in orderto achieve the desired fast dynamics, near the ac-tuator limit, without overshoot, which could let thewheel slip to the unstable region of the tire frictioncurve. Integral effect is already inherent to the sys-tem. Figure 7 displays the controlled step responseof the original system and shows that it was possi-ble to eliminate the overshoot whilst maintaining avery fast response.

0 5 · 10−2 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time (s)

Pres

sure

(MPa

)

pref

pcal(Original)pcal(PID)

Figure 1.10: Step response with PD controller for pcal.

9

Figure 7: Step response with PD controller for pcal.

3.3. Wheel Slip Controller

The proposed PID-fuzzy controller resorts on azero-order Takagi-Sugeno-Kang (TSK) FIS withtwo input variables for the slip error eSX and itsvariation ∆eSX . The output variable is selected to

be the reference pressure variation ∆prefb , since theoptimal pressure varies with velocity, and also toensure adaptivity of the designed controller to pa-rameter uncertainty and disturbances that, other-

4

PressureController

pbref

BrakeLine

Pressure Sensor

ep u

pb

-

PWMConversion

HydraulicModulator

[uin uout] p BrakeLine

pcal Tb

BrakePedal

Master Cylinder

Fbp Fmc pmc

Brakeline Dynamics

Figure 5: Schematic of brakeline dynamics and pressure controller.

wise, would require an adaptive controller.As depicted in Fig. 8 before entering the FIS,

both input variables are multiplied by the gains Ke

and K∆e, which equivalent to the proportional andderivative gains of a PID controller. On the otherside, the output of the FIS is multiplied by the gainK∆p before integration.

Figure 8: PID-type fuzzy controller scheme [4].

For the TSK fuzzy inference system, a set of fuzzyrules is defined in the form

Rij : If eSX is Ai and ∆eSX is Bj

then yij = fk = Ck (8)

where the correspondence between the rule outputyij and the name of the function fk is given byTab. 1. The corresponding constant value Ck isthen listed in Tab. 2.

HHHHHAi

Bj Decreasing Steady Increasing

Negative If Is HZero Is H Ds

Positive H Ds Df

Table 1: Fuzzy rules for wheel slip controller.

Similar membership functions (MFs) for each ofthe fuzzy input sets are defined by Gaussian curves(Eq. 9) and evenly distributed within the normal-ized interval [−1 1] (universe of discourse), as de-picted in Fig. 9. Doing so allows the controllerdynamics to be quantitatively adjusted solely bytuning the PID gains outside the FIS, whereas thequalitatively aspect is ensured inside the FIS.

k Function Name Ck

1 Decrease-fast (Df) -12 Decrease-slow (Ds) -0.53 Hold (H) 04 Increase-slow (Is) 0.55 Increase-fast (If) 1

Table 2: TSK consequent function parameters foroutput ∆prefb .

µAi = µBj = Gaussian(x; c, σ) = e−(x−c)2

2σ2 (9)

− 1 − 0.8 − 0.6 − 0.4 − 0.2 0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1Negative Zero Positive

eSx

Degreeofmembership

− 1 − 0.8 − 0.6 − 0.4 − 0.2 0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1Decreasing Steady Increasing

∆ eSx

Degreeofmembership

Negative Positive

IncreasingDecreasing

Figure 9: Membership functions of FIS.

The final output is computed as a weighted aver-age of the output yk of each rule:

∆prefb =

∑ij wij .yij∑

ij wij(10)

where the firing strength wij of the rule is calculatedusing min as the AND (T-norm) operator [18]. Thenon-linear output surface of the designed FIS dis-played in Fig. 10. An almost linear relation betweeninputs and output is achieved when the wheel slip

5

error and its variation are near zero, i.e., the sys-tem is within the desired region depicted in Fig. 1.Outside this region the controller experiences satu-ration phenomena, as the reference pressure varia-tion ∆prefb is limited to prevent excessive overshootwhen the system is evolving far from the referenceslip or within/towards the stable/unstable region ofthe friction curve.

Figure 10: Output surfaces of FIS models for wheelslip controller.

Input and output gains Ke, K∆e are tuned sothe signal entering the FIS would range within thenormalized interval [−1 1]. As for the output gainK∆p, the value is chosen according to the actuatorlimits of pressure increase/decrease rate. Fine tun-ing is then performed until the desired controllerresponse is achieved. Finally, different values forreference wheel slip Sref

X near the peak longitudi-nal force are iteratively tested for minimum brakingdistance.

3.4. Wheel Slip Estimator

Recalling Eq. 1, wheel slip computation depends onboth wheel velocity ωW , available through direct,yet noisy measurements from the equipped wheelencoders, and vehicle inertial velocity vX , whichcannot be directly measured without impracticaland expensive equipment [19]. Hence, the prob-lem of estimating the longitudinal wheel slip SX iscrucial for an effective implementation of an ABScontroller.

Additionally to the controllers, a wheel slip esti-mator is designed using a complementary filter, i.e.,a Wiener filter equivalent to the stationary Kalmanfilter solution (Fig. 11). The estimation mergesinformation provided by sensors over distinct, yetcomplementary frequency regions [20]. In this par-ticular case, low frequency measurements providedby the mounted accelerometer are combined with

high frequency data from the wheel speed encoders:

ˆvS = K (ωmRe − (vS + vm)) (11)

where

ωW,m ≡ ωm = ωW + wω (12)

vX,m ≡ vm = ∫(aX + wa) (13)

Figure 11: Block diagram of complementary filter.

As depicted in Fig. 12, a good estimation isachieved despite the noisy measured data. More-over, different choices of gain K translate in a com-promise between fast convergence and steady-statesmoothness. Higher values favor high frequencymeasurements from wheel encoders and thus a nois-ier response is attained.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

Times (s)

v S(m/s)

Real (vS = −5 m/s)MeasuredEstimated (K = 100)

Estimated (K = 10)

Figure 1.16: Slip velocity vS estimation results with complementary filter.

16

Figure 12: Slip velocity vS estimation results withcomplementary filter.

The estimate longitudinal slip ratio is then cal-culated as:

SX =vS∫ am

(14)

3.5. ABS Triggers

To ensure the correct operation of the ABS, thefollowing on and off thresholds are defined:

Minimum brake pressure Ensures that theABS is on when, and only when the brake

6

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40

10

20

30

Velo

city

(m/s

)

ω.RLvX

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40

10

20

30

40

xE

(m/s)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4−40

−20

0

20

aX

(m2/s

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4−1

−0.8

−0.6

−0.4

−0.20

SX

SrefX

SX

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40

5

10

15

20

Time (s)

pca

l(M

Pa) pref

pmcpcal

Figure 1.1: Comparative simulation results with (solid) and without (dashed) ABS.

4

Figure 13: Comparative simulation results for constant µ, with (solid) and without (dashed) ABS.

pedal is stepped. A threshold of 0.1 MPa forthe caliper pressure pcal is chosen.

Wheel slip and wheel slip rate trigger Ifwheel slip is beyond reference slip (unstableregion) or decreasing at a high rate (to-wards lock-up), ABS should start the controlcycle. Otherwise, normal brake applies.Trigger values are defined as SX = −0.1 and∆SX = −2.

Low velocity trigger Since limvX→0 SX = 0−00 ,

a minimum velocity of 10 km/h is defined, be-low which the ABS is switched off avoids in-creasingly bigger oscillations around the refer-ence wheel slip that exceedingly degrade ABSefficiency and could even drive the system to-wards instability.

4. Simulation Results

The performance of the proposed ABS controlscheme is tested on the aforementioned full vehicleand tire model with a series of simulations on theSimulink environment. Data for the FST 05e pro-totype is taken from experimental measurements orCAD tools, and introduced in the model for a ac-curate simulation.

4.1. Straight Line Hard Braking

The first set of simulations is performed for a simplehard braking manoeuvre: initially the vehicle trav-els at constant velocity vX = 100 km/h; at t = 0.5 s,the driver steps hard on the pedal brake (so thatTb � FXT .RL) and keeps it until the vehicle stops.

With a constant road coefficient of friction µ =0.80, results over time with and without ABS arecompared in Fig. 13. Up to t = 0.5 s wheel tangen-tial velocity ω.RL equals the vehicle velocity, i.e.,

7

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40

10

20

30

Velo

city

(m/s

)

ω.RLvX

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40

10

20

30

40

xE

(m/s

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4−40

−20

0

20

aX

(m2/s

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4−1

−0.8

−0.6

−0.4

−0.20

SX

SrefX

SX

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40

5

10

15

20

Time (s)

pca

l(M

Pa) pref

pmcpcal

Figure 1.2: Simulation results with varying µ.

1.2.2 Varying µ

To test the adaptivity of the proposed ABS controller to real-time variations of the surface type, the

following values for the road coefficient of friction µ have been introduced as input parameter on the

Pacejka tire model (Section ??):

µ = 0.8 for t < 0.75s

µ = 0.5 for 0.75 ≤ t < 1.25s

µ = 0.8 for t ≥ 1.25s

(1.1)

The resulting plots on Fig. 1.2 evidence a fast adaptation of the controller with varying coefficient

5

Figure 14: Comparative simulation results for varying µ, with (solid) and without (dashed) ABS.

−60 −55 −50 −45 −40 −35 −30 −25 −20 −15 −10 −5 0 5 10−50

−45

−40

−35

−30

−25

−20

−15

−10

−5

0

5

10

15

20

xE (m)

y E(m

)

Figure 1.9:

11

(a) Without ABS.

−60 −55 −50 −45 −40 −35 −30 −25 −20 −15 −10 −5 0 5 10−50

−45

−40

−35

−30

−25

−20

−15

−10

−5

0

5

10

15

20

xE (m)

y E(m

)

Figure 1.10:

12

(b) With ABS.

Figure 15: Brake pedal actuation within lap perimeter.

wheel is in free-rolling condition with SX = 0. Forthe case without ABS (dashed line), when the driverstarts braking the wheel rapidly decelerates, until

it locks-up at SX = −1. If the vehicle is equippedwith the designed ABS controller, wheel velocitydecreases until the reference slip Sref

X = −0.16 is

8

matched and efficiently tracked.Regarding the aX plot, it is noticeable that for

both cases a minimum acceleration is soon reached.As expected, this occurs when SX = Sref

X , i.e.,when SX crosses the optimal value for which peaklongitudinal force is attained , as shown in Fig. 1.However, the ABS is able to keep this minimumthenceforth, thus resulting in a steeper velocityplot, and a smaller braking time and distance(Tab. 3).

Observing the pressure plot for a braking withoutABS, pressure on the caliper quickly matches thepressure on the master cylinder (a certain dynamiccan be noticed). On the other hand, the ABS man-aged to initially increase pressure at the maximumrate allowed by the actuator. The increase rate isthen slowed to avoid overshot when the wheel slipapproaches the reference value.

Distance (m) Time (s)Without ABS 21.87 1.63

With ABS 15.21 1.17∆ -6.66 -0.46

%∆ -30.5 % -28.2 %

Table 3: Braking distance and time results for con-stant road coefficient of friction, with and withoutABS.

4.2. Varying Road µ

To test the adaptivity of the designed ABS con-troller to real-time road-type changes, varying val-ues for the road coefficient of friction µ have beenintroduced as input parameter on the Pacejka tiremodel according to Eq. 15.

µ = 0.8 for xE < 22 m

µ = 0.5 for 22 ≤ xE < 28 m

µ = 0.8 for xE ≥ 28 m

(15)

The resulting plot on figure 14 evidence a fastadaptation of the controller with varying coefficientof friction. At x = 22 m, the wheel slip controllermanages to compensate for the sudden decrease inthe available tire force, which lead the wheel to-wards lock-up, by immediately diminishing the ref-erence pressure. After a brief acceptable undershot,reference pressure stabilizes and the reference wheelslip is restored. At x = 28 m, the opposite phe-nomenon occurs and brake pressure can again bemoderately increased to maximize tire force. Evenfor varying conditions, the effectiveness of the ABSon reducing braking distance is once again demon-strated in Tab. 4.

Distance (m) Time (s)Without ABS 24.61 1.77

With ABS 18.42 1.37∆ -6.19 -0.40

%∆ -25.2 % -22.6 %

Table 4: Braking distance and time results for vary-ing road coefficient of friction, with and withoutABS.

4.3. Complete Lap

The absolute advantage of the ABS on a FS proto-type is more evident across a complete lap in a typ-ical FS circuit. As described in Sec. 1, the absenceof an ABS may lead to wheel lock, i.e., SX = −1,for which no lateral force is developed by the tiresand longitudinal force is suboptimal. To reproducethis effect, an optimal GG diagram of the FST 05eis narrowed in the pure braking and combined ac-celeration regions. Figures 15 display a sequenceof curves made with and without ABS. In the for-mer case, one can noticed that the driver is able tobrake later and while cornering. At the end of thelap, results indicate a reduction of 4 s.

Without ABS With ABS ∆ %∆52.17 s 48.31 s 3.86 s 7.4%

Table 5: Lap time results with and without ABS.

5. Conclusions

This paper reported the design and implementationof an ABS solution for a FS prototype. The con-trol scheme resorts on a cascade control architecturecontaining PID-type fuzzy system for wheel slip reg-ulation, a conventional PD controller for brake pres-sure reference tracking. The ABS is designed andtested in a 14-DOFs vehicle model integrated witha Pacejka tire friction model and wheel slip estima-tor.

A set of hard braking simulations evidence theeffectiveness of the proposed approach on reducingthe braking time and distance. Results exhibit afast response and smooth tracking a reference wheelslip value. Furthermore, the robustness of the con-troller against sudden variations of the road coef-ficient of friction µ, both high-to-low and low-to-high, was also successfully demonstrated. A finalsimulation within a complete lap around an typicalFS circuit conclusively confirmed the competitiveadvantage of an ABS-equipped FS prototype.

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5.1. Future work and research

As stated in Sec. 1, further implementation of theproposed controller is a concern. Having that mindduring the modelling and design phases, specialattention has already been given to some impor-tant practical issues: sampling time, PWM conver-sion and noise filtering were all considered. Hence,hardware-in-the-loop validations and hardware re-alization on a real FS prototype are the naturalsteps to follow.

Further research is also recommended on the is-sues of stability over asymmetric µ surfaces, andcontrol complementary of ABS and the equippedelectric motor or an active suspension system.

References

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[4] C.-K. Chen and M.-C. Shih, “PID-Type FuzzyControl for Anti-Lock Brake Systems withParameter Adaptation,” JSME InternationalJournal Series C, vol. 47, pp. 675–685, 2004.

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