surface prediction and control algorithms for anti-lock brake system

Transportation Research Part C 21 (2012) 181–195

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Transportation Research Part C

journal homepage: www.elsevier .com/locate / t rc

Surface prediction and control algorithms for anti-lock brake system

Rishabh Bhandari, Sangram Patil, Ramesh K. Singh ⇑Department of Mechanical Engineering, Indian Institute of Technology Bombay, Mumbai, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 29 May 2010Received in revised form 8 September 2011Accepted 8 September 2011

Keywords:Anti-lock brake systemSurface identificationABS controlSliding mode controller

0968-090X/$ - see front matter � 2011 Elsevier Ltddoi:10.1016/j.trc.2011.09.004

⇑ Corresponding author.E-mail address: [email protected] (R.K. Singh

Anti-lock brake system (ABS) has been designed to achieve maximum deceleration by pre-venting the wheels from locking. The friction coefficient between tyre and road is a nonlin-ear function of slip ratio and varies for different road surfaces. In this paper, methods havebeen developed to predict these different surfaces and accordingly control the wheel slip toachieve maximum friction coefficient for different road surfaces. The surface predictionand control methods are based on a half car model to simulate high speed braking perfor-mance. The prediction methods have been compared with the results available in the lit-erature. The results show the advantage of ABS with surface prediction as compared toABS without proper surface identification. Finally, the performance of the controller devel-oped in this paper has been compared with four different ABS control algorithms reportedin the literature. The accuracy of prediction by the proposed methods is very high witherror in prediction in a range of 0.17–2.4%. The stopping distance is reduced by more than3% as a result of prediction for all surfaces.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Anti-lock brake system or ABS is a safety system that preserves the tyre–road rolling friction by preventing the wheelsfrom locking up. Therefore, the driver can control the direction of the vehicle by steering input, which is extremely difficultwith the locked wheels. While ABS typically decreases the stopping distances on dry and slippery surfaces, it can also in-crease the braking distance under certain conditions. Specifically, ABS is known to increase the braking distance on loosesurfaces such as snow and gravel. However, this shortcoming could potentially be mitigated by developing new control strat-egies for ABS systems. In order to develop effective control strategies, the vehicle dynamics and the tyre–surface interactionneed to be considered.

A number of vehicle dynamics models for braking simulations have been reported in the literature. Notable among themare quarter car model used by Baslamisli et al. (2007, p. 217–232) and half car model used by Harifi et al. (2008, p. 731–741).A full car model can be used to predict braking on a curved path, i.e., it can capture the effect of side slip and lateral loadtransfer. However, high speed braking and slipping is generally observed in straight line braking. Consequently, a half carmodel would suffice in such cases.

Various tyre models have been proposed in the literature for longitudinal motion of the car. Harifi et al. (2008, p.731–741) have proposed a model that describes the coefficient of friction between tyre and road as a function of slip ratioand velocity. Park and Lim (2008, p. 290–295) used a simplified version of this model and expressed coefficient of friction asa function of slip ratio only. Anwar (2006, p. 1101–1117) has used piecewise linear approximation as a tyre model and Onizet al. (2007, p. 90–95) have also used a model with coefficient of friction expressed in terms of slip ratio.

Different control strategies have been employed to control the slip ratio of the vehicle. Anwar (2006, p. 1101–1117) hasproposed a nonlinear sliding mode type controller for slip regulation. Harifi et al. (2008, p. 731–741) have used a sliding

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Nomenclature

g acceleration due to gravitya distance from CG of front axelb distance from CG of rear axelH height of massmtot total mass of vehiclem massmf front unsprung mass of vehiclemr rear unsprung mass of vehicleJ moment of inertia of wheelR radius of tyreT torquek slip ratiol coefficient of frictionv velocity of vehiclex wheel rpmx longitudinal displacementFx longitudinal forceFz normal forceK constant

SubscriptsF frontR rearS sprungD desiredB brakingP proportionalI integralD derivative

182 R. Bhandari et al. / Transportation Research Part C 21 (2012) 181–195

mode control along with an integral switching surface for chatter reduction. Neural networks and Fuzzy logic has also beenimplemented to control the slip ratio in ABS control strategies in various cases.

Braking under changing road conditions requires a methodology for identifying the transition and adequately modifyingthe control strategy. Zhang et al. (2008, p. 493–497) have proposed a neuro-adaptive unit to compensate external distur-bances and uncertainty in vehicle dynamics due to the variation in road conditions. They have studied the braking perfor-mance of the proposed control strategy under changing road conditions.

Various methods have been reported in the literature for prediction of road conditions. Choi (2008, p.996–1003) used aProportional Differential (PD) controller with a phase lag on rear wheel to predict the slip ratio for maximum friction coef-ficient and the reference angular velocity for the front wheel. Once the angular velocity is determined, a PD control is used tocontinuously control the front wheel velocity. Note that the above method involves oscillation about the peak for checkingthe peak which results in loss of braking force during prediction. Oniz et al. (2007, p. 90–95) used a Grey predictor along witha sliding mode control to predict the road conditions. Such methods are computationally intensive and need higher process-ing power. Tanelli et al. (2009, p.891) have proposed some other prediction methods and presented results for various sur-face conditions.

Based on the above discussion, it can be seen that limited work has been reported in the literature for a vehicle transi-tioning from one road condition to another during braking. Consequently, this paper is focused on implementing computa-tionally efficient algorithms for identification of surface condition followed by an effective anti-lock braking control strategyfor changing surface conditions. This paper presents the vehicle dynamics and tyre–surface interaction models followingwhich the methods to identify surface characteristics are described. Based on these models, the control strategies requiredto implement the ABS system have been developed. Finally, the results obtained from the simulations are presented andcompared with experimental and simulation results reported in other studies on anti-lock brake systems (see Fig. 1).

2. Vehicle dynamics

The development of a vehicle and tyre model is the first step towards the development of ABS strategy. Once the model isready, a sliding mode control is used to control the wheel slip ratio at a desired position. The desired position varies for dif-ferent surfaces and this position is determined by the prediction algorithm. This gives the desired slip ratio for maximum

Dynamics and Tyre Model

Prediction Algorithm Controller Response

Fig. 1. Steps used to develop ABS strategy with prediction.

R. Bhandari et al. / Transportation Research Part C 21 (2012) 181–195 183

friction coefficient (kmax) via a PID controller as input to the sliding mode controller as shown in Fig. 2. The sliding modecontrol then gives braking torque for front and rear wheels (TBf and TBr). Detailed description of the dynamics and controlshas been given in following sections.

2.1. Half car model

As mentioned previously, half car model is adequate for straight line braking. It also takes into account the longitudinalweight transfer. As shown in Fig. 3, the half car or bicycle model divides the car into two parts and therefore does not takeinto account side slip and cornering forces. The model has been used by Harifi et al. (2008, p. 731–741) for the developmentof ABS strategies. This model assumes longitudinal braking conditions and steering effects are neglected.

The equations demonstrating the vehicle dynamics involved in the half car model are,

_v ¼ �glðkf Þm1 þ lðkrÞm2

mtot � lðkf Þm3 þ lðkrÞm3ð1Þ

_xf ¼�TBf þ lðkf Þm1Rg � lðkf Þm3R _v

2Jfð2Þ

_xr ¼�TBf þ lðkrÞm2Rg þ lðkrÞm3R _v

2Jfð3Þ

where R is the radius of wheels and Jf and Jr are the moment of inertia of front and rear wheels respectively.

m1 ¼bmtot

ðaþ bÞ ; m2 ¼amtot

ðaþ bÞ ; m3 ¼ðmf hf þmshs þmrhrÞ

ðaþ bÞ ð4Þ

kf ¼ 1�xf Rv ; kr ¼ 1�xrR

v ð5Þ

2.2. Friction coefficient as a function of wheel slip

The model used by Park and Lim (2008, p. 290–295) is based on a tyre friction model developed by Burckhardt andReimpell (1993). It gives the friction coefficient between the tyre road as a function of wheel slip as:

lðkÞ ¼ ðC1ð1� e�c2kÞ � C3k ð6Þ

where C1, C2 and C3 are constants for characterizing different road conditions. C1 is the maximum value of the friction curve,C2 gives the friction curve shape and C3 represents the difference between the maximum value of the friction curve and thevalue when slip ratio is one. The values of the constants for various surfaces are given in Table 1. These values have beenobtained from the data used by Harifi et al. (2008, p. 731–741) and some additional values have been used for the same mod-el by Oudghiri et al. (2007, p. 13–28). Fig. 4 shows a plot of the friction coefficient according to this model for all the surfacesmentioned in Table 1.

TBf , TBr

Vehicle dynamicsPrediction Algorithm

Prediction (PID) Control

(Sliding Mode Control)

Controller

λ max

Fig. 2. Flow chart of representing prediction and control loop.

Fig. 3. Half car model, free body diagram (Harifi et al., 2008, p. 731–741).

Table 1Values of constants for various surfaces (Harifi et al., 2008, p. 731–741, (2007, p. 13–28)).

Surface conditions C1 C2 C3

Dry asphalt 1.2801 23.99 0.52Wet asphalt 0.857 33.822 0.347Dry concrete 1.1973 25.168 0.5373Cobble wet 0.4004 33.708 0.1204Cobble dry 1.3713 6.4565 0.6691Snow 0.1946 94.129 0.0646Ice 0.05 306.39 0

Coe

ffic

ient

of

fric

tion

(μ)

Slip Ratio (λ)

Fig. 4. Friction coefficient as a function of slip ratio for different surfaces.


3. Surface prediction algorithms

Prediction algorithms are required to identify the surface conditions. A robust ABS controller has been proposed byBaslamisli et al. (2007, p. 217–232) using the quarter car model. The proposed method uses the half car model and priorinformation (shown in Fig. 4) about the nature of friction curves for prediction of the surface the vehicle is treading on ata given instance. The first step is to move to predetermined values of slip ratios and then compare the measured coefficientof friction with the value of coefficient of friction based on tyre friction model given in Fig. 4 to determine the existing surfaceconditions. The two methods have been explained in detail below.

3.1. One point prediction method

In the one point prediction method, a reference value of slip ratio (k0) is selected initially. At this value of slip ratio, thedeceleration of the vehicle is measured using an accelerometer. This deceleration at a particular slip ratio is then related todifferent surfaces. Once the surface is known, the slip ratio for maximum friction coefficient is also known.


There are three major constraints while selecting a reference wheel slip ratio, k0. Firstly, it should be close to the range ofslip ratio for maximum friction coefficient of all surfaces. This ensures that the loss of braking during prediction is minimalirrespective of road surface. Secondly, the prediction of slip ratio for maximum friction coefficient (kmax) should result inminimal error for all surfaces. Finally, since there is a sharp drop in coefficient of friction before the slip ratio for maximumfriction coefficient in all curves (except dry cobblestone), therefore, it should be ensured that the selected value of slip ratio isgreater than that for all curves. If the example of dry asphalt is considered then, for a slip ratio value which is 0.1 less than theoptimal the reduction in friction is 14.5% as compared to only 2.7% reduction when the slip ratio value is 0.1 more thanoptimal.

Using the data for surfaces available in the tyre model and considering the above constraints a suitable range of slip ratio(0.2–0.25) has been selected. The values of friction coefficient for this range are computed for all the road surfaces. Thesefriction coefficients when related to the slip ratio for peak friction coefficient give a linear relation for each value of k asshown in Fig. 5. A best fit line is then used to find the value of k with minimum standard deviation. This value is used ask0. The standard deviations for the given range are shown in Table 2.

3.1.1. Relations between kmax and deceleration at k0

Practical measurement of acceleration is pretty straightforward with the available accelerometers in any vehicle. Thisdeceleration can be easily related to l0 using the vehicle dynamics model (Eqs. (1)–(3)).

€x ¼ �glðkf Þm1 þ lðkrÞm2

mtot � lðkf Þm3 þ lðkrÞm3ð7Þ

Setting both kf and kr as k0, the equation simplifies to

€x ¼ �glðk0Þ ð8Þ

By assuming linear variation between the values of l0 observed at k0 for different surfaces and the peak coefficient of fric-tion lmax for these surfaces a linear relation of the form lmax = A � l0 + B (Eq. (9)) can be fit, where the constants A and B arefound by fitting a best fit line to the data points. Similarly, a second linear relation (Eq. (10)) can be obtained for each of theindividual surfaces, which relates the peak coefficient of friction to the peak slip ratio.

The following relations can be obtained for the four surfaces by assuming linear variation

lmax ¼ A � l0 þ B ð9Þlmax ¼ C � kmax þ D ð10Þ

where l0 is the coefficient of friction at k0, lmax is the maximum coefficient of friction for a given surface and kmax is the slipratio at which this coefficient of friction is achieved. The constants obtained after fitting a linear best curve for the four sur-faces are: A = 0.9973, B = 0.0111, C = 8.9430 and D = �0.3516.

By combining the Eqs. (8)–(10), slip ratio for peak coefficient of friction can be related to deceleration as:

kmax ¼ Aoþ Bo €x; where A0 ¼ 0:041 and B0 ¼ �0:0114 ð11Þ

The prediction method works well for most surfaces, however for surfaces like, cobblestone; the prediction method is notsuitable. Considering that anti-lock brakes are useful at high speeds, the method is very useful. Table 3 shows the accuracyof prediction by the algorithm for all surfaces.

As shown in Fig. 6, the range of the slope of the best fit equation with variation in the reference slip ratio used for onepoint prediction is 0.0024 which corresponds to a percentage change of 2.15% which is not significant. Hence it can be said

Slip

rat

io a

t pea

k co

effi

cien

t of

fric

tion

(λm

ax)

Coefficient of friction at λ = 0.2 (μ0)

Fig. 5. Slip ratio at peak coefficient of friction vs. coefficient of friction at k0.

Table 2Standard deviation from best fit line with different slip ratios.

k 0.2 0.21 0.22 0.23 0.24 0.25Std. deviation 0.0041 0.0046 0.0051 0.0054 0.0057 0.006

Table 3Predicted values using one point prediction method.

Surface conditions k For peak l Predicted value of k Error in prediction (%) Peak l l At predicted k Error in l (%)

Dry asphalt 0.1700 0.1716 0.94 1.1700 1.1700 0.01Dry concrete 0.1600 0.1623 1.44 1.0900 1.0900 0.00Wet asphalt 0.1308 0.1291 1.30 0.8013 0.8013 0.00Cobble wet 0.1400 0.0830 40 0.3800 0.3660 3.66Cobble dry 0.4000 0.1361 65.97 1.0000 0.7107 28.93Snow 0.0600 0.0613 2.17 0.1900 0.1900 0.00

Fig. 6. Sensitivity analysis on the slope of the best fit line.


that the choice of the reference slip ratio at 0.2 is relatively robust. The slope referred to in Fig. 6 corresponds to the curveparameter A in Eq. (9).

Although, this method could suffice for practical applications but a generalized approach which could account for all sur-faces needs to be developed. This is because the method does not work well on surfaces like wet and dry cobblestone. Thenext section presents a generalized prediction methodology based on three point prediction.

3.2. Three point prediction method

To overcome some of the limitations of the one point method, a three point method has been developed. This algorithmtests the deceleration at three different predefined slip ratios and uses these values to predict the surface characteristics.

Each surface can be defined by the equation as shown in the tyre model by Burckhardt and Reimpell (1993).

lðkÞ ¼ ðC1ð1� e�C2kÞ � C3kÞ ð12Þ

If the values of these three constants are known, the peak slip can be located. At the maxima and hence the peak of theslip ratio curve

_lðkmaxÞ ¼ ðC1C2e�C2kmax Þ � C3 ¼ 0 ð13Þ

Therefore, slip ratio for maximum friction coefficient is given by

Table 4Predicted values using three-point prediction method.

Surface conditions k For peak l Predicted value of k Error in prediction (%) Peak l l At predicted k Error in l (%)

Dry asphalt 0.1700 0.1683 1.00 1.1700 1.1700 0.00Dry concrete 0.1600 0.1587 0.81 1.0900 1.0900 0.00Wet asphalt 0.1308 0.1299 0.69 0.8013 0.8013 0.00Cobble wet 0.1400 0.1396 0.29 0.3800 0.3800 0.00Cobble dry 0.4000 0.4096 2.4 1.0000 0.9998 0.02Snow 0.0600 0.0601 0.17 0.1900 0.1900 0.00


kmax ¼lnðC1C2=C3Þ

C2ð14Þ

For large values of C2k, the exponential term in can be neglected, hence the first two points are chosen at high slip ratiovalues. The values of k1 and k2 chosen for simulation are 0.6 and 0.5 respectively.

C1 ¼lðk1Þk2� lðk2Þk1

k2� k1ð15Þ

C3 ¼lðk1Þ � lðk2Þ

k2� k1ð16Þ

Now the third point is chosen at a small value of k and is used to evaluate C2. The value of k3 used for simulations is 0.1.A lower value needs to be chosen for k3 to capture the effect of the exponential term in Eq. (12).

C2 ¼ln C1

C1�lðk3Þ�C3k3

k3ð17Þ

where l(k1), l(k2) and l(k3) are coefficient of friction at the respective slip ratios.Using this method, prediction can be done for all surfaces; however the accuracy of prediction depends upon the values of

k1, k2 and k3 chosen. Table 4 presents the range of errors in prediction for the above selection of k for all surfaces.Note that this prediction methodology yields better prediction than one point method for all surfaces. The three point

method yields significantly reduced prediction errors (e.g. 40–0.29% for cobblestone wet and 65.97–2.4% for cobblestonedry). The only drawback is that the number of cycles used for prediction results in a loss in braking and hence in instanceswhen surface does not change there may be an unnecessary loss of braking while checking for the surface change. The nextsection deals with the development of controllers used to achieve the desired values of slip ratio.

4. Control algorithms

There are two loops for the control algorithm and prediction algorithm as shown in the Fig. 7. The goal of the predictionloop is to converge and gather data at the desired slip ratio set points based on which method is being used. The accelerationvalues used in the prediction loop during simulations are obtained from the vehicle dynamics model. Switching between thetwo loops is generally based on time. Initially, the controller starts the prediction algorithm and if the error in slip ratio is lessthan a desired value, then it predicts the surface and switches to the control loop. Otherwise the prediction loop continuesfor a predetermined number of cycles and predicts the surface. Once the surface prediction is completed, a new value for kmax

is determined. This value is now used as the desired value for the slip ratio in the control loop. Note that the slip ratios inboth the loops are determined by solving the system of equations (Eqs. (1)–(3))) which govern the vehicle dynamics.

The selection of the number of cycles for which the prediction and control loop are executed is very important. The pre-diction loop should take as fewer cycles as possible. The minimum value of prediction cycles (M) is restricted by the settlingtime required.

A higher number of control cycles ensures that braking efforts are not reduced due to regular predictions. However, a verylarge number may delay the prediction and even when the surface conditions change it might go undetected for some time.Thus, a number is chosen that results in acceptable loss in braking due to prediction but does not delay the prediction loop bymore than a certain time interval. For this braking distance of a car going at 30 m per second when brakes are applied iscompared as shown in Table 6. The vehicle used for all the simulations has the following parameters.

Based on the results in Table 6, a smaller value of control cycles (N), results in better braking during road change, how-ever, there is a loss in braking when there is no road change. 0.20% Loss in braking is not big trade off as compared to theadvantages that come with quick predictions. Therefore, a smaller number of control cycles can be used during braking.

Now there is a need to develop control strategies for both prediction loop and control loop. The following flowchart rep-resents the entire process. Sliding mode control has been widely used for ABS applications but it demands knowledge ofvehicle dynamics. Therefore, it cannot be used for prediction but can be used in the control loop.

x

x

Fig. 7. Prediction and control loop.

Table 5Parameters of the vehicle used for simulations.

Symbol Quantity Value

G Acceleration due to gravity 9.8 m/s2

A Distance from centre of gravity to front axle 1.186 mB Distance from centre of gravity to rear axle 1.258 mha Height of sprung mass 0.6 mhf Height of front unsprung mass 0.3 mhr Height of rear unsprung mass 0.3 mmtot Total mass of vehicle 1500 kgma Sprung mass of the vehicle 1285 kgmf Front unsprung mass of vehicle 96 kgmr Rear unsprung mass of vehicle 119 kgJf Moment of inertia of front wheel 1.7 kg m2

Jr Moment of inertia of rear wheel 1.7 kg m2

R Radius of tyre 0.326 mTmax Constraint to the brake torque 8000 N m


4.1. Sliding mode control

Sliding mode control, or SMC, is a form of variable structure control (VSC). It is a nonlinear control method that alters thedynamics of a nonlinear system by application of a high-frequency switching control. The state-feedback control law is not acontinuous function of time. Instead, it switches from one continuous structure to another based on the current position inthe state space. The motion of the system as it slides along these boundaries is called a sliding mode. For the case of con-trolling the wheel slip, first a sliding surface is defined

S ¼ ðk� kdÞ ð18Þ_s ¼ �gðsÞ ð19Þ

where g is the convergence factor

sgnðsÞ ¼1 for s > 00 for s ¼ 0�1 for s < 0

8><>: ð20Þ

Also assuming kd is constant, _kd ¼ 0.

Table 6Effect of control cycles on the braking distance.

Value ofN

Delay between consecutiveprediction (s)

Braking distance withprediction

Braking distance withoutprediction

Loss in braking with prediction(%)

Dry asphalt for 30 m and then wet asphalt500 0.07 43.51 43.87 �0.834000 0.42 43.57 43.87 �0.69No change500 0.07 39.27 39.19 0.204000 0.42 39.21 39.19 0.05


Now this value is replaced to get the brake torque in terms of sliding mode parameters. There are various ways to reducechattering; one of them is to introduce the sat function instead of the sgn function. Therefore,

_s ¼ �g satðs=uÞ ð21Þ

where g is the convergence factor and u is the boundary layer thickness (u > 0)

satðs=uÞ ¼sgn s for jSj > uS=u for jSj 6 u

�ð22Þ

The major advantage of sliding mode for ABS applications is that it makes the system robust with respect to systemparameter variations, un-modelled dynamics and external disturbances (Fig. 8).

The control strategy can be modelled based on the sliding surface defined in the previous section and as given in byTanelli et al. (2007, p. 593–611) .

_S ¼ _k� _kd ð23Þ

Here _kd is the desired slip ratio, which being constant gives _kd ¼ 0.As defined earlier in the section,

_k ¼ �g satðs=/Þ ð24Þ

Also,

kf ¼v �xf R

v ð25Þ

_kf ¼ �Rxf

v_xf

xf�

_vv

� �ð26Þ

Putting _x and _v from the half car model into the above equation gives us

_kf ¼ �Rxf

v

�TBf þ lðkf Þm1Rg � lðkf Þm3R _v2Jf xf

þgv

lðkf Þm1þlðkrÞm2

mtot�lðkf Þþm3lðkrÞm3

0BBBB@

1CCCCA ð27Þ

Now, using _k ¼ �g satðs=uÞ; TBf in terms of sliding control parameter is found to be

TBf ¼ lðkf ÞRðm1 �m3€xÞ þ 2Jf g_vv þ

2vJf gR

satkd � k

/

� �ð28Þ

Similarly,

TBr ¼ lðkrÞRðm1 �m3€xÞ þ 2Jrxrg_vv þ

2vJrgR

satkd � k

/

� �ð29Þ

These equations can now be used to control the slip ratio at the predicted values. The variable sliding mode parametersare / and g. These can be tuned to get optimum results. However, unless the surface is not known, applying sliding modecontrol is not possible as the desired torque value of Eq. (29) depends on the coefficient of friction. Consequently, PID controlis used for predicting of surface and settling time is minimized to ensure minimal loss of braking.

4.2. PID control

As mentioned previously, a relatively simple PID algorithm is used for predicting the surface upfront and then slidingmode controller will be used. The following graphs give an overview of the settling time and loss in braking due to PID

Fig. 8. Sgn and sat function.

10 20 30 40 50 60 70 80 90 100

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Time (*10-4 Seconds)

Slip

Rat

io

PID ControlSMC Control

Fig. 9. Slip ratio vs. time for PID and SMC control.

Table 7Comparison of braking and settling time for PID and sliding mode control.

PID SMC

Braking distance 4.553 m 4.373 mBraking time 1.006 s 0.9 sSettling time 85 cycles 22 cycles


control being used in prediction over SMC control. On the same vehicle parameters, brakes are applied when the vehicle is ata speed of 10 m/s (see Fig. 9 and Table 7).

The major disadvantage of PID control is the settling time. Since the prediction can only be done once the slip ratio hassettled around a desired value, and the number of cycles needed for PID to settle is much higher than SMC, the predictiontime increases and therefore there is a loss in braking. The loss is not only because of the use of PID over SMC as the brakingdistance here shows. It is also because during prediction, the vehicle is at a slip ratio at which the braking forces available aremuch lesser as compared to the peak braking forces.

4.3. Parameter selection

The parameters for the PID and SMC control loops have been chosen by a manual tuning strategy to achieve a reasonablygood performance in terms of rise time and settling time for a step input. For PID tuning, corresponding to each particularvalue of the proportional gain, the integral and derivative gains were varied in a suitable range and the performance wasstudied. As mentioned earlier for the sliding mode Eq. (21), g is the convergence factor and u is the boundary layer thickness(u > 0). It was found that similar performance results are obtained for g// values of the same order of magnitude. For highervalues of g// (around 50,000) a lot of chattering is observed as the saturation function acts almost like the sgn function. On


the other hand for lower g// values (around 1000) the saturation function is weak and does not have a noticeable effect onthe braking torque to achieve the desired convergence of slip ratio.

5. Results and discussion

First, the advantage of ABS over normal braking is shown. Then the results obtained by the proposed methods are com-pared to the results available in literature for both prediction and overall braking performance to benchmark theperformance.

5.1. Comparison of braking performance with and without ABS

First, the results of ABS with sliding mode control are compared to a car without ABS. A clear advantage of ABS over lockedwheels is visible. When brakes are applied to a car running at 20 m/s, the following results are obtained.

As shown in Fig. 10, ABS reduces the braking distance substantially by not allowing the wheels to lock. In the above cases,the braking distance is reduced in a range of 14.5% for wet cobblestone to 30% for wet asphalt. Similarly, it is found that thestopping time is reduced in a range of 16.9% for wet cobblestone to 32.8% for wet asphalt.

5.2. Evaluation of surface identification methods

To benchmark the results, the first task is to compare the surface prediction and/or identification methods. Tanelli et al.(2009, p.891), Savaresi and Tanelli (2010) have presented results for their methods. The prediction has been done on noisysimulation data. Variances have been included in the velocity and wheel speed values to indicate estimation errors and mea-surement noise respectively. For noisy conditions it was found that a combination of the one point and three point methodsperforms better as compared to the individual algorithms. The one point prediction is found to be relatively accurate for sur-faces with peak slip ratio less than 0.25. In the combination method, the one point prediction is used if both the predictionsare below 0.25; otherwise the value predicted by the three point method is used. Figs. 11 and 12 compare the prediction bythe combined method in noisy conditions to those obtained in Tanelli et al. (2009, p.891).

Figs. 11 and 12 clearly show that the combined method gives good results for most of the surfaces.

5.3. Sensitivity analysis for testing robustness of control algorithms

The vehicle parameters that need to be considered for sensitivity analysis are those that can vary during each run or thosewhose estimated values can be erroneous. The values that have been considered are total mass, height of CG and moment ofinertia of wheel which have nominal values of mtot = 1500 kg, ha = 0.6 m, J = 1.8 kg m2 as indicated in Table 5. ±10% Variationin each of these parameters has been considered and the effect on stopping distance for each case has been compared. Thevariance observed in the stopping distances for one point and three point prediction methods have been shown in Table 8.The initial velocity for this analysis is assumed to be 30 m/s.

5.4. Benchmarking the control strategy

Results of the models presented in this paper are compared to the results presented by Harifi et al. (2008, p. 731–741) tobenchmark the control strategies. The surface prediction method coupled with SMC control could potentially reduce thebraking distance as compared to most of the other ABS strategies. The advantage of prediction methods over other controlstrategies can be seen from Fig. 13.

One point method gives slightly better results (0.6–4.3%) than three point method since there is a substantial loss in brak-ing during the prediction cycles of three point method and since the road conditions do not change, the accurate prediction

Bra

king

Dis

tanc

e (m

etre

s)

Fig. 10. Comparison of braking distance with and without ABS.

Fig. 11. Comparison of slip ratio prediction by different methods.

Fig. 12. Percentage error in slip ratio prediction.

Table 8Sensitivity analysis based on variation in vehicle parameters.

Varying parameter (±10%) Variance in distance for each prediction method

One point prediction Three point prediction

Total mass 1.6 � 10�6 1.96 � 10�4

CG height 4.9 � 10�9 4.71 � 10�6

Wheel moment of inertia 5.13 � 10�7 9.71 � 10�5

Dis

tanc

e (m

etre

s)

Fig. 13. Comparison of braking with different control strategies used in ABS.


Fig. 14. Simulation results for a surface change of dry to wet asphalt at 10 m.

Fig. 15. Comparison of braking distance for different ABS methods.


gives no major advantage in case of three point prediction method. Advantages of the proposed prediction methods over selflearning fuzzy-sliding mode and neural hybrid network are evident from Fig. 13. However, SMC and genetic adjusted fuzzymethod can reduce the braking distance as in case of snow by about 4.3% and 3.6% respectively. They can also increase by3.4% and 7.3% for the same methods in case of Dry asphalt. However, the real advantage of prediction methods is visible incase of changing road conditions.

5.5. Single surface change during braking

Figs. 14 and 15 compare results of one point and three point prediction methods with the method proposed by Harifi et al.(2008, p. 731–741) when surface changes from dry to wet asphalt, dry concrete to snow and dry to wet cobblestone. When

Fig. 16. Wheel velocity, vehicle velocity and stopping distance for all surfaces.

Fig. 17. Slip ratio for braking at all surfaces using three point method.


brakes are applied to a vehicle moving at 20 m/s, the results obtained if road condition changes after 10 m of braking can beseen in Figs. 14 and 15.

This clearly highlights the advantages of the prediction methods. Since the number of prediction cycles is less in one pointmethod, it gives better results as compared to both three point prediction method and simple SMC control. The stoppingdistance in case of dry to wet asphalt for three point method is 4.7% more and for dry concrete to snow it is 6.6% more thanone point prediction. The same for SMC is 1% more and 3.7% more respectively. However, since the prediction by one pointmethod is not accurate for cobblestone, the braking distance increases in case of one point method. The three point methodwith its accurate prediction gives better results in case of cobblestone and the stopping distance reduces by 19%. The stop-ping distance for one point method in the first two cases is almost the same as that achieved in perfect maximal braking. Inthe third case, the stopping distance of the three point method is 6.4% more than the distance in perfect braking.


5.6. Braking under changing road conditions

Finally, the utility and accuracy of prediction in ABS systems can be highlighted by considering a situation when brakesare applied to a car running at 40 m/s and it is exposed to all road conditions as shown in Figs. 16 and 17. For this simulation,the vehicle under braking first comes across

(1) Wet cobblestone for 10 m.(2) Dry concrete for the next 20 m.(3) Dry asphalt for the next 20 m.(4) Snow for 10 m.(5) Wet asphalt for 10 m.(6) Dry cobblestone till it stops.

Such a simulation has been used to test the function of the three point prediction method under any change in conditionit encounters. The results obtained show the robustness of the prediction method. The prediction starts for cobblestone wetgives 0.57% error in estimation of peak slip ratio. The error in estimation of peak slip ratio of the subsequent surfaces is 0.25%for dry concrete, 0.24% of dry asphalt, 7% for snow, 6.8% for wet cobblestone and 2.5% for dry cobblestone. Even under suchchanges, the accuracy of prediction does not suffer much as compared to the original single surface change predictions by thethree point method. As shown in Fig. 16, the stopping distance under such conditions for a vehicle running at 40 m/s is just98 m.

6. Conclusions

From the above results it is clear that for off-road conditions, three point method is suitable and one point method isappropriate for normal driving conditions. A combination of both with the ability to switch as per the drivers requirementscan also be used and would be the best as it can have the advantages of fast prediction (one point method) in normal drivingconditions and better prediction (three point method) in off road driving condition. Following specific conclusions can bedrawn from the current work:

� Three point method give best predictions with an average error of around 2% for the surfaces used during simulation. Byany other method, the average error in prediction for all surfaces is more than 6%.� One point method is optimized for working with normal road conditions that are regularly faced. The prediction for such

surfaces by one point method also gives an average error of around 2% and it reduces the braking distance by 1–5% ascompared to any other method.� One point method has serious shortcomings for unconventional roads. For such conditions, three point method has been

proposed which reduces the braking distance by 19% as compared to one point method when surface changes from dry towet cobblestone.� A combination of PID control for prediction and SMC control to achieve the predicted slip ratio for maximum friction coef-

ficient, braking distances can be reduced by more than 3% if any surface change is encountered during braking.

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surface prediction and control algorithms for anti-lock brake system

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