optimal motorcycle braking

8/18/2019 optimal motorcycle braking

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Control Engineering Practice 16 (2008) 644–657

On optimal motorcycle braking

Matteo Cornoa, Sergio Matteo Savaresia,, Mara Tanellia, Luca Fabbrib

aDipartimento di Elettronica e Informazione, Politecnico di Milano, Piazza L. da Vinci, 32, 20133 Milano, ItalybPiaggio Group—Aprilia Brand, Via Galileo Galilei, 1, 30033 Noale, Venice, Italy

Received 21 March 2007; accepted 1 August 2007

Available online 21 September 2007

Abstract

The optimal braking strategy in a high-performance motorbike is discussed. First, the control strategy using the front brake only isanalyzed, highlighting the role of aerodynamics and studying how to select and modify the control objective during braking. The

importance of the brake modulation in the very first part of the braking maneuver is also discussed. The role played by the rear brake is

then analyzed. Finally, the attention is shifted to the damping ratio of the front suspension, which is probably the single motorbike

parameter with the largest impact on vehicle dynamics during braking. A possible policy for semi-active suspension control during

braking is sketched.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Motorbike; Brake control; Semi-active suspension; Vehicle dynamics; Automotive systems

1. Introduction and motivation

In this work a simulation-based study of the brakingmaneuver in high-performance (racing) motorcycles is

presented.

The starting point of this work is the analysis of a typical

hard-braking maneuver performed by a professional

driver. An example is given in Fig. 1, where the measured

front-wheel speed, front-brake pressure and suspensions

elongation are displayed (the measurements are made on

the MY05 Aprilia RSV1000 Factory). During this braking

maneuver, the driver makes no use of the rear brake.

Notice that the brake pressure has a ramp (of about

250 ms), followed by a small brake release (probably due to

an over-slip phenomenon sensed by the driver); then thebrake pressure is kept almost constant until the end of

the braking maneuver. By inspecting the elongation of the

front and rear suspensions, it is interesting to observe that

the suspension hard-limit is reached in both cases; notice

that this means that the rear-wheel is at the contact limit;

apparently, this may induce the driver to avoid turning the

vehicle before the braking maneuver is ended.

Among pilots and race engineers it is common opinion

that the braking phase is the most critical and sensitive

maneuver. The ability of a driver to achieve an ‘‘optimalbraking’’ can make the difference on the lap-time. Even few

milliseconds per braking hence can be crucial. In this work,

optimality simply means minimum time to decelerate the

motorbike from the initial speed to a target speed. A

broader range of performance parameters will be discussed

in Section 4, where the sensitivity to the front-suspension

damping will be analyzed.

The objective of this paper is to deeply analyze a single

braking maneuver, trying to understand what the optimal

maneuver should be, and the main parameters which can

influence it.

The analysis is developed in the following setting:

A pure braking maneuver is assumed, on a straight line(no lateral forces are engaged).

An ideal driver is assumed; the concept of the idealdriver can be recast into an automatic closed-loop

control system, following a reference signal (e.g., a

target slip or a target load). The design of the reference

signal is clearly a key issue for the optimality (‘‘ideality’’)

of the driver.

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0967-0661/$- see front matter r 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.conengprac.2007.08.001

Corresponding author. Tel.: +39 02 23993545; fax: +39 02 23993412.

E-mail address: [email protected] (S.M. Savaresi).

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The main challenge in the development of the ‘‘optimal

maneuver’’ is due to the fact that a complete analytical

model of a motorcycle is very complex, and it can hardly beused for the direct closed-form calculation of the optimal

solution. On the other hand, reduced order models can

indeed offer analytic solutions, but they cannot be really

useful to find the optimum for the real target vehicle. This

classical dilemma is solved according to the objective and

the perspective of the specific research work; in this paper,

since the objective is to analyze the very details of a braking

maneuver, the simulator-based approach has been the

natural choice (Fig. 2).

The analysis developed in this work is based on a full-

fledged motorcycle simulator (the Mechanical Simulation

Corp. BikeSims simulation environment, based on the

AutoSim symbolic multi-body software (Sharp, Evangelou,

& Limebeer, 2004, 2005), which takes into account all the

motorcycle dynamics, and accurately models the road–tire

interaction forces (Sharp et al., 2004). See also Cossalter

and Lot (2002), Donida, Ferretti, Savaresi, Schiavo, and

Tanelli (2006), Frezza and Beghi (2005), Lin, Chyuan-Yow,

and Tsai-Wen (2006), Sayers (1999), and Sharp and

Limebeer (2001) for other state-of-the-art simulation

environments suitable for this kind of analysis. The

complete set of parameters used in the simulator is listed

in Appendix A.

Solving the problem of the ‘‘optimal’’ maneuver is very

attractive, since it allows to decouple the intrinsic

performance of the vehicle from the driver behavior. In

2-wheel vehicles, these two aspects are so strictly inter-

leaved that it is hard to predict the effect of a parameter

change on the vehicle. This is a well-known problem, which

sometimes has the picturesque effect of transforming the

tuning of a racing motorbike from a rigorous model-based

procedure into a sort of fine (or ‘‘magic’’) art.

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0 1 2

0

100

200

[ k m / h ]

Front-wheel speed

0 1 2

0

5

10

15

20

[ b a r ]

Front brake pressure

0 120

10

20

30

40

[ m m ]

Rear suspension extension

0 1 2

0

50

100

time [s]

[ m m ]

Front suspension extension

START BRAKING

250ms

Temporary brake release Steady-state braking pressure

Rebound hard-limit

Compression hard-limit

Fig. 1. Typical hard-braking maneuver (measured on the MY05 Aprilia RSV1000 Factory). The convention used for the suspension extension is that the

maximum value is taken at full compression, the minimum at full extension.

F z f F zr F x r F x f

wheelbase p

b

h

Fig. 2. Notation and GUI of the hypersport-class motorcycle used in the

SW simulator Bikesim.

M. Corno et al. / Control Engineering Practice 16 (2008) 644–657 645


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The optimal-maneuver problem is a very challenging

task, which has been studied in the last decade by different

research groups, from different perspectives (see e.g.,

Cossalter, Doria, & Lot, 1999; Cossalter, Lot, & Maggio,

2002; Frezza, Beghi, & Saccon, 2004; Hauser & Saccon,

2006; Hauser, Saccon, & Frezza, 2004; Limebeer, Sharp, &

Evangelou, 2001; Sharp, 1971, 2001 for an overview of theliterature on this topic). The specific problem of optimal

braking has been considered in Cossalter, Lot, and Maggio

(2004), where the focus was on steady-state conditions

during a bend. Another paper dealing with the braking

maneuver is Cossalter, Doria, and Lot (2000), where the

design of the suspension for a scooter is discussed. The

present work focuses on the dynamic aspects of a pure

braking maneuver on a straight line.

In this paper the problem of the optimal braking

maneuver is discussed following this path:

Section 2. As first step, it is assumed that only thefront brake is used (this is quite common, even inracing). Under this assumption the optimal control

target is discussed; first this analysis is made by

considering quasi-stationary conditions; then the fast

transient of the very first part of the braking maneuver is

analyzed.

Section 3. The role and the benefits of rear braking arediscussed.

Section 4. Since the front-suspension damping is verycritical for the overall braking performance, the

influence of this parameter on the braking maneuver is

analyzed.

The approach used in this paper differs from the classical

automatic-control design path (system modeling, para-

meter identification, control algorithms design, perfor-

mance evaluation). A mix of analysis of vehicle dynamics

and closed-loop control architectures is proposed; as a

matter of fact the primary goal of this paper is not the

complete design of an automatic braking control system,

but the understanding of the main dynamic phenomena

and reference signals to be considered in the design of an

optimal braking control strategy.

2. Optimal front-braking

In a high-performance motorbike it is customary to use

only the front brake during a hard-braking maneuver.

Since no front–rear coupling phenomena must be taken

into account, and almost all the vehicle load is transferred

onto the front wheel, the braking dynamics can be well-

approximated with a single-corner model (Cossalter, 2002;

Savaresi, Tanelli, & Cantoni, 2007; Solyom, Rantzer, &

Lu ¨ demann, 2004):

J _o ¼ rF x T ;

m_

v ¼ F x;( (1)

where o is the angular speed of the wheel (it is expressed in

rad/s; o40 is assumed), v is the longitudinal speed of the

vehicle body, T the braking torque which plays the role of

control/input variable, F x the longitudinal road–tire con-

tact force, J , m and r are the rotational inertia of the wheel,

the corner mass and the wheel radius, respectively.

The dynamic behavior of the system is hidden in theexpression of F x, which depends on the state variables v

and o. The most general expression of F x is quite

complicated, since F x depends on a large number of

features of the road, tire and suspension; however, it can be

well-approximated as F x ¼ F zmðl; btÞ, where F z is thevertical force at the tire–road contact point and l is

the longitudinal slip, given by l ¼ ðor vÞ=v ¼ ro=v 1(the sign convention of negative slip when braking is used);

bt is the side-slip angle of the wheel. Since it is assumed that

the braking maneuver is performed along a straight line

(bt ¼ 0), the dependence of m( ) on bt will be omitted.

The function m(l) is called friction coefficient. It can be

modeled using many empirical or semi-empirical para-

metric models, among which the most famous is probably

the Pacejka ‘‘magic’’ formula (Pacejka, 2002; Savaresi,

Tanelli, Langthaler, & Del Re, 2006). The curve m(l) is

characterized by a peak, which is typically around

lE0.15; this non-monotonic characteristic is responsible

for the well-known fact that (for constant braking torque

values) the equilibria associated to slip values beyond the

peak of m(l) are unstable. In Fig. 3, the road–wheel friction

curve of the front tire used in the simulator is depicted, for

different vertical contact forces F z. Notice that, at different

vertical loads, this curve slightly changes. In the rest of the

paper it is assumed that the maximum friction forceis achieved at ¯ l ¼ 0:12. This is rigorously true if F z ¼ 2000N ; since the total mass of the vehicle and driver

is about 270 kg—see Appendix A—this vertical load

corresponds to a significant load transfer onto the front

axle. Notice that for higher vertical loads the peak position

slightly changes, but by keeping the target slip at ¯ l ¼ 0:12the loss of performance is negligible (see details on Fig. 3).

The shape of the friction curve provides a seemingly simple

solution to the optimal braking control target: in order to

maximize the longitudinal force, the longitudinal slip l should

be simply regulated at the peak value of the friction curve,

whatever is the vehicle speed and the vertical load.

Fig. 4 shows the results of a braking maneuver, using

only the front brake and a target-slip control policy. The

results displayed in Fig. 4 are obtained using a well-tuned

PID slip regulator, applied to the Bikesim simulator (see

Buckholtz, 2002; Johansen, Petersen, Kalkkuhl, & Lu ¨ demann,

2003 for details on the design of such a slip controller). The

results displayed in Fig. 4 clearly show that—unfortu-

nately—in a motorcycle the target-slip policy cannot be

implemented, since the maximum braking torque can be

higher than the overturn torque. This is a very peculiar

feature of motorbikes (this problem does not exist in racing

cars); it is clearly due to the high ratio between the height

of the center of mass and the wheelbase (see Fig. 2).

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By inspecting Fig. 4, it is worth pointing out that:

the braking maneuver starts at a very high longitudinalspeed (300 km/h);

the PID-based automatic-control system is able to trackvery accurately (with both dynamic and static precision)

the reference slip target ð¯ l ¼ 0:12Þ; the vertical load at the rear wheel ðF zr Þ rapidly decreases,

while the pitch angle (y) increases; after 2.2 s from the

braking start, at the speed of about 200 km/h and at the

pitch angle of about 101, the rear wheel starts losing

contact with the road;

the front-wheel target-slip is kept at ¯ l ¼ 0:12 evenafter the loss of contact; this causes an overturn of the

vehicle (also called a ‘‘stoppie’’ in the motorcycle

jargon—see photo in Fig. 4).

In order to better understand the reason of this

phenomenon, a simplified model of the in-plane dynamics

of the motorcycle can be used. Note that this model does

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0 0.1 0.12 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

−λ

µ ( λ )

0.1 0.121.32

1.34

1.36

1.38

1.4

1.42

1.44

1.461.48

1.5

Fig. 3. Front-tire friction coefficient for different values of F z.

-0.2

0

0.2

λ f

0

1000

2000

F z r [ N ]

0

10

20

θ

[ ° ]

4 4.5 5 5.5 6 6.5 70

100

200

300

time [s]

4 4.5 5 5.5 6 6.5 7

time [s]

4 4.5 5 5.5 6 6.5 7

time [s]

4 4.5 5 5.5 6 6.5

time [s]

v [ k m / h ]

Rear-wheel loss of contact

Constant slip target (-0.12)

Fig. 4. Braking maneuver at constant target slip ¯ l ¼ 0:12: the ‘‘stoppie’’ (from top: front-slip; rear vertical load; vehicle pitch angle; forward speed).



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not consider the suspension dynamics; hence, it is valid for

quasi-static conditions. The model is the following (sub-

scripts ‘‘ f ’’ and ‘‘r’’ indicate front and rear parameters):

m €x ¼ F x C D _x2;

m€z ¼ F zr þ F zf mg þ C L _x2;

J y€y ¼ F zrb F zf ð p bÞ þ F xh C P p _x2;

8>>>: (2)where

p, b, h are the wheelbase, the distance between theprojection of the center-of-mass on the road and the

rear-wheel contact point, and the height of the center-of-

mass, respectively (see Fig. 2);

{x, z, y} are the longitudinal, vertical and pitch coordi-nates of the vehicle center-of-mass, respectively;

m, J y are the vehicle mass and its pitch inertia (aroundthe center-of-mass), respectively;

{C D, C L, C P } are the drag, lift and pitch aerodynamiccoefficients, respectively.

Using the simple model (2), two useful quantities, as

functions of the forward speed _x, can be derived.

The first is the deceleration €xloss which causes the loss of

contact of the rear wheel. It can be obtained by imposing€y ¼ 0, €z ¼ 0, F zr ¼ 0. The result is the following:

€xloss ¼ mgð p bÞ C Lð p bÞ _x

2 þ C P p _x2

mh þ C D _x

2. (3)

The second is the deceleration €xlock which causes the wheel

lock-up; it can be obtained by imposing €y ¼ 0, €z ¼ 0,

F x ¼ mmax F zf , where mmax ¼ mð0:12Þ. It is straightfor-ward to show that when braking only with the front brake,

this deceleration is related to the forward speed _x as

follows:

€xlock ¼ mmax

ð p mmaxhÞmðmgb C L _x

2b C P _x2Þ þ C D _x

2. (4)

The two variables (3)–(4) are plotted in Fig. 5. The

parameter values used in (3) and (4) have been extracted by

the Bikesim model, in order to obtain consistent results.

Note that in (3) and (4) the sign of €x differs from (2), since

the €xloss and €xlock are decelerations (they are positive when

the vehicle speed decreases—see also Fig. 5). The results,

although somehow intuitive, are interesting: at high speed,

thanks to the large aerodynamic forces, the wheel-lock

limit is dominant; this means that all the friction capabilityof the front wheel can be used, and the slip-target policy is

optimal. When the speed decreases, so do the aerodynamic

forces, and the rear-wheel contact limit becomes active (in

the simulated vehicle this limit is about 200 km/h, see also

Fig. 4). That speed is the ‘‘critical-speed’’ from the rear-

wheel contact point of view. Below that speed value, in

order to minimize the stopping distance while maintaining

the vehicle lateral stability, the controlled variable must be

switched from the front wheel slip to the rear vertical load.

Also notice that, at low speed, €xloss rapidly decreases, and

the friction capability of the tires is significantly under-

exploited.

The optimal braking strategy in a motorbike hence is

twofold: at high speed a slip-control must be used, the

reference signal being the peak of the friction curve; below

the critical speed this control policy must be disengaged

and replaced with a controller of the rear-wheel vertical

load; in practice, since the critical speed is hard to estimate,

the switching variable is the rear-wheel load. The control

target of the load controller must be ideally ¯ F zr ¼ 0þ; in

practice, the zero-target has to be replaced with a small

load (in the following, a load reference value of ¯ F zr ¼ 100

has been used) to guarantee a minimum amount of

lateral contact force on the rear wheel. This control

architecture is pictorially described in Fig. 6. The resultsof this control policy are displayed in Fig. 7. The two

SISO controllers of Fig. 6 are implemented with a standard

PID architecture with anti-windup. Notice from Fig. 7

that the switch is seamless, since when the load-control

PID is activated the state of its PI-part is not zero but it

exactly equals the output of the slip controller at the

switching time (this is implemented with the standard

automatic-manual commutation algorithm for PID). In

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0 50 100 150 200 250 3000.8

1

1.8

1.6

1.4

1.2

2

forward speed [km/h]

l o n g i t u d i n a l d e c e l e r a t i o n [

g ]

Rear-wheel contact limit x loss

Front-wheel-lock limit x lock

Fig. 5. Deceleration limits at different speeds.



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this way the switching guarantees a continuous behavior of

the control variable.

From Fig. 7 notice that, when the rear-load controller is

engaged, the front slip is progressively reduced, in order to

avoid the loss of contact, while maximizing the braking

performance. The braking maneuver of Fig. 7 can be

considered optimal when only the front brake is used. Also

notice that—as expected—the switch of the target occurs at

a forward speed slightly higher than the critical speed, since¯ F zr40.

In the first part of this analysis the focus was on the

general architecture of the control strategy. However, due

to the complex dynamics of the vehicle and the tire, the

very first part of the braking maneuver (say the first

200 ms) is particularly critical. This issue is now analyzed in

detail.

The starting point of this analysis is the behavior of the

automatic slip controller. This controller has been designed

to provide the maximum achievable dynamic performance

(maximum closed-loop bandwidth—see Savaresi et al.,

2007). The behavior of the control variable (front-braking

torque) in the first 200 ms of the braking maneuver, when

the target slip is set at ¯ l ¼ 0:12, is displayed in Fig. 8.Notice that the controller initially requires a ‘‘pulse’’ of

torque (about 10 ms long), due to the derivative part of the

PID; then the torque is set to a small value and gradually

increased again with a ramp-like behavior, until its steady-

state value is reached. The ramp lasts about 150 ms. Notice,

ARTICLE IN PRESS

-

-

+

+

Vehicle

dynamics

Slip

controller

Load

controller

switch

T f

f

F zr F zr

Fig. 6. Optimal control architecture for the front brake.

4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.50

100

200

300

time [s]

v [ k m / h ]

4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5-0.2

-0.1

0

0.1

0.2

time [s]

λ

4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.50

500

1000

1500

2000

time [s]

F z r [ N ]

front-slip target

rear-load target

slip-target control policy

load-target control policy

Fig. 7. Braking maneuver using the mixed slip-load control policy with only the front brake (from top: vehicle speed; front-wheel slip; rear-wheel vertical

load).



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however, that the control strategy implemented in Fig. 8

exceeds the actuator limit (saturation), and, in practice, the

initial pulse is cut-off.

Starting from this ideal behavior, a sensitivity analysis

on the effects of three different control strategies in this

initial phase of the braking maneuver has been carried out.

The three initial control strategies are (see the top sub-plotof Fig. 9):

Control strategy 1. This control strategy simply mimicsthe behavior of the closed-loop slip controller: an initial

short (10 ms) pulse, using the maximum available

torque, is followed by a 140 ms ramp which starts from

a value of about 40% of the final steady-state value.

Control strategy 2. This control strategy mimics thebehavior of a ‘‘real-driver’’ (see Fig. 1): the braking

torque is increased from 0% to 100% of the steady-state

value, with a 150 ms ramp (notice that a very fast-reactingdriver has been assumed; in the measurement displayed in

Fig. 1 the driver takes 250 ms to make the ramp).

Control strategy 3. The idea of this control strategy isto immediately set the braking torque to the right

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3.95 4 4.05 4.1 4.15 4.20

1000

2000

3000

4000

5000

6000

7000

8000

9000

time [s]

T [

N m ]

Front Braking Torque

Actuator limit(saturation)

Initial pulse(withoutsaturation)

Torque ramp Steady-state torque

Fig. 8. Initial behavior of the control variable (front braking torque) using a large-bandwidth slip controller.

4 4.05 4.1 4.15 4.2 4.250

1000

2000

3000

time [s]

4 4.05 4.1 4.15 4.2 4.25

time [s]

4 4.05 4.1 4.15 4.2 4.25

time [s]

4 4.05 4.1 4.15 4.2 4.25

time [s]

T f [ N m ]

-0.2

-0.1

0

λ f

0

1000

2000

F z r [ N

]

285

290

295

300

v [ k m / h ]

Control strategy 1

Control strategy 2

Control strategy 3

Performance difference: about 30ms

Fig. 9. Sensitivity analysis to initial transient control strategy (from top: front braking torque; front-wheel slip; rear-wheel vertical load; vehicle speed).



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steady-state value; in practice this would cause wheel

lock, since the load cannot be transferred to the front-

wheel instantaneously, due to the suspension dynamics;

hence, in practice, the braking torque is set to about

80% of the steady-state value, followed by a 150 ms

ramp which closes the 20% gap.

The results of these three different control policies are

condensed in Fig. 9, where the longitudinal slip, rear load,

and forward speed of the vehicle are displayed. The

following remarks are due:

Even if the control strategies 1 and 3 show a remarkablydifferent behavior in the longitudinal slip, their overall

performances are very similar, in terms of vehicle

deceleration. This can be explained with the fact that

in the very initial part of the braking maneuver

(20–30 ms) the load has not yet been fully transferred

to the front wheel; thus, during this phase, the sensitivity

to different values of wheel slip is comparatively small.

The control strategy 2 shows significantly lowerperformance than 1 and 3. This is clearly due to the

fact that the friction capability of the tire is under-

exploited, since the magnitude of the wheel slip is

increased too slowly.

It is interesting to observe that the control strategy 2 is the

worst from the performance point of view, but it is the

most natural and easiest for a ‘‘human’’ controller (see

Fig. 1). On the other hand, control strategies 1 and 3

provide higher performance, but they can hardly be

managed by a human driver. Note that the difference of performance in the 300-290 km/h deceleration is about

30ms; it is non-negligible, considering that all this

difference is condensed in the very first part of the braking

maneuver. Hence, this implicitly confirms the potentially

beneficial effect of an electronic brake assistant.

3. Optimal front–rear braking

In racing motorcycles the role and the importance of rear

braking is very debated as, in practice, during a hard

braking on a straight line, most of the drivers make no use

at all of the rear brake. This brake is typically used for

better managing the attitude and the stability of the

motorbike during a bend. The reason is threefold:

Due to the (almost) total load transfer to the front tire atmid-low speed in a hard-braking maneuver, the braking

capability of the rear tire is comparatively small.

The simultaneous optimal management of the front andrear brake is a very hard task even for a professional

driver.

If the engine is engaged during braking, the load torqueof the engine (especially on high-performance high-

compression 4-stroke engines) in many cases is high

enough to lock the rear wheel; this phenomenon can be

alleviated using mechanical devices called ‘‘anti-hop-

ping’’ clutches, which can be considered as a raw anti-

lock braking system acting on the engine-induced

braking torque.

In order to simplify the analysis and to give a deep

understanding of the role of the rear brake, it is assumedthat the engine is disengaged during braking, and that all

the rear-braking torque is provided by the rear brake.

As a starting point, it is interesting to analyze the

behavior of the rear-wheel slip when the front-brake mixed

slip-load control strategy illustrated in Figs. 6 and 7 (see

the previous section) is used. The rear and front slip are

displayed in Fig. 10.

From Fig. 10 it is interesting to point out that, at the rear

wheel, the slip is positive, namely the rear wheel provides a

traction torque, not a braking torque. This fact can be

somewhat surprising at a first glance, but it can be easily

explained by observing that the rear wheel has a kinetic

energy given by 12J rð ¯ orÞ2, where ¯ or is the rear wheel

rotational speed at the beginning of the braking maneuver,

and J r the inertia of the rear wheel. If the engine is

disengaged during braking and only the front brake is used,

this energy is not dissipated but it is transformed into a

traction torque.

When the rear brake is used, the control architecture of

Fig. 6 must be changed into the control architecture of

Fig. 11. The mixed slip-load strategy at the front brake is

kept unchanged; the rear wheel is simply managed by

regulating its slip lr to the peak value of the rear friction

curve.

The results obtained with this control architecture aredisplayed in Fig. 12. These results are compared with those

obtained using the front brake only. The major difference

is, obviously, in the slip of the rear wheel. As already

noticed, this slip is positive (namely it provides traction

torque) if no rear brake is used; if the rear brake is used, the

rear slip is regulated to the same target ð¯ l ¼ 0:12Þ usedfor the front slip. Notice that the regulation of the rear slip

is a difficult control task, due to the very small load on the

rear wheel which causes a significant cross-disturbance of

the front-brake controller onto the rear-slip dynamics. This

non-perfect tracking of the rear wheel slip, however, has

little effect on the overall performance.

This section is concluded by observing that in the long

300-80 km/h braking maneuver, the difference of perfor-

mance is very large, about 300 ms. This difference is largely

due to the fact that the ‘‘traction’’ torque has been replaced

by a ‘‘braking’’ torque at the rear wheel Although the

braking torque at the rear wheel is rather small compared

to the front-wheel one, on a strong braking maneuver the

effect of the rear brake can be clearly appreciated. This is

true, in particular, if the front-brake controller is able to

maintain a small but non-zero load on the rear tire.

For the sake of completeness it is worth mentioning that,

in practice, this performance improvement is significantly

reduced if the engine is engaged and the vehicle is equipped

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4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5

-0.2

-0.1

0

0.1

0.2

time [s]

λ

Front λ

Rear λ

Target front λ

slip-target control policy

load-target control policy

Fig. 10. Front- and rear-wheel slip using a front-brake mixed slip-load control policy.

-+

-+

Vehicle

dynamics

f

f Slip

controller

Load

controller

switch

T f

-

+ Slipcontroller

r

r

F zr F zr

Fig. 11. Optimal control architecture of the front and rear brakes.

4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5

100

150

200

250

300

v [ k m / h ]

4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5

4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5

4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5

-0.2

0

0.2

λ f

-0.2

0

0.2

λ r

0

1000

2000

time [s]

F z r [ N ]

Front Brake Only

Both Brakes

Performance difference: about 300ms

= -0.12

= -0.12

Fig. 12. Brake maneuver using the complete mixed slip-load control policy using the front and the rear brake (from top: longitudinal speed; front slip; rear

slip; rear vertical load).



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with anti-hopping clutch. From this point of view, the anti-

hopping clutch can be interpreted as a very raw automatic

rear slip controller. Obviously enough, a genuine electronic

slip controller (acting on an electronically-controlled clutch

or brake) can guarantee much better and more consistent

slip-tracking performances.

This analysis demonstrates the potential benefits of anautomatic control strategy for the rear brake: the braking

performance is maximized while guaranteeing the road

contact of the rear wheel. A somehow futuristic but very

attractive way of achieving such high performance is the

by-wire electro-mechanical brake (see e.g., Savaresi et al.,

2007).

4. Sensitivity to front-suspension damping

In this section the effect of the damping ratio of the front

suspension (fork) during a braking maneuver is discussed

(see also Cossalter et al., 2000). As a matter of fact, amongthe many tunable parameters of a motorbike, the damping

ratio of the fork is one of the most critical, since the vehicle

dynamics are very sensitive to this parameter, and there is a

strong trade-off in its tuning (see below). On this

parameter, drivers and race engineers are always struggling

in the search for the best compromise among different

conflicting requirements.

The first part of the analysis has been developed in the

frequency domain; more specifically, two transfer functions

have been experimentally estimated using the simulator:

the transfer function from the front braking torque ( T f )to the front wheel longitudinal slip (l f ); this transferfunction is useful in the design of slip controllers;

the transfer function from the front braking torque ( T f )to the front wheel vertical contact force ðF z f Þ; this

transfer function is useful to investigate the minimiza-

tion of vertical-load variations, which is a classical

prerequisite to achieve high (and constant) longitudinal

and lateral contact forces.

Since the braking maneuver is a non-equilibrium condition

(due to vehicle deceleration), the computation of such

transfer functions cannot be managed with traditionallocal-linearization tools, which typically require a well-

defined steady-state condition; thus, an ad-hoc small-

perturbation approach has been used (see e.g., Guarda-

bassi & Savaresi, 2001; Savaresi & Spelta, 2007).

Fig. 13 shows a single step of this procedure: starting

from a vehicle in straight running with a speed of 200 km/h,

a constant front braking torque is applied; this constant

torque generates (after a short transient) a steady-state

front-wheel slip. Then a small sinusoidal perturbation is

superimposed to the constant torque (a 2 Hz perturbation

is depicted in Fig. 13); the corresponding slip is affected by

a sinusoidal oscillation at the same frequency; by compar-ing the amplitude and phase relationships between the two

sinusoidal oscillations, a single point of the estimated

frequency response can be computed. This procedure has

been repeated for a large number of frequencies (from 0.5

to 15 Hz, with a frequency resolution of 0.25Hz).

The magnitudes of the estimated transfer functions are

displayed in Fig. 14, for three different values of damping

(low: 1000 Ns/m; medium: 3000 Ns/m; high: 5000 Ns/m).

The following remarks can be made.

In all cases, a strong trade-off is clearly visible: at low-

damping all the transfer functions are characterized by alow-frequency (2.5 Hz) poorly-damped resonance; by

increasing the damping ratio this resonance phenomen-

on first disappears (for medium damping) and then

reappears as a poorly-damped resonance located at

about 7 Hz.

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1 2 3 4 5 6 7 8 9 10

0

100

200

300

400

T f [ N m ]

1 2 3 4 5 6 7 8 9 10-0.04

-0.02

0

λ f

1 2 3 4 5 6 7 8 9 10

100

150

200

time [s]

v [ k m / h ]

2Hz sinusoidal perturbation on Tf

2Hz sinusoidal response on λ f

Fig. 13. Example of single-tone perturbation for frequency–response estimation.



11/14

The resonance in the slip response is critical for thecontrol of the wheel slip (both from a human-driver

perspective and from an automatic-controller perspec-

tive). The resonance in the load-response to the braking

torque is critical for the behavior of the contact forces.

In both cases, for low (below 5 Hz) and high (beyond

9 Hz) frequencies, the high-damping configuration is thebest. The low-damping configuration outperforms it in

the mid-range frequency range (5–9 Hz) only. This

trade-off is well known for the vertical movement of

the sprung and unsprung masses; it is interesting to see

that this behavior also holds for the slip dynamics of

the tire.

The frequency-domain interpretation has been comple-

mented with a time-domain analysis. To this end, in Fig. 15

the time responses of vertical contact force (F z), long-

itudinal contact force (F x) and fork elongation (Dx), to a

step in the front braking torque, are displayed for the three

different levels of damping of the front suspension. All

these responses refer to the front axle. By analyzing thesestep responses, additional insight in the influence of the

damping ratio during braking can be gained. Some

remarks are due.

The first remark refers to the initial step response of thevertical contact force, in case of low damping. Notice a

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100

100

101

101

-90

-85

-80

-75

-70

g a

i n [ d B ]

g a i n [ d B ]

Estimated transfer function from front-braking-torque (Tf ) to front-tire slip (λ f )

-10

0

10

20

30

Estimated transfer function from front-braking-torque (Tf ) to vertical load at the front-tire contact point (Fzf )

Best damping: high

Best damping: high

Best damping: high

Best damping: high

Best damping: low

Best damping: low

Frequency [Hz]

Low damping

Medium damping

High damping

Fig. 14. Frequency-domain analysis of the sensitivity of braking torque responses to fork damping.

1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3

1500

2000

2500

F z [ N ]

1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3

0

500

1000

1500

2000

2500

F x [ N ]

1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3

0

20

40

60

80

time [s]

∆ x [ m m ]

Low dampingMedium dampingHigh damping

Fast compression rate

Slow compression rate

Fig. 15. Responses (from top: vertical and longitudinal contact forces; fork elongation) to a step on the front braking torque.



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seemingly strange behavior: when a braking torque step

is applied, the contact force initially decreases; then it

increases and reaches its steady-state value. This

behavior is not intuitive, since a load transfer from the

rear to the front wheel is expected, whatever the

damping ratio of the fork is. This unusual phenomenon

is due to the fact that the angle with respect to the roadsurface of the front suspension of a motorbike

significantly differs from 901. The angle between the

fork and a vertical line—called caster angle—for a sport

motorbike is usually about 251 (26.11 in our case). When

a step on the braking torque is applied, a non-zero

longitudinal braking force is rapidly generated; due to

the non-zero caster angle, this force is not perpendicular

to the front suspension; hence, the component of this

force projected onto the fork axis tends to compress the

suspension, unloading the tire (see Fig. 16). This

phenomenon is particularly visible if the damping ratio

is low. Obviously, this phenomenon is rapidly compen-

sated by the load transfer from the rear to the front

wheel. This remark shows that, at the very beginning of

the braking maneuver, the best setting for the damping

ratio is ‘‘high’’.

From the stopping-time point of view, the mostrepresentative variable in a braking maneuver is the

longitudinal contact force F x, since it is the actual force

which decelerates the vehicle. Hence, this force must be

maximized at all times. By comparing the F x responses

at low and high damping, it is interesting to observe

that, in both cases, a resonance phenomenon occurs. In

the case of low damping this phenomenon occurs at low

frequency; in the case of high damping, at a higherfrequency. This obviously confirms the frequency

domain analysis. It is interesting to point out that the

overall braking performance is indeed very similar with

both damping values: a careful inspection of the two

responses shows that the overall balance of the long-

itudinal force is approximately even, whatever the

damping ratio value is. Notice that this substantial

independence of the braking performance from the

damping ratio no longer holds if a dynamic adjustment

of the damping can be done (semi-active damping

control—e.g., Savaresi, Bittanti, & Montiglio, 2005;

Savaresi & Spelta, 2007). It is easy to see that a

‘‘high–medium–high’’ (or even ‘‘high–low–high’’) damp-ing switch during the first 300 ms of the braking

maneuver would maximize the overall longitudinal

forces. This brake-oriented semi-active control strategy

(which is out of the scope of the present work) is

currently under development.

Another important characteristic to be carefully consid-

ered during braking is the behavior of the fork compression,

since this variable is directly ‘‘sensed’’ by the driver. By

inspecting its step response for different damping ratios, it is

clear that, with the low damping setting, the fork rapidly

reaches its steady-state value. However, this fast transient is

paid for overshoot and under-damped oscillations. On the

other hand, the high damping setting guarantees no

overshoot and no oscillations. In this case, the price to be

paid is a slower response: the fork takes almost 500 ms to

reach its steady-state condition. In a short braking

maneuver followed by a bend, this slow transient is

considered as a drawback, since the driver typically wants

to ‘‘feel’’ a steady-state condition before engaging a turn.

Also in this case, semi-active damping can help to alleviate

the trade-off: notice that the ‘‘high–medium–high’’ switch-

ing policy above described can also speed up the transient

of the high damping setting, while maintaining the

attractive properties of no overshoot, no oscillations andno initial unload of the front wheel.

5. Conclusions and future work

In this work the problem of high-performance braking

has been studied. The focus is on optimizing the braking

performance, in terms of stopping time. All the main

aspects of the problem have been considered and discussed:

the effect of aerodynamics on the loss of contact of the rear

wheel; the wheel-lock problem; the best torque modulation

strategy at the beginning of the braking maneuver; and the

role of the rear brake. Moreover, the sensitivity of the

braking performance to fork damping has been studied.

The results of this analysis provide some useful insights

on the problem of optimal braking. Moreover, this analysis

shows the potential benefits of electronically assisted

brakes: these benefits could be further exploited by joint

design of braking control and semi-active suspension

damping control.

Acknowledgments

This work has been supported by MIUR PRIN project

‘‘identification and adaptive control of industrial systems’’.

Thanks are due to Mario Santucci, Andrea Strassera and

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F x f

Casterangle

Fig. 16. A pictorial representation of the fork compression due to the

braking force and the caster angle.



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Onorino di Tanna of Piaggio Group, Carlo Cantoni and

Roberto Lavezzi of Brembo, for enlightening discussions

on the topic of braking control.

Appendix A. Motorcycle geometric and mechanical

properties

Wheelbase 1.370 m

Center of mass height (vehicle with rider) 0.621 m

Center of mass distance from front wheel 0.560 m

Caster angle 26.11

Rear arm length 0.250 m

Mass of vehicle with rider 274.2 kg

Front wheel mass 12.7 kg

Rear wheel mass 14.7 kg

Frame pitch inertia 22 kgm2

Front wheel moment of inertia 0.484 k gm2

Rear wheel moment of inertia 0.638 kgm2

Frontal cross section area 0.6 m2

Drag coefficient 0.52Lift coefficient 0.085

Pitch coefficient 0.205

Rear suspension properties

Spring stiffness 40,000 N/m

Spring pre-load 2080 N

Spring travel 0.2 m

Damping coefficient 10,000 Ns/m

End stroke stiffness 600,000 N/m

Swing arm mass 8 kg

Front suspension properties

Spring stiffness 40,000 N/m

Spring pre-load 933 N

Spring travel 0.12 m

Damping coefficient 2500 Ns/m

End stroke stiffness 100,000 N/m

Front fork mass 7.25 kg

Tire properties

Radial stiffness of front tire

130,000 N/m

Radial stiffness of

rear tire

141,000 N/m

Paceika parameters

of front tire

120/70, (Sharp, Evangelou,

& Limebeer, 2004)

Paceika parameters

of rear tire

180/55, (Sharp, Evangelou,

& Limebeer, 2004)

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