Roll angle estimation in two-wheeled vehicles

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Published in IET Control Theory and Applications

Received on 12th February 2008

Revised on 29th May 2008

doi: 10.1049/iet-cta:20080052

ISSN 1751-8644

Roll angle estimation in two-wheeled vehiclesI. Boniolo

1S.M. Savaresi

1M. Tanelli

1,2

1Dipartimento di Elettronica e Informazione, Politecnico di Milano, Piazza L. da Vinci, 32, 20133 Milano, Italy2Dipartimento di Ingegneria dellInformazione e Metodi Matematici, Universitadegli studi di Bergamo, Via Marconi 5,

24044 Dalmine (BG), Italy

E-mail: [email protected]

Abstract:An innovative method for estimating the roll angle in two-wheeled vehicles is proposed. The roll angle is

a crucial variable in the dynamics of two-wheeled vehicles, since it greatly affects the behaviour of the tire-road

contact forces. Hence, the capability of providing in real time a reliable measure of such quantity allows us to

evaluate the dynamic properties of the vehicle and its tyres, and represents the enabling technology for the

design of advanced braking, traction and stability control systems. The method proposed is based on a low-

cost sensor configuration, suitable for industrial purposes. The validity of the proposed approach is assessed in

a multi-body motorbike simulator environment and also on an instrumented test vehicle.

1 Introduction and problemstatement

Nowadays, four-wheeled vehicles are equipped with manydifferent active control systems which enhance driver andpassengers safety, some of which such as the ABS (anti-lock braking system) have recently become a standard onall cars (see e.g.[1, 2]).

In the field of two-wheeled vehicles, instead,electronic systems for the control of vehicle dynamics arestill in their infancy: only a few motorbikes are equipped

with ABS systems; traction control, drive-by-wire andsemi-active suspensions are mostly confined to advancedR&D prototypes; electronic stability control systems arestill far from an industrial application and only preliminarystudies have been done on this topic (see e.g. theEU-sponsored REGINS project SAFEBIKE http://safebike.jku.at).

More specifically, the ABS systems today available on themarket are expected to work at their best only when panicbrakes occur at in-plane conditions (zero roll angle).However, it is well known that a two-wheeled vehicle is

characterised by large values of roll angle (Fig. 1), whichcan reach the astonishing value of 508 558 using high-performance racing tyres. This angle is the inclination of

the vehicle with respect to the vertical direction and itrepresents the amount of inclination that the bike needs inorder to ensure the force balance on the curve. These largeroll angles obviously play a major role in the overall vehicledynamics and make the motorcycle dynamics very different(and much more complicated to be modelled andcontrolled) from the car dynamics.

Hence, to move a step further in active control systemsfor two-wheeled vehicle dynamics, the enabling technologycomes from a system capable of estimating the roll anglein a reliable way and in real time. Moreover, to suitindustrial cost constraints, such system should rely on a

low-cost sensor configuration.

Besides its usefulness in control system design, a reliableon-line measure of the roll angle might be useful in theracing context to assess the tyre performance with respectto this fundamental variable. In fact (see e.g. [313]), theroll angle has a major impact in determining the lateraltyre-road contact forces.

At a first glance, the problem of estimating the roll angleof a motorbike may appear trivial, using a standardinclinometer. However, since the stability of a motorcycle is

based on the equilibrium between gravitational andcentrifugal forces, it is easy to understand that a standardbody-fixed inclinometer is useless.

20 IET Control Theory Appl., 2009, Vol. 3, No. 1, pp. 2032

& The Institution of Engineering and Technology 2009 doi: 10.1049/iet-cta:20080052

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A more sound way of measuring the roll angle is thenumerical integration of the rotational speed along the rollaxis, measured with a gyrometer. However, numericalintegration is highly sensitive to DC measurement errors,

which induce a drift in the integrated signal (see Fig. 2,where this simple approach is applied on a set of real

measurements). This method can be used only if theintegration procedure is used over short time windows, anda robust re-initialisation procedure is available. In amotorcycle, none of these two conditions hold, and thissimple and naive approach cannot be consistently used.

In practice, no production motorcycle today is equippedwith a system for the roll-angle estimation. Roll-estimationsystems are used only for prototyping purposes and inracing applications. Two systems are typically used:

Global positioning systems (GPS or differential GPS),which allow the measurement of the complete trajectoryand derive from this information the roll angle as aby-product.

Inertial measurement units (IMU) characterised by threeaccelerometers and three gyros, which reconstruct the

whole vehicle attitude by gyros integration and byaccelerometer low-frequency components.

These approaches, however, can hardly be used on standardproduction motorcycles: GPS systems are reliable only whenopen-sky conditions are guaranteed, offers a comparativelylow precision and are better suited for off-line processing;complete IMU systems are still too expensive for industrialapplications.

In the open scientific literature, very little has been publishedon this topic so far. In [14], an interesting approach forestimating the whole vehicle trajectory is proposed; it employsa vision system made by cameras complemented with microelectro-mechanical systems (MEMS) accelerometers.

Something more can be found in the recent patent literature.Forexample, in [1518] someapproaches are described, whosecommon purpose is to devise robust estimation methods withlow-cost (and small size) equipment. All the above

approaches, nonetheless, differ from that proposed in thiswork. Specifically,[18]focuses on roll-over detection mainlytailored to four-wheeled vehicles, whereas [1517] focus onaccelerometer-based estimation algorithms.

The proposed approach, instead, is based on angular speedmeasurements provided by gyroscopes installed in a specialconfiguration, and on the longitudinal vehicle speedmeasurement, provided by wheel encoders. The mainadvantage of employing gyroscopes instead of accelerometersis that the formers provide measurements which are notaffected by gravity.

The key idea proposed herein is that of splitting the inputsignals into high-frequency (HF) and low-frequency (LF)components. Then, the HF part of the roll angle isestimated by direct integration of the HF component ofthe roll gyroscope, whereas two different approaches for theestimation of the LF roll angle are proposed. The HF andLF estimates are finally added to get the final estimate ofthe roll angle.

Figure 1 Roll angle of a two-wheeled vehicle

Figure 2 Real roll angle and roll angle estimated with a simple integration of the roll rotational speed (real measurements)

IET Control Theory Appl., 2009, Vol. 3, No. 1, pp. 20 32 21

doi: 10.1049/iet-cta:20080052 & The Institution of Engineering and Technology 2009

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As for the expected estimation performance, we seek for anestimation algorithm capable of providing peak estimationerrors of +58, which is considered to be an acceptableperformance for industrial production, if the main goal ofthe roll estimation system is to be included into a braking,traction or stability control system.

The outline of the paper is as follows: in Section 2, the basicnotation, the simulation environment and the experimentalset-up is introduced, whereas Section 3 illustrates anddiscusses the proposed approach and shows the results bothin simulation and in experimental data. In Section 4, analternative method for the estimation of the low-frequencycomponent of roll angle is proposed and discussed. Thepaper ends with some concluding remarks and an outlookon future work.

2 Notation, simulationenvironment and experimentalset-up

To improve the readability of the paper, we first introduce thenotation and reference frames employed in this work for thegyroscope measurements, the simulation environment usedto test the algorithms and the experimental set-up used forfinal validation.

2.1 Notation and reference frames

The four gyroscopes used by the estimation algorithms aremounted along four axes, indicated as x, z, y0 and z0.Notice that the axes x, y, z constitute the main body-reference frame fixed to the vehicle chassis. The two otheraxes y0 and z0 are obtained by rotating the body-referenceframe of 458 around x. The gyroscopes along x and z areused by the main algorithm presented in this work (Section3); the additional gyroscopes installed along y0 and z0 areused for an advanced estimation algorithm (Section 4).Finally, the absolute reference frame is indicated with xa,ya, za.

The roll, pitch and yaw angles will be indicated with the

symbols w, q and c, respectively (using a more compactnotation: a w q c T); vx, vy, vz are the correspondingbody-reference angular velocities.

InFig. 3, all the angular velocities used in this work aredisplayed (note that the absolute angular velocities areindicated with the symbols Vxa, Vya, Vza). More specifically,the following vectors of angular velocities will be used

w vxvyvz

2

4

3

5; w0

vxvy0vz0

2

4

3

5; W

Vxa

Vya

Vza

2

4

3

5 (1)

The roll-pitch-yaw (RPY) rotation matrix, which correlatesthe absolute and body-fixed angular velocities, is thus given

by (for the sake of conciseness, we indicate c1 cos(1) and

s1 sin(1))

RRPY a cqcc swsqsc cwsc sqcc swcqsccqsc swsqcc cwcc sqsc swcqcc

cwsq sw cwcq

24 35 (2)

Considering the motorcycle dynamics, to avoid singularityproblems in the rotation matrices within the admissibleranges of variation for the roll and pitch angles (see e.g.[19]), it is more appropriate to employ the rotation matrixRPY, which is defined as

RRPY a RZ c RX w RY q (3)

where RX(w), RY(q) and RZ(c) represent the elementaryrotations around the Xaxis of an angle w, around the Yaxisof an angle qand around theZaxis of an angle c, respectively.

Accordingly, the geometrical relationships between thebody-reference measurement axes and the absolute ones are

W RRPY a w RRPY a RRPY458

0

0

0B@

1CAw0

w RT

RPY a W

w0 RRPY a RRPY458

0

0

0B@

1CA

264

375

T

W RTRPY458

0

0

0B@

1CAw (4)

According to the adopted RPY convention, the absoluteangular velocities can be expressed as functions of theconsidered attitude angles in the form

Vxa

Vya

Vza

24 35 cc _w cwsc_qsc _w cwcc_q

_c sw_q2664 3775 (5)

Figure 3 Directions of the rotational axes used for the

angular velocities



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2.2 Simulation environment

For the development and analysis of roll-angle estimationalgorithms, the availability of a suitable motorcyclesimulator is mandatory, in order to analyse and test theestimation algorithms before moving to in-vehicle tests. Inthis work, the motorbike simulator presented in [13, 20,21] has been used. The model is developed in Modelica,

within the Dymola environment (see e.g. [2224]) and itis characterised by 11 degrees of freedom (d.o.f.) (Fig. 4):the chassis has the 6 d.o.f. of a floating body, 2 rotationald.o.f. are introduced for the steer and for the front hub,

whereas 1 translational d.o.f. is introduced for the frontsuspension; other 2 d.o.f. are introduced for the swingingarm and of the rear hub. The suspensions are modelled asspring-damper systems, with nonlinear characteristics ([11,25]). The exogenous variables (inputs) of the model are thetorques applied to the steer and to the front and rear wheelhubs. The tire-road forces are computed according to a roadmodel. The longitudinal and lateral forces models are basedon [26]. Further, the motorcycle model is complemented

with a virtual driver (see e.g. [4, 5, 20, 26, 27]), whichallows to follow a reference trajectory.

In what follows, the performed simulations have beencarried out by providing as inputs to the simulator a desiredroll angle profile and a speed profile. The virtual driver isable to perform the desired manoeuvre at the desired speed.

The most important customisation of a simulator tailoredto roll angle estimation is the installation on the motorcyclemodel of a set of virtual sensors, which mimic the availablemeasurements. The chosen simulation environment providesa library of virtual sensors, which can be placed ontranslational and rotational joints in order to measure linearand angular quantities (i.e. accelerations, velocities andangles).

Fig. 5 shows the wheel, the hub, the point of contact(POC) and the world reference frames, which are involvedin the computation of the reference roll, the pitch and the

yaw angle. Specifically, the roll angle wis computed as

w arctan kyPOC,zwheellkzwheel,zPOCl

(6)

The pitch angle q is computed using the wheel and hubframes (note that these two reference frames are used sinceonly one of them the hub frame is affected by pitchmotion)

q arctan kxhub,zwheellkxhub,xwheell

(7)

Finally, the yaw angle cis computed as

c arctan kxwheel,yworldlkxwheel,xworldl

(8)

where theworldframe is the absolute reference frame.

2.3 Experimental set-up

The proposed estimation algorithm has been tested on a realvehicle (a MY05 Aprilia RSV1000 Factory), as shown in

Fig. 6.

Figure 4 Modular scheme of the motorcycle modelFigure 5 Reference frames of the wheel, hub, point of

contact (POC) and world of the simulator

Figure 6 Instrumented Aprilia RSV1000 Factory used for the

experimental tests (courtesy of Piaggio Group)



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In the test vehicle, the following set of sensors and data-logging systems have been used:

four one-axis gyroscopes, with a cut-off frequency of10 Hz;

a 100-step wheel encoder;

two optical distance sensors;

a real-time board for the acquisition and data logging ofthe whole set of measurements.

Fig. 7shows how to the actual roll angle from the opticalsensors is measured, namely (the meaning ofz1,z2and L isclearly indicated inFig. 7)

w arctan z1

z2

L

(9)

This kind of non-contact distance sensors allows us to obtaina very accurate measure of the roll angle, with a precisionsuitable for our purposes (i.e. the measurement error is on average less than 18).

As illustrated inFig. 6, the gyroscopes were placed on agyro-box on top of the bike tank. The assembling andmounting of the gyroscopes are rather critical, since it is

difficult to eliminate all the possible misalignment errors.To partially overcome such problems, an on-line tuning ofthe mean value of the gyroscopes has been implemented.Each test was always started with a 10 s-long phase ofconstant speed on a straight road, in order to calibrate thegyroscopes.

3 Proposed estimation approach

The estimation approach proposed herein has the generalarchitecture shown in Fig. 8. As mentioned above, themain idea is that of splitting the input (measured) signals(the two angular speeds vx and vz, and the vehiclelongitudinal speed v) into high-frequency (index HF) andlow-frequency (index LF) components. Such signalcomponents are processed independently after having been

split by the frequency separation block and then at eachsampling instant the LF estimate wLF and the HFestimate wHFare added to build the final estimate wof theroll angle.

The internal structure of the frequency separation block isillustrated inFig. 9. It is constituted by a standard first-orderlow-pass digital filter (with a pole inaand DC-gain equal to1) and a summing junction which splits the signal into theLF and HF components. The parameter a is the onlytuning knob. In principle, the value of a should be assmall as possible, in order to eliminate only the DC

component of the noise, which is the main cause of thedrift in the integrated signal. However, as the rollgyroscope noise has a frequency spectrum which exhibitsnon-zero components also in the neighbourhood of theDC frequency, in order to avoid the integration of suchnoise LF components, a trade-off in the choice of the

value ofamust be managed. Experiments have shown thata sensible range for such parameter is a[ [0.1, 0.4]Hz.

The final value of a 0.15 Hz has been found via rootmean square error minimisation computed for all themeasured tests.

Notice that the LF estimation block receives as inputs

the LF components of the measured body-fixed yaw rate

Figure 7 Schematic diagram of the roll-angle computation

from the optical sens ors (lef t) and the sens ors mounted

on the test vehicle (right)

Figure 8 High-level architectural view of the proposed estimation algorithm



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vz and the longitudinal speed v (the additional gyroscopemeasurements vz0 and vy0 will be used only in thealternative method presented in Section 4); instead, theHF estimation block only uses the HF component ofthe body-fixed roll rate vx. In the rest of this section,the HF and LF estimation algorithms will be presentedand discussed.

3.1 HF estimation

Since the roll anglewis simply related to the body-fixed rollrate vx by the relationship _w vx, in principle it can bereconstructed by a simple numerical integration of themeasured vx. However, as already briefly illustrated inSection 1 (Fig. 2), the main problem is that vx is affectedby noise: vx vx nx, where vx is the true roll rate andnx is the measurement noise. Hence, the numericalintegration procedure applied to vxprovides the followingresult (DTis the sampling interval)

w(t) DT1 z1 vx(t)

DT

1 z1 vx(t) DT

1 z1nx(t) w(t) nw(t) (10)

The main problem is that the measurement noisenxcontainsLF components (and, in particular, DC offsets). Even if theLF components are small, they cause a quick drift in the

integrated signal. The numerical integration hence can beapplied only to the HF part of the signal, namely

wHF(t) DT

1 z1 vxHF(t) DT

1 z11 a

1 az1 vx(t)

DT1 z1

1 a1 az1 vx(t)

DT1 z1

1 a1 az1nx(t) (11)

Using algorithm (9), the measurement noise is depurated ofthe LF component, and the drift problem is removed. Onthe other hand, this algorithm also removes the LFcomponent of the true roll rate vx: the estimated roll anglewHFhence carries no information on the LF component ofthe roll angle, which must be reconstructed following adifferent path.

In Fig. 10, the results of a 5-min long experimentare displayed. The estimated HF roll angle is compared

with the HF true angle (measured with the opticalsensors). Apparently, the HF estimation is very accurateand the drift problem (see Fig. 2 for comparison) iscompletely removed. The corresponding error-to-signal-ratio is about 0.05 (namely, the error variance is 5% of thesignal variance).

3.2 LF estimation

The problem of estimating the LF component of the rollangle is much more tricky. In order to develop theestimation algorithm, it is useful to analyse the mainforces acting on the vehicle, when negotiating a curve.Such forces are concisely and pictorially represented inFig. 11.

Consider now the following set of assumptions:

Figure 9 Internal structure of the frequency separator block

Figure 10 HF-components: true roll angle (thin line) and estimated roll angle (bold line)



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the vehicle is in a quasi-stationary condition (i.e. it is notsubject to longitudinal acceleration, the roll angle is constantand the curve radius is constant);

the gyroscopic effects are negligible;

the tire has null cross-section.

In this case, it is easy to see that a balance equation whichallows to compute the roll angle can be easily worked out,namely

w arctan vVzag

(12)

where Vza is the absolute yaw velocity, v is the forwardvehicle speed and g is the gravitational acceleration.

Unfortunately, the absolute yaw velocity Vza cannot bedirectly measured. The measurable variable is thebody-fixed yaw velocity vz, which is related with Vza bythe following expression (see Section 2.1 for the notation)

vz sqcc swcqsc

Vxa sqsc swcqcc

Vya

cwcqVza (13)

Under the above set of assumptions, (13) can be stronglysimplified. As a matter of fact Vxa Vya 0, since the

vehicle has no roll or pitch velocity [see also (4)]. Noticethat this condition is fulfilled rigourously if the vehicle isnegotiating a curve and it is in a steady-state condition;however, it still holds even if approximately if the

vehicle is not at steady-state but we consider the LFcomponent of the signals. Moreover, under the assumptionof null (or small) pitch angle, another simplification can beapplied, namelycq cos(q) 1.

Equation (11) hence can be simplified as

vz cwVza (14)

This relationship is simply expressed by the pictorialrepresentation of Fig. 12. The correction term for theestimation of Vza from vz is given by cos(w): it isnegligible for small roll angles, but it must be accounted for

when the motorbike has large roll motions.

It is interesting to observe that this problem is complicatedby another small distortion, due to the effects of a non-nulltire cross-section on the roll angle. To this end, let usassume that the tire cross-section has diameter equal to 2ct:it can be shown (see [6, 28]) that the error say Dwc induced on the estimation of the roll angle is given by thefollowing expression

Dwc arcsinctsin arctan v

2=gR

h ct

(15)

wherehis the height of the vehicle centre of gravity measuredalong the vehicle symmetry plane, and R is the curvatureradius of the trajectory. This error is usually small, but itcan take values up to 28.

Another subtle but critical source of distortion as alreadyremarked in Section 2 is because of misalignments in theinstallation of the sensors on the vehicle. This can bestrongly reduced in an industrial product, but on prototypeinstallations this is a key issue.

Hence, the overall estimation algorithm for the LF

component is summarised in the block scheme ofFig. 13.

Notice that the LF signals vzLFand vLFare first fed into thefunction (12), whereVzais replaced byvzLF. The output say

Figure 11 Force balance on a curve

Figure 13 Structure of the LF estimation algorithm

Figure 12 Absolute and body-fixed yaw rates



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~wLF of this function is then fed into a warping function.Notice that this warping function, if (14) is the only source ofdistortion, would be simply wLF arcsin(tan( ~wLF)).However, since many other distortion factors must beaccounted for, in practice it is easier to estimate the warpingfunction directly from measurements on the vehicle. For theestimation of the warping function, several tests have beenperformed once the final gyroscopes installation was fixed tothe test bike. These tests were constituted by steady-statecornering conditions at fixed vehicle speed and fixed

curvature radii taken on a steering-pad-like track, which wererepeated for different roll-angle values. The estimated rollangle was compared to the one measured with the opticalsensor to provide the data points in Fig. 14b. In view ofindustrial applicability, note that the estimated compensationcurve would depend on the considered motorbike only fordiscrepancies in the mounting phase on different

vehicles rather than for the specific bike geometry andloading. So, once the installation process has been defined, asingle warping function fora whole production line can be used.

In Fig. 14, an example of estimation of the warping function

is displayed. This function is estimated using standardnonlinear system identification tools (see e.g. [2830]).Specifically, parametric polynomial functions have been used.Notice that the warping function estimated on the simulatoris almost perfectly overlapped with the ideal warpingfunction wLF arcsin(tan( ~wLF)), whereas, when using realdata, the estimated function slightly differs from the idealcurve, due to non-null tyre cross-section and installation errors.

3.3 Overall estimation algorithm:simulation and experimental results

According to the scheme ofFig. 8, the final estimation of the rollangle can be simply obtained by adding the two independently

estimated HF and LF components: w wHF wLF. InFig. 15, the results obtained with the estimation algorithmspresented in this Section are illustrated, both in the simulationenvironment and on the real vehicle. It is easy to see that insimulation the real and estimated roll angles are almostindistinguishable; however, also in the (much morechallenging) real experiment, the residual error is very small(peak errors of less than 58, and ESR of about 6%). Noticethat the magnitude of the LF and HF components are similar;this is a further indication that the choice of the frequency

split is correct.

4 Alternative method forLF estimation

The algorithm presented in the previous section has shown toprovide accurate results, with a comparatively simple set ofsensors. The only significant flaw of the proposed methodis its sensitivity to road inclination, which affects the LFcomponent of the estimation algorithm.

More specifically, assuming that the vehicle is negotiatinga road with a slope rx(measured along the longitudinal axisof the vehicle) and a banking ry (measured along thetransversal axis of the vehicle) from (12) (14) it can beseen that under the usual assumption of steady-stateconditions the true roll angle (calculated with respect tothe road surface) can be written as

w arctan vvzgcos (w ry)cos(rx u)

! ry (16)

Notice that [apart from the presence of the term w in theright-hand side of (14)], the estimation of the roll angle isdistorted by the pitch angle and the road slope and

Figure 14 Warping functions (continuous lines) estimated starting from virtual measurements on the simulator (left) and

from real measurements on the vehicle (right)

The dashed line is the ideal compensation curve w LF arcsin (tan(wLF))



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banking, if these angles are not measured and explicitly takeninto account. Whereas the pitch angle can be considerednegligible in LF (or near-steady-state) conditions, theinfluence of the road slope and banking cannot beremoved; when the road inclination is large, a significantdistortion is introduced in the estimated roll angle.

In order to cope with this problem, an alternative methodfor the estimation of the LF component of the roll angle is

now proposed, discussed and compared with the methodpresented in the previous section.

Consider now the extended set of four gyroscopes, asintroduced in Section 2; they measure the angular ratesalong the axesx,z,y0and z0(seeFig. 3). By exploiting the

geometrical relationships between the measurementreference frames, the four measured rotational velocities canbe written as functions of the absolute velocities and

Figure 15 Simulation and experimental results of the overall estimation algorithm

a Real and estimated roll angles (top) and estimation error (bottom) simulation experimentb Real and estimated roll angles (top), estimation error (middle) and HF/LF components (bottom)



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attitude angles as follows

vx cqcc swsqsc

Vxa cqsc swsqcc

Vya cwsqVza

vy0 ffiffiffi

2p

2 sqcc swcqsc cwsc Vxah sqsc swcqcc cwcc

Vya cwcq sw

Vza

i

vz0 ffiffiffi

2p

2 sqcc swcqsc cwsc

Vxa

h sqcc swcqcc cwcc

Vya cwcq sw

Vza

ivz swcqsc sqcc

Vxa swcqcc sqsc

Vya cwcqVza

(17)

Recalling that the reference framexyz is rotated to 458withrespect to the framexy0z0, the angular velocities in these twoframes are linked as

vy0 ffiffiffi

2p

2 vy vz

vz0 ffiffiffi

2p

2 vy vz (18)

By manipulating (5) and (17), the following expression canbe found

vy

vz

_q sw _csq _w cwcq_c

(19)

Under the only assumption of steady-state conditions (noticethat no other assumptions on the vehicle attitude or roadinclination are made), (19) can be strongly simplified byplugging in _w _q 0 (Vxa Vya 0), so obtaining

vyLF

vzLF tan(wLF)

1

cos q (20)

In (20), an error term due to the pitch angle q is present.However, under the working assumption of steady-statecurve, the pitch angle is negligible, so that in what follows

we considerq 0.

Since vy is not an available measurement, using (18) the

relationship (20) can finally be rewritten as

wLFarctan 2ffiffiffi

2p vz0LF

vzLF 1

;vzLF 0

arctan 2ffiffiffi

2p vy0LF

vzLF 1

;vzLF, 0

8>>>: (21)

From (21), one could notice that, in principle, the roll-angleestimation could be done using only two gyroscopes (inaddition to the gyroscopic measurement vx, used for the

HF component). The additional measurement ismandatory in order to avoid singularity problems whenthe roll angle is around 458. From (21), it can be observedthat the gyroscopic measurement vz is always used,

whereas vz0 is used on the right turns only, and vy0 onthe left turns only. The rationale behind this choice canbe better appreciated by the pictorial representation ofFig. 16.

As already remarked, the main advantage of using thisalternative method is that it is not directly affected byroad inclination. This can be appreciated in Fig. 17,

where using the motorbike simulator the results ofan experiment made on a non-negligible 108 slope aredisplayed. Clearly, the algorithm presented in the previoussection is strongly affected by the road inclination,

whereas the above method is almost insensitive to thisproblem.

We conclude this section with a brief comparative discussionof the two algorithms for the estimation of the LFcomponent of the roll angle. We call speed-based thealgorithm presented in Section 3, and gyro-based thealgorithm presented above.

Figure 16 Pictorial representation of the alternative LF estimation method



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The main evaluation and comparison parameters are thefollowing:

Equipment cost (number and cost of the sensors): Fromthis point of view, the speed-based algorithm issignificantly better. As a matter of fact it requires two gyrosonly, and a wheel speed measure which is always alreadyavailable on the motorbike. On the other hand, the gyro-based method does require four gyrometers.

Robustness to null roll angles: Also from this perspective,the speed-based algorithm is the best. As a matter of fact itcan seamlessly estimate the whole range of angles, withoutsingularity problems. On the other hand, from (21) it isapparent that the gyro-based algorithm works as long asthe vz velocity is non-null, namely when the roll angleis non-null. When vz 0, the algorithm as is cannotbe used.

Sensitivity to road inclination: As already stressed, thespeed-based algorithm cannot reject the distortions becauseof slopes and banking [see (16)], whereas the gyro-based

algorithm is almost insensitive to road inclination, thanksto the fact that it is based on the ratio between tworotational speeds and not on their absolute values [see (21)].

The above remarks clearly show that the features of the twoalgorithms are somehow complementary. Roughly speaking,the speed-based algorithm is preferable (for its robustnessand low-cost) when the estimation-error requirements arenot particularly tight. For more demanding applications,the gyro-based algorithm must be used. In this case,however, the best approach is to combine the twoalgorithms in order to exploit their best features; as a

matter of fact the longitudinal speed measurement is alwaysavailable, and the speed-based algorithm uses a subset ofthe gyroscopes employed by the gyro-based method.

Hence, when the gyro-based algorithm is used, the speed-based algorithm can be implemented with no additionalsensor cost. In this case, the problem of mixing andblending the two algorithms obviously arises; this problemhowever is out of the scope of this paper, and it is an issuecurrently under study.

5 Concluding remarks andfuture work

In this work, an innovative method for estimating the rollangle of a two-wheeled vehicle has been proposed. Suchalgorithm allows us to evaluate the dynamic properties ofthe vehicle and its tyres and it represents the enablingstep to move towards a new generation of active controlsystems for motorcycles. The proposed estimationalgorithms have been shown to provide a reliable roll-angleestimation based on a low-cost sensor configuration, whichmay be suitable for large-scale production purposes. The

validity of the proposed approach has been assessed bothvia a detailed multi-body simulation environment and onan instrumented test vehicle.

The current research activity on this topic is focussedon the development of a blending algorithm that exploitsthe benefits of both the proposed LF methods, whosecomplementarities have been shown and discussed.

6 Acknowledgments

This work has been partially supported byMIUR project Newmethods for Identification and Adaptive Control for IndustrialSystems. Thanks are due to Enrico Silani and Vittorio

Gariboldi for their help in the preliminary part of this workand to Luca Fabbri and Lorenzo Nardo of Aprilia for theconstructive discussions and for their support in the

Figure 17 Simulation experiment on a 108 slope

The true roll angle (with respect to the road surface) is compared with the roll angle estimated with the algorithm based on the vehiclespeed (Section 3), and with the algorithm based on additional gyros (Section 4)



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experimental testing. Thanks are also due to Filippo Preziosiof Ducati Corse, to Carlo Cantoni and Roberto Lavezzi ofBrembo, and to Mario Santucci and Onorino di Tanna ofPiaggio, for stimulating discussions on this topic.

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