nonlinearity modelling in an...

Nonlinearity Modelling in an ElectromechanicalBraking System for Development of a SmartCaliper

Reza Hoseinnezhad and Alireza Bab-Hadiashar

Abstract In electromechanical brakes, central controllers require accurate informa-tion about the clamp force between brake pad and disc as a function of pad dis-placement. This function is usually denoted as characteristic curve of the caliper. Ina typical electromechanical braking system, clamp force measurements vary withactuator displacements in a hysteretic manner. Due to ageing, temperature and otherenvironmental variations, the hysteretic characteristic curve of calliper varies withtime. Therefore, automatic caliper calibration in real-time is vital for high perfor-mance braking action and vehicle safety. Due to memory and processing powerlimitations, the calibration technique should be memory efficient and of low com-putational complexity. This chapter investigates the hysteresis as a nonlinear effectin the electromechanical brakes, and describes a technique to prametrically modelthis effect. This technique is a simple and memory-efficient real-time calibrationmethod in which a Maxwell-slip model is fitted to the data samples around eachhysteresis cycle. Experimental results from the data recorded in various tempera-tures show that this technique results in clamp force measurements with less than0.7% error over the range of clamp force variations. It is also shown that by usingthese measurements, the characteristic curve can be accurately calibrated in real-time.

1 Introduction

Electromechanical brakes (EMBs) replace traditional mechanical and hydrauliclinkages with electric actuators and computer control systems. Many vehicle manu-facturers are engaged in research or collaborative development programs focussed inthis area. The requirements for reliability and safety in automotive braking systemsare just as stringent as the ones they are replacing. On the other hand, EMB compo-

RMIT UniversityVictoria, Australia, e-mail: {rezah,abh}@rmit.edu.au

1

2 Reza Hoseinnezhad and Alireza Bab-Hadiashar

nents should be cost-wise competitive with conventional technologies. Therefore,there are always limited scope for hardware redundancy which means the EMBsshould be inherently safe and reliable [4, 8, 11, 12, 15].

To achieve accurate and stable control of the vehicles equipped by EMBs, weneed to attain accurate models for various nonlinear static and dynamic processesinvolved in such braking systems. Using our earlier works [1–3, 6, 14], we presentone of the most important nonlinearities existing in electric calipers. Before focusingon the nonlinear phenomena, we present a quick review of general structure andcomponents of common EMB system designs.

A general diagram of a brake-by-wire system is shown in Fig. 1 [5]. A typicalsystem includes four principal components: a central brake controller (also calledcentral control unit or CCU), a sensing/measurement apparatus for driver’s brakedemand, brake units in four corners of the vehicle, and a communication network.More specifically, any EMB system includes electro-mechanical brake calipers (e-calipers) with embedded brake torque controllers at each vehicle corner, wheelspeed and vehicle motion sensors, a central controller unit, and a human-machine in-terface, such as an instrumented brake pedal, all communicating via a fault-tolerantcommunications network [1–3, 11, 13].

Measurement & Estimation in Brake-By-Wire

Brake Request Sensing

Apparatus

Comm. Network

Central Brake Controller

ECU and other

Electronic Systems and

Sensors

Controller

Sensors/Observers

Actuator

Caliper Brake Unit FR

Controller

Sensors/Observers

Actuator

Caliper Brake Unit RR

Controller

Sensors/Observers

Actuator

Caliper Brake Unit FL

Controller

Sensors/Observers

Actuator

Caliper Brake Unit RL

Fig. (1). General structure of a brake-by-wire system.

The central brake controller (which in some cases is embedded within the

vehicle’s Electronic Control Unit - ECU) is responsible for generating brake

commands to implement high level stability control systems such as antilock braking

system (ABS), traction control (TC), vehicle stability control (VSC), electronic

stability protocol (ESP) and the like. Since the stability control related systems have

been already developed for electro-hydraulic brakes, the major challenges involved in

development of central brake controllers for brake-by-wire systems is with

modification and adaptation of those techniques for EMBs. A few novel methods and

techniques have also appeared in brake-by-wire patent literature [1-4].

The communication network is responsible for transmitting signals between the

different sensors, processors and actuators in the system. Because of safety criticality

of the system, the network needs to be fault tolerant. For this purpose, a number of

Fig. 1 General structure of a brake-by-wire system [5].

Figure 2 shows a block diagram of the e-caliper control system which includesthe connections between the central control unit (CCU) and one of the e-calipers inthe EMB system. The CCU is the central vehicle dynamic control unit and generatesthe brake commands required to perform high-level braking tasks such as anti-skidbraking (ABS), vehicle stability control (VSC) or traction control (TC). These com-mands are sent to the four e-calipers via a communication network. These brakecommands are in the form of the desired clamp force to be generated by each e-

Title Suppressed Due to Excessive Length 3

caliper. Such commands are usually generated in CCU by processing the clampforce and displacement measurements in the calipers and the wheel speed measure-ments. A local controller in each caliper regulates the electric current that drives thebrake actuator.

CentralControl

Unit

*cf

LocalController Actuator

PlanetaryGear- Set Ball-Screw

Load Sleeve

Communication Network

,

Clamp ForceSensor

Resolver

ˆ,ˆ

ˆ,ˆcf

cf

e-caliper

Fig. 2 Block diagram of the e-caliper control system in a typical EMB design [6, 14].c©[2008] IEEE

A schematic diagram of an e-caliper developed by PBR Australia [6,14] is shownin Fig. 3. In this design, the rotational displacement of the brake actuator is convertedto transitional displacement of a ball-screw through a planetary gear-set. This causesthe load sleeve to push the brake pad toward the brake disc and generate the clampforce.

Measurement of the position and speed of the actuator and the resulting clampforce in the caliper are safety-critical tasks in an EMB system, because thosemeasurement are the key variables used by the CCU to generate the brakecommands.

The position and speed of the actuators are measured by resolvers (as shownin Fig. 2). The techniques required for obtaining accurate and robust estimates ofposiiton and speed as well as automatic calibration of the resolver have already beendeveloped [1, 3]. Both techniques are efficient in terms of their accuracy, memoryusage and computational complexity, and can be implemented in real-time. Thus,we assume that reliable and accurate measurements for the position and speed ofthe actuators in Figures 2 and 3 would be available.


1 stator field winding 7 resolver location2 brake pads 8 permanent rotor magnet location3 ball-screw 9 load distribution plate4 planetary gear-set 10 nut5 thrust bearing 11 caliper bridge6 clamp force sensor location

Fig. 3 The diagram of the electro-mechanical brake caliper [2]. c©[2008] IEEE

The CCU and the caliper local controllers require accurate knowledge of thecharacteristic curve of the calipers, i.e. the profile of the clamp force versus paddisplacement. In addition, accurate characteristic curve of e-calipers can be utilisedto calculate clamp force estimates from the displacements measured by resolvers.Fusion of the direct force measurements given by the sensors with their alterna-tive estimates from the actuator position can result in more reliable clamp forcemeasurements which enhances the performance of the brake control and systemsafety [13, 16, 17].

In the above design, rotational displacement of the caliper actuator is transformedto pad movement via the planetary gear-set and ball-screw and based on the kinemat-ics of this transformation, actuator and pad displacements are almost proportional.Therefore, without noteworthy loss of accuracy, we study the profile of clamp forcemeasurement against the actuator position in place of the pad movement.

2 The Hysteresis Effect

Characteristic curve varies with ageing and environmental conditions (e.g. tempera-ture and humidity) and should be accurately calibrated in real-time. Such a calibra-tion can only be performed by utilising recent samples of the measured forces andtheir corresponding displacements. Therefore, the accuracy of e-caliper calibrationsignificantly depends on the accuracy of clamp force measurements.

Since the stress in the load sleeve is almost uniformly distributed over the crosssection, for the purpose of measurement of the very large loads experienced in abrake caliper, the load sleeve can be considered as an axially loaded spring element.


Thus, load cells are used to measure the clamp force. Fig. 4 shows the arrangementof the strain gauges in a typical load cell on the load sleeve and their electricalcircuitry. In a load cell, adjacent strain gauges are connected in opposite bridgearms to remove bending strains resulted from off-axis or transverse componentsof forces. Moreover, the strain gauges are oriented transversely to desensitise thebridge output to temperature changes [18].

Fig. 4 Arrangement of thestrain gauges in each loadcell around the load sleeveand Strain gauge electricalcircuitry [6]. c©[2008] IEEE

Vo VDC

1

23

4

1,2 3,4

1,(3)

2,(4)

+ +

As shown in [6], when the load cell measurements are plotted against the actua-tor displacement, the result involves hysteresis around the true characteristic curveof the caliper similar to the one shown in Fig. 5. This hysteresis is caused by thepresliding component of the friction that exists between the key (placed to preventthe load sleeve from rotating with the ball-screw) and its keyway inside the housingof the load sleeve - see Fig. 6.

Fig. 5 A hysteretic behaviouris observed when the clampforce measurements are plot-ted versus actuator displace-ment [6]. c©[2008] IEEE

0 1 2 3 4 5 60

5

10

15

20

25

30

Actuator Displacement (Mechanical Revolutions)

Cla

mp

For

ce (

kN)

at a

roun

d 44

°C

Given by internal clamp force measurements Given by external clamp force measurements

To obtain accurate clamp force measurements for caliper calibration and control,the hysteretic friction component of the force measurements provided by the loadcells should be detected. The friction modelling and estimation procedure appliedby the system should be simple and efficient in terms of its required memory andcomputational power. This is because of the limits of the processing power of centralcontrol unit (CCU in Fig. 2) and the available memory in the system. In addition,besides the identification of the hysteresis part and removing it from the measuredclamp force (to obtain a reliable measurement) and calibration of the characteris-tic curve, there are many other complicated processing jobs to be performed by


Fig. 6 A picture of the ball-screw and the load sleevewith the key on it [6].c©[2008] IEEE

KeyLoad sleeve Ball-screw

the CCU using its available memory and computational power. Some examples arevehicle state estimation, ABS, VSC, and TC.

In the next sections, a memory and computational efficient technique is presentedfor identification of the hysteresis part of the measurements, extracting reliable es-timates of clamp forces, and real-time calibration of characteristic curve using theestimated clamp forces. In Section 3 different hysteresis models are reviewed andan appropriate model (with the desired accuracy and computational complexity) isselected to be applied for extracting the true clamp force from the hysteretic mea-surements. A memory and computationally efficient technique for automatic tuningof the characteristic curve of the caliper is explained in Section 4. Experimentalresults are presented in Section 5, and Section 6 concludes the chapter.

3 Hysteresis Modelling

As it is observed in the example shown in Fig. 5 [6], the clamp force measured bythe strain gauges (hereafter, the set of the six load cells are called internal clampforce sensor or internal sensor for short) is comprised of two parts: The real clampforce that causes the axial load on the load sleeve and changes the resistance of thestrain gauges, and the sliding friction force between the key on the load sleeve andthe keyway in its housing. To obtain an accurate measurement of the clamp force,the friction part should be estimated and removed. Since this friction force causesthe hysteresis phenomenon, a hysteresis model can be used to estimate the frictionforce.

Hsu and Ngo [7] have introduced a Hammerstein configuration, which includes aHammerstein-based dynamic model for hysteresis. This model includes a nonlinearstatic block followed by a linear dynamic block, and is applied to model the rate-dependent and temperature-dependent hysteresis phenomenon. Li and Tan [9] haveapplied a neural network to estimate the influence of hysteresis for adaptive controlof a nonlinear system which involves hysteresis. The above two approaches are toocomplicated to be implemented in real-time brake-by-wire systems.

Oh et al. [10] have analysed the Dahl, LuGre, and Maxwell-slip friction modelsas Duhem hysteresis models, classifying each model as either a generalised or asemilinear Duhem model. Here, we follow their unified treatment of Duhem-based


friction models to investigate the friction-induced hysteresis in e-calipers. Throughsome experiments (explained in Section 5), it was shown in [6] that by using theMaxwell-slip model to capture the hysteresis part of the load cell measurements,clamp force estimates with sufficient accuracy1 can be obtained.

3.1 Maxwell-Slip Model

In Maxwell-slip model the hysteretic slippage is modelled as M zero-mass elasto-slip elements connected in parallel as shown in Figures 7 and 8. Each element ofthis model is characterised by its stiffness Ki, position xi(t), spring deflection δi(t) =x(t)− xi(t) and maximum spring deflection ∆i (before the element i starts to slip).The input displacement x(t) is common to all elements. The total hysteretic frictionforce f f is given by the summation of all operators spring forces:

f f (t) =M

∑i=1

Ki∆iδ i(t) (1)

where δ i(t) is given by:

δ i(t) =

{δi(t)∆i

if |δi(t)|< ∆i (stick)sgn(δi(t)) if |δi(t)| ≥ ∆i (slip)

(2)

and the dynamics of the element position xi(t) is as follows:

xi(t +1) ={xi(t) if |δi(t)|< ∆i (stick)x(t)− sgn(δi(t))∆i if |δi(t)| ≥ ∆i (slip). (3)

Figure 9 shows the friction force f f given by the above model plotted versusthe displacement x. We observe that the model results in a hysteretic friction forcevarying within [−∑

Mi=1 Ki∆i,+∑

Mi=1 Ki∆i], and the more the number of elements M

are, the smoother the curve is. The centre of the hysteresis curve along the x axisdepends on the location of the actuator at the beginning of the hysteresis cycle [6].

To model the hysteresis in the e-caliper application, it is assumed that the Mpoints are evenly distributed over the maximum sticking displacement ∆max:

∆i =i

M∆max ; i = 1, . . . ,M. (4)

The maximum sticking displacement can be calculated from some previously recordedforce-displacement measurements: It is the displacement at which the maximum de-flection from the characteristic curve is observed. This deflection is caused by the

1 The minimum accuracy of clamp force measurement - required by high level braking functionsin CCU - is around 99% accuracy over the range of clamp force variations 0-40KN.


Fig. 7 Modelling the hys-teretic slippage as a paral-lel combination of saturat-ing elasto-slip elements [6].c©[2008] IEEE

)(1 tx

)(txi

)(txM

)(tx

11 ,K

iiK ,

MMK ,

Fig. 8 Stiffness curves of thespring elements in the modelof Fig. 7 [6]. c©[2008] IEEE

)()( txtx i

)(tFi

0 i

i

iiK

iiK

pre-sliding friction and for larger displacements, the measured clamp force followsa path almost parallel with the characteristic curve.

4 Characteristic Curve Calibration: Algorithm

In our previous works [13] we have shown that the characteristic curve of the e-caliper can be modelled by a third-order polynomial. The measurements given bythe internal clamp force sensor are also modelled as the sum of two parts: Theclamping component and the friction component. The clamping component is a thirdorder polynomial function of the displacement (given by the characteristic curvemodel) and the friction component is given by Maxwell-slip model as described insubsection 3.1:

fc(t) = Ax(t)3 +Bx(t)2 +Cx(t)+D+M

∑i=1

Ki∆iδ i(t). (5)

The parameters {∆1, . . . ,∆M} are assumed to be known a priory by using equa-tion (4). Characteristic curve parameters A, B, C and D, and the linear coefficientsof the hysteresis model {K1, . . . ,KM} are determined by fitting an ensemble of datato the above model simply by using least-squares technique.


x

cf

12

22

22 M

12 M

M2

i

M

i

iK

1

)2( 1

2

11 i

M

i

iKK

)2( 2

3

2211 i

M

i

iKKK

)2( 2

1

222211 iM

M

Mi

iMM KKKK

)2( 1112211 MMMMM KKKK

i

M

i

iK

1

0

Fig. 9 Force versus displacement plot for a general Maxwell’s slip model [6]. c©[2008] IEEE

An appropriate ensemble of data samples {(x(i), fc(i))} should contain pointsaround a full hysteresis cycle on the force-displacement plot. As the data samplesare sequentially received by the CCU through the communication network, the CCUshould be able to detect the starting point of a hysteresis cycle so as to start recordingdata samples till the end of the cycle (before the next starting point). Before a newhysteresis cycle begins, the position coordinate x is decreasing with time (data pointsare moving on the lower half of the current hysteresis cycle as shown in Fig. 9)and as soon as a new cycle starts, the x coordinate begins to increase with time.Therefore, it was suggested in [6] that a starting point is detected as below:

x(i)> x(i−n0)x(i−n0)< x(i−2n0)

}⇒ x(i) is a starting point. (6)

The parameter n0 prevents the incorrect detection of a starting point due to the fluc-tuations of displacement signals (caused by noise). However, there is a trade-off, asa large n0 would result in late detection of the starting point of hysteresis cycles. Anappropriate value for n0 depends on the application specific factors such as samplingrate, signal to noise ratio of the position measurements, and nominal and maximumactuator speeds, and can be determined by trial and error through experiments.

Because of memory limitations, recording of all data samples in a hysteresis cy-cle is not feasible. Assume that only L samples of {(x(i), fc(i))} pairs out of the datasamples in each hysteresis cycle can be recorded for determining the parameters ofthe model (5). An iterative method is needed for optimal selection and recording ofthe data samples as they are consecutively received by the central controller via the


communication network. The recorded data samples should be distributed aroundthe hysteresis cycle as evenly as possible. For this purpose, the mutual distances be-tween the recorded samples in the force-displacement plane need to be maximised.The following iterative method to perform this maximisation was used in [6] whilechoosing the data samples for model fitting.

When a starting point is detected, the next first L data samples {(x(i), fc(i))} arerecorded and denoted by {xtemp(1), . . . ,xtemp(L)} and { ftemp(1), . . . , ftemp(L)}.Upon receiving the next data sample by the CCU, that sample is also recorded anddenoted by xtemp(L+1), ftemp(L+1). A normalised geometric distance betweentwo consecutive data samples is defined as:

dk =

√√√√[xtemp(k+1)− xtemp(k)

Xmax−Xmin

]2

+

[ftemp(k+1)− ftemp(k)

Fmax−Fmin

]2

(7)

where 1≤ k ≤ L and Fmax and Fmin are the upper and lower bounds of clamp forcevariations, and Xmax and Xmin are similar quantities for displacement and in practice,they can be determined off-line. Let d j be the smallest distance among the L mu-tual distances which can be easily recorded and updated iteratively as new samplesarrive. If j = L, then the new sample (the (L+1)-th sample) is too close to its pre-vious sample and is not recorded. If j < L, then the normalised geometric distancebetween the j-th and ( j+1)-th samples is the smallest distance. Therefore, the lastL− j data samples are left-shifted in the memory and the new sample is stored asthe L-th location:

First left-shift:ftemp( j+1)→ ftemp( j)xtemp( j+1)→ xtemp( j)

Second left-shift:ftemp( j+2)→ ftemp( j+1)xtemp( j+2)→ xtemp( j+1)

...L-th left-shift:

ftemp(L+1)→ ftemp(L)xtemp(L+1)→ xtemp(L)

The above scheme is repeated until the cycle finishes and the starting point of a newcycle is detected.

As it is shown in Fig. 5, a full cycle may include hysteretic variations around asmall part of the whole characteristic curve of the caliper. Therefore, the use of leastsquares for fitting the model (5) to the data recorded from a single cycle will onlylocally enhance the characteristic curve. To resolve this issue, considering memorylimitations, it was suggested in [6] to select N + 1 points (N is assumed to pre-determined based on the available memory space) with their x coordinates evenly


distributed over the whole range of variations of displacements [Xmin,Xmax], andthose are called principal fitting points or PF points for short. The displacementcoordinates of PF points are given by: {Xmin,Xmin + δ ,Xmin + 2δ , . . . ,Xmax} withδ = (Xmax − Xmin)/N. The force coordinates of PF points are calculated using athird order polynomial model with the last updated values of A,B,C and D as itsparameters. When a local hysteresis cycle finishes and the next one starts, a new setof parameters, say A′,B′,C′ and D′ are estimated. For each of the PF points whosedisplacement coordinate is between the minimum and maximum range of the localhysteresis cycle, the force coordinate is replaced with the measure given by the newparameters A′,B′,C′ and D′. Then a third order polynomial is fitted to the PF pointsand the parameters A,B,C and D are updated. Fig. 10 demonstrates an example ofthis part of parameter updating process.

Fig. 10 An example of char-acteristic curve parameterupdating upon the terminationof a local hysteresis cycle andstart of the next cycle [6].c©[2008] IEEE

0 1 2 3 4 5 6 7

0

10

20

30

40

50C

lam

p F

orce

(kN

)


0

2

4

6

Previous characteristic curve and regression pointsNew regression points from a local hysteresis cycleUpdated characteristic curveLocal hysteresis cycle

A detailed flowchart of the complete algorithm for real-time calibration of calipercharacteristic curve is shown in Fig. 11. There is an Initialisation block in which aninitial set of model (and other required) parameters of the proposed technique areinputted, the first hysteresis cycle is detected and the locations of hysteresis elements{x0(1), . . . ,x0(M)} are initialised. Then, in the Iterative Parameter Updating block,the next hysteresis cycles are detected and the characteristic curve parameters areupdated iteratively, as next data samples become available to the CCU.

5 Experimental Results

The performance of the proposed real-time calibration technique was examinedthrough a series of experiments conducted using the e-caliper of the EMB systemdeveloped at PBR Australia [2,6,14]. The e-caliper was placed in an environmentalchamber which provided a controlled temperature and humidity. Fig. 12 shows a pic-ture of the experimental setup. The PC is running Vector CANape under WindowsXP. It also controls the e-Caliper and records the position, temperature and clampforce measurements provided by caliper sensors. CANape also controls the 42Vpower supply (a Delta Elektronika SM70-45D power supply for the e-caliper actua-


Input: ,,,,,,,,,,,, maxminmaxminmax0 FFXXnLMDCBA

Read the first

Ln02 data

samples

DCXBXAXF

XXX

23

maxmin ]::[

12 0ni

Is )(ix a cycle

starting point?

Lni 02

1ii

Read the

next data

sample

Yes

Yes

No

No

max0

max0

max0

start

)()(

/2)()2(

/)()1(

ixMx

Mixx

Mixx

ii

Lii

Liixx

Liiff c

))1(:(

))1(:(

temp

temp

Read the

next data

sample

Is )(ix a cycle

starting point

andstartiLi ?

iistart

Yes

))(),...,1(,,(delMaxwell_Mo

)](),...,1(,',',','[

00temptemp

00

Mxxfx

MxxDCBA

''''''''

::'

)(max

)(min

23

maxmin

max

min

DxCxBxAf

xxx

kxx

kxx

k

k

temp

temp

Fit a 3rd order polynomial

to ]'[ xX and ]'[ fF

Update A, B, C and D

1ii

Read the

next data

sample

No

)(minarg

)(

:,...,1

)()1(

)()1(

kdj

kd

Lk

ixLx

ifLf

k

c

end

compute

for

temp

temp

Is Lj ?

end

for

temptemp

temptemp

)1()(

)1()(

:,...,

kxkx

kfkf

Ljk

1ii

No

Yes

Initialisation

DCXBXAXF

XXX

23

maxmin ]::[

Iterative

Parameter

Updating

Fig. 11 A flowchart for the proposed automatic calibration technique using the hysteresismodel (5) [6]. c©[2008] IEEE


tor and brake-by-wire circuitry) via a standard National Instruments DAQ break outbox connected to a PCI-MIO16E4 PCI card installed in the PC. An external forcesensor has been also used to measure and record the true clamp force between thebrake pad and brake disc.

Fig. 12 A picture of theexperimental setup [6, 14].c©[2008] IEEE

Programmable Environmental

Chamber

e-Caliper Under Test (inside the chamber)

Vector CAN Card

National Instruments

DAQ Interface

Power Supply (42V)

PC Running CANape Graph.

In each experiment, the measurements provided by the internal and external forcesensors and the position sensor (resolver) were recorded in a specific temperature.In a total number of fourteen experiments the temperature varied between 28◦C and54◦C (with 2◦C increments). In each experiment, the e-caliper was commanded byCANape to follow a number of consecutive sinusoidal displacements with increas-ing amplitudes, as shown in Fig. 13. The force-displacement plot shown in Fig. 5includes the data recordings at 40◦C.

To show the variations of characteristic curve with temperature, in Fig. 14, thetrue clamp forces are plotted against actuator displacements in three different tem-peratures. By using the proposed technique to estimate the friction force and correctthe hysteretic part of the clamp force measurements provided by the internal sen-sors, the true clamp force is estimated from the internal measurements, and used tocalibrate the characteristic curve.

Fig. 15 shows the true clamp forces (given by external measurements) and the es-timates from the internal measurements, plotted versus time. It was observed that theforce estimates obtained from the internal measurements by the proposed technique


Fig. 13 Position of the actua-tor during the experiment [6].c©[2008] IEEE

0 50 100 150 200 250 300−1

0

1

2

3

4

5

6

7

8

9

Time (seconds)

Act

uato

r D

ispl

acem

ent (

Mec

hani

cal R

evol

utio

ns)

242 243 244 245 246 2474.5

5

5.5

6

28°C 30°C 32°C 34°C 36°C 38°C 40°C 42°C 44°C 46°C 48°C 50°C 52° 54°

Fig. 14 Variation ofcharacteristic curve [6].c©[2008] IEEE

0 1 2 3 4 5 6 7 8−10

0

10

20

30

40

50

−10

0

10

20


Cla

mp

For

ce (

kN)

28 ° C

40 ° C

54 ° C

Fig. 15 Clamp force mea-surements by the externaland internal sensors duringthe experiment: The inter-nal sensor measurement hasbeen corrected by calculatingand removing its hysteresispart [6]. c©[2008] IEEE

0 50 100 150 200 250 3000

10

20

30

40

50

60

Time (seconds)

Cla

mp

For

ce (

kN)

External measurementsEstimated from internal measurements

33

35

37

29

31


Fig. 16 Error of the forceestimates with respectto their true values [6].c©[2008] IEEE

0 50 100 150 200 250 300−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Time (seconds)

Cla

mp

For

ce E

stim

atio

n E

rror

(kN

)

closely follow the external clamp force measurements. To quantify the accuracy ofclamp force estimates, the error of the force estimates is calculated with respect totheir true values and plotted versus time as shown in Fig. 16. It was observed in [6]that the clamp force measurement error does not exceed 0.27KN. According to thetechnical specifications of the EMB design developed at PBR Australia [1, 2, 6, 14],the high level braking modules require clamp force measurements with a maximumerror of 1% over the range 0-40KN. This error limit was devised by EMB designexperts through multiple tests of the EMB prototype in various road conditions andbraking scenarios. A maximum error of 0.27KN achieved in those experiments [6]was equivalent to 0.7% error over the range 0-40KN and therefore, the presentedmethod is suitable for clamp force measurement and caliper calibration in an EMBsystem.

6 Conclusions

This chapter introduced a hysteretic nonlinearity existing in the relationship be-tween force and position measurements. A real-time calibration technique for EMBcalipers, based on parametrically modelling the hysteresis and adaptively estimat-ing the model parameters, was presented. The proposed method is computationallyinexpensive and memory-efficient and can be easily implemented into an electro-mechanical braking system. In this method, upon the starting of each hysteresiscycle, a clamp force model is fitted to the data samples recorded from the previoushysteresis cycle. The clamp force model includes a Maxwell-slip model for the hys-teresis caused by friction. As a result, a set of model parameters estimates for thecharacteristic curve are obtained. Then, this model is applied to update the charac-teristic curve over the whole range of force-displacement variations. In a series ofexperiments, a brake-by-wire caliper was controlled to follow a sinusoidal displace-ment pattern in different temperatures and the displacement data and force sensorreadings were recorded, along with the true clamp force measured by an external


force sensor. The proposed technique was applied to extract the true clamp forcefrom the hysteretic internal force sensor readings and to update the characteristiccurve in real-time. The results showed a clamp force measurement error of less than0.7% over the range of 0-40KN, and using these measurements, the characteristiccurve was automatically calibrated in real-time with desirable accuracy.

References

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