clamping force control based on dynamic model estimation for...

Review article

Proc IMechE Part DJ Automobile Engineering2018 Vol 232(14) 2000ndash2013 IMechE 2017Article reuse guidelinessagepubcomjournals-permissionsDOI 1011770954407017738394journalssagepubcomhomepid

Clamping force control based ondynamic model estimation forelectromechanical brakes

Giseo Park and Seibum B Choi

AbstractThe electromechanical brake (EMB) is expected to be utilized for future brake systems due to its many advantages Inthis paper keeping commercialization of the EMB in mind a new EMB clamping force controller is proposed to over-come the limitations of the existing controller namely the extra cost for sensor installation and response delay Todesign the controller both mechanical parts and electrical parts in the EMB have to be mathematically analyzed Alsodynamic models clamping force and friction torque are estimated to generate some feed-forward terms of the control-ler With an estimation of the contact point where brake pads start to come into contact with a disk wheel the clampingforce is expressed as a polynomial curve versus the motor angle The estimated clamping force is evaluated in compari-son with measured values by a load cell The proposed controller is based on an adaptive sliding mode control methodwith an adaptive law reducing errors of the friction torque model Lastly the performance of the entire control systemis compared with that of the existing controller on a test bench

KeywordsElectromechanical brake estimation clamping force friction torque adaptive sliding mode control

Date received 29 March 2017 accepted 28 September 2017

Introduction

Influenced by technological advances interest in brake-by-wire systems (BBWs) has increased steadily in theautomotive industry BBWs can provide both superiorperformance and convenience for drivers12

Electromechanical brake (EMB) systems are one kindof BBW that have replaced some of the complexhydraulic components of conventional braking systemswith electrical components such as a brake pedal sen-sor a pedal simulator and control units3 A brakecommand transmitted through these electrical compo-nents drives a motor which generates brake torque Asbrake pads vertically come into contact with a diskwheel on both sides clamping force and braking torqueare simultaneously generated

The EMB has some advantages over conventionalhydraulic brake systems such as faster response forshorter braking distance lighter weight and betterspace efficiency1 Also it can be more easily connectedto some vehicle chassis control systems such as antil-ock braking systems (ABSs) and electronic stabilitycontrol systems (ESCs)45 These advantages haveattracted many researchers and industry experts to

study EMBs for commercialization1ndash10 Some research-ers have expected that EMBs can be utilized for thebrake systems of future vehicles136 A vehicle equippedwith an EMB must control it with high accuracy forimplementation of high-level control systems such asABSs and ESCs6

So far most EMB systems have been controlled by aclamping force feedback controller which has cascadedclamping force motor velocity and motor current con-trol loops78 However a load cell measuring the clamp-ing force is too expensive to be installed in productionvehicles and is not accurate at high temperatures7 Sothe force feedback from the load cell has both robust-ness and reliability issues To solve these problems aneffective method by which the clamping force is esti-mated without a load cell is required Schwarz et al9

proposed a basic algorithm for clamping force

Department of Mechanical Engineering KAIST Republic of Korea

Corresponding author

Seibum B Choi Department of Mechanical Engineering KAIST 291

Daehak-ro Yuseong-gu Daejeon 305-701 Republic of Korea

Email sbchoikaistackr

estimation by which the curve of the clamping forceversus the motor angle is described Based on this pro-posed method various methods of clamping force esti-mation have been proposed1610

In this paper an estimation algorithm extending theearlier studies is proposed which is based on elabo-rated curve fitting with a contact point estimationmethod The contact point the value of the motorangle when the brake pads start to come into contactwith the disk wheel has to be exactly estimated for theaccuracy of clamping force estimation All of the esti-mation algorithms in this paper are carried out in afeedback control system shown in Figure 1(a) using amotor angle controller as the outer control loop Theencoder can be replaced with sensorless speed and posi-tion estimators11

Furthermore the developed clamping force estima-tor can be utilized to design a new controller includingsome feed-forward terms related to the EMB dynamicmodels It is anticipated that the newly proposed con-troller can avoid response phase lag occurring fre-quently in the existing controller For this purpose anestimation of the other dynamic model friction torqueis essential If these estimated dynamic models haveacceptable differences in actual values they can have apositive effect and improve control performances Anoverall algorithm for EMB controller design is illu-strated by the block diagrams in Figure 1(b) As shownin these diagrams prior to completion of the controllerdesign the feedback control system is performed only

during an initial step to develop the dynamic modelestimators Consequently the EMB is expected to becontrolled by an adaptive sliding mode control (SMC)method This method is applied to EMB control forthe first time there were already some methods forEMB control such as adaptive proportionalndashintegralndashderivative (PID) control5 and model predictive control(MPC)7

The remainder of this paper is organized as followsMathematical models of both electrical and mechanicalparts are introduced in the second section Estimationalgorithms for the contact point clamping force andfriction torque are developed in the third section In thefourth section the new clamping force controllerwhich uses the outcomes of the developed estimators ispresented with a formulation of the adaptive SMCmethod Some experiments for verification of the newcontroller are performed on an EMB test bench andthe results are analyzed in the fifth section Finally thesixth section concludes the paper

Modeling of electromechanical brake

Figure 2(a) illustrates the overall structure of an EMBcomposed of primary components such as a permanentmagnet synchronous motor (PMSM) Some reductiongears and one planetary gear set reducing the angularvelocity of the PMSM are connected to the PMSM inseries Then this rotational motion is converted intothe linear motion of the piston through the ball screw

Figure 1 Electromechanical brake (EMB) block diagrams (a) feedback control system (b) overall algorithm for EMB controllerdesign ECU engine control unit SMC sliding mode control

Park and Choi 2001

While the linear motion presses the disk between twopads the outer caliper body clamps the disk at the sametime The stiffness of this caliper body accounts for themain portion of overall brake stiffness9 Modeling ofthe EMB is simply divided into the electrical andmechanical parts Also the EMB structure can beexpressed as a bond graph as shown in Figure 2(b) thebond graph theory is well explained by Ma et al12

Modeling of the electrical part

The PMSM is suitable to control the EMB due to itshigh precision in both angular velocity and motor cur-rent control1314 In the PMSM three-phase vectors ofa voltage or a current are transformed to two-phasevectors which are represented with stator orthogonalcoordinates the a b axis15 This is called the Clarketransform

fafb

=

1 12 1

2

0ffiffi3p

2 ffiffi3p

2

fafbfc

24 35 eth1THORN

Here f is the voltage or the current Subsequently thePark transform is performed to convert these statororthogonal coordinates into others the d q axiswhich rotates synchronously with the rotor13ndash16 Theformula of the Park transform is expressed as follows

fdfq

=

cos us sin us

sin us cos us

fafb

eth2THORN

where us is the electrical rotor angle Voltage equationswith d and q axes are described in following equa-tions113ndash17

vd =Rid +Lddiddt wsLqiq eth3THORN

Figure 2 Electromechanical brake structure (a) block diagram (b) bond graph PMSM permanent magnet synchronous motor

2002 Proc IMechE Part D J Automobile Engineering 232(14)

vq =Riq +Lqdiq

dt+ws(Ldid +caf) eth4THORN

ws = npwm eth5THORN

where vd and vq are the voltages id iq are the currentsLd Lq are the inductances of each axis wm ws are themechanical and electrical angular velocity respectivelyR is the stator resistance caf is the flux linkage and isnp the number of pole pairs The torque of the interiorpermanent magnet synchronous motor (IPMSM) usedin this research is given as

Tm =3

2npfrac12cafiq +(Ld Lq)idiq eth6THORN

To generate maximum motor torque the IPMSM iscontrolled such that id becomes zero13ndash16 Thereforethe motor torque which is linearly proportional to themotor current can be simply expressed as direct cur-rent (DC) motor torque as follows

Tm =3

2npcafiq =Kmiq eth7THORN

where Km represents the motor torque constant

Modeling of the mechanical part

The motor angular velocity is reduced by passingthrough a series of reduction gears and the planetarygear set The relations between two gears connected toeach other are described as

w1 =wm

w2 =w1=(sect13GR1)

w3 =w2=(sect23GR2)

w4 =w3=(sect33GR3)

wcarrier =w4=(sect43GRplanetary)

eth8THORN

where GR and sect are the gear ratio and the gear effi-ciency respectively Also each subscript in equation (8)denotes the number of each gear in Figure 2(a) Thenthe rotational motion of the carrier gear is convertedinto the linear motion of the ball screw Eventually thedisplacement of the ball screw is linearly proportionalto the motor angle59 It is described as

xscrew =1

GRtot

p

2pum = kscrewum eth9THORN

where

GRtot= secttot3GR13GR23GR33GRplanetary

secttot= sect13sect23sect33sect4

Here GRtot is the total gear ratio of all rotationalparts which is calculated by the multiplication of allgear ratios Also secttot denotes the total gear efficiencyThen p and kscrew are the pitch of the ball screw andtranslating gain respectively Jtot the total inertia ofthe system from the perspective view of the motorangle is expressed as

Jtot = Jmotor + JGR1 +JGR2

(sect13GR1)2

+ JGR3=(sect13sect23GR13GR2)2

+ fJGR4 + (Jplanetary+ Jscrew)=(sect43GRplanetary)2g=

(sect13sect23sect33GR13GR23GR3)2

eth10THORN

where each subscript of J corresponds to each rota-tional component The motor torque in Equation (7) isequal to the sum of TL the load torque Ti the inertiatorque that is proportional to the motor angular accel-eration with the total inertia in equation (10) and Tfthe friction torque35ndash10

TL frac14kscrew

sectsFcl frac14 kclFcl eth11THORN

Tm =TL + Jtotd2um

dt2+Tf eth12THORN

where

Tf (um)= sgn(wm)(gFcl +Tf0)

Here sects and kcl are the efficiency of the ball screw andthe gearing gain respectively Also g and Tf0 are thecoefficient of Coulomb friction and the offset termrespectively6

Dynamic model estimation

Knowing the dynamic models such as the clampingforce and friction torque is very valuable for control-ling the EMB However as mentioned in the first sec-tion using a load cell to measure the clamping force isnot recommended due to extra cost and problems asso-ciated with installation135 Also directly measuring thefriction torque with any production sensor is impossi-ble Instead both Fcl and Tf can be described as func-tions of the motor angle9 Thus based on only onedegree of freedom um the new EMB controller in thefourth section can be easily controlled

Contact point estimation

Estimation of the contact point takes priority overclamping force estimation In developing an algorithmfor contact point estimation both the motor currentand the motor angle are individually measured by cor-responding sensors during the special brake operationas shown in Figure 3 This brake operation is dividedinto pressing and releasing the brake pedal whichmeans moving the EMB ball screw forward and back-ward respectively In particular when a car is startedthis operation can be quickly performed Many kindsof vehicle are equipped with a shift-lock system thatprevents a driver from starting a vehicle without usingthe brake18 Thus most drivers can naturally imple-ment the brake operation during starting a vehicle

Park and Choi 2003

In Figure 3 when the motor current becomes greaterthan the constant threshold i0 the motor angle at thattime can be regarded as the contact point u0

19 Duringthe forward operation that makes the ball screw moveforward the polynomial curve on the basis of the rela-tion between iq and um starting at u0 is described asfollows

iqfor(um)=afor(um u0)2 +bfor(um u0)+ i0 eth13THORN

A recursive least square (RLS) method used for everycurve fitting in this paper is introduced in the followingexplanation The RLS method has some advantagessuch as much faster convergence and less computationthan those of the least square method It is suitable fora particular curve fitting that is desired to be completedquickly The basic idea of the RLS method is to find anestimated parameter w which makes the sum of thesquares of the error between approximated data fT(i)wand measured data y(i) minimal17 This sum the costfunction V(w k) is defined as follows

V( bw k)= 1

2

Xki=1

lki(y(i) fT(i) bw)2 eth14THORN

where k denotes the number of samples In the case ofequation (13) y(i) is the motor current iq i0 and fT(i)the motor angle um u0 at the ith sample in the for-ward operation Also w denotes the coefficients of thesecond-order polynomial curve afor and bfor To findoptimal w V(w k) is differentiated by w Consequentlythe following forms are derived2021

w(k)= w(k 1)+L(k)fy(k) fT(k)w(k 1)g eth15THORN

where

L(k)=P(k)f(k)

=P(k 1)f(k)flf +fT(k)P(k 1)f(k)g1

P(k)= fI L(k)fT(k)gl1f P(k 1)

=1

lffP(k 1) P(k 1)f(k)fT(k)P(k 1)

lf +fT(k)P(k 1)f(k)g

Here L(k) and P(k) are the update gain and theerror covariance matrix respectively and lf is the for-getting factor It determines the weighting of the

influence of old samples which are not strongly relatedto newly estimated w(k) In the calculation of w(k) thesmaller lf is the greater the influence of the compara-tively recent sample is Generally the value of lf is cho-sen in the interval [09 1]20

Clamping force estimation

Assuming that both the backlash of the gears and elas-tic hysteresis of the brake pads are negligible the rela-tionship between Fcl and um can be expressed as apolynomial with no hysteresis3910 The polynomialfunction is an appropriate mathematical model for therelationship

From the preceding analysis in Table 1 showing theRMS errors between the actual and estimated clampingforces a second-order polynomial is considered anoptimum In Figure 3 during the backward operationthe motor current iqback can be expressed as a second-order polynomial as in equation (13) It is described as

iqback(um)=aback(um u0)2 +bback(um u0) i0

eth16THORN

To find aback and bback the raw data iqback + i0 andum u0 are utilized to perform the RLS method in thesame way as equations (13)ndash(15) Depending on thedirection of each brake operation the clamping forcecan be defined from the torque balance equation (12)

Forward Fcl =1

kcl(Kmiqfor Jtot

d2umfordt2

+Tf)

Backward Fcl =1

kcl(Kmiqback Jtot

d2umbackdt2

Tf)

eth17THORN

Generally Tf in the EMB is regarded as a Coulombfriction torque whose sign is changed according toeach brake operation392223 When the motor angle isthe same it is reasonable to assume that the absolutevalue of the friction torque is the same regardless of thedirection of the brake operation Schwarz et al9 pro-posed that the friction torque can be cancelled out dueto this characteristic Equations (13) and (16) can beused to substitute for iqfor and iqback in equation (17)respectively As shown in Figure 4(a) clamping force is

Figure 3 Curve fitting with the contact point

Table 1 The root mean square (RMS) errors of polynomialcurve fitting

Order of a polynomial RMS error (N)

1 75493812 11009493 11009334 11009305 11009568 1100968


estimated by taking the average of the formulas inEquation (17) over the whole range of the motor angleThe signal noises of d2um=dt

2 are removed by utilizingthe low pass filter

Fcl=Km

2kcl(iqfor+ iqback)

Jtot2kcl

d2

dt2(umfor + umback)

eth18THORN

Lastly by using the RLS method again this estimatedclamping force can be expressed as a polynomial func-tion of the motor angle The function starting at thecontact point is described as

Fclfitting(um)=K2(um u0)2 +K1(um u0) eth19THORN

where K2 and K1 are coefficients calculated by the RLSmethod Figure 4(b) compares the actual and estimatedclamping force versus the motor angle its range up to30 rad is a primary operational range for a midsize vehi-cle (Hyundai LF Sonata) The estimated curve is quiteclose to the actual curve which is based on measuredvalues by a load cell

To verify the accuracy of the estimation an apprai-sal standard is set up in this study Firstly it is assumedthat EMBs are installed in all wheels of the midsizevehicle (see the specifications in Table 2) Also onlylongitudinal motion of vehicle is considered The braketorques of the front and rear wheels are as follows

Front wheel Tb f =2mfrfFcl

Rear wheel Tb r =2mrrrFcl

eth20THORN

The dynamic equation of the vehicle wheel in the brak-ing situation is represented as follows

Jw _w=RwFx Tb

FxrsquoTb

Rw(Tb Jw _w)

eth21THORN

From the sum of tire longitudinal forces the clampingforce can be derived as a function of vehicle longitudi-nal acceleration ax It is assumed that Fcl in every wheelis the same

X4ifrac141

Fxi frac142ethTbf thorn TbrTHORN

Rwfrac14 max

)Fcl frac14Rwmax

4ethmfrf thorn mrrrTHORN

eth22THORN

A low deceleration of less than 003g in a vehicle repre-sents smooth and comfortable riding performance thatis difficult to notice by a driver24 From equation (22)the clamping force of approximately 039 kN to eachwheel is required to decelerate the midsize vehicle to003g So the error tolerance of clamping force estima-tion or control in this paper is set to 039 kN As theresults in Figure 5(a) show with a full pad thickness of

Figure 4 Characteristic curve fitting (a) estimated clamping force versus the motor angle (b) comparison with the actual clampingforce

Table 2 The specifications of a midsize vehicle (LF Sonata)

Parameter Quantity Value

m Vehicle mass + people 1560 kgrf Effective radius of the front disc 1309 mmmf Nominal pad friction coefficient at the front brake pad 038rr Effective radius of the rear disc 1240 mmmr Nominal pad friction coefficient at the rear brake pad 038Rw Effective rolling radius of tire 330 mm

Park and Choi 2005

13 mm the maximum error is less than 039 kN Evenwhen the pad thickness decreases to 6 mm due to padwear the accuracy is still maintained in that range asseen in Figure 5(b) Therefore according to theseresults the estimated clamping force including the con-tact point estimation presented in the Contact pointestimation section is suitable to be part of the newclamping force controller Also to manage the changeof the pad thickness the curve fitting process (equa-tions (13)ndash(19)) is desired to be periodically performed

Friction torque estimation

On the surfaces of some components such as themotor the gears and the ball screw a lumped frictiontorque resisting the external torque TE (=Tm TL) isgenerated synthetically In some previous papers39 thekinetic friction torque during slipping motions of allrotational components is expressed as a function of theclamping force (refer to Tf (um) in equation (12)) Inthis paper for elaborate modeling the load dependencyof friction torque is newly expressed as a second-orderpolynomial function of the motor angle

Tf (um)=

sgn(wm)ff2(umu0)2+f1(um u0)+Tf0g if um5u0

Tf0sgn(wm) else

8gtltgteth23THORN

Here f1 and f2 are the load-dependent coefficientsand Tf0 is the offset term

After many brake maneuvers the brake pad can beover heated and the heat of the pad is concurrently pro-pagated into the nearby components925 This remark-able change of temperature is the main parameteraffecting the change of friction in an EMB actuatorTherefore although nominal values of f1 f2 and Tf0 inequation (23) are given at the beginning they aredesired to be updated to enhance the robustness Hereh is defined as a varying scale of load termf2(um u0)

2 + f1(um u0) (initial h = 1) and then the

nominal Tf(um) can be adaptively changed by the vari-able h While the estimated h converges to real h by anadaptation law the error of friction torque ~Tf(um) isreduced26 The adaptation law is introduced in the nextsection

Controller design

Generally a brake actuator should have a fast responsewithout time delay The faster the reaction of the motorangle control in the EMB the faster the generation ofthe clamping force is and the shorter the braking dis-tance is In this section a controller is proposed that isbased on the clamping force and friction torque estima-tors in the third section The block diagram of the con-trol scheme is as shown in Figure 6

Based on the inverse of equation (19) the desiredclamping force Fcldes can be converted to the desiredmotor angle umdes Since equation (19) is a strictlyincreasing function each value of Fcldes(50) ismatched one-to-one with the correspondingumdes(5u0)

umdes=( K1 +ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK2

1 +4K2Fcldes

q)=2K2 + u0

eth24THORN

To determine the control input the adaptive SMCmethod is used Firstly the torque balance equation(12) is rearranged as the following formula for theangular acceleration

euroum =(Kmiq kclFcl Tf)=Jtot eth25THORN

The tracking error e and variable s which are boundedphysically are separately defined as

e= um umdes

s= _e+ leeth26THORN

where the positive constant l is the only tuning para-meter Let _s be the saturation function of s setsat(x)= x=e as xj j4e and sat(x)= sgn(x) as xj j ewhere e is a very small parameter

Figure 5 Clamping force estimation (a) with 13 mm pad thickness (b) with 6 mm pad thickness


_s= lsat(s) eth27THORN

The time derivative of s is written as

_s= euroe+ l _e= euroum euroumdes+ l _e eth28THORN

The error dynamics can be written by substitutingequation (28) into equation (27)

euroum euroumdes+ l _e+ lsat(s)=0 eth29THORN

Substituting equation (25) into euroum yields

(Kmiq kclFcl Tf)=Jtot euroumdes+ l _e+ lsat(s)=0

eth30THORN

Lastly Fcl and Tf can be replaced with equations (19)and (23) respectively At this point it is assumed thatthe estimated Fcl is actual Therefore the motor currentin equation (30) is regarded as the control input iqdeswhich is designed as

iqdes =1

Km(kclFcl + Tf + Jtoteuroumdes)

lJtotKm

( _e+sat( _e+ le))

eth31THORN

where

Tf =Tf0sgn(wm)+ h(f2(um u0)2 + f1(um u0))sgn(wm)

Here equation (31) is the reference of the inner loopcurrent PI feedback controller (see Figure 6) To provethe stability of the proposed algorithm by usingBarbalatrsquos lemma consider a Lyapunov function candi-date V that is lower bounded (V 0) and positivedefinite

V=1

2s2 +

1

2ka~h2 eth32THORN

where

~h=h h

Here ka is a positive constant adaptation gainAssuming that the real h varies slowly the time deriva-tive of V is obtained as

_V= _ss+1

ka~h( _h _h)

= ((Kmiq kclFcl Tf)=Jtot euroumdes + l _e)s 1

ka~h _h

eth33THORN

By substituting equation (31) into iq _s and _V can berewritten as

_s=~hsgn(wm)(f2(um u0)

2 + f1(um u0)) l sat(s)

Jtot

_V= ls sat(s)

Jtot

~h_h

ka+

sgn(wm)(f2(um u0)2 + f1(um u0))

Jtots

eth34THORN

The adaptive law can be naturally obtained as follows

_h= kasgn(wm)(f2(um u0)2 + f1(um u0))s=Jtot

eth35THORN

Hence _V in equation (34) can be expressed as negativesemidefinite that is

Figure 6 Block diagram of the proposed electromechanical brake (EMB) controller ECU engine control unit

Park and Choi 2007

_V= ls sat(s)=Jtot40 eth36THORN

From the following equation (37) it is verified that euroV isuniformly continuous in time

euroV= l2(sat(s))2=Jtot eth37THORN

From equations (32)ndash(37) it is verified by usingBarbalatrsquos lemma that s 0 as t lsquo2627 This impliesthat the tracking error e in equation (26) converges tozero Also since both s and _s converge to 0 ~h con-verges to 0 (refer to equation (34))

Since the perfect clamping force estimation withoutthe load cell is quite difficult the modeling error ~Fcl isinevitable in the experimental results However it isexpected that a sufficiently large gain l can cancel outthis modeling error

Experiments

Experimental set-up

All experiments in this paper are performed using anEMB test bench (see Figure 7) All of the data are com-municated by a controller area network (CAN) Figure7(b) illustrates how the desired commands Fcldes andumdes from the control desk are transmitted to the EMBplant Also the transmission of data from the EMB plantis shown um and wm are measured by an encoder whileia ib and ic are measured by current sensors Using aMicro-Autobox all of the processes are monitored at a 1ms sampling rate28 The feedback control system in Figure1(a) for developing the estimators is downloaded onto theEMB engine control unit (ECU) through USB MultilinkThe outer control loop consisting of the inversed charac-teristic curve friction torque estimator and the adaptiveSMC is built in Micro-Autobox (see Figure 6) Thus thecontrol input is calculated in Micro-Autobox and is trans-mitted to the motor current feedback controller down-loaded onto the EMB ECU The system parameters arepresented in Table 3

Experimental results

The controller design in the fourth section is evaluatedin this section First of all an experiment with the feed-

forward control input iqff only that is with a zerofeedback gain (l=0) in equation (31) and no adapta-tion law in equation (35) is performed

The results of this experiment with the feed-forwardcontrol are shown in Figure 8 The clamping forcecommand is applied as a step input (see Figure 8(b))As shown in Figure 8(a) to follow this desired com-mand Fcldes iqdes is calculated by the sum of the termsreflecting each dynamic model the load torque frictiontorque and the inertia torque The measured clampingforce Fclmeasured obtained from the load cell and actualclamping force Fclact (the output of the controller) aresimultaneously generated (see Figure 6) Fclmeasured

from a load cell is for the purpose of monitoring onlyand are not used for the controls if the clamping forceestimation is perfect Fclact will be equal to FclmeasuredFclact quickly reacts to the desired commands andpromptly follows them These results confirm the con-tribution of the feed-forward terms for the control per-formance improvement

To validate the performance of the entire controllerproposed in Figure 6 some experiments with variousclamping force commands are performed The stepcommands with various magnitudes the triangularcommand and sinusoidal command are individuallyapplied into the proposed controller (see Figure 9)Regardless of the command the acceptable controlerrors are shown As well as the tracking errors betweenthe desired and actual those between the desired andmeasured are under the error tolerance of 039 kN Asa result it is confirmed that the proposed controllerbased on dynamic model estimation can be applied tothe various braking situations

From the previous research the adaptive law hasthe following characteristics26 (a) it increases the totalsystem order (b) due to the model error and signalnoise it may cause the instability of system Thereforethe adaptive law in equation (35) is activated only whenthe steady-state error between the desired and actual islarge In Figures 9 12(b) and 13(a) since each controlresult has an acceptable steady-state error the adaptivelaws are deactivated This implies that the friction tor-que models are well estimated (h=1hrsquo1)

To validate the performance of the adaptive law theinitial scale is intentionally set to an incorrect valueand the real scale is approximately 1 (h=057hrsquo1)As shown in Figure 10 the adaptation law is activatedby the large error between desired and actual clampingforces in about 3 s It is confirmed that the control errorin Figure 10(a) converges to 0 as the estimated para-meter converges to the real scale in Figure 10(b)

Schwarz et al2 suggested a typical EMB clampingforce controller which was based on feedback errors withproportional (P) and integral (I) gains (see Figure 11)Figure 12 compares the performance of the proposedcontroller with that of the existing controller for the samestep input of 8 kN While the existing controller recordsthe settling time within 2 error of 0395 s the proposedcontroller records 0175 s As shown in Figure 12(a) a

Table 3 The main parameters of the electromechanical brake

Symbol Parameter Value

caf Flux linkage 000402 WbLd Motor inductance (d-axis) 788 mHLq Motor inductance (q-axis) 935 mHR Motor resistance 0137 mOKm Motor torque constant 0047 NmAnp Number of pole pairs 3Jtot Total inertia moment 31843105kg m2

GRtot Total gear ratio 391p Pitch of ball screw 5 mmsects Efficiency of ball screw 68


Figure 7 Electromechanical brake (EMB) test bench (a) all equipments (b) block diagram ECU engine control unit CANcontroller area network

Figure 8 Only feed-forward control without feedback and adaptation (a) control input (b) the desired actual and measuredclamping force

Park and Choi 2009

noticeable amount of overshoot and signal noise are seenIn Figure 12(b) although the steady-state errors betweenthe desired and measured values are seen due to theclamping force estimation error the root mean square(RMS) error is only 027 kN which is under the errortolerance

In addition the control inputs that is the motorcurrent values in each controller are shown inFigure 13 Since the motor current in the existing con-troller is forced to oscillate largely in order to track theclamping force command it is distinctly larger thanthat in the proposed controller Because the motor cur-rent is reduced in the proposed controller the energyefficiency of the actuator can be improved Also thereduced number of tuning parameters of the proposed

controller is beneficial from the viewpoint of tuningsimplicity There is a summary comparison betweenexisting and proposed controllers in Table 4 The con-troller proposed in this paper shows some advantagesthe sensorless system cost-effective system less com-plex gain scheduling higher energy efficiency andfaster response

Conclusion

In this paper using the readily available motor currentand motor angle signals the clamping force and thefriction torque are estimated The clamping force isexpressed as a polynomial with the estimated coeffi-cients The contact point algorithm plays a major role

Figure 9 Various clamping force commands (a) step input 1 (b) step input 2 (c) triangular input (d) sinusoidal input

Figure 10 Validation of the adaptation law (a) control results (b) adaptation of parameter in friction torque


in clamping force estimation With respect to applic-ability these proposed algorithms distinguish them-selves from previous estimation algorithms Thelumped friction torque occurring in some componentsof the EMB is modeled for slipping motion

Based on the estimation of the dynamic models anew EMB clamping force controller is proposed Since

it uses the estimated dynamic models that can be peri-odically updated and compensated by the adaptive lawit can maintain robustness to changes in parametersThe experimental results of the proposed controller areevaluated and compared to an existing typical feedbackcontroller Consequently the proposed controllerbased on the adaptive SMC method has following

Figure 11 Block diagram of the existing feedback controller EMB electromechanical brake

Figure 12 Controller comparison by same step input (a) existing controller (b) proposed controller

Table 4 Comparison between existing and proposed controllers

Feature Existing Proposed

Sensors Load cell current sensor encoder Current sensorEncoder

Cost Too high LowType Cascaded feedback force velocity current Adaptive SMC + Feedback (current)Tuning gains Three pairs of PI gains l ka

One pair of PI gainsEnergy efficiency Low

(large motor current)High(small motor current)

Transient state Large overshootLong settling timeHigh tracking performance

Small overshootShort settling timeHigh tracking performance

Steady state Small errorNoise due to load cell

Acceptable errorNo signal noise

PI proportionalndashintegral SMC sliding mode control

Park and Choi 2011

advantages (a) it is a cost-effective and sensorless algo-rithm for commercialization of the EMB (b) due to thereduced number of tuning parameters it requires lesscomplex gain scheduling than the existing controller(c) it has higher energy efficiency of the actuator (d) byusing the feed-forward terms it has a faster response

This study has demonstrated that the new design ofthe EMB controller can make a valuable contributionto the development of an outstanding controller forBBWs As a future work this controller needs to becombined with the ABS and ESC for vehicle safety

Declaration of conflicting interests

The author(s) declared no potential conflicts of interestwith respect to the research authorship andor publi-cation of this article

Funding

The author(s) disclosed receipt of the following finan-cial support for the research authorship andor publi-cation of this article This work was partly supportedby the Hyundai Motor Company the MSIP (Ministryof Science ICT and Future Planning) Korea underthe ITRC (Information Technology Research Center)support program (IITP-2017-2012-0-00628) supervisedby the IITP (Institute for Information amp communica-tions Technology Promotion) the Technological

Innovation RampD program of SMBA (S2341501) theNational Research Foundation of Korea (NRF) grantfunded by the Korea government (MSIP) (grant num-ber 2017R1A2B4004116) and the BK21+ programthrough the NRF funded by the Ministry of Educationof Korea

References

1 Ki Y Lee K Cheon J et al Design and implementationof a new clamping force estimator in electro-mechanical

brake systems Int J Automot Technol 2013 14 739ndash7452 Schwarz R Isermann R Bohm J et al Modeling and

control of an electromechanical disk brake SAE paper980600 1998

3 Saric S Bab-Hadiashar A and Hoseinnezhad R

Clamp-force estimation for a brake-by-wire system a

sensor-fusion approach IEEE Trans Veh Technol 200857 778ndash786

4 Hoseinnezhad R and Bab-Hadiashar A Calibration of

resolver sensors in electromechanical braking systems a

modified recursive weighted least-squares approachIEEE Trans Ind Electron 2007 54 1052ndash1060

5 Jo C Hwang S and Kim H Clamping-force control for

electromechanical brake IEEE Trans Veh Technol 2010

59 3205ndash32126 Hoseinnezhad R Bab-Hadiashar A and Rocco T Real-

time clamp force measurement in electromechanical

brake calipers IEEE Trans Veh Technol 2008 57 770ndash

777

Figure 13 Control performance comparison (a) control results in the proposed controller (b) control results in the existingcontroller (c) motor current in the proposed controller (d) motor current in the existing controller


7 Line C Manzie C and Good M Electromechanicalbrake modeling and control from PI to MPC IEEE

Trans Control Syst Technol 2008 16 446ndash4578 Line C Manzie C and Good M Control of an electro-

mechanical brake for automotive brake-by-wire systemswith adapted motion control architecture SAE paper012050 2004

9 Schwarz R Isermann R Bhm J et al Clamping forceestimation for a brake-by-wire actuator SAE paper990482 1999

10 Saric S Bab-Hadiashar A and Walt J Estimating clampforce for brake-by-wire systems thermal considerationsMechatronics 2009 19 886ndash895

11 Allotta B Pugi L Paolucci L et al Development ofsensorless PM servo-system for harsh environments In2015 AEIT international annual conference Naples Italy14ndash16 October 2015 New York IEEE

12 Ma S Zhang T Liu G et al Bond graph-based dynamicmodel of planetary roller screw mechanism with consider-ation of axial clearance and friction Proc Inst Mech Eng

C J Mech Eng Sci 2017 0 1ndash1313 Zhong L Rahman M Hu W et al Analysis of direct

torque control in permanent magnet synchronous motordrives IEEE Trans Power Electron 1997 12 528ndash536

14 Prokop L Stulrajter M and Radhostem R 3-phase

PMSM vector control using the PXS20 and tower systemRoznov pod Radhostem Czech Republic TX FreescaleSemiconductor Application Note AN4537 2012

15 Pillay P and Krishnan R Modeling simulation andanalysis of permanent-magnet motor drives part I thepermanent-magnet synchronous motor drive IEEE

Trans Ind Appl 1989 25 265ndash27316 Park H and Choi SB The development of a sensorless

control method for a self-energizing brake system usingnoncircular gears IEEE Trans Control Syst Technol 201321 1328ndash1339

17 Jo C Lee S Song H et al Design and control of anupper-wedge-type electronic brake Proc IMechE Part D

J Autom Eng 2010 224 1393ndash140518 OrsquoHem T Brake operated shift lock Patent 4187935

USA 198019 Park G Choi SB and Hyun D Clamping force estima-

tion based on hysteresis modeling for electro-mechanicalbrakes Int J Automot Technol 2017 18 883ndash890

20 Choi M and Choi SB Linearized recursive least squaremethods for real-time identification of tire-road frictioncoefficient IEEE Trans Veh Technol 2013 62 2906ndash2918

21 Rajamani R Vehicle dynamics and control New YorkSpringer 2006

22 Karnopp D Computer simulation of stick-slip friction inmechanical dynamic systems J Dyn Sys Meas Control

1985 107 100ndash10323 Olsson H Astrom K Wit C et al Friction models and

friction compensation Eur J Control 1998 4 176ndash19524 Chang K Hedrick J Zhang W et al Automated highway

system experiments in the PATH program J Intell Trans-

port Syst 1993 1 63ndash87

25 Han M Lee C Park T et al Coupled thermo-mechanicalanalysis and shape optimization for reducing uneven wearof brake pads Int J Automot Technol 2017 18 1027ndash1035

26 Ioannou P and Sun J Robust adaptive control 2nd edNew Jersey Prentice-Hall 1996

27 Khalil H Nonlinear system 3rd ed New Jersey Prentice-Hall 2002

28 Pugi L Malvezzi M Tarasconi A et al HIL simulationof WSP systems on MI-6 test rig Vehicle Syst Dyn 200644(Suppl 1) 843ndash852

Appendix

Notation

us um electrical rotor angle and motorangle

vd vq motor voltages of d-axisq-axisid iq motor currents of d-axisq-axisLdLq self inductances of d-axisq-axiswmws motorsynchronous angular

velocitiesR stator resistancecaf flux linkagenp number of pole pairsTm motor torqueKm motor torque constantp pitch of ball screwGR gear ratiosect gear efficiencyxscrew displacement of ball screwkscrew translating gain of ball screwkcl gearing gainJtot total inertiaFcl clamping forceTL load torqueTf friction torqueg coefficient of Coulomb frictionu0 contact pointi0 threshold of motor currentiqfor(um) motor current versus motor angle

during forward operationaforbfor coefficients of iqfor(um)iqback(um) motor current versus motor angle

during backward operationabackbback coefficients of iqback(um)Fclfitting(um) characteristic curveK1K2 coefficients of Fclfitting(um)f1 f2 load-dependent coefficients of

friction torqueTf0 offset term of friction torque

Park and Choi 2013

estimation by which the curve of the clamping forceversus the motor angle is described Based on this pro-posed method various methods of clamping force esti-mation have been proposed1610

In this paper an estimation algorithm extending theearlier studies is proposed which is based on elabo-rated curve fitting with a contact point estimationmethod The contact point the value of the motorangle when the brake pads start to come into contactwith the disk wheel has to be exactly estimated for theaccuracy of clamping force estimation All of the esti-mation algorithms in this paper are carried out in afeedback control system shown in Figure 1(a) using amotor angle controller as the outer control loop Theencoder can be replaced with sensorless speed and posi-tion estimators11

Furthermore the developed clamping force estima-tor can be utilized to design a new controller includingsome feed-forward terms related to the EMB dynamicmodels It is anticipated that the newly proposed con-troller can avoid response phase lag occurring fre-quently in the existing controller For this purpose anestimation of the other dynamic model friction torqueis essential If these estimated dynamic models haveacceptable differences in actual values they can have apositive effect and improve control performances Anoverall algorithm for EMB controller design is illu-strated by the block diagrams in Figure 1(b) As shownin these diagrams prior to completion of the controllerdesign the feedback control system is performed only

during an initial step to develop the dynamic modelestimators Consequently the EMB is expected to becontrolled by an adaptive sliding mode control (SMC)method This method is applied to EMB control forthe first time there were already some methods forEMB control such as adaptive proportionalndashintegralndashderivative (PID) control5 and model predictive control(MPC)7

The remainder of this paper is organized as followsMathematical models of both electrical and mechanicalparts are introduced in the second section Estimationalgorithms for the contact point clamping force andfriction torque are developed in the third section In thefourth section the new clamping force controllerwhich uses the outcomes of the developed estimators ispresented with a formulation of the adaptive SMCmethod Some experiments for verification of the newcontroller are performed on an EMB test bench andthe results are analyzed in the fifth section Finally thesixth section concludes the paper

Modeling of electromechanical brake

Figure 2(a) illustrates the overall structure of an EMBcomposed of primary components such as a permanentmagnet synchronous motor (PMSM) Some reductiongears and one planetary gear set reducing the angularvelocity of the PMSM are connected to the PMSM inseries Then this rotational motion is converted intothe linear motion of the piston through the ball screw

Figure 1 Electromechanical brake (EMB) block diagrams (a) feedback control system (b) overall algorithm for EMB controllerdesign ECU engine control unit SMC sliding mode control

Park and Choi 2001




fafb

=

1 12 1

2

0ffiffi3p

2 ffiffi3p

2

fafbfc

24 35 eth1THORN


fdfq

=

cos us sin us

sin us cos us

fafb

eth2THORN





vq =Riq +Lqdiq


ws = npwm eth5THORN


Tm =3



Tm =3





w1 =wm

w2 =w1=(sect13GR1)

w3 =w2=(sect23GR2)

w4 =w3=(sect33GR3)


eth8THORN


xscrew =1

GRtot

p


where





(sect13GR1)2




eth10THORN


TL frac14kscrew


Tm =TL + Jtotd2um

dt2+Tf eth12THORN

where







Park and Choi 2003





V( bw k)= 1

2

Xki=1




where

L(k)=P(k)f(k)



=1









eth16THORN


Forward Fcl =1

kcl(Kmiqfor Jtot

d2umfordt2

+Tf)

Backward Fcl =1

kcl(Kmiqback Jtot

d2umbackdt2

Tf)

eth17THORN





1 75493812 11009493 11009334 11009305 11009568 1100968




Fcl=Km

2kcl(iqfor+ iqback)

Jtot2kcl

d2

dt2(umfor + umback)

eth18THORN







eth20THORN


Jw _w=RwFx Tb

FxrsquoTb

Rw(Tb Jw _w)

eth21THORN


X4ifrac141


Rwfrac14 max

)Fcl frac14Rwmax


eth22THORN






Park and Choi 2005




Tf (um)=


Tf0sgn(wm) else

8gtltgteth23THORN





Controller design




1 +4K2Fcldes

q)=2K2 + u0

eth24THORN




e= um umdes

s= _e+ leeth26THORN











eth30THORN


iqdes =1


lJtotKm

( _e+sat( _e+ le))

eth31THORN

where



V=1

2s2 +

1

2ka~h2 eth32THORN

where

~h=h h


_V= _ss+1

ka~h( _h _h)


ka~h _h

eth33THORN



2 + f1(um u0)) l sat(s)

Jtot

_V= ls sat(s)

Jtot

~h_h

ka+


Jtots

eth34THORN



eth35THORN



Park and Choi 2007






Experiments

Experimental set-up


















Park and Choi 2009




Conclusion





















Park and Choi 2011





Funding



References















777
































Appendix

Notation







Park and Choi 2013




fafb

=

1 12 1

2

0ffiffi3p

2 ffiffi3p

2

fafbfc

24 35 eth1THORN


fdfq

=

cos us sin us

sin us cos us

fafb

eth2THORN





vq =Riq +Lqdiq


ws = npwm eth5THORN


Tm =3



Tm =3





w1 =wm

w2 =w1=(sect13GR1)

w3 =w2=(sect23GR2)

w4 =w3=(sect33GR3)


eth8THORN


xscrew =1

GRtot

p


where





(sect13GR1)2




eth10THORN


TL frac14kscrew


Tm =TL + Jtotd2um

dt2+Tf eth12THORN

where







Park and Choi 2003





V( bw k)= 1

2

Xki=1




where

L(k)=P(k)f(k)



=1









eth16THORN


Forward Fcl =1

kcl(Kmiqfor Jtot

d2umfordt2

+Tf)

Backward Fcl =1

kcl(Kmiqback Jtot

d2umbackdt2

Tf)

eth17THORN





1 75493812 11009493 11009334 11009305 11009568 1100968




Fcl=Km

2kcl(iqfor+ iqback)

Jtot2kcl

d2

dt2(umfor + umback)

eth18THORN







eth20THORN


Jw _w=RwFx Tb

FxrsquoTb

Rw(Tb Jw _w)

eth21THORN


X4ifrac141


Rwfrac14 max

)Fcl frac14Rwmax


eth22THORN






Park and Choi 2005




Tf (um)=


Tf0sgn(wm) else

8gtltgteth23THORN





Controller design




1 +4K2Fcldes

q)=2K2 + u0

eth24THORN




e= um umdes

s= _e+ leeth26THORN











eth30THORN


iqdes =1


lJtotKm

( _e+sat( _e+ le))

eth31THORN

where



V=1

2s2 +

1

2ka~h2 eth32THORN

where

~h=h h


_V= _ss+1

ka~h( _h _h)


ka~h _h

eth33THORN



2 + f1(um u0)) l sat(s)

Jtot

_V= ls sat(s)

Jtot

~h_h

ka+


Jtots

eth34THORN



eth35THORN



Park and Choi 2007






Experiments

Experimental set-up


















Park and Choi 2009




Conclusion





















Park and Choi 2011





Funding



References















777
































Appendix

Notation







Park and Choi 2013

vq =Riq +Lqdiq


ws = npwm eth5THORN


Tm =3



Tm =3





w1 =wm

w2 =w1=(sect13GR1)

w3 =w2=(sect23GR2)

w4 =w3=(sect33GR3)


eth8THORN


xscrew =1

GRtot

p


where





(sect13GR1)2




eth10THORN


TL frac14kscrew


Tm =TL + Jtotd2um

dt2+Tf eth12THORN

where







Park and Choi 2003





V( bw k)= 1

2

Xki=1




where

L(k)=P(k)f(k)



=1









eth16THORN


Forward Fcl =1

kcl(Kmiqfor Jtot

d2umfordt2

+Tf)

Backward Fcl =1

kcl(Kmiqback Jtot

d2umbackdt2

Tf)

eth17THORN





1 75493812 11009493 11009334 11009305 11009568 1100968




Fcl=Km

2kcl(iqfor+ iqback)

Jtot2kcl

d2

dt2(umfor + umback)

eth18THORN







eth20THORN


Jw _w=RwFx Tb

FxrsquoTb

Rw(Tb Jw _w)

eth21THORN


X4ifrac141


Rwfrac14 max

)Fcl frac14Rwmax


eth22THORN






Park and Choi 2005




Tf (um)=


Tf0sgn(wm) else

8gtltgteth23THORN





Controller design




1 +4K2Fcldes

q)=2K2 + u0

eth24THORN




e= um umdes

s= _e+ leeth26THORN











eth30THORN


iqdes =1


lJtotKm

( _e+sat( _e+ le))

eth31THORN

where



V=1

2s2 +

1

2ka~h2 eth32THORN

where

~h=h h


_V= _ss+1

ka~h( _h _h)


ka~h _h

eth33THORN



2 + f1(um u0)) l sat(s)

Jtot

_V= ls sat(s)

Jtot

~h_h

ka+


Jtots

eth34THORN



eth35THORN



Park and Choi 2007






Experiments

Experimental set-up


















Park and Choi 2009




Conclusion





















Park and Choi 2011





Funding



References















777
































Appendix

Notation







Park and Choi 2013





V( bw k)= 1

2

Xki=1




where

L(k)=P(k)f(k)



=1









eth16THORN


Forward Fcl =1

kcl(Kmiqfor Jtot

d2umfordt2

+Tf)

Backward Fcl =1

kcl(Kmiqback Jtot

d2umbackdt2

Tf)

eth17THORN





1 75493812 11009493 11009334 11009305 11009568 1100968




Fcl=Km

2kcl(iqfor+ iqback)

Jtot2kcl

d2

dt2(umfor + umback)

eth18THORN







eth20THORN


Jw _w=RwFx Tb

FxrsquoTb

Rw(Tb Jw _w)

eth21THORN


X4ifrac141


Rwfrac14 max

)Fcl frac14Rwmax


eth22THORN






Park and Choi 2005




Tf (um)=


Tf0sgn(wm) else

8gtltgteth23THORN





Controller design




1 +4K2Fcldes

q)=2K2 + u0

eth24THORN




e= um umdes

s= _e+ leeth26THORN











eth30THORN


iqdes =1


lJtotKm

( _e+sat( _e+ le))

eth31THORN

where



V=1

2s2 +

1

2ka~h2 eth32THORN

where

~h=h h


_V= _ss+1

ka~h( _h _h)


ka~h _h

eth33THORN



2 + f1(um u0)) l sat(s)

Jtot

_V= ls sat(s)

Jtot

~h_h

ka+


Jtots

eth34THORN



eth35THORN



Park and Choi 2007






Experiments

Experimental set-up


















Park and Choi 2009




Conclusion





















Park and Choi 2011





Funding



References















777
































Appendix

Notation







Park and Choi 2013



Fcl=Km

2kcl(iqfor+ iqback)

Jtot2kcl

d2

dt2(umfor + umback)

eth18THORN







eth20THORN


Jw _w=RwFx Tb

FxrsquoTb

Rw(Tb Jw _w)

eth21THORN


X4ifrac141


Rwfrac14 max

)Fcl frac14Rwmax


eth22THORN






Park and Choi 2005




Tf (um)=


Tf0sgn(wm) else

8gtltgteth23THORN





Controller design




1 +4K2Fcldes

q)=2K2 + u0

eth24THORN




e= um umdes

s= _e+ leeth26THORN











eth30THORN


iqdes =1


lJtotKm

( _e+sat( _e+ le))

eth31THORN

where



V=1

2s2 +

1

2ka~h2 eth32THORN

where

~h=h h


_V= _ss+1

ka~h( _h _h)


ka~h _h

eth33THORN



2 + f1(um u0)) l sat(s)

Jtot

_V= ls sat(s)

Jtot

~h_h

ka+


Jtots

eth34THORN



eth35THORN



Park and Choi 2007






Experiments

Experimental set-up


















Park and Choi 2009




Conclusion





















Park and Choi 2011





Funding



References















777
































Appendix

Notation







Park and Choi 2013




Tf (um)=


Tf0sgn(wm) else

8gtltgteth23THORN





Controller design




1 +4K2Fcldes

q)=2K2 + u0

eth24THORN




e= um umdes

s= _e+ leeth26THORN











eth30THORN


iqdes =1


lJtotKm

( _e+sat( _e+ le))

eth31THORN

where



V=1

2s2 +

1

2ka~h2 eth32THORN

where

~h=h h


_V= _ss+1

ka~h( _h _h)


ka~h _h

eth33THORN



2 + f1(um u0)) l sat(s)

Jtot

_V= ls sat(s)

Jtot

~h_h

ka+


Jtots

eth34THORN



eth35THORN



Park and Choi 2007






Experiments

Experimental set-up


















Park and Choi 2009




Conclusion





















Park and Choi 2011





Funding



References















777
































Appendix

Notation







Park and Choi 2013








eth30THORN


iqdes =1


lJtotKm

( _e+sat( _e+ le))

eth31THORN

where



V=1

2s2 +

1

2ka~h2 eth32THORN

where

~h=h h


_V= _ss+1

ka~h( _h _h)


ka~h _h

eth33THORN



2 + f1(um u0)) l sat(s)

Jtot

_V= ls sat(s)

Jtot

~h_h

ka+


Jtots

eth34THORN



eth35THORN



Park and Choi 2007






Experiments

Experimental set-up


















Park and Choi 2009




Conclusion





















Park and Choi 2011





Funding



References















777
































Appendix

Notation







Park and Choi 2013






Experiments

Experimental set-up


















Park and Choi 2009




Conclusion





















Park and Choi 2011





Funding



References















777
































Appendix

Notation







Park and Choi 2013



Park and Choi 2009




Conclusion





















Park and Choi 2011





Funding



References















777
































Appendix

Notation







Park and Choi 2013




Conclusion





















Park and Choi 2011





Funding



References















777
































Appendix

Notation







Park and Choi 2013

















Park and Choi 2011





Funding



References















777
































Appendix

Notation







Park and Choi 2013





Funding



References















777
































Appendix

Notation







Park and Choi 2013






























Appendix

Notation







Park and Choi 2013

clamping force control based on dynamic model estimation for...

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