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does not accommodate the unique behavior of pre-sliding faithfully. In fact, the friction force showshysteresis behavior with nonlocal memory in thatregime (Prajogo, 1999, Swevers et al., 2000). TheLeuven model succeeded in including this type of hysteresis, but introduced some modellingcomplexities and difficulties (Swevers et al., 2000,Lampaert et al., 2002). Finally, the GMS modelmanages to overcome those difficulties by modelingfriction as a Maxwell-Slip model where the slipelements satisfy a certain, new state equation. As aresult, the GMS model is able to predict not only thefriction behavior in pre-sliding regime with nonlocalmemory hysteresis, but also the friction force insliding regime, which behaves in a similar way tothat in the LuGre model.An important feature in the friction identificationprocedure in this paper is that we will use only asingle set of experiments to identify all the unknownparameters together, using a suitable optimizationmethod, namely the Nelder-Mead Simplex algorithm.

Once the friction models have been optimized,position control incorporating friction compensationis performed. For this purpose, the inertial force andfriction behavior are compensated for using afeedforward control, while a simple (PID) feedbackpart is included to track set-point changes and tosuppress unmeasured disturbances.In the following, section 2 formulates the DC motortorque balance, and outlines the friction models usedin this investigation. Section 3 describes theexperimental apparatus, while section 4 discusses theidentification procedure. The identification resultsare discussed in section 5. Thereafter, the frictioncompensation scheme is sketched in section 6, whilethe compensation results are discussed in section 7.Finally, appropriate conclusions are drawn in Section8.

2. MOTOR TORQUESIn general, a DC motor can be viewed as a black boxwith two inputs: current and load torque, and anoutput angular displacement (or velocity). Torquebalance for a DC motor can be written as:

m = i + f + 0 , (1)where m is the motor torque generated by theamplifier current, i is the inertial torque from motorarmature and shaft, f is the friction torque and 0 isthe load torque.Due to the limitation of the amplifier current in themotor, it can be shown that the motor torque m = Km

(i,isat); and the inertial torque mi J= . Motortorque is bounded due to the current saturation limitof the servo amplifier. The saturation function isrepresented by (i,isat), where it has a constant

value for |i| > isat; and it has a slope of 1 for |i| isat.As for the friction torque f in equation (1), which isour main concern, several models were used asdescribed below.

1.1 Coulomb FrictionThe classical Coloumb model of friction is describedby a discontinuous relation between the friction forceand the relative velocity between the rubbingsurfaces. In this model, when the mass that issubjected to friction is slipping, the friction force willremain constant until the motion is reversed.1.2 Stribeck Friction

The Stribeck friction consists of (i) a function s(v)that is decreasing in the velocity and bounded by anupper limit at zero velocity equal to the static frictionforce Fs, and a lower limit equal to the Coulombforce Fc., and (ii) a viscous friction part. In thisapproach, the constant portion of the Coulomb modelis replaced by Stribeck function. Moreover, in orderto overcome the jump discontinuity of the Coloumbmodel, at v=0, that jump is replaced by a line of finiteslope, up to a very small threshold, as shown inFigure 1.

Figure 1. Friction curve of Stribeck models

1.3 LuGre FrictionThe above two methods have proven to beimpractical for friction compensation at motion stop

and inversion, where the worst effects due to friction,namely stick-slip motion, could arise. Motion neverstarts or stops abruptly and pre-sliding displacementsare actually observed (Prajogo, 1999, Swevers et al.,2000). Consequently, two different friction regimescan be distinguished, i.e. the pre-sliding and the grosssliding regimes, as explained in section 1. The LuGremodel (Canudas de Wit et al., 1995) was the firstformulation that could effect a smooth transitionbetween those two regimes, i.e. without recourse toswitching functions. It, furthermore, accounts forother friction characteristics such as the breakawayforce and its dynamics. The model achieves this byintroducing a state variable, representing the averagedeflection of elastic bristles (representing surfaceasperities) under the action of a tangential force,together with a state equation, governing thisvariables dynamics and friction equation.The LuGre model is very popular in the domain of control and simulation of friction due to its simplicityand the integration it affords of pre-sliding andsliding into one model. However, it has beensubjected to important criticism (Swevers et al.,2000) in regard to its failure to model pre-sliding/pre-rolling hysteresis with nonlocal memory. The latter

authors proposed an extension in form of the Leuvenmodel and a subsequent improvement (Lampaert etal., 2002), however, not without introducing furtherdifficulties.

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1.4 GMS FrictionThe Generalized Maxwell-Slip (GMS) friction model(Al-Bender et al., 2004, Lampaert et al., 2003) is aqualitatively new formulation of the rate-stateapproach of the LuGre and the Leuven models. TheGMS model retains the original Maxwell-slip modelstructure (see e.g., Iwan, 1966), which is a parallelconnection of different elementary slip-blocks and

springs, but replaces the simple Coulomb lawgoverning each block, by another state equation toaccount for sliding dynamics. Thus, the friction forceis given as the summation of the outputs of the Nelementary state models. The dynamic behavior of each elementary block is represented by one of twoequations (in whichF i is the elementary frictionforce,k i is the elementary spring constant, andv isthe velocity): If the model is sticking: vk

dtdF

ii = ; (2)

If the model is slipping:

+=

)s(FC)(

dtdF

vvsgn ii

i (3)

Each elementary model (or block) corresponds to ageneralised asperity in the contact surface, where itcan stick or slip. The asperity will slip if theelementary friction force equals the maximum valueW i = i s(v) that it can sustain. In this regime,equation (3) will characterize the friction behavior.Once the model is slipping, it remains slipping untilthe direction of movement is reversed or its velocityapproaches zero value. When the asperity is stickingthe elementary model will act as a spring withstiffness of k i.

3. EXPERIMENTAL SETUPExperimental identification of friction in a DC motorwas performed on ABB motor type M19-S, withmaximum rated torque of 0.49 Nm/amp and armatureinertia of 0.001 kgm2. To allow a straightforwardposition and velocity data collection on the motor, anincremental angular encoder was connected to theshaft through a toothed belt and pulleys. The pulleysgive a reduction ratio of 1:3 to increase the sensitivityof the encoder, whose resolution is 5000 pulses perrevolution. Figure 2 shows a schematic of theexperimental setup.

Figure 2. Schematic of the setup

4. IDENTIFICATION OF FRICTION MODELSThe identification experiment was carried out byapplying an input velocity signal. A special inputcommand was designed for this purpose. Thiscommand signal is composed of a band limitedrandom signal with cutoff frequency of 4 Hz, whichis enveloped by a certain signal. This envelope signalis composed of the three first non-zero terms of the

Fourier series of a rectangular signal, which hasperiod of 10 seconds. The purpose of enveloping thisinput signal is to emphasize the behavior of presliding friction in the experiment, whichcorresponds to the friction at low velocity and lowdisplacement. The signal and its correspondingenvelope signal are shown in Figure 3. These signalswere applied through a dSPACE-1104 acquisitioncard to the servo amplifier. The real current input tothe motor, which is assumed to correspond to thetorque was measured and recorded using the sameacquisition unit.

0 1 2 3 4 5 6 7 8 9 10-30

-20

-10

0

10

20

30

time (sec)

velocity (rad/sec)

velocity inputenvelope signal

Figure 3. Velocity input signal

5. IDENTIFICATION RESULTS

Fifty thousand points, at one millisecond timesampling, were collected for each test. The first 10seconds were used as a training set and the remaining40 seconds were used as testing set. The Nelder-Mead Simplex algorithm was used for optimizing theparameters in each model.Since the system being identified here is (highly)nonlinear, error quantification techniques pertainingto linear systems, such cross-correlation between

input, output and residuals, etc., cannot be used.Instead, as a measure of performance, the normalizedmean square error was used, which is defined by:

=

=

N

N)()(MSE

1

2100

iii yyy 2

y (4)

where y is the output (in this case is the frictionforce), y2 is its variance and the caret denotes anestimated quantity.The performance values can be seen in Table 1. Thevalues between brackets represent the peak-to-peakerrors that are normalized by the RMS value of theactual torques. In this table the performances aredivided into two groups, one for high velocities andthe other for low velocities. Each group is calculatedbased on its envelope signal. High velocity

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performance values are calculated at the instanceswhen the envelope velocity is high, and low velocityperformance values are calculated when the envelopevelocity is low.The estimated torques of the modeling technique andthe measured torque are shown in Figure 4, forCoulomb, Stribeck-Coulomb, LuGre and GMS (with4 elements), from left to right and top to bottom

respectively. It can be clearly seen that theidentification results of the classical friction modelsfall far from the actual value in some regions,especially in the low velocity regions (approximatelybefore time 3 seconds), which have lower angulardisplacement, i.e. more pre-sliding regime portion.

2 2.2 2.4 2.6 2.8 3-5

0

5GMS

4

time (s )2 2.2 2.4 2.6 2.8 3-5

0

5LuGre

time (s )

Torque (Nm)

2 2.2 2.4 2.6 2.8 3-5

0

5Coloumb

Torque (Nm)

2 2.2 2.4 2.6 2.8 3-5

0

5Stribeck-Coloumb

Figure 4. Measured and estimated torques for different models. Dotted lines represent the measured torques, and

solid lines are the estimated torques.

2 3 4-5

0

5Error of LuGre

time (s)

Error (Nm)

2 3 4-5

0

5Error GMS

4

time (s)

2 3 4-5

0

5Error of Coloumb

Error (Nm)

2 3 4-5

0

5Error of Stribeck-Coloumb

Figure 5. Error of the estimated torques

2 2.5 3 3.5 4-40

-20

0

20

40

time (s )

velocity (rad/sec)

Figure 6. Velocity signal in the validation set

On the other hand, the results of the two recentmodels, LuGre and GMS, show much betterperformance. They give significant improvements,especially in the low velocity region, for which theclassical models could not account well.The most recent friction model, GMS, gives thebest results in this experiment. In particular, strongimprovement is achieved in the low velocity region.The ability to estimate the friction behavior in pre-

sliding regime is shown as the superiority of theGMS model, while it does not lose its ability toestimate the friction in the gross sliding regime.These results are summarized in Table 1.As a comparison, another identification utilizingGMS model with higher number of Maxwell-Slip

elements was conducted. In this identification, 10Maxwell-Slip elements were used, instead of 4.Although this gives a slightly better result, theimprovement is not significant as shown in the lastrow of Table 1. Increasing the Maxwell-Slip by 6elements means adding twice times six (=12)parameters in the optimization process, which is ahigh price for the slight improvement.The error of the corresponding models and thevelocity input signal are shown in Figure 5 andFigure 6 respectively.

Table 1. Performances of identification result forhigh frequencies experiment

High velocityMSE (max. err.)

Low velocity MSE (max. err.)

Coulomb 4.00% (0.6757) 17.92% (1.2993)Stribeck 0.55% (0.2762) 9.59% (1.2604)LuGre 0.41% (0.2440) 4.30% (0.6466)GMS 4 0.40% (0.2350) 1.39% (0.5711)GMS 10 0.38% (0.2300) 1.19% (0.5177)

6. FRICTION COMPENSATION SCHEMEIn this section, position control incorporating

friction compensation using the aforementionedmodels in the feedforward will be described. Forthis purpose, the inertial force and the frictionforce, as modeled in the previous section, are

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included in the feedforward loop. The controlscheme is depicted in Figure 7.The feedback loop, which is required to track set-point changes and to suppress unmeasureddisturbances, is chosen to be a PID controller withgains Kp = 200 Nm/rad, Ki = 0 Nm/rad.s, and Kd =0.15 Nm.s/rad. In order to obtain a consistentcomparison, these gains were optimized for one of

the control schemes, namely that employing LuGrecompensation, and used with the same values for allthe other models. As a numerical quantification of the validation criterion, the peak-to-peak trackingerror will be used for different reference inputsignals.

Figure 7. The feedforward/feedback control scheme

for friction compensation

7. COMPENSATION RESULTS

Two different reference signals were employed tovalidate the friction compensation. The firstreference position signal is a filtered random signalwith a very small stroke in order to emphasize thepre-sliding regime of the friction torque. Thereference signal was generated using a random

signal generator, which is passed through a lowpass, 4th order digital Butterworth filter with 1 Hzcutoff frequency. The tracking error of each frictioncompensation model can be seen in Figure 8. (Allthese results were obtained with the same PIDcontroller parameters). The numerical RMS valuesare given in Table 2.Friction compensation based on the Coloumbmodel yields large errors, and is almost identical tothe scheme that has no feedforward compensation.Since the reference signal has a very small strokeand it is emphasizing the presliding regime, theCoulomb model is merely playing on its thresholdregion. Thus, the Coulomb model will act only as again to the reference signal. The GMS modelcompensation, in contrast, gives significantimprovement in comparison with the other models.It shows superiority at the reversal points comparedto the LuGre model. The jumps in the tracking errorof the LuGre model, which correspond to thesticking problem, are significantly reduced in theGMS compensation model.

Table 2. Performances of friction compensation fora low stroke random input

Compensation RMS (rad x1000)No Feedforward 5.6

Coulomb 5.3LuGre 1.6GMS 0.6

0 5 10 15 20 25-0.04

-0.02

0

0.02

Amplitude (rad)

Low Stroke Random Reference Signal

0 5 10 15 20 25-4

-2

0

2

4

Tracking Error (rad)

x1000

No FF

0 5 10 15 20 25-4

-2

0

2

4Coloumb

0 5 10 15 20 25-4

-2

0

2

4

Tracking Error (rad)

x1000

time (sec)

LuGre

0 5 10 15 20 25-4

-2

0

2

4

time (sec)

GMS4

Figure 8. The tracking errors for low stroke filtered

random reference input. The top-left figureshows the tracking error for a system with nofeedforward friction compensation. In the top-right figure, the Coulomb friction was used as afeedforward friction compensation, while thebottom figures show the friction compensationresults using the LuGre and the GMS model.

0 5 10 15 20 25-4

-2

0

2

Amp

litude (rad)

High Stroke Random Reference Signal

0 5 10 15 20 25-6

-4

-2

0

2

4

6

Tracking Error (rad)

x1000

No FF

0 5 10 15 20 25-6

-4

-2

0

2

4

6Coloumb

0 5 10 15 20 25-6

-4

-2

0

2

4

6

Tracking Error (rad)

x1000

time (sec)

LuGre

0 5 10 15 20 25-6

-4

-2

0

2

4

6

time (sec)

GMS 4

Figure 9. The tracking errors for high stroke filtered

random reference input. The top-left figureshows the tracking error for a system with nofeedforward friction compensation. In the top-right figure, the Coulomb friction was used as afeedforward friction compensation, while thebottom figures show the friction compensationresults using the LuGre and the GMS model.

The second reference signal used in this experimentis a filtered random signal similar to the signal inthe previous experiment, except that this signal haslarger amplitude. The maximum stroke of this

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random signal was set to rad (or half of theshafts rotation). In this case, the pre-sliding frictionis not expected to dominate the overall behavior.

The tracking errors for each compensation schemeare depicted in Figure 9. For this desired positionsignal, the GMS model again gives the bestperformance. Nevertheless, according to theperformance values in Table 3, it does not give asignificant improvement compared to the LuGremodel, even though the spikes in the tracking error,which are indicated by the arrows in the figure, areobviously reduced. These results areunderstandable, since the system was excited with ahigh stroke, i.e. more in sliding regime than in pre-sliding regime. In this case, the zero crossings of the velocity, where the pre-sliding regime frictionarises, will occur only over a very short portionduring the whole motion.

Table 3. Performances of friction compensation for

a high stroke random inputCompensation RMS (rad x1000)No Feedforward 10.5

Coulomb 5.7LuGre 4.7GMS 4.6

8. CONCLUSIONThe following conclusions can be drawn from thisinvestigation: Friction identification using a single experiment

is possible to conduct, even for the morecomplex and more nonlinear models. Frictionidentification utilizing the most recent GMSmodel, which incorporates two regimes of friction, was also possible to conduct using asingle experiment. However, selection of theexcitation signal plays an important role foridentification using single experiment.

For motions with high velocities, the classicalmodels such as Coulomb friction and theStribeck friction model give satisfactory resultscomparable to the advanced models. However,at low velocities, which emphasize the pre-

sliding regime, the classical models fail to givea satisfactory estimation of the overall friction.For a good estimation in pre-sliding, modelsthat incorporate hysteresis behavior arenecessary.

The ability to estimate the friction behavior inpre-sliding regime is the superiority of GMSmodel, while it does not lose its ability toestimate the friction in sliding regime.Therefore the GMS model can capture frictionbehavior for any working range of displacementand velocity.

The nonlinear effect of friction in a high load

torque DC motor is successfully compensatedfor in a feedforward based control experiment.In all of the validation cases, the GMS modelyields the best results. This can be understoodsince this (most recent model) accommodates

the hysteresis behavior with non-local memoryfor the presliding regime friction, whilemodeling the gross sliding also.

ACKNOWLEDGEMENTThe authors wish to acknowledge the partialfinancial support of this study by the

Volkswagenstiftung under grant No. I/76938.

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Armstrong-Hlouvry, B. (1991). Control of Machines with friction, Kluwer AcademicPublishers.

Canudas de Wit, C., H. Olsson, K.J. strm and P.Lischinsky (1995). A new model for control of systems with friction.IEEE Trans. Automat.Contr . 40 , 3, 419-425.

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