Optimizing Karnopp friction model parameters of a pendulum using RSM

Optimizing Karnopp friction model parameters of a pendulum using RSM

Sabri Bicakci a,n, Davut Akdas a, Aslan Deniz Karaoglan b

a Department of Electrical and Electronics Engineering, Balikesir University, 10145 Balikesir, Turkeyb Department of Industrial Engineering, Balikesir University, 10145 Balikesir, Turkey

a r t i c l e i n f o

Article history:Received 28 February 2014Accepted 4 April 2014

Keywords:PendulumKarnopp friction modelResponse surface methodologyFriction parameter identification

a b s t r a c t

Accurate mathematical models of physical systems are essential for understanding the behaviour ofactual systems under different operating conditions and for designing control systems. In mechanicalsystems, difficulty of the exact determination of friction force parameters adversely affects the accuracyof the models. In this study, Karnopp friction model is chosen in order to model the friction parametersof the bob of a pendulum. The parameters are determined by sectioning the speed regions and thenusing “Response Surface Methodology” (RSM) by sectioning the speed regions. Proposed method hasproduced accurate yet simple model of the friction parameters.

& 2014 European Control Association. Published by Elsevier Ltd. All rights reserved.

1. Introduction

Control methods for mechanical systems can be divided intotwo main classes: model based [25,6] and non-model basedcontrol methods [27]. The model based control method is themost commonly used technique whose performance is directlydependant on accuracy of the mathematical model employed.A mathematical model is a function of parameters that consists ofmasses, the distance between the centres of masses and joint axes,moments of inertia and friction forces. Depending on the accuracyof these parameters, the mathematical model will represent theactual system. Determination of the friction force parameters ispossibly the most difficult part of these models. Since the frictionforces depend on the nature of surfaces moving relative to eachother, a full model of the friction has yet to be determined.

Numerous researchers had carried out studies about frictionforce modelling over the last decades and most of them werepresented in Ref. [3]. In general, a two-term classical friction modelhas been used, where one of the terms is a speed dependentparameter and the other one is a constant [4]. This classical modelproduces sufficient results at higher speeds; however, it is insuffi-cient at lower speeds and cannot model dry friction which mayexist around zero-speed region. Exclusion of the dry friction causesundesired oscillations in the control of mechanical systems such asan inverted pendulum [28]. To overcome this kind of shortcomings,improved friction models at lower speeds are required. In order todo that, models similar to Dahl [8] and LuGre [9] have beendeveloped. Zhao et al. [28] have devoted studies to model the

friction between the cart of a pendulum and the surface by usingGauss [28] and LuGre [9] friction models and lower oscillation levelsin control of the pendulum were observed. However, these modelswere complex and computationally expensive which may result ininstabilities. As an alternative to the model developed by Zhao et al.[28], another method was proposed by Karnopp [16] which pro-vided faster calculations and better control performance. However,performance of the control system does not exactly represent theaccuracy of the friction model. The accuracy of the friction modelscan be evaluated more realistically by considering the naturalmovement of the mechanismwithout the control action. Pendulummechanism are considered as convenient platforms for this purposesince the rotational joint of the pendulum, which can move freely,rests in its final position after a number of oscillations whenreleased from a certain height. During the free motion of apendulum, there are no external forces in effect alongside thenatural factors as gravitational and frictional forces. The frictionforces in oscillating mechanisms, such as in pendulums, weremostly neglected in the previous studies and no detailed studyregarding friction force modelling have been given. As a first steptowards developing a full friction model, the aim of this study is toobtain a mathematical model of the rotating part of a single stagependulum, which will later be used as an experimental platform forcontrol systems. Due to its simplicity and better performancecharacteristics than others [24], Karnopp friction model has beenused as the base friction model for the study.

For an acceptable friction model performance, the modelparameters should be determined accurately. Liu et al. [18] useda statistical parameter determination method for the Karnoppmodel. Their method was entirely based on algebraic calculationsand it was fairly easy to obtain the parameters. However, noadjustment was done between real data and the calculatedparameters. Fayez et al. [1] have used LuGre friction model [9]

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journal homepage: www.elsevier.com/locate/ejcon

European Journal of Control

http://dx.doi.org/10.1016/j.ejcon.2014.04.0010947-3580/& 2014 European Control Association. Published by Elsevier Ltd. All rights reserved.

n Corresponding author. Fax: þ902666121257.E-mail addresses: [email protected] (S. Bicakci),

[email protected] (D. Akdas), [email protected] (A. Deniz Karaoglan).

Please cite this article as: S. Bicakci, et al., Optimizing Karnopp friction model parameters of a pendulum using RSM, European Journalof Control (2014), http://dx.doi.org/10.1016/j.ejcon.2014.04.001i

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for a DC motor control in automotive industry and claimed to havedetermined the incalculable parameters of the model using meansquare error (mse) method. However, limited detailed informationabout calculation procedure was given. Elhami et al. [12] havedetermined the parameters of the friction model consideringCoulomb and viscous frictions using the nonlinear filtering para-meter determination method of Detchmendy and Sridhar [10]. Inthis method, parameters are constantly updated in real timeaccording to the results of the solution of nine differential equationsin every sampling interval. The difference between the real torquevalue and simulations has been minimized with every update.Numerous calculations were done during this process. Garcia [13]compared eight of the existing friction modelling techniques incontrol of a valve and tested the accuracy of the friction models Itwas concluded that Karnopp [16], LuGre [9] and Kano [15] modelswere more suitable than other models in the problem chosen. It wasalso stated that more compatible results could be obtained usingmore successful parameter determination methods. In their latterstudy [14], they have carried out experiments and acquired resultsusing some statistics methods for determination of the frictionparameters. In another study, Kim et al. [17] have used Karnoppfriction model in the modelling of an elevator with hydraulicactuators and tried to determine the parameters using trial anderror experiments.

Determination of the friction parameters naturally starts fromexperimental evaluations. Since most of the parameters cannot becalculated directly, fine tuning of the parameters due to measuringand production errors is required using the experimental results.Response surface methodology (RSM) is one of the well-knownexperiment design techniques widely used for modelling mathe-matical relations between the experimental inputs (factors) andthe outputs (responses) with minimum number of experiments[20]. RSM is an effective modelling technique which can beimplemented for modelling nonlinear relations using second orderpolynomial equations. In this paper, RSM was used to find theoptimum factor levels those best-fits between the experimentaldata and simulation results for the given factors (Viscous FrictionConstant—Mw, Coulomb Friction Constant—Mc, Stiction Constant—Ms, Karnopp Velocity—DV, and Inertial Moment of bob of pendu-lum—I).

Rest of the paper is organized as follows. Next section describesthe experiment system. The numerical solution of second orderdifferential equation is explained in Section 3. Karnopp frictionmodel and RSM are introduced in Sections 4 and 5, respectively.Section 6 contains the identification of the parameters (Mw,Mc,Ms,DV, I). And finally the last section contains conclusions.

2. Experimental system

The pendulum mechanism used in the present study is shownin Fig. 1. Pendulum is a mechanical system which can make a freerotational motion in the frontal plane. It is mounted on a cart thatcan make a linear motion in a single axis. In order to fully determinethe effects of friction moment that only exists in the rotating part ofthe pendulum on the system, the cart of the pendulum is anchoredto the base. Therefore, at this stage the mathematical model of thesystem considers only the rotating part.

Mathematical model of this pendulum is given in Eq. (1) whichis derived using free-body diagram the system

0¼ ðml2þ IÞ€θþmgl sin θþMfriction ð1Þ

wherem is the mass of pendulum, l is the distance from the centreof gravity of the pendulum to the rotation axis, I is the moment ofinertia of the pendulumwith respect to the centre of mass, g is the

gravitational constant, θ is the rotation angle of the pendulum andMfriction is the friction moment.

Steady state resting position of the pendulum is taken as zerodegree and counter-clockwise rotating direction is taken positive.The pendulum is designed using a computer-aided-design (CAD)software (SolidWorks 2011, USA) and manufactured using computer-aided-manufacturing (CAM) methods. Mass of the pendulum ismeasured using a scale with 0.0001g sensitivity. The distance fromthe centre of gravity to the rotation axis and moment of inertiahas been determined using Mass Properties tool of the software

Fig. 1. Pendulum mechanism.

Table 1Pendulum parameters.

m 0.1466084 kgl 0.1562177 mI 0.00284252 kg m2

g 9.81 m/sn2

Pendulum Mechanism

Linear Potentiometer Oscilloscope Computer

Fig. 2. Schematic of the experimental system.

0 5 10 15 20 25 30 35 40 45 50-1.5

-1

-0.5

0

0.5

1

1.5

2Position - Time

Time (s)

Pos

ition

(rad

)

Fig. 3. Recorded position signal of free falling pendulum.

S. Bicakci et al. / European Journal of Control ∎ (∎∎∎∎) ∎∎∎–∎∎∎2





(SolidWorks 2011, USA). Parameters of the pendulum are given inTable 1. Due to production and round-up errors of the CAD software,calculated moment of inertia value did not exactly match with theactual value. Therefore, the moment of inertia is re-determined usingRSM, in Section 6.

Rotation of the pendulum is measured by a Vishay Model 157servo potentiometer which is directly connected to the pendulumshaft. Position of the pendulum is measured from the potentiometerusing a PicoScope 3424 model digital oscilloscope. Measured valuesof position are filtered using a second degree Butterworth filter.Natural oscillation movements are created by releasing the pendu-lum from certain angles and recorded directly. Schematic of theexperimental system is shown in Fig. 2 and a typical recordedposition signal is shown in Fig. 3.

3. Numerical solution of second order differential equation

Nonlinear second order differential equation of the mathema-tical model should be solved in order to carry out simulation of thependulum. Fourth order Runge–Kutta Method (FORKM) is used forthe solution of the differential equation [22].

FORKM can be applied by converting Eq. (1) into two first orderdifferential equations as in Eqs. (2) and (3)

_θ¼w ð2Þ

_w¼ �ðMfrictionþmgl sin θÞðml2þ IÞ

¼ f ðθ;w; tÞ ð3Þ

For these two first order differential equations, FORKM equa-tions are as follows.

θnþ1 ¼ θnþ16ðk1aþ2k2aþ2k3aþk4aÞ ð4Þ

wnþ1 ¼wnþ16ðk1bþ2k2bþ2k3bþk4bÞ ð5Þ

k1a ¼wnΔt ð6Þ

k1b ¼ f ðθn;wn; tÞΔt ð7Þ

k2a ¼ wnþk1b2

� �Δt ð8Þ

k2b ¼ f θnþk1a2;wnþk1b

2; tþΔt

2

� �Δt ð9Þ

k3a ¼ wnþk2b2

� �Δt ð10Þ

k3b ¼ f θnþk2a2;wnþk2b

2; tþΔt

2

� �Δt ð11Þ

k4a ¼ ðwnþk3bÞΔt ð12Þ

k4b ¼ f ðθnþk3a;wnþk3b; tþΔtÞΔt ð13ÞThe simulation of the pendulum's mathematical model is

performed by applying the acquired equations iteratively.

4. Karnopp friction model

Friction forces are divided into three basic sections while buildingthe friction model: viscous, Coulomb and dry friction [3]. Viscousfriction is a velocity dependent parameter while Coulomb friction isspeed independent but nonexistent around zero speed region. Dryfriction, also known as stiction, is a parameter considered to beexistent at points where speed is zero. According to ANSI, dry friction

is described as “The resistance to the start of motion, usuallymeasured as the difference between the driving values required toovercome static friction upscale and downscale” [2]. Various math-ematical models have been developed to model dry friction and it isconcluded that they can be separated into two basic groups as staticand dynamic models. Dynamic models can model dry friction moreefficiently compared to static models but they contain too muchoperational burden and their parameters are more difficult todetermine [19]. Static models are easier to apply and can produceacceptable results in performance evaluation. One of the mostcommonly used static models is Karnopp friction model [16].

Static friction models generally require zero speed point to beprecisely measured to produce acceptable results. In practicalapplications, measurement of the zero speed region is usuallynot possible when sampling speeds and noise effect is taken intoaccount. In Karnopp friction model, lower speed regions areconsidered zero speed instead of an absolute zero speed value.Karnopp friction model can be expressed as in Eq. (14)

Mfriction ¼Mw �wþMc � sgnðwÞ ; jwjZDV

minðMeðtÞ;MsÞ ;MeðtÞZ0maxðMeðtÞ; �MsÞ ;MeðtÞo0; jwjoDV

8><>: ð14Þ

where Mw is viscous friction coefficient, Mc is Coulomb frictioncoefficient, Ms is dry friction coefficient, Me(t) is external moment,DV is Karnopp zero speed range value and w is the velocity of themoving mechanism.

If w is greater than DV, friction moment is made up of twocomponents: speed dependent viscous friction and constantCoulomb friction whose direction is dependent on direction ofthe speed. If w is less than DV, friction moment only consist of dryfriction that depends on external moment. In this way, frictionstate just before the system begins to move or stops is expressed.Karnopp friction model can be expressed graphically as in Fig. 4.

5. Response surface methodology (RSM)

RSM is a well-known design of experiment (DOE) technique. It is acombination of statistical and mathematical techniques and widelyused for reducing the number of experiments for optimizationprocesses. RSM has been used extensively in experimental studies inorder to examine and characterize problems in which the inputvariables (factors) influence the outputs (response) of the process.RSM can provide process optimization by simultaneous testing ofnumerous factors in a limited number of experiments. This consumesless time and effort.

Mfriction

w

+DV−DV

Fig. 4. Karnopp friction model.

S. Bicakci et al. / European Journal of Control ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 3





In addition, the RSM provides the mathematical relation betweeninputs and outputs of the system including the interactions betweenthe factors, which is difficult to obtain using heuristic optimizationtechniques especially for multivariable optimization. The RSM wasproposed by Box and Wilson for finding the input combination thatminimizes the output of a real non-simulated system [5,7,20]. Eq.(15) shows the general second-order polynomial response surfacemathematical model (full quadratic model) for the experimentaldesign [5,7,11,20]

Y ¼ β0þ ∑n

i ¼ 1βiXiþ ∑

n

i ¼ 1βiiX

2i þ ∑

n

io jβijXiXjþe ð15Þ

where Y is the corresponding response, Xi are coded values of the ithinput parameters, terms, β0, βi, βii and βij are the regressioncoefficients, i and j are the linear and quadratic coefficients, and eis the residual experimental error (random error) [23,26]. In terms ofthe observations, the model may be written in matrix notation as:

Y¼ βXþε ð16Þwhere, Y is the output matrix and X is the input matrix, and ε is theresiduals (random error term). The least square estimator of βmatrixthat composes of coefficients of the regression equation calculated bythe given formula in Eq. (17), by ignoring ε

β¼ ðXTXÞ�1XTY ð17Þwhere XT is the transpose of X.

6. Simulation and parameter identification

In order to adjust the actual value of the moment of inertia andparameters of the friction model using the RSM, initial value rangeof each parameter and criteria that represents the relevancy of theparameters to reality should be determined. These criteria (shownin Fig. 5) are chosen as the differences of zero crossing numbers(eZC), the differences of positive and negative peak numbers (ePM,eNM) and the mean square errors of angle and speed values (mseA,mseV) of both the experimental and simulated data. They repre-sented the error between pendulum's real motion and its simu-lated motion. The errors have been minimized by adjusting theparameters with the RSM. The parameter values that drive theerrors to their minimum have been taken as the most suitablevalues.

Starting value range for the moment of inertia (I) is chosenaround the value that was calculated with the CAD software.

Simulation is performed by heuristically chosen random values forthe remaining parameters (Mw, Mc, Ms, DV) in the first stage. Thestarting value range for the parameters have been roughlyadjusted according to acquired results. Even when this adjustmentis being made it is seen that every parameter has very differenteffects on the evaluation criteria. Heuristic studies without anysystematic base could be extremely time consuming in order todetermine the values that could give the best results.

In the presented study, central composite face centered designis used. Initial factor levels used in the design are given in Table 2.Accepted orthogonal series for five variables in literature is L32(requires minimum 32 experiments) [20]. This orthogonal seriesrequires six repeats in centre points (medium values of thevariables). Because of having the non-random responses obtainedfrom the simulations, only one center point is used instead of six.Therefore, a total of 27 experimental runs is adopted in order to fitthe simulation results to the experimental results, see Table 3.

A statistical software package (Minitab 16, USA) is used to findthe β matrix and establish the mathematical models for predictingthe responses. According to the experiments presented in Table 3,mathematical models based on the RSM (presented in Eq. (15)with its general representation) for the responses mseA, mseV, ePM,eNM, eZC have been established with 95% confidence. After theoptimization process by response optimizer module of Minitab forthe target value of 0 for each response, the optimum factor levelsare calculated as presented in Table 4.

For the optimum factor levels, the response values are obtainedas mseA¼5.8208e�04, mseV¼0.0179, ePM¼1, eNM¼2, eZC¼3 fromthe simulation results. Fig. 6 demonstrates the comparison ofactual data and the simulated results.

It is clear from Fig. 6 that the obtained parameter combinationprovides improvement on the process. The chosen criteria forestablishing acceptable parameters is to make ePM¼0, eNM¼0,eZC¼0 and to minimize mseA and mseV. Optimum factor levels areselected as the medium factor level and a new design is used toimprove the performance further. This new design is constructedon narrowed factor level range. The new parameter set for thesecond design is given in Table 5.

After the optimization process for the second design case, theoptimum factor levels are calculated as presented in Table 6.

For the optimum factor levels, the response values are obtainedas mseA¼2.0995e�04, mseV¼0.0079, ePM¼0, eNM¼0, eZC¼0 fromthe simulation results. Fig. 7 shows the comparison of the real dataand the simulated results. Results indicate that the calculatederrors (responses) are decreased.

The parameter combination provides improvement on theprocess as shown in Fig. 7, but there is still a small shift on thetime scale. In order to decrease the error and provide an accur-ate fit, a new design is needed. To construct the new design theoptimum factor levels are selected as the medium factor level andthe feasible solution region is narrowed by focusing on the regionof interest. The parameter set for the third design and the designof experiment matrix are given in Tables 7 and 8, respectively.

0 5 10 15 20 25 30 35 40 45-1.5

-1

-0.5

0

0.5

1

1.5Position - Time

Time (s)

Pos

ition

(rad

)

Simulation DataExperimental Data

Negative PeakValues

Positive PeakValues

Zero Cross

Fig. 5. Evaluation criteria.

Table 2List of initial actual values of friction model parameters.

Parameter Symbol Level

Low Medium High

Viscous friction (N-m s/rad) Mw 0.00010 0.00055 0.00100Coulomb friction (N-m) Mc 0.00010 0.00055 0.00100Stiction (N-m) Ms 0.0010 0.0055 0.0100Karnopp velocity (rad/s) DV 0.01 0.03 0.05Inertial moment (kg m2) I 0.0026 0.0028 0.0030






The responses are formulated by using the RSM as giventhrough Eqs. (18)–(22) in 95% confidence level by R2 values of99.96%, 99.96%, 98.25%, 97.87%, and 99.78%.

mseA ¼ 22:528194þ14033:1Mwþ2803:71Mcþ13:68Ms

þ0:421228DV�20193:53Iþ2492169Mw2þ117904Mc

2

�13:96Ms2�0:151637DV2þ4549294:77I2þ1042163:91MwMc

þ4750:3MwMsþ125:44MwDV�6458354:64MwIþ1228:96McMsþ1:42McDV�1307208:23McIþ28:29MsDV�6833:08MsI�191:46DV � I ð18Þ

mseV ¼ 770:69þ480327:16Mwþ96913:75Mcþ466:7Msþ14:34DV

�691038:47Iþ85503755:07Mw2þ3827817:84Mc

2

�364:78Ms2�5:22DV2þ1557167198:56I2

þ35965714:18MwMcþ160917:57MwMsþ4471:81MwDV

�221177987:49MwIþ40528:44McMsþ313:96McDV�45042303:76McIþ965:41MsDV�232698:92MsI�6624:14DV � I ð19Þ

Table 3Design of experiments matrix showing uncoded actual values and observed responses.

Mw Mc Ms DV I mseA mseV ePM eNM eZC

0.00010 0.00010 0.0010 0.01 0.0030 0.851394 24.5198 �1 0 �10.00100 0.00010 0.0010 0.01 0.0026 0.172923 5.902288 �5 �5 �100.00010 0.00100 0.0010 0.01 0.0026 0.420202 13.79541 �4 �3 �70.00100 0.00100 0.0010 0.01 0.0030 0.024684 0.83359 15 15 300.00010 0.00010 0.0100 0.01 0.0026 0.748657 22.94686 �2 �2 �40.00100 0.00010 0.0100 0.01 0.0030 0.037481 1.249698 1 2 30.00010 0.00100 0.0100 0.01 0.0030 0.527503 16.03793 �2 �2 �40.00100 0.00100 0.0100 0.01 0.0026 0.170452 5.726086 17 17 340.00010 0.00010 0.0010 0.05 0.0026 0.74871 22.94693 �2 �2 �40.00100 0.00010 0.0010 0.05 0.0030 0.037696 1.265339 �4 �4 �80.00010 0.00100 0.0010 0.05 0.0030 0.527562 16.03839 �2 �2 �40.00100 0.00100 0.0010 0.05 0.0026 0.169584 5.725135 15 15 300.00010 0.00010 0.0100 0.05 0.0030 0.850958 24.51124 �1 0 �10.00100 0.00010 0.0100 0.05 0.0026 0.174283 5.939797 2 3 50.00010 0.00100 0.0100 0.05 0.0026 0.419574 13.78747 �4 �3 �70.00100 0.00100 0.0100 0.05 0.0030 0.025338 0.834071 16 17 330.00010 0.00055 0.0055 0.03 0.0028 0.620959 19.04629 �2 �2 �40.00100 0.00055 0.0055 0.03 0.0028 0.058951 2.008085 10 11 210.00055 0.00010 0.0055 0.03 0.0028 0.158014 5.321195 �4 �4 �80.00055 0.00100 0.0055 0.03 0.0028 0.008474 0.277822 7 7 140.00055 0.00055 0.0010 0.03 0.0028 0.053817 1.815587 �4 �4 �80.00055 0.00055 0.0100 0.03 0.0028 0.051149 1.722521 �1 �1 �20.00055 0.00055 0.0055 0.01 0.0028 0.053532 1.805356 �3 �2 �50.00055 0.00055 0.0055 0.05 0.0028 0.051252 1.728908 �3 �2 �50.00055 0.00055 0.0055 0.03 0.0026 0.040183 1.389426 �2 �2 �40.00055 0.00055 0.0055 0.03 0.0030 0.23898 7.8004 �3 �3 �60.00055 0.00055 0.0055 0.03 0.0028 0.052539 1.772103 �3 �2 �5

Table 4Optimum parameter level for the 1st design.

Parameter Symbol Optimum value

Viscous friction (N-m s/rad) Mw 0.00054Coulomb friction (N-m) Mc 0.0007Stiction (N-m) Ms 0.006Karnopp velocity (rad/s) DV 0.04Inertial moment (kg m2) I 0.0027

5 10 15 20 25 30 35 40 45-1.5

-1

-0.5

0

0.5

1

1.5Position - Time

Time(s)

Pos

ition

(rad

)


38.5 39 39.5 40 40.5

-0.04

-0.02

0

0.02

0.04

12.9 13 13.1

-0.7

-0.65

-0.6

Fig. 6. Comparison of real data and simulated results for the first design.

Table 5New parameters set for actual values of friction model parameters (2nd design).


Low Medium High


Table 6Optimum parameter level for the 2nd design.








ePm ¼ 1524:88þ754387:63Mwþ1766130:05Mcþ13099:75Ms

þ1015:66DV�1778061:87I�909090909:09Mw2

0 5 10 15 20 25 30 35 40 45-1.5

-1

-0.5

0

0.5

1

1.5Position - Time

Time (s)

Posi

tion

(rad)

12.9 13 13.1

-0.7

-0.65

-0.6

40 41-0.04

-0.02

0

0.02

0.04 Simulation DataExperimental Data

Fig. 7. Comparison of real data and simulated results for the second design.

Table 7New parameters set for actual values of friction model parameters (3rd design).


Low Medium High


Table 8Design of experiments matrix showing uncoded actual values and observed responses for new parameters (3rd design).

Mw Mc Ms DV I mseA mseV ePM eNM eZC

0.00052 0.00068 0.0010 0.045 0.00272 0.005323 0.178887 �2 �1 �30.00056 0.00068 0.0010 0.045 0.00268 0.006282 0.219974 0 0 00.00052 0.00072 0.0010 0.045 0.00268 0.001221 0.043418 �1 0 �10.00056 0.00072 0.0010 0.045 0.00272 0.00029 0.01126 1 1 20.00052 0.00068 0.0020 0.045 0.00268 0.000791 0.027965 �2 �1 �30.00056 0.00068 0.0020 0.045 0.00272 0.000201 0.007459 0 0 00.00052 0.00072 0.0020 0.045 0.00272 0.00324 0.107903 �1 0 �10.00056 0.00072 0.0020 0.045 0.00268 0.008422 0.294349 1 2 30.00052 0.00068 0.0010 0.055 0.00268 0.000764 0.027009 �2 �1 �30.00056 0.00068 0.0010 0.055 0.00272 0.000231 0.008513 0 0 00.00052 0.00072 0.0010 0.055 0.00272 0.003362 0.11211 �1 0 �10.00056 0.00072 0.0010 0.055 0.00268 0.008207 0.287249 1 1 20.00052 0.00068 0.0020 0.055 0.00272 0.005033 0.168945 �2 �1 �30.00056 0.00068 0.0020 0.055 0.00268 0.006582 0.230216 0 1 10.00052 0.00072 0.0020 0.055 0.00268 0.00133 0.047181 0 0 00.00056 0.00072 0.0020 0.055 0.00272 0.000289 0.011291 1 1 20.00052 0.00070 0.0015 0.050 0.00270 0.000684 0.022663 �1 �1 �20.00056 0.00070 0.0015 0.050 0.00270 0.002044 0.073151 0 1 10.00054 0.00068 0.0015 0.050 0.00270 0.000272 0.009973 �1 0 �10.00054 0.00072 0.0015 0.050 0.00270 0.000557 0.020501 0 1 10.00054 0.00070 0.0010 0.050 0.00270 0.000352 0.013214 0 0 00.00054 0.00070 0.0020 0.050 0.00270 0.000375 0.014015 0 0 00.00054 0.00070 0.0015 0.045 0.00270 0.000361 0.013486 0 0 00.00054 0.00070 0.0015 0.055 0.00270 0.000366 0.013665 0 0 00.00054 0.00070 0.0015 0.050 0.00268 0.00332 0.117046 0 0 00.00054 0.00070 0.0015 0.050 0.00272 0.001054 0.034939 0 0 00.00054 0.00070 0.0015 0.050 0.00270 0.000362 0.013542 0 0 0

�909090909:09Mc2þ545454:55Ms

2þ545:55DV2

þ340909090:91I2�156250000MwMc�6250000MwMs

�625000MwDVþ156250000MwIþ6250000McMs

�625000McDV�156250000McIþ25000MsDV

�6250000MsI�625000DV � I ð20Þ

eNm ¼ �1792:49�110378:88Mw�1344318:18Mcþ28313:13Ms

þ1238:64DVþ1409217:17I�227272727:27Mw2

þ1022727272:73Mc2�363636:36Ms

2�3636:36DV2

�227272727:27I2þ0:00000047MwMcþ12500000MwMs

�0:000000021MwDV�312500000MwIþ0:000000894McMs

�1250000McDV�0:0000103McI�0:00000000041MsDV

�12500000MsIþ0:0000000259DV � I ð21Þ

eZC ¼ �267:61þ1858175:51Mwþ421811:87Mcþ41412:88Ms

þ2254:29DV�368844:7I�1136363636:36Mw2

þ113636363:64Mc2þ181818:18Ms

2þ1818:18DV2

þ113636363:64I2�156250000MwMcþ6250000MwMs

�625000MwDV�156250000MwIþ6250000McMs

�625000McDV�156250000McIþ25000MsDV

�18750000MsI�625000DV � I ð22Þ

Fig. 8 represents the Minitab response optimizer module out-puts for the optimization results of the third design. Ms and DV areonly effective around zero velocity and do not have a significantimpact on the overall performance within the third design valuerange. It is seen in the fourth design of experiment that extendingthe lower value for Ms and the higher value for DV does not haveany distinguishable improvement on the process. Thereforeexperiments for parameter determination process is stopped afterthird iteration.

The optimum factor levels are presented in Table 9.






For the optimum factor levels, the response values are calculatedas mseA¼1.164e�04, mseV¼0.0046, ePM¼0.2817, eNM¼0.1723, eZC¼0.454 by using Eqs. (18)–(22). Observed results obtained from thesystem are better than the expected values calculated from mathe-matical equations. Fig. 9 shows the comparison of experimental and

the simulated results. Comparative study has indicated that thecalculated errors (responses) has decreased and also they are in avery close agreement with the desired target response values (targetvalues were 0 for all five responses).

7. Conclusions

The more accurate the mathematical model of the mechanismthe better the performance of the control system. One of the mostdifficult part of modelling the mechanical systems is the frictionforce that is acting on the mechanism. For this reason, a modelbased on Karnopp friction model is used to model the bob of apendulum which has low friction moments. Also, response surfacemethodology (RSM) is used for predicting the optimum levels ofparameters (factors) of Karnopp friction model. The resultsdemonstrated in the present study shows that Karnopp frictionmodel is suitable for modelling mechanisms such as bob ofpendulum and RSM is an effective tool for optimizing the para-meters of Karnopp friction model. In near future work, success ofthe Karnopp friction model and the RSM will be evaluated usingdifferent starting angles and rotation directions.

The highlights of this study can be summarized as follows:

� Mathematical model of the bob of a pendulum is built usingKarnopp friction model. Optimum values of the parameters aredetermined using the RSM for the first time in the literature.

� Results indicated that the Karnopp friction model providesaccurate results even for the mechanisms which have lowfriction forces.

� The RSM can be used as an effective modelling and optimiza-tion method for optimizing the Karnopp friction model para-meters with less effort. The RSM provides researchers theoptimum combination of Mw, Mc, Ms, DV, and I parameters byusing the proposed responses of mseA, mseV, ePM, eNM and eZCwith minimum number of experiments, even for the numerouscombinations of experimental results.

By using the experimental design demonstrated in this study,effective second order full quadratic RSM models are obtainedwith fewer experiments. It is expected that the response of variouscombinations of experiments could be predicted accurately with a95% confidence interval with time minimization.

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0 5 10 15 20 25 30 35 40 45-1.5

-1

-0.5

0

0.5

1

1.5Position - Time

Time (s)

Pos

ition

(rad

)

12.95 13 13.05 13.1-0.7

-0.68

-0.66

-0.64

40.5 41 41.5-0.04

-0.02

0

0.02


Fig. 9. Comparison of real data and simulated results for the third design.

Fig. 8. Minitab response optimizer module outputs for the optimization results of3rd design.

Table 9Optimum parameter level for the 3rd design.





http://refhub.elsevier.com/S0947-3580(14)00030-2/sbref1


























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Optimizing Karnopp friction model parameters of a pendulum using RSM