research article dynamic friction parameter...

Research ArticleDynamic Friction Parameter Identification Method withLuGre Model for Direct-Drive Rotary Torque Motor

Xingjian Wang Siru Lin and Shaoping Wang

School of Automation Science and Electrical Engineering Beihang University Beijing 100191 China

Correspondence should be addressed to Xingjian Wang wangxjbuaaeducn

Received 2 December 2015 Revised 22 February 2016 Accepted 6 March 2016

Academic Editor Roque J Saltaren

Copyright copy 2016 Xingjian Wang et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Attainment of high-performance motionvelocity control objectives for the Direct-Drive Rotary (DDR) torque motor should fullyconsider practical nonlinearities in controller design such as dynamic friction The LuGre model has been widely utilized todescribe nonlinear friction behavior however parameter identification for the LuGre model remains a challenge A new dynamicfriction parameter identification method for LuGre model is proposed in this study Static parameters are identified through aseries of constant velocity experiments while dynamic parameters are obtained through a presliding process Novel evolutionaryalgorithm (NEA) is utilized to increase identification accuracy Experimental results gathered from the identification experimentsconducted in the study for a practical DDR torque motor control system validate the effectiveness of the proposed method

1 Introduction

The torque motor especially Direct-Drive Rotary (DDR)torque motor has been widely utilized in modern industryapplications and features rotation blockage soft mechanicalproperties and a wide speed range [1 2] Advantages ofthe motor include high power density torqueweight ratioefficiency rapid response and small torque ripple [3 4]However designing a high-performance positionvelocitytracking controller for the DDR torque motor [5] remains achallenge as multiple factors affecting control precision anddynamic nonlinear friction must be considered [6]

Dynamic friction nonlinearity may be addressed with aproperly designed friction compensation controller Effec-tiveness of the friction compensation controller is largelydependent on the frictionmodel and accurate friction param-eters [7]The current frictionmodel is divided into two typesthe static friction model and the dynamic friction model [8]The static model reflects the relationship between frictionforce and the relative movement speed consisting of severaldistinct parts including static friction Coulomb frictionviscous friction and the Stribeck curve effect Effect of fric-tion when relative velocity between the two contact surfacesis zero cannot be described by the static friction model

The dynamic model features superior practical applicationvalue as it reflects the relationship between the friction forceand both the speed and displacement reflecting friction phe-nomenon more accurately Primary dynamic friction modelsare the Dahl model [9] the LuGre model [10 11] the Leuvenmodel [12 13] the Generalized Maxwell-Slip (GMS) model[14 15] and two-state friction model [16] The Dahl model[9] is derived from the original Coulomb friction model thatincludes the Stribeck effect and the static friction torque [17]Canudas de Wit et al combined the Dahl model with thebristles model to propose a new approach referred to as theLuGre frictionmodel [10 11 18]The LuGremodel accuratelydescribes the complex process of static and dynamic proper-ties in friction including presliding displacement memorialfriction variable static friction viscous friction and Stribeckeffect The LuGre model describes the asperities betweentwo contact surfaces by elastic bristles and true nature ofdynamic friction is considered as the result of the averagedeflection of these bristles In advance of the LuGre modelthe Leuvenmodel [12 13] and GMSmodel authors of [14 1519] modeled the hysteresis behavior of the dynamic frictionwithin presliding regime Two-state dynamic friction model[16] was also proved to be capable of capturing the hysteresisbehavior However the modeling of the hysteresis behavior

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 6929457 8 pageshttpdxdoiorg10115520166929457

2 Mathematical Problems in Engineering

complicates the friction models significantly and increasesthe difficulties of implementation in the real-time controls[20] In addition the hysteresis behavior is not the essentialphenomenon in normal positionvelocity control of DDRtorque motorTherefore the LuGre model is still widely usedin motor control systems for dynamic friction compensation

LuGre model operates based on a group of complexnonlinear functions with six parameters that is four staticparameters and two dynamic parameters difficult to identifyas coupling exists among the six parameters [21ndash23] Diffi-culty also exists for LuGre parameter identification as theinternal state of the model is immeasurable and depends onthe previously mentioned unknown friction parameters Tra-ditional parameter identification methodology is challengedin deriving accurate values of these six parameters in theLuGre model

A new dynamic friction parameter identification methodis proposed in this study for the LuGre model and anidentification experiment is conducted for a practical DDRtorque motor control system Static parameters are obtainedthrough a series of constant velocity experiments whiledynamic parameters are obtained by presliding processNovel evolutionary algorithm [24] (NEA) is utilized toincrease identification accuracy and optimization speed byemploying Time Variant Mutation (TVM) operator Theproposed method is applied to the practical DDR torquemotor control system and the LuGre parameters are obtainedby using the proposed identification method Experimentalresults validate the effectiveness of the proposed method

2 Experimental Setup andMathematical Modeling

21 Experimental Setup and Modeling of DDR Torque MotorThe experimental setup of DDR torquemotormotion controlsystem for dynamic friction parameter identification is firstpresented (Figure 1)

The torque motor studied is a current-controlled DDRtorque motor D143M by Danaher driven by a commercialdigital servo amplifier S620 by Danaher A Heidenhainhigh-resolution rotary encoder ECN113 with Heidenhain PCcounter card IK220 is installed to measure the motor rotarydisplacement Rotary velocity can then be calculated by thederivative of rotary displacement signal Rotary displacementand velocity signals may be utilized here to identify dynamicfriction parameters and online estimate friction torque Orig-inal designed real-time control software based on RTX real-time operating system and LabWindowsCVI is applied tocontrol and monitor the torque motor system with samplingfrequency selected as 119891

119904= 2 kHz

Frequency response bandwidth of the motor amplifieris typically higher than 1000Hz however the mechanicaldynamics of theDDR torquemotor system generally does notexceed 100Hz Disregarding the electrical dynamics of theamplifier in normal operating conditions of the DDR torquemotor is then reasonable Input saturation may then alsobe ignored when the torque motor operates under normalconditions The relationship of the electromagnetic torque

Rotary encoderECN113

DDR torque motorD143M

Figure 1 Experimental setup of DDR torque motor control system

119879em and input control voltage 119906 to the motor amplifierwith the above simplifications may be represented by thefollowing equation [25]

119879em = 119870119898119906 (1)

where 119870119898is a proportional coefficient from input control

voltage 119906 to electromagnetic torque 119879emConsidering friction torque and external disturbances

the dynamics of torque motor may be described as

119879em = 119870119898119906 = 119869119898119898

+ 119861119898120596119898

+ 119879119889(119905) + 119879

119891 (2)

where 119869119898is total inertia of motor rotator and output shaft

119861119898is damping coefficient 120596

119898is rotational velocity of torque

motor 119879119889(119905) represents lumped effect of external distur-

bances 119879119891represents the combination of dynamic friction

effects and will be formulated laterMotor parameter identification experiments are then

performed to attain nominal values of system parametersIdentification results are119870

119898= 377NmV 119869

119898= 0045 kgsdotm2

and 119861119898

= 216Nm(rads)

22 Dynamic Friction Description with LuGre Model Thedynamic friction 119879

119891will be formulated in this subsection

by utilizing the LuGre friction model a nonlinear dynamicfriction model widely utilized in mechanical and servosystems Elastic bristles are employed by the LuGre model toderive asperities between the two contact surfaces on amicro-scopic scale with the dynamic effects of friction resultingfrom the average deflection of these bristles (Figure 2) TheLuGremodelmore accurately describes sliding displacementmemorial friction variable static friction viscous frictionand Stribeck curve effects synchronously

The mathematical formulation of LuGre model is asfollows [10]

119879119891= 1205900119911 + 1205901 + 1205902120596119898

(3)

= 120596119898

minus

10038161003816100381610038161205961198981003816100381610038161003816

119892 (120596119898)119911 (4)

1205900119892 (120596119898) = 119891119888+ (119891119904minus 119891119888) 119890minus(120596119898V119904)

2

(5)

where 119911 is the internal state of LuGre friction modelrepresenting the average deflection of bristles between two

Mathematical Problems in Engineering 3

Bristles

Contact surface

Contact surface

Figure 2 Deflection of bristles between two contact surfaces

120596m z = 120596m minus|120596m| z

zTf = 1205900z + 1205901z + 1205902120596m

1

sg(120596m)

1205900g(120596m) = fc + (fs minus fc)eminus(120596119898119904)

2

Figure 3 The internal relationship of LuGre model

contact surfaces 119891119888 119891119904 1205902 and V

119904are four static parame-

ters representing Coulomb friction stiction friction viscouscoefficient and Stribeck velocity respectively 120590

0and 120590

1are

two dynamic parameters and 1205900is the average stiffness coef-

ficient of bristles while 1205901is the average damping coefficient

of bristles respectively 1205900119892(120596119898) describes Stribeck effect

According to the mathematical expressions of LuGre model(3)ndash(5) the internal relationship of LuGre model may be aspresented in Figure 3

The average deflection of bristles will obviously reach arelatively stable state (meaning = 0) when relative velocityexceeds a certain value The steady-state bristle averagedeformation 119911ss may be described as

119911ss = 119892 (120596119898) sgn (120596

119898) (6)

where sgn(120596119898) is the sign function and may be expressed as

sgn (120596119898) =

1 120596119898

gt 0

0 120596119898

= 0

minus1 120596119898

lt 0

(7)

Thus in the steady state friction torque may be derivedby the following formula

119879119891119904

= [119891119888+ (119891119904minus 119891119888) 119890minus(120596119898V119904)

2

] sgn (120596119898) + 1205902120596119898 (8)

The LuGremodel as detailed by the LuGremodel expres-sion includes four static parameters and twodynamic param-eters However identification of these parameters presentsa challenge with increased difficulty for identification ofdynamic parameters an issue solved in the following section

Individual evaluation

Crossover operation

Mutation


Alternate generation

Best solution

Initial population

Final condition

Parent

Offspring

Yes

No

Figure 4 New evolution algorithm flow chart

3 NEA Based ParameterIdentification Technique

The NEA based parameter identification technique will bepresented in this section to solve the friction parameteridentification issue for the LuGremodel in the practical DDRtorque motor control system

31 Novel Evolutionary Algorithm Evolutionary algorithmis a widely utilized optimization method [18 26] NEA isthe development of evolutionary algorithm by utilizing thedynamic TVM operator to improve algorithm speed andprecision [24] TVM operator may produce rapid changesat initial stages of evolution and precise changes at finalstages therefore it is believed that TVM could improve theefficiency of NEA and guarantee rapid convergence

The basic flow chart of NEA is presented in Figure 4 andthe specific steps are described as follows

(1) Initial Population According to empirical statistics aninitial population 119875(0) can be selected with 119899 individuals fortarget variable set Ω These 119899 individuals are generated by arandom function within the desired domain of Ω Followingevaluation of these 119899 individuals with an objective functionthe initial population 119875(0) is divided into 119904 subpopulationswith each subpopulation featuring 119899119904 individualsThe initialpopulation 119875(0) is named as parent population for the nextgeneration

(2) Individual Evaluation For target variable set Ω the indi-vidual which makes objective function minimum is selectedas the first individual of the 119894th subpopulation and is named aselite individual Θ119905

119894mom (mother) at 119905th generation expressedas

Θ119905119894mom = [119891119905

119888(119894max) 119891119905

119904(119894max) 120590119905

2(119894max) V119905

119904(119894max)] (9)


The average of other individuals then in the 119894th subpopulationexcludingΘ119905

119894mom is defined asmean individualΘ119905119894dad (father)

expressed as

Θ119905119894dad = [119891119905

119888(119894mean) 119891119905

119904(119894mean) 120590119905

2(119894mean) V119905

119904(119894mean)] (10)

(3) Crossover Operation Two offspring 1205751199051198941

and 1205751199051198942

aregenerated in this step by crossover operation for each sub-population as

1205751199051198941

= 1205721Θ119905119894mom + (1 minus 120572

1)Θ119905119894dad = [120572

1119891119905119888(119894max)

+ (1 minus 1205721) 119891119905119888(119894mean) 1205722119891

119905

119904(119894max)

+ (1 minus 1205722) 119891119905119904(119894mean) 1205723120590

119905

2(119894max)

+ (1 minus 1205723) 1205902(119894mean) 1205724V

119905

119904(119894max) + (1 minus 1205724) V119905119904(119894mean)]

1205751199051198942

= (1 minus 1205721)Θ119905119894mom + 120572

1Θ119905119894dad = [(1 minus 120572

1) 119891119905119888(119894max)

+ 1205721119891119905119888(119894mean) (1 minus 120572

2) 119891119905119904(119894max)

+ 1205722119891119905119904(119894mean) (1 minus 120572

3) 1205901199052(119894max)

+ 12057231205902(119894mean) (1 minus 120572

4) V119905119904(119894max) + 120572

4V119905119904(119894mean)]

(11)

where 120572119896 119896 = 1 2 3 4 is a random coefficient in [0 1] and

will be recreated for each individualFor each subpopulation the elite individual (mother) and

the mean individual (father) are chosen and combined Thisoperation ensures that genes of the elite individual exhibitsuperior opportunity for the next generation

(4) Mutation Mutation is a genetic operator which maintainsthe genetic diversity and acts as a random variation in acertain range for each subpopulation

120575119905lowast

1198941 120575119905lowast

1198942= [120575119905

lowast

119891119888

120575119905lowast

119891119904

120575119905lowast

1205902

120575119905lowast

V119904

] = [120575119905119891119888

+ 1205731120582 (119905) 120575

119905

119891119904

+ 1205732120582 (119905) 120575

119905

1205902

+ 1205733120582 (119905) 120575

119905

V119904

+ 1205734120582 (119905)]

(12)

where 120575119905lowast

1198941 120575119905lowast

1198942are two new offspring generated by the TVM

operator and120573119896 119896 = 1 2 3 4 is a random coefficient in (0 1)

and regenerated for each TVM operation 120582(119905) is a generatingfunction in TVM operation given as

120582 (119905) = [1 minus 119903(1minus119905119879)120574

] (13)

where 119903 is a random coefficient in [0 1] 119879 is the maximumnumber of evolutionary generations and 120574 is a real-valuedparameter to determine the degree of dependency

(5) Individual Assessment Each offspring is evaluated by theobjective function after mutation operation

(6) Alternate Generation The 119899119905minus1th parental generation andthe 119899119905th subgeneration at this stage are integrated togetherand reordered according to objective function with best 119899individuals selected as the next generation parent

(7) Check for Final Condition Evolution is ended when thefinal condition is met and the optimal parameter estimatesare found if not evolution continues

32 Identification of Static Parameters in LuGre Model Thestatic parameters of LuGre model are obtained by identifyingthe Stribeck curve of theDDR torquemotorThe digital servoamplifier S620 is set to current closed-loop mode with PIcontroller in this experiment A speed closed-loop with PIcontroller is also developed to controlmotion speed of torquemotor in the real-time control software to ensure normalmovement of the torque motor without experiencing stick-slip phenomenon The velocity-friction torque curve may beobtained that is Stribeck curve through a series of constantvelocity experiments then the static parameters of LuGremodel may be identified by fitting the Stribeck curve

Considering the constant speed motion control of torquemotor and disregarding the disturbance 119879

119889(119905) the friction

in torque motor under constant velocity motion steady statemay be described as

119879119891asymp 119870119898119906 (14)

Then according to (14) the friction torque 119879119891

may becalculated during a series of constant speed motions

A series of constant velocity motion experiments areconducted and the speed values are recorded as 120596

119898119873119894=1

thenthe corresponding friction torques 119879

119891119873119894=1

may be calculatedaccording to (14)The number of constant speed experimentsin positive rotation direction was chosen in this study tobe 119873 = 50 with 25 experiments in the low-speed rangeIdentification accuracy of the Stribeck effect was improvedby conducting these 25 experiments in low-speed rangeThe same operations are conducted for negative rotationdirection

By fitting above 100 groups of experimental data therelationship curve between velocity and friction torque canbe obtained that is Stribeck curve

Define the desired static parameter set as

Ω119904= [119891119888 119891119904 1205902 V119904] (15)

Then define the fitting error to be

119890119904(Ω119904 120596119898) = 119879119891(119894) minus 119879

119891119904(Ω119904 120596119898 119894) (16)

where 119879119891(119894) is the measured friction torque (14) at the

119894th constant speed motion experiment 119879119891119904(Ω119904 120596119898 119894) is the

calculated friction torque of steady-state LuGre model (8) byusing the parameter set Ω

119904 that is the friction torque of the

Stribeck curve determined by Ω119904at the 119894th speed point

Define an objective function of static parameter identifi-cation as

119869119904=

1

2sum[119890119904(Ω119904 120596119898)]2

(17)

By applyingNEAoptimization andminimizing the objec-tive function 119869

119904 the Stribeck curve may be identified as

demonstrated in Figure 5 with the best estimate of four staticparameters which characterize the Stribeck effect presentedin Table 1


Table 1 Static parameters identification results

Parameter Unit Value119891119888

Nm 6975119891119904

Nm 85581205902

Nm(rads) 1819V119904

rads 6109 times 10minus2

0

90

70

Fric

tion

torq

ue (N

m) 80

Experimentalresults

Identifiedresults

minus70

minus80

minus90minus02minus06 minus04minus08 02 0604 080Velocity (rads)

Figure 5 Identified Stribeck curve for static parameter identifica-tion

33 Identification of Dynamic Parameters in LuGre ModelThe process from a static state to a distinct movementbetween the two contact surfaces is referred to as the pres-liding process LuGre dynamic friction model also explainsthe presliding process No obvious movement exists betweenthe two contact surfaces in the presliding process thoughlack of obvious movement is not indicative of displacementabsence as even miniscule displacement between the twocontact surfaces may be considered as bristles deflectionFriction in the presliding process is mainly composed of twoparts that is 120590

0119911 and 120590

1 according to (3) The presliding

process in DDR torquemotormay be applied then to identifythe two dynamic parameters in LuGre model that is 120590

0and

1205901The effect of dynamic parameters 120590

0and 120590

1is strength-

ened in the presliding process in this experiment as the testedtorquemotor is controlled by themotor driverrsquos inner currentloop only Input voltage signal 119906(119905) to the motor driver is setto be a slowly varying ramp signal

119906 (119905) = 119896119888119905 (18)

where 119896119888

gt 0 is a miniscule gradient coefficient and 119905is time So under such an input voltage signal the motordriver will generate a slowly varying electromagnetic torque119879em in the motor The motor is in the presliding motionstate before 119879em exceeds the starting torque of torque motorExperimental data of input voltage signal and output rotarydisplacement in this presliding process experiment may beutilized for the identification of two dynamic parameters 120590

0

and 1205901 Four identified static parameters of LuGre model are

also employed for the identification of dynamic parameters

Dynamic parameters 1205900and 120590

1are also optimized uti-

lizing the NEA method Define a desired set of dynamicparameters as

Ω119889= [1205900 1205901]119879

(19)

Then define the identification error

119890119889(Ω119889 119905119894) = 120579119898(119905119894) minus 120579119889(Ω119889 119905119894) (20)

where 120579119898(119905119894) is an angle output of the torque motor at the

119905119894moment 120579

119889(Ω119889 119905119894) is an angle output of the parameter

identification at the 119905119894moment

Application of NEA to identify the static parametersallows straightforward estimation of the four initial param-eters from the Stribeck curve shape Estimating the initialvalue of dynamic parameters 120590

0and 120590

1 however from the

presliding process curve of torque motor in the identificationprocess of dynamic parameters is difficult Amore reasonableinitial estimate for dynamic parameters is then necessary tooptimize dynamic parameters by NEA Realistically in thepresliding process of the torque motor the input voltagesignal is less than the starting torque of torque motor andchanges only gradually thus it is reasonable to disregard theacceleration of motor rotor and the change rate of LuGremodel internal state System dynamics and LuGremodel maybe transformed into

119896119898119906 asymp 119879119891asymp 1205900119911 (21)

1205900119892 (120596119898) = 119891119904

(22)

= 120596119898

minus

10038161003816100381610038161205961198981003816100381610038161003816

119892 (120596119898)119911 (23)

Suppose this experiment is starting from zero position(ie 119911(0) = 0) so (23) can be rewritten as

=119889119911

119889119905= 120596119898

minus119879119891

119891119904

10038161003816100381610038161205961198981003816100381610038161003816 (24)

where rotary velocity120596119898can be calculated by differencing the

rotary displacement signal of the encoder and friction torquemay be calculated by (21) A slowly varying ramp signal 119906(119905) =119896119888119905 is input to the torque motor system until the motor starts

When 120596119898

gt 0 the average deflection of the bristles betweenthe contact surfaces may be obtained by integrating (24) as

119911 (119905) = 120579119898(119905) minus 120579

119898(0)

+119896119898119896119888

2119891119904

(120579119898(119905) 119905 + int

119905

0

120579119898(120591) 119889120591)

(25)

A data group of calculated 119911(119905) can be computed forspecified time interval (0 119879) and the vectors composed of119911(119905) are marked as Z while the vectors composed of 119906(119905) aremarked asUThen the initial value of the dynamic parameters1205900 by averaging this data group may be calculated by

12059000=Z119879UZ119879Z

(26)


The initial value of 1205900as 12059000

= 2000Nmrad is obtainedby applying the above method to the practical torque motorsystem

The presliding process of torque motor is selected to esti-mate the reasonable initial value of other dynamic parameters1205901 No obvious movement of motor rotor existed in this

case however there exists a slight deflection of the bristlesbetween two contact surfaces with the average deflection ofthe bristles equal to the rotary displacement of motor rotorthus 120579

119898asymp 119911 and 120596

119898asymp where 120579

119898could be measured by

the high-resolution rotary encoder and120596119898could be obtained

by the differential of 120579119898 When the torque motor is close to

zero position system dynamics and LuGre model may beexpressed as

119898119898

+ (1205901+ 1205902) 119898

+ 1205900120579119898

= 119906 (119905) (27)

Then the transfer function of (27) is

120579119898(119904)

119906 (119904)=

1

1198981199042 + (1205901+ 1205902) 119904 + 120590

0

(28)

where 119904 is a Laplace variable As reported in [21] it is suitableto describe the combination parameter (120590

1+ 1205902) by applying

the concept of damping coefficient The initial value 12059001of

dynamic parameter 1205901 as a result may be obtained from (28)

as

12059001= 2120585119898radic

1205900

119898minus 1205902 (29)

where 120585 is optimal damping ratioApplying the above method to the practical torque motor

system the initial value of 1205901parameter is obtained as 1205900

1=

40NmsradThe objective function of the dynamic parameter identi-

fication is defined as

119869119889= 1199021

119873

sum119894=1

[119890119889(Ω119889 119905119894)]2

+ 1199022max 119890

119889(Ω119889 119905119894) (30)

where 1199021and 119902

2are two weight coefficients and the iden-

tification goal is to minimize 119869119889 Using the initial values

of dynamic parameters (1205900012059001) and static parameters Ω

119904

previously obtained NEA method is also applied to derivethe optimal values of dynamic parameters 120590

0and 120590

1by

minimizing 119869119889

Experimental and identification results of the preslidingprocess are revealed in Figure 6 According to these resultsandNEA optimization the dynamic parameter identificationresults are 120590

0= 2750Nmrad and 120590

1= 452Nmsrad

4 Online Friction Estimation Experiments

In order to further illustrate the proposed parameter identi-fication method two online friction estimation experimentsare carried out in which the LuGre model with the identifiedparameters is used The DDR torque motor is working in theangle control model with a position servo controller

1

2

Inpu

tRo

tary

0

3

Experimentalresults

Identificationresults

Input voltage

500 30 4010 20Time (s)

volta

ge (V

)ve

loci

tylowast100

(rad

s)

Figure 6 Experimental and identification results of presliding proc-ess

0 05 15 2010Time (s)

05 10 15 200Time (s)

minus10

minus5

0

5

10

Fric

tion

torq

ue (N

m)

minus40

minus20

0

20

40

z (r

ad)

times10minus3

Figure 7 Online friction estimation along with internal state in thefirst experiment

Two experiments are executed In the first one the desiredoutput angle for the DDR torque motor is given by a 10Hzsinusoidal signal with 004 rad amplitudeThe online frictionestimation results are shown in Figure 7 which is calculatedby using the LuGre model with the identified parameters

In the second experiment the desired output angle for theDDR torque motor is given by a 05Hz sinusoidal signal with015 rad amplitude The online friction estimation results areshown in Figure 8

These two friction estimation experiments illustrate thatthe LuGre model is effective to estimate the dynamic frictionin the practical system and this online estimation could befurther utilized for the purpose of friction compensationExperimental results also validate the effectiveness of theproposed parameter identification method for the LuGremodel

5 Conclusion

A new dynamic friction parameter identification method isproposed in this study for LuGre model NEA technique isutilized to increase identification accuracy and an identifi-cation experiment is conducted for a practical DDR torque


times10minus3

minus15

minus10

minus5

05

1015

Fric

tion

torq

ue (N

m)

minus40

minus20

0

20

40

z (r

ad)

20 40 60 80 1000Time (s)

0 20 40Time (s)

60 80 100

Figure 8 Online friction estimation along with internal state in thesecond experiment

motor control system The four static parameters of LuGremodel are identified through a series of constant velocityexperiments while two dynamic parameters of LuGre modelare obtained through the presliding process Experimentalresults validate the effectiveness of the proposed method

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (Grant no 51305011) the NationalBasic Research Program of China (973 Program) (Grant no2014CB046402) and Program 111 of China

References

[1] M Rashed P F A MacConnell A F Stronach and P AcarnleyldquoSensorless indirect-rotor-field-orientation speed control of apermanent-magnet synchronous motor with stator-resistanceestimationrdquo IEEE Transactions on Industrial Electronics vol 54no 3 pp 1664ndash1675 2007

[2] J Peng S Li and Y Fan ldquoModeling and parameter identifica-tion of the vibration characteristics of armature assembly in atorque motor of hydraulic servo valves under electromagneticexcitationsrdquo Advances in Mechanical Engineering vol 6 ArticleID 247384 2014

[3] X Wang S Wang and P Zhao ldquoAdaptive fuzzy torque controlof passive torque servo systems based on small gain theoremand input-to-state stabilityrdquo Chinese Journal of Aeronautics vol25 no 6 pp 906ndash916 2012

[4] J Yao Z Jiao andDMa ldquoRISE-based precisionmotion controlof DC motors with continuous friction compensationrdquo IEEETransactions on Industrial Electronics vol 61 no 12 pp 7067ndash7075 2014

[5] Z Li D Fan and S Fan ldquoLuGre-model-based friction com-pensation in direct-drive inertially stabilization platformsrdquo inProceedings of the 6th IFAC Symposium onMechatronic Systemspp 636ndash642 Hangzhou China April 2013

[6] M Ruderman ldquoTracking control of motor drives using feed-forward friction observerrdquo IEEE Transactions on IndustrialElectronics vol 61 no 7 pp 3727ndash3735 2014

[7] L Alvarez J Yi R Horowitz and L Olmos ldquoDynamic fric-tion model-based tire-road friction estimation and emergencybraking controlrdquo Journal of Dynamic SystemsMeasurement andControl vol 127 no 1 pp 22ndash32 2005

[8] Y Sang H Gao and F Xiang ldquoPractical friction modelsand friction compensation in high-precision electro-hydraulicservo force control systemsrdquo Instrumentation Science and Tech-nology vol 42 no 2 pp 184ndash199 2014

[9] P R Dahl ldquoMeasurement of solid friction parameters of ballbearingsrdquo in Proceedings of the 6th Annual Symposium onIncremental Motion Control Systems and Devices pp 49ndash60University of Illinois May 1977

[10] C Canudas de Wit H Olsson K J Astrom and P LischinskyldquoA new model for control of systems with frictionrdquo IEEETransactions on Automatic Control vol 40 no 3 pp 419ndash4251995

[11] C Canudas de Wit and P Lischinsky ldquoAdaptive frictioncompensation with partially known dynamic friction modelrdquoInternational Journal of Adaptive Control and Signal Processingvol 11 no 1 pp 65ndash80 1997

[12] J Swevers F Al-Bender C G Ganseman and T Prajogo ldquoAnintegrated friction model structure with improved preslidingbehavior for accurate friction compensationrdquo IEEE Transac-tions on Automatic Control vol 45 no 4 pp 675ndash686 2000

[13] V Lampaert J Swevers and F Al-Bender ldquoModification ofthe Leuven integrated friction model structurerdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 47 no 4 pp 683ndash687 2002

[14] V Lampaert F Al-Bender and J Swevers ldquoA generalizedMaxwell-slip friction model appropriate for control purposesrdquoin Proceedings of the 1st International Conference Physics andControl (PhysCon rsquo03) vol 4 pp 1170ndash1177 IEEE Saint Peters-burg Russia August 2003

[15] F Al-Bender V Lampaert and J Swevers ldquoThe generalizedMaxwell-slip model a novel model for friction simulation andcompensationrdquo IEEE Transactions on Automatic Control vol50 no 11 pp 1883ndash1887 2005

[16] M Ruderman and T Bertram ldquoTwo-state dynamic frictionmodel with elasto-plasticityrdquo Mechanical Systems and SignalProcessing vol 39 no 1-2 pp 316ndash332 2013

[17] J Y Yoon andD L Trumper ldquoFrictionmodeling identificationand compensation based on friction hysteresis and Dahl reso-nancerdquoMechatronics vol 24 no 6 pp 734ndash741 2014

[18] X Wang and S Wang ldquoHigh performance adaptive con-trol of mechanical servo system with LuGre friction modelidentification and compensationrdquo Journal of Dynamic SystemsMeasurement and Control vol 134 no 1 Article ID 011021 2012

[19] M Boegli T De Laet J De Schutter and J Swevers ldquoAsmoothed GMS friction model suited for gradient-based fric-tion state and parameter estimationrdquo IEEEASME Transactionson Mechatronics vol 19 no 5 pp 1593ndash1602 2014

[20] L Lu B Yao Q Wang and Z Chen ldquoAdaptive robust controlof linear motors with dynamic friction compensation usingmodified LuGre modelrdquo Automatica vol 45 no 12 pp 2890ndash2896 2009


[21] R H A Hensen M J G van de Molengraft andM SteinbuchldquoFrequency domain identification of dynamic friction modelparametersrdquo IEEE Transactions on Control Systems Technologyvol 10 no 2 pp 191ndash196 2002

[22] D D Rizos and S D Fassois ldquoFriction identification basedupon the LuGre and Maxwell slip modelsrdquo IEEE Transactionson Control Systems Technology vol 17 no 1 pp 153ndash160 2009

[23] R Zglimbea V Finca E Greaban and M Constantin ldquoIden-tification of systems with friction via distributions using themodified friction LuGre modelrdquo in Proceedings of the 13thWseas International Conference on Systems-Recent Advancesin Systems pp 579ndash584 World Scientific and EngineeringAcademy and Society Athens Greece July 2009

[24] K Watanabe and M Hashem Evolutionary ComputationsNew Algorithms and Their Applications to Evolutionary RobotsSpringer 2012

[25] X Wang S Wang and B Yao ldquoAdaptive robust torque con-trol of electric load simulator with strong position couplingdisturbancerdquo International Journal of Control Automation andSystems vol 11 no 2 pp 325ndash332 2013

[26] M B Nazir and W Shaoping ldquoOptimization based on con-vergence velocity and reliability for hydraulic servo systemrdquoChinese Journal of Aeronautics vol 22 no 4 pp 407ndash412 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


complicates the friction models significantly and increasesthe difficulties of implementation in the real-time controls[20] In addition the hysteresis behavior is not the essentialphenomenon in normal positionvelocity control of DDRtorque motorTherefore the LuGre model is still widely usedin motor control systems for dynamic friction compensation

LuGre model operates based on a group of complexnonlinear functions with six parameters that is four staticparameters and two dynamic parameters difficult to identifyas coupling exists among the six parameters [21ndash23] Diffi-culty also exists for LuGre parameter identification as theinternal state of the model is immeasurable and depends onthe previously mentioned unknown friction parameters Tra-ditional parameter identification methodology is challengedin deriving accurate values of these six parameters in theLuGre model

A new dynamic friction parameter identification methodis proposed in this study for the LuGre model and anidentification experiment is conducted for a practical DDRtorque motor control system Static parameters are obtainedthrough a series of constant velocity experiments whiledynamic parameters are obtained by presliding processNovel evolutionary algorithm [24] (NEA) is utilized toincrease identification accuracy and optimization speed byemploying Time Variant Mutation (TVM) operator Theproposed method is applied to the practical DDR torquemotor control system and the LuGre parameters are obtainedby using the proposed identification method Experimentalresults validate the effectiveness of the proposed method

2 Experimental Setup andMathematical Modeling

21 Experimental Setup and Modeling of DDR Torque MotorThe experimental setup of DDR torquemotormotion controlsystem for dynamic friction parameter identification is firstpresented (Figure 1)

The torque motor studied is a current-controlled DDRtorque motor D143M by Danaher driven by a commercialdigital servo amplifier S620 by Danaher A Heidenhainhigh-resolution rotary encoder ECN113 with Heidenhain PCcounter card IK220 is installed to measure the motor rotarydisplacement Rotary velocity can then be calculated by thederivative of rotary displacement signal Rotary displacementand velocity signals may be utilized here to identify dynamicfriction parameters and online estimate friction torque Orig-inal designed real-time control software based on RTX real-time operating system and LabWindowsCVI is applied tocontrol and monitor the torque motor system with samplingfrequency selected as 119891

119904= 2 kHz

Frequency response bandwidth of the motor amplifieris typically higher than 1000Hz however the mechanicaldynamics of theDDR torquemotor system generally does notexceed 100Hz Disregarding the electrical dynamics of theamplifier in normal operating conditions of the DDR torquemotor is then reasonable Input saturation may then alsobe ignored when the torque motor operates under normalconditions The relationship of the electromagnetic torque

Rotary encoderECN113

DDR torque motorD143M

Figure 1 Experimental setup of DDR torque motor control system

119879em and input control voltage 119906 to the motor amplifierwith the above simplifications may be represented by thefollowing equation [25]

119879em = 119870119898119906 (1)

where 119870119898is a proportional coefficient from input control

voltage 119906 to electromagnetic torque 119879emConsidering friction torque and external disturbances

the dynamics of torque motor may be described as

119879em = 119870119898119906 = 119869119898119898

+ 119861119898120596119898

+ 119879119889(119905) + 119879

119891 (2)

where 119869119898is total inertia of motor rotator and output shaft

119861119898is damping coefficient 120596

119898is rotational velocity of torque

motor 119879119889(119905) represents lumped effect of external distur-

bances 119879119891represents the combination of dynamic friction

effects and will be formulated laterMotor parameter identification experiments are then

performed to attain nominal values of system parametersIdentification results are119870

119898= 377NmV 119869

119898= 0045 kgsdotm2

and 119861119898

= 216Nm(rads)

22 Dynamic Friction Description with LuGre Model Thedynamic friction 119879

119891will be formulated in this subsection

by utilizing the LuGre friction model a nonlinear dynamicfriction model widely utilized in mechanical and servosystems Elastic bristles are employed by the LuGre model toderive asperities between the two contact surfaces on amicro-scopic scale with the dynamic effects of friction resultingfrom the average deflection of these bristles (Figure 2) TheLuGremodelmore accurately describes sliding displacementmemorial friction variable static friction viscous frictionand Stribeck curve effects synchronously

The mathematical formulation of LuGre model is asfollows [10]

119879119891= 1205900119911 + 1205901 + 1205902120596119898

(3)

= 120596119898

minus

10038161003816100381610038161205961198981003816100381610038161003816

119892 (120596119898)119911 (4)

1205900119892 (120596119898) = 119891119888+ (119891119904minus 119891119888) 119890minus(120596119898V119904)

2

(5)

where 119911 is the internal state of LuGre friction modelrepresenting the average deflection of bristles between two


Bristles

Contact surface

Contact surface


120596m z = 120596m minus|120596m| z

zTf = 1205900z + 1205901z + 1205902120596m

1

sg(120596m)


2





0and 120590

1are






119911ss = 119892 (120596119898) sgn (120596

119898) (6)


sgn (120596119898) =

1 120596119898

gt 0

0 120596119898

= 0

minus1 120596119898

lt 0

(7)


119879119891119904

= [119891119888+ (119891119904minus 119891119888) 119890minus(120596119898V119904)

2

] sgn (120596119898) + 1205902120596119898 (8)



Crossover operation

Mutation



Best solution

Initial population

Final condition

Parent

Offspring

Yes

No









Θ119905119894mom = [119891119905

119888(119894max) 119891119905

119904(119894max) 120590119905

2(119894max) V119905

119904(119894max)] (9)




expressed as

Θ119905119894dad = [119891119905

119888(119894mean) 119891119905

119904(119894mean) 120590119905

2(119894mean) V119905

119904(119894mean)] (10)


and 1205751199051198942


1205751199051198941

= 1205721Θ119905119894mom + (1 minus 120572

1)Θ119905119894dad = [120572

1119891119905119888(119894max)

+ (1 minus 1205721) 119891119905119888(119894mean) 1205722119891

119905

119904(119894max)

+ (1 minus 1205722) 119891119905119904(119894mean) 1205723120590

119905

2(119894max)

+ (1 minus 1205723) 1205902(119894mean) 1205724V

119905

119904(119894max) + (1 minus 1205724) V119905119904(119894mean)]

1205751199051198942

= (1 minus 1205721)Θ119905119894mom + 120572

1Θ119905119894dad = [(1 minus 120572

1) 119891119905119888(119894max)

+ 1205721119891119905119888(119894mean) (1 minus 120572

2) 119891119905119904(119894max)

+ 1205722119891119905119904(119894mean) (1 minus 120572

3) 1205901199052(119894max)

+ 12057231205902(119894mean) (1 minus 120572

4) V119905119904(119894max) + 120572

4V119905119904(119894mean)]

(11)





120575119905lowast

1198941 120575119905lowast

1198942= [120575119905

lowast

119891119888

120575119905lowast

119891119904

120575119905lowast

1205902

120575119905lowast

V119904

] = [120575119905119891119888

+ 1205731120582 (119905) 120575

119905

119891119904

+ 1205732120582 (119905) 120575

119905

1205902

+ 1205733120582 (119905) 120575

119905

V119904

+ 1205734120582 (119905)]

(12)


1198941 120575119905lowast




120582 (119905) = [1 minus 119903(1minus119905119879)120574

] (13)









119879119891asymp 119870119898119906 (14)




119898119873119894=1


119891119873119894=1




Ω119904= [119891119888 119891119904 1205902 V119904] (15)


119890119904(Ω119904 120596119898) = 119879119891(119894) minus 119879

119891119904(Ω119904 120596119898 119894) (16)







119869119904=

1

2sum[119890119904(Ω119904 120596119898)]2

(17)







Nm 6975119891119904

Nm 85581205902

Nm(rads) 1819V119904


0

90

70

Fric

tion

torq

ue (N

m) 80

Experimentalresults

Identifiedresults

minus70

minus80




0119911 and 120590



0and


0and 120590

1is strength-


119906 (119905) = 119896119888119905 (18)

where 119896119888


0






Ω119889= [1205900 1205901]119879

(19)


119890119889(Ω119889 119905119894) = 120579119898(119905119894) minus 120579119889(Ω119889 119905119894) (20)


119905119894moment 120579




0and 120590

1 however from the


119896119898119906 asymp 119879119891asymp 1205900119911 (21)

1205900119892 (120596119898) = 119891119904

(22)

= 120596119898

minus

10038161003816100381610038161205961198981003816100381610038161003816

119892 (120596119898)119911 (23)


=119889119911

119889119905= 120596119898

minus119879119891

119891119904

10038161003816100381610038161205961198981003816100381610038161003816 (24)



When 120596119898


119911 (119905) = 120579119898(119905) minus 120579

119898(0)

+119896119898119896119888

2119891119904

(120579119898(119905) 119905 + int

119905

0

120579119898(120591) 119889120591)

(25)


12059000=Z119879UZ119879Z

(26)






119898asymp 119911 and 120596






119898119898

+ (1205901+ 1205902) 119898

+ 1205900120579119898

= 119906 (119905) (27)


120579119898(119904)

119906 (119904)=

1

1198981199042 + (1205901+ 1205902) 119904 + 120590

0

(28)





as

12059001= 2120585119898radic

1205900

119898minus 1205902 (29)



1=



119869119889= 1199021

119873

sum119894=1

[119890119889(Ω119889 119905119894)]2

+ 1199022max 119890

119889(Ω119889 119905119894) (30)

where 1199021and 119902




119904


0and 120590

1by



0= 2750Nmrad and 120590

1= 452Nmsrad



1

2

Inpu

tRo

tary

0

3

Experimentalresults


Input voltage

500 30 4010 20Time (s)

volta

ge (V

)ve

loci

tylowast100

(rad

s)


0 05 15 2010Time (s)

05 10 15 200Time (s)

minus10

minus5

0

5

10

Fric

tion

torq

ue (N

m)

minus40

minus20

0

20

40

z (r

ad)

times10minus3





5 Conclusion



times10minus3

minus15

minus10

minus5

05

1015

Fric

tion

torq

ue (N

m)

minus40

minus20

0

20

40

z (r

ad)

20 40 60 80 1000Time (s)

0 20 40Time (s)

60 80 100



Competing Interests


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Bristles

Contact surface

Contact surface


120596m z = 120596m minus|120596m| z

zTf = 1205900z + 1205901z + 1205902120596m

1

sg(120596m)


2





0and 120590

1are






119911ss = 119892 (120596119898) sgn (120596

119898) (6)


sgn (120596119898) =

1 120596119898

gt 0

0 120596119898

= 0

minus1 120596119898

lt 0

(7)


119879119891119904

= [119891119888+ (119891119904minus 119891119888) 119890minus(120596119898V119904)

2

] sgn (120596119898) + 1205902120596119898 (8)



Crossover operation

Mutation



Best solution

Initial population

Final condition

Parent

Offspring

Yes

No









Θ119905119894mom = [119891119905

119888(119894max) 119891119905

119904(119894max) 120590119905

2(119894max) V119905

119904(119894max)] (9)




expressed as

Θ119905119894dad = [119891119905

119888(119894mean) 119891119905

119904(119894mean) 120590119905

2(119894mean) V119905

119904(119894mean)] (10)


and 1205751199051198942


1205751199051198941

= 1205721Θ119905119894mom + (1 minus 120572

1)Θ119905119894dad = [120572

1119891119905119888(119894max)

+ (1 minus 1205721) 119891119905119888(119894mean) 1205722119891

119905

119904(119894max)

+ (1 minus 1205722) 119891119905119904(119894mean) 1205723120590

119905

2(119894max)

+ (1 minus 1205723) 1205902(119894mean) 1205724V

119905

119904(119894max) + (1 minus 1205724) V119905119904(119894mean)]

1205751199051198942

= (1 minus 1205721)Θ119905119894mom + 120572

1Θ119905119894dad = [(1 minus 120572

1) 119891119905119888(119894max)

+ 1205721119891119905119888(119894mean) (1 minus 120572

2) 119891119905119904(119894max)

+ 1205722119891119905119904(119894mean) (1 minus 120572

3) 1205901199052(119894max)

+ 12057231205902(119894mean) (1 minus 120572

4) V119905119904(119894max) + 120572

4V119905119904(119894mean)]

(11)





120575119905lowast

1198941 120575119905lowast

1198942= [120575119905

lowast

119891119888

120575119905lowast

119891119904

120575119905lowast

1205902

120575119905lowast

V119904

] = [120575119905119891119888

+ 1205731120582 (119905) 120575

119905

119891119904

+ 1205732120582 (119905) 120575

119905

1205902

+ 1205733120582 (119905) 120575

119905

V119904

+ 1205734120582 (119905)]

(12)


1198941 120575119905lowast




120582 (119905) = [1 minus 119903(1minus119905119879)120574

] (13)









119879119891asymp 119870119898119906 (14)




119898119873119894=1


119891119873119894=1




Ω119904= [119891119888 119891119904 1205902 V119904] (15)


119890119904(Ω119904 120596119898) = 119879119891(119894) minus 119879

119891119904(Ω119904 120596119898 119894) (16)







119869119904=

1

2sum[119890119904(Ω119904 120596119898)]2

(17)







Nm 6975119891119904

Nm 85581205902

Nm(rads) 1819V119904


0

90

70

Fric

tion

torq

ue (N

m) 80

Experimentalresults

Identifiedresults

minus70

minus80




0119911 and 120590



0and


0and 120590

1is strength-


119906 (119905) = 119896119888119905 (18)

where 119896119888


0






Ω119889= [1205900 1205901]119879

(19)


119890119889(Ω119889 119905119894) = 120579119898(119905119894) minus 120579119889(Ω119889 119905119894) (20)


119905119894moment 120579




0and 120590

1 however from the


119896119898119906 asymp 119879119891asymp 1205900119911 (21)

1205900119892 (120596119898) = 119891119904

(22)

= 120596119898

minus

10038161003816100381610038161205961198981003816100381610038161003816

119892 (120596119898)119911 (23)


=119889119911

119889119905= 120596119898

minus119879119891

119891119904

10038161003816100381610038161205961198981003816100381610038161003816 (24)



When 120596119898


119911 (119905) = 120579119898(119905) minus 120579

119898(0)

+119896119898119896119888

2119891119904

(120579119898(119905) 119905 + int

119905

0

120579119898(120591) 119889120591)

(25)


12059000=Z119879UZ119879Z

(26)






119898asymp 119911 and 120596






119898119898

+ (1205901+ 1205902) 119898

+ 1205900120579119898

= 119906 (119905) (27)


120579119898(119904)

119906 (119904)=

1

1198981199042 + (1205901+ 1205902) 119904 + 120590

0

(28)





as

12059001= 2120585119898radic

1205900

119898minus 1205902 (29)



1=



119869119889= 1199021

119873

sum119894=1

[119890119889(Ω119889 119905119894)]2

+ 1199022max 119890

119889(Ω119889 119905119894) (30)

where 1199021and 119902




119904


0and 120590

1by



0= 2750Nmrad and 120590

1= 452Nmsrad



1

2

Inpu

tRo

tary

0

3

Experimentalresults


Input voltage

500 30 4010 20Time (s)

volta

ge (V

)ve

loci

tylowast100

(rad

s)


0 05 15 2010Time (s)

05 10 15 200Time (s)

minus10

minus5

0

5

10

Fric

tion

torq

ue (N

m)

minus40

minus20

0

20

40

z (r

ad)

times10minus3





5 Conclusion



times10minus3

minus15

minus10

minus5

05

1015

Fric

tion

torq

ue (N

m)

minus40

minus20

0

20

40

z (r

ad)

20 40 60 80 1000Time (s)

0 20 40Time (s)

60 80 100



Competing Interests


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












expressed as

Θ119905119894dad = [119891119905

119888(119894mean) 119891119905

119904(119894mean) 120590119905

2(119894mean) V119905

119904(119894mean)] (10)


and 1205751199051198942


1205751199051198941

= 1205721Θ119905119894mom + (1 minus 120572

1)Θ119905119894dad = [120572

1119891119905119888(119894max)

+ (1 minus 1205721) 119891119905119888(119894mean) 1205722119891

119905

119904(119894max)

+ (1 minus 1205722) 119891119905119904(119894mean) 1205723120590

119905

2(119894max)

+ (1 minus 1205723) 1205902(119894mean) 1205724V

119905

119904(119894max) + (1 minus 1205724) V119905119904(119894mean)]

1205751199051198942

= (1 minus 1205721)Θ119905119894mom + 120572

1Θ119905119894dad = [(1 minus 120572

1) 119891119905119888(119894max)

+ 1205721119891119905119888(119894mean) (1 minus 120572

2) 119891119905119904(119894max)

+ 1205722119891119905119904(119894mean) (1 minus 120572

3) 1205901199052(119894max)

+ 12057231205902(119894mean) (1 minus 120572

4) V119905119904(119894max) + 120572

4V119905119904(119894mean)]

(11)





120575119905lowast

1198941 120575119905lowast

1198942= [120575119905

lowast

119891119888

120575119905lowast

119891119904

120575119905lowast

1205902

120575119905lowast

V119904

] = [120575119905119891119888

+ 1205731120582 (119905) 120575

119905

119891119904

+ 1205732120582 (119905) 120575

119905

1205902

+ 1205733120582 (119905) 120575

119905

V119904

+ 1205734120582 (119905)]

(12)


1198941 120575119905lowast




120582 (119905) = [1 minus 119903(1minus119905119879)120574

] (13)









119879119891asymp 119870119898119906 (14)




119898119873119894=1


119891119873119894=1




Ω119904= [119891119888 119891119904 1205902 V119904] (15)


119890119904(Ω119904 120596119898) = 119879119891(119894) minus 119879

119891119904(Ω119904 120596119898 119894) (16)







119869119904=

1

2sum[119890119904(Ω119904 120596119898)]2

(17)







Nm 6975119891119904

Nm 85581205902

Nm(rads) 1819V119904


0

90

70

Fric

tion

torq

ue (N

m) 80

Experimentalresults

Identifiedresults

minus70

minus80




0119911 and 120590



0and


0and 120590

1is strength-


119906 (119905) = 119896119888119905 (18)

where 119896119888


0






Ω119889= [1205900 1205901]119879

(19)


119890119889(Ω119889 119905119894) = 120579119898(119905119894) minus 120579119889(Ω119889 119905119894) (20)


119905119894moment 120579




0and 120590

1 however from the


119896119898119906 asymp 119879119891asymp 1205900119911 (21)

1205900119892 (120596119898) = 119891119904

(22)

= 120596119898

minus

10038161003816100381610038161205961198981003816100381610038161003816

119892 (120596119898)119911 (23)


=119889119911

119889119905= 120596119898

minus119879119891

119891119904

10038161003816100381610038161205961198981003816100381610038161003816 (24)



When 120596119898


119911 (119905) = 120579119898(119905) minus 120579

119898(0)

+119896119898119896119888

2119891119904

(120579119898(119905) 119905 + int

119905

0

120579119898(120591) 119889120591)

(25)


12059000=Z119879UZ119879Z

(26)






119898asymp 119911 and 120596






119898119898

+ (1205901+ 1205902) 119898

+ 1205900120579119898

= 119906 (119905) (27)


120579119898(119904)

119906 (119904)=

1

1198981199042 + (1205901+ 1205902) 119904 + 120590

0

(28)





as

12059001= 2120585119898radic

1205900

119898minus 1205902 (29)



1=



119869119889= 1199021

119873

sum119894=1

[119890119889(Ω119889 119905119894)]2

+ 1199022max 119890

119889(Ω119889 119905119894) (30)

where 1199021and 119902




119904


0and 120590

1by



0= 2750Nmrad and 120590

1= 452Nmsrad



1

2

Inpu

tRo

tary

0

3

Experimentalresults


Input voltage

500 30 4010 20Time (s)

volta

ge (V

)ve

loci

tylowast100

(rad

s)


0 05 15 2010Time (s)

05 10 15 200Time (s)

minus10

minus5

0

5

10

Fric

tion

torq

ue (N

m)

minus40

minus20

0

20

40

z (r

ad)

times10minus3





5 Conclusion



times10minus3

minus15

minus10

minus5

05

1015

Fric

tion

torq

ue (N

m)

minus40

minus20

0

20

40

z (r

ad)

20 40 60 80 1000Time (s)

0 20 40Time (s)

60 80 100



Competing Interests


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Nm 6975119891119904

Nm 85581205902

Nm(rads) 1819V119904


0

90

70

Fric

tion

torq

ue (N

m) 80

Experimentalresults

Identifiedresults

minus70

minus80




0119911 and 120590



0and


0and 120590

1is strength-


119906 (119905) = 119896119888119905 (18)

where 119896119888


0






Ω119889= [1205900 1205901]119879

(19)


119890119889(Ω119889 119905119894) = 120579119898(119905119894) minus 120579119889(Ω119889 119905119894) (20)


119905119894moment 120579




0and 120590

1 however from the


119896119898119906 asymp 119879119891asymp 1205900119911 (21)

1205900119892 (120596119898) = 119891119904

(22)

= 120596119898

minus

10038161003816100381610038161205961198981003816100381610038161003816

119892 (120596119898)119911 (23)


=119889119911

119889119905= 120596119898

minus119879119891

119891119904

10038161003816100381610038161205961198981003816100381610038161003816 (24)



When 120596119898


119911 (119905) = 120579119898(119905) minus 120579

119898(0)

+119896119898119896119888

2119891119904

(120579119898(119905) 119905 + int

119905

0

120579119898(120591) 119889120591)

(25)


12059000=Z119879UZ119879Z

(26)






119898asymp 119911 and 120596






119898119898

+ (1205901+ 1205902) 119898

+ 1205900120579119898

= 119906 (119905) (27)


120579119898(119904)

119906 (119904)=

1

1198981199042 + (1205901+ 1205902) 119904 + 120590

0

(28)





as

12059001= 2120585119898radic

1205900

119898minus 1205902 (29)



1=



119869119889= 1199021

119873

sum119894=1

[119890119889(Ω119889 119905119894)]2

+ 1199022max 119890

119889(Ω119889 119905119894) (30)

where 1199021and 119902




119904


0and 120590

1by



0= 2750Nmrad and 120590

1= 452Nmsrad



1

2

Inpu

tRo

tary

0

3

Experimentalresults


Input voltage

500 30 4010 20Time (s)

volta

ge (V

)ve

loci

tylowast100

(rad

s)


0 05 15 2010Time (s)

05 10 15 200Time (s)

minus10

minus5

0

5

10

Fric

tion

torq

ue (N

m)

minus40

minus20

0

20

40

z (r

ad)

times10minus3





5 Conclusion



times10minus3

minus15

minus10

minus5

05

1015

Fric

tion

torq

ue (N

m)

minus40

minus20

0

20

40

z (r

ad)

20 40 60 80 1000Time (s)

0 20 40Time (s)

60 80 100



Competing Interests


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119898asymp 119911 and 120596






119898119898

+ (1205901+ 1205902) 119898

+ 1205900120579119898

= 119906 (119905) (27)


120579119898(119904)

119906 (119904)=

1

1198981199042 + (1205901+ 1205902) 119904 + 120590

0

(28)





as

12059001= 2120585119898radic

1205900

119898minus 1205902 (29)



1=



119869119889= 1199021

119873

sum119894=1

[119890119889(Ω119889 119905119894)]2

+ 1199022max 119890

119889(Ω119889 119905119894) (30)

where 1199021and 119902




119904


0and 120590

1by



0= 2750Nmrad and 120590

1= 452Nmsrad



1

2

Inpu

tRo

tary

0

3

Experimentalresults


Input voltage

500 30 4010 20Time (s)

volta

ge (V

)ve

loci

tylowast100

(rad

s)


0 05 15 2010Time (s)

05 10 15 200Time (s)

minus10

minus5

0

5

10

Fric

tion

torq

ue (N

m)

minus40

minus20

0

20

40

z (r

ad)

times10minus3





5 Conclusion



times10minus3

minus15

minus10

minus5

05

1015

Fric

tion

torq

ue (N

m)

minus40

minus20

0

20

40

z (r

ad)

20 40 60 80 1000Time (s)

0 20 40Time (s)

60 80 100



Competing Interests


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










times10minus3

minus15

minus10

minus5

05

1015

Fric

tion

torq

ue (N

m)

minus40

minus20

0

20

40

z (r

ad)

20 40 60 80 1000Time (s)

0 20 40Time (s)

60 80 100



Competing Interests


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article dynamic friction parameter...

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