friction model of industrial robot joint with temperature...

Research ArticleFriction Model of Industrial Robot Joint with TemperatureCorrection by Example of KUKA KR10

Maksim N Nevmerzhitskiy 1 Boris S Notkin12

Andrey V Vara1 and Konstantin V Zmeu1

1School of Engineering Far Eastern Federal University Vladivostok 690950 Russia2Institute of Automation and Control Processes Far Eastern Branch of the Russian Academy of Sciences Vladivostok 690950 Russia

Correspondence should be addressed to Maksim N Nevmerzhitskiy maximnevmergmailcom

Received 27 October 2018 Revised 24 November 2018 Accepted 17 December 2018 Published 6 January 2019

Academic Editor Shahram Payandeh

Copyright copy 2019 Maksim N Nevmerzhitskiy et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The quality of industrial robots essentially depends on the properties of their kinematic couples This research has involvedconducting an experimental study of the friction torque in a joint of the KUKA KR10 industrial robot and building its model Ithas been established that the largest impact on friction in the joint is caused by its axial load and velocity as well as the temperatureof the mechanism which is generally not homogeneous It is not possible to measure temperature fields in the joints of a serialindustrial robot directly This study has set forth a method to estimate friction torque taking into account the temperature factorindirectly For this we have used the motor temperature available for measuring in combination with special periodical motionsperformed by the robot during which we estimated the actual friction torque in the joint and calculated a temperature correctionbased on our findings

1 Introduction

The mechanical rigidity of an industrial robot-manipulatoris largely determined by the rigidity of its joints To increaserigidity andminimize backlash the joints of industrial robotsare designed to have high kinematic precision and highload capacity but increased friction on the downside Theauthors of the studies [1 2] point out a number of factorsaffecting the friction force in robot joints joint designmanufacturing and assembly precision material of the jointrsquoscomponents lubricants used geometry and permanency ofcontact between the mating surfaces wear and tear of thejointrsquos parts backlash vibration etc

Taking friction forces into account may considerablyimprove the efficiency of controlling the robot [3 4] andallows assessing the state of lubricant in the joint [5] themechanical wear and tear predicting its service life [1]and implementing force-torque control algorithms using therobotrsquos axial torque data [6]The literature contains referencesto many identification techniques [6ndash9] and friction models

Both experimental [10 11] and theoretical friction forcedetermination methods [2 12 13] are available

The most comprehensive friction models include depen-dencies based on such variables as angular velocity loadand temperature [5 10 12] It has been experimentallydemonstrated [10 14] that temperature has a substantial effecton friction torque However it cannot always be measuredsince the jointrsquos interior is not accessible due to its design[2 5 12 15] In this case the jointrsquos friction area temperaturecan be obtained either by indirect measurements or throughestimation In the study [10] the temperature of the lubricantwas used as the jointrsquos own temperature When estimatingfriction area temperature with the aid of state observersmotorwinding temperature [12] or joint housing temperature[5] can be used The observers are based on various heattransmission models based in turn on the law of conservationof thermal energy and taking into account the motorrsquosthermal output [12 15] Moreover in some cases lubricantviscosity can be taken into account as well as thermal effectof friction on it [5] Temperature estimation methods based

HindawiJournal of RoboticsVolume 2019 Article ID 6931563 11 pageshttpsdoiorg10115520196931563

2 Journal of Robotics

onmodelsmay have limited accuracywhich deteriorates withtime due to various unmeasured factors [4]

This study explores friction in one of the joints of theKUKA KR10 R900 sixx robot-manipulator [16] Its controlsystem allows measuring the motor temperature only Toaccount for the thermal factor influence on friction in therobotrsquos joint a technique is proposed based on introducinga motor temperature correction which is regularly calcu-lated using the friction torque registered when special axialmotions are performed by the robot

2 Overview

21 Friction Function and Its Components Axial controltorque u of the industrial robot can be described by

u = M (120593) + C (120593 ) + L minus sign () 120591 (1)

where M(120593) is inertia matrix C(120593 ) is vector of centrifugaland Coriolis forces 120593 and are angular positionsvelocities and accelerations in the robotrsquos axles respectivelyL is axial load torque 120591 is friction torque counter-directed tothe axial velocity

Friction is a very complex process and to some degreeunique to every mechanism [14] The studies dedicated totribology use a wide range of friction function formalizations[2 17] One of the fullest descriptions of the friction torque 120591in a robot-manipulator joint is presented in the study [10] Itestablishes a dependence of friction torque 120591 on load torqueL angular velocity and the jointrsquos temperature T and canbe presented as follows

120591 (L T) = 120591L (L) + 120591s () + 120591v (T) + 120591c (2)

where 120591L is friction component dependent on load 120591s isStribeck friction 120591v is viscous friction 120591c is Coulomb frictionLet us consider these components in greater detail

Coulomb friction 120591c is related to the force of intermolec-ular interaction of the mating surfaces and is a constant [14]

Theway load L has an effect on the friction component 120591L is linked with the pressing force of the mating surfaces theirroughness the presence of lubricant between them and thematerial properties of the parts [14] Axial load L of the robotconsists of holding torque lg(120593) (caused by the gravitationalforce and dependent on the robotrsquos kinematics) and loadtorque l(t) (caused by the external forces applied to the robotat the current time t)

L = lg (120593) + l (t) (3)

To describe the dependence of friction on load both linear[5 7 10] and nonlinear functions [2] can be used Most oftenthis dependence is described as follows

120591L (L) = 120572 |L| 120572 gt 0 (4)

where 120572 is a friction coefficientThe viscous friction component 120591v arises when the jointrsquos

mechanical elements move in viscous fluid This dependenceis often expressed linearly in terms of velocity [6 9 15

17] in some cases a nonlinear function approximation isused power [12] trigonometrical [18] and exponential [11]functions The value and characteristics of viscous frictionare substantially dependent on the viscosity of lubricant used[14] which in turn depends on temperature For this reasona more general description of viscous friction involves thejointrsquos both velocity and temperature T In the studies[2 5 10] this dependence is formalized as follows

120591v (T) = (120573 + 120574eminusT120575+1205750) 10038161003816100381610038161003816100381610038161003816 120573 120574 120575 1205750 gt 0 (5)

where 120573 120574 are coefficients to characterize viscosity 120575 1205750 arecoefficients to take temperature into account

The nature of the Stribeck effect 120591s is related to thecatching of peaks on rough surfaces at themicrolevel througha layer of liquid lubricant The effect depends on the finishof the surfaces lubricant viscosity and quantity and it isregistered at low velocities and canmanifest itself to a varyingdegree The Stribeck effect is identified in the mechanismsexplored in the studies [3 18 19] where it is described asfollows

120591s () = 120582eminus|s|120583 10038161003816100381610038161003816100381610038161003816 (6)

where s ndash Stribeckrsquos velocity 120582 and 120583 ndash coefficients tocharacterize the amplitude and slope of the nonlinearity ofStribeck friction respectively The expression (6) reflects thedependence of Stribeck friction on velocity only howeverthere are studies which to describe this component furtherhave taken into account the influence of the jointrsquos tempera-ture and load [10] There are also a number of joint designsin which the Stribeck effect has been either not detected orneglected by the researchers [2 4 5 12 13 20]

22 FrictionMeasurement Methods Themost common solu-tion for estimating friction torque 120591 in robot-manipulatorjoints is separating it using measurable control torque uwhere the contribution of the remaining components of theequation (1) is calculated with due regard to the mass-inertialcharacteristics of the robotrsquos links and then eliminated[12 19ndash22] If these characteristics are unknown variousadditional computational approaches andmethodologies canbe used to identify them [2ndash4 6ndash9 13]

The study [1] proposes an alternative method that doesnot require identifying mass-inertial characteristics of therobotrsquos links In a nutshell it consists in paired counter-directed motions performed by one robot axle at constantvelocity || = const without external load l = 0 At constantspeed there is no influence of inertia forces Centrifugal andCoriolis forces are directed radially from considered jointand create torque acting only on subsequent joints and linksThen according to (1) allow the components of the inertiaforcesM(120593) = 0 and those of centrifugal and Coriolis forcesC(120593 ) = 0 while the axial control torque u counteractsonly the friction forces 120591 and the load torque L(120593) = lg(120593)caused by the gravitational forces Both components dependon the angle 120593 Notably the gravitational component of theload torque lg(120593) does not depend on the motion directionsign( 120593) whilst the friction torque component 120591 changes signas the direction reverses

Journal of Robotics 3

Inputsvelocity = [1 2 119898119886119909] vector of velocity values degsu control torque N sdotm120593 joint angular position degsOutputs joint angular velocity degsu+(120593) control torque until mowing clockwise N sdotmuminus(120593) control torque until mowing counterclockwise N sdotm120591(120593) friction torque N sdotmlg(120593) load torque N sdotmTm motor temperature ∘C CYCLE OF CHANGING VELOCITY VALUESfor i = 1 to length(velocity) step 1 = velocity[i] assigning the velocity value to the robot joint velocity

Moving to 120593 = 1800 joint clockwise motionuminus(120593) = u assigning u+ until mowing clockwiseMoving to 120593 = 00 joint counterclockwisemotion

u+(120593) = u assigning uminus until mowing counterclockwise120591(120593) and lg(120593) calculating by eq (8)Tm data measuring

end for

Algorithm 1 Pseudocode for experiments robot motion on different velocities

gt 0 u+ (120593) = lg (120593) minus 120591 (120593) lt 0 uminus (120593) = lg (120593) + 120591 (120593) (7)

where u+ and uminus are control torques registered at therobot axis moving in the forward and reverse directionrespectively Then the friction torque 120591(120593) and load torquelg(120593) are

120591 (120593) = uminus (120593) minus u+ (120593)2 lg (120593) = uminus (120593) + u+ (120593)2 (8)

This method and its variations [11] do not require a prioriknowledge of the robotrsquosmass-inertial characteristics and theidentified dependences lg(120593) and 120591(120593) have high repeatability[18 19 23]

23 Experiment Preparation Figure 1 presents an overallview of the industrial robot-manipulator KUKA KR 10 R900sixx and its kinematic structure with six degrees of freedomThe robotrsquos load capacity is 10 kg withmaximum reach of 900mm

The friction model identification experiments are per-formed based on the robotrsquos second axle To identify frictiontorque 120591 and holding torque lg the method (8) is used involv-ing paired motions in counter side directions as shown inFigure 2 For load torque L the study uses gravitational torqueL = lg To take into account the thermal factor the studyuses motor temperature available from the robotrsquos controlsystem additionally the robotrsquos joint housing temperature

x

yz

Axis 1

Axis 2

Axis 3Axis 4

Axis 5

Axis 6

+-170deg

+45deg-190deg

+156 deg-120deg

+-185deg

+-120deg

+-350deg

Figure 1 Kinematic structure of robot-manipulator KUKA KR 10R900 sixx

is measured In the study the friction model is consideredas the expression (2) Then the friction torque components120591(L T) dependent on load L velocity and temperature Tare identified consecutively

3 Measurement Results

31 Dependence of Friction Torque upon Load Torque Toestimate the friction torque component 120591L(L) which isdependent on load torque L = lg in the joint a series ofexperiments are carried out where according to the method(8) and scheme in Figure 2 robot motions can be reviewed atvarious velocities (Algorithm 1)

The experiment results are presented in Figure 3 Takinginto account that each experimental curve is obtained for =


+ -

Figure 2 Experiment design

plots of approximating functionexperiment data

100

20

30

40

50

60

70

80

90

Fric

tion

(Nm

)

0 20 40 60 80Load L (Nmiddotm)

minus80 minus40minus60 minus20

L(L) + C( T)

=90 degs T=31 Co

=1012 degs T=33 Co=675 degs T=33 Co

=338 degs T=32 Co

=191 degs T=32 Co


Figure 3 Dependence of friction 120591 upon load L at various velocities and temperatures T

const and assuming that during a single experiment (pair ofcounter-directed motions) T = const the expression (2) canbe reduced to

120591 (L T) = 120591L (L) + G (T) (9)

where G(T) = 120591c + 120591s() + 120591v(T) = const if =const and T = const Accordingly the effects of the com-ponent 120591L(L) on friction 120591 are linked with the shape of thecurves presented in Figure 3 Further their different butinvariable within the same experiment vertical shifts G(T)are determined by the other components in the expression(2) Assuming that 120591L(0) = 0 for each curve it is possible toidentify its vertical shift G(T) and define the law of the total

0 100 200 300 400

20

30

40

50

60

70Warming Cooling

Time t (min)

Tem

pera

ture

(C)o

Tm

T＝

Figure 4Motor temperatureTm and jointrsquos housing temperatureTcin the process of heating up and cooling down

quadratic dependence of friction torque 120591L upon the load L inthe joint

120591L (L) = 120572L2 (10)

In Figure 3 the experimental dependences are juxtaposedwith their approximation obtained in accordance with (10)and plotted with various shifts G(T) The revealed depen-dence (10) is expressly nonlinear which is in contrast with itstraditionally linear shape (4)

32 Viscous Friction To identify the viscous friction com-ponent 120591v(T) let us consider experiments at differentvelocities at varying temperature T Algorithm 2 presentsthe pseudocode for experiments motion

Figure 4 presents time plots of motor temperature Tmand jointrsquos housing temperature Tc including two phasesheat-up through intensive motions at maximum velocityand natural cool-down As temperature changed in orderto accumulate experimental data motions were regularlyperformed by the robot at four velocities The obtainedexperimental dependences 120591 plotted for L = 0 (pointscorresponding to curves minimum shown on Figure 3)in relation to motor temperature Tm and joint housingtemperature Tc are presented in Figures 5 and 6 respectively

In all of the presented experiments a considerable reduc-tion in friction torque 120591 with the rise of temperature couldbe observed However it should be noted that dependencesof friction torque 120591 on temperatures registered during heat-up and cool-down are differ That difference is true for bothmotor Tm and housing Tc temperatures being the hottestand coolest points in the jointrsquos design respectively and anyof their linear combination that reflects the temperature insome middle point of the jointrsquos mechanism (not shown inthe plot) The observed ambiguity can be explained by thejointrsquos complex design where friction concurrently occurson many mating surfaces that are within a heterogeneous


Inputsvelocity = [12 3 4] vector of velocity values degsTm motor temperature ∘Cu control torque NmTemp max = Tm temperature parameter ∘C120593 joint angular position degsOutputs joint angular velocity degsu+(120593) control torque until mowing clockwise N sdotmuminus(120593) control torque until mowing counterclockwise N sdotm120591(120593) friction torque N sdotmlg(120593) load torque N sdotmTm motor temperature ∘C CYCLE OF CHANGING VELOCITY VALUES IN JOINT HEATING-UP MODEwhile Tm lt 65∘C Cycle of changing velocity values

for i = 1 to 4 step 1 = velocity[i] assigning the velocity value for the robot joint velocityMoving to 120593 = 1800 joint clockwise motion

uminus(120593) = u assigning u+ until mowing clockwiseMoving to 120593 = 00 joint counterclockwisemotion

u+(120593) = u assigning uminus until mowing counterclockwise120591(120593) and lg(120593) calculating by eq (8)Tm data measuringend forTemp max = Temp max + 1 changing temperature parameter to 1∘C up

Heating-up cycle by 1∘C making intensive motions at maximum velocitywhile Tm lt Temp max = 119898119886119909 assigning the velocity maximum value for the robot joint velocity

Moving to 120593 = 1800 joint clockwise motionMoving to 120593 = 00 joint counterclockwisemotion

end whileend while CYCLE OF CHANGING VELOCITY VALUES IN JOINT COOLING-DOWNMODEwhile Tm gt 25∘C Cycle of changing velocity values



u+(120593) = u assigning uminus until mowing counterclockwise120591(120593) and lg(120593) calculating by eq (8)Tm data measuringend forTemp max = Temp max ndash 1 changing temperature parameter to 1∘C downwait for (Tm lt= Temp max) waiting for natural cooling to 1∘C down

end while

Algorithm 2 Pseudocode for experiments robot motion on different velocities and temperatures

temperature fieldThus the experiments presented in Figures5 and 6 on the one hand demonstrate a significant effectof joint temperature on friction and on the other hand thecomplexity of the task of its objective monitoring

For further experiments let us perform identificationof the viscous friction component 120591v in terms of meantemperature Tlowast and mean friction torque 120591lowast

Tlowast (t) = Tc (t) + Tm (t)2 120591lowast (Tlowast) = 120591+ (Tlowast) + 120591minus (Tlowast)2 (11)

where t ndash time 120591+ and 120591minus ndash friction torques registered in heat-up and cool-down modes respectively


20 25 30 35 40 45 50 55 60 6520

30

40

50

60

70

80

90

100

110

warmingcooling

oMotor temperature Tm ( C)

Fric

tion

(Nm

)

1=191 degs

4=1012 degs

3=675 degs

2=338 degs

Figure 5 Dependence of friction torque120591 upon motor temperatureTm for various velocities for the load L = 0

20

30

40

50

60

70

80

90

100

110

Fric

tion

(Nm

)

warmingcooling

20 25 30 35 40oBody temperature T＝ ( C)

4=1012 degs

3=675 degs

2=338 degs1=191 degs

Figure 6 Dependence of friction torque 120591 upon housing tempera-ture Tc for various velocities for the load L = 0

Figure 7 presents mean experimental data and theirapproximation obtained in accordance with the simplifiedexpression (2)

120591lowast (L Tlowast) = 120591c + 120591v (Tlowast) (12)

where the components excluded whose effects according tothe terms of the experiment are either nonexistent (120591L(L) = 0for L = 0) or small are eliminated (the Stribeck friction 120591s()component manifests itself only at low velocities which arenot used in the experiment in Figure 7)

25 30 35 40 45 502020

30

40

50

60

70

80

90

100

110

plots of approximating functionaverage experiment data

Aver

age f

rictio

nlowast

(Nm

)

oAverage temperature Tlowast ( C)

1=191 degs

2=338 degs

3=675 degs

4=1012 degs

Figure 7 Dependence of mean friction torque 120591lowast upon meantemperature Tlowast for various velocities for the load L = 0

The viscous friction 120591v component in the study is identi-fied according to the formula120591v (T) = (120573 + 120574eminusT120575) 1003816100381610038161003816100381610038161003816100381612 (13)

which is obtained based on the expression (5) but differs fromthe latter by a nonlinear dependence on velocity identifiedfor the mechanism being studied

33 Stribeck Effect Consider the component 120591s() of frictiontorque (2) caused by the Stribeck effect This effect occurs atlow velocities 120593 at the pass between friction at rest and frictionof motion The effect manifests itself in increased friction atthe beginning of motion and is characterized by the fact thatas velocity rises total friction 120591 does not increase as itfollows for example from (12) but decreases

To identify the Stribeck component 120591s() of frictiontorque letrsquos turn again to the experiment presented inFigure 3 where successive motions are considered in a widerange of velocities starting from 23 degs In the process ofconducting the experiment the jointrsquos mechanism heated upinevitably The temperature factor has a substantial influenceon viscous friction torque however the magnitude of thisinfluence was identified and formalized as (13) and hence canbe accounted for by this correction

120591v (T0T) = 120591v (T0) minus 120591v (T) (14)

which at a given velocity reflects the change in viscousfriction torque 120591v caused by the jointrsquos temperature changingfrom T0 to T This allows identifying and eliminating theinfluence of uncontrolled temperature changes from theexperimental data obtaining an estimate of friction torquedependence upon velocity for any temperature T0120591 (L T0) = 120591 (L T) + 120591v (T0T) (15)

where 120591(L T) is friction torque at actual temperature T


20 40 60 80 100 120 140 160 180 200

20

40

60

80

100

120

0

140

o

plots of approximating function

Fric

tion

(Nm

)

T0=20 C

Angular velocity (degs)

experiment data + Δexperiment data

Figure 8 Dependence of friction torque 120591 upon velocity atconstant temperature T0 = 20∘ and load L = 0

Figure 8 presents two dependences of friction torque onvelocity at constant temperature T0 = 20∘b and loadL = 0 experimental and model ones The experimentaldependence is built using the expression (15) where fortemperature T the motor temperature Tm is used The modeldependence is presented by the expression (13) which wasobtained earlier by approximation of experimental data atvelocities ranging from 191 to 1012 degs As can be seenfrom Figure 8 the model dependence provides an acceptableaccuracy of describing the experimental data over the wholerange of working velocities including in the proximity of = 0 Therefore we can conclude that the experimentaldependence at low velocities has no manifestations such asthe Stribeck effect So when describing friction in the mecha-nism under study this effect can be neglected accepting that120591s = 034 Mean Friction Torque Function Combining the compo-nents of friction torque included in (2) dependent upon loadtorque L (10) velocity and temperature T (13) as well astaking into account Coulomb friction component 120591c we getan expression for friction torque in the second joint of theKUKA KR 10 robot

120591 (L T) = 120591c + 120572L2 + (120573 + 120574eminusT120575) 1003816100381610038161003816100381610038161003816100381612 (16)

where 120591c = 1057 120572 = 00014 120573 = 348 120574 = 1566 120575 =0056It is noteworthy that the expression (16) with presented

parameters is obtained using mean values (12) under theconditions of established temperature ambiguity and is aqualitative description of friction torque in the first placeThe following section of this paper discusses the use of (16)for quantitative estimates of friction torque

35 Friction Function Temperature Parameter AdjustmentThe effects of temperature on friction in the robot joint aresubstantial In the case (see Figures 5 and 6) temperature maybe responsible for up to half the amount of friction torque soits effects must not be neglected However the study showedthat it is impossible to establish a clear relationship betweenfriction torque and temperature in a single point of the jointrsquosmechanism (the study considered motor and joint housingtemperatures as well as a set of their linear combinations)This can be explained by the jointrsquos complex design wherefriction occurs simultaneously between many paired partswhich operate in contact with each other under differentconditions including temperature Building a friction modelbased on distributed temperature is difficult due to thecomplex task of registering thermal fields inside the closedjointrsquos mechanism

Let us review a technique that allows a practical esti-mation of friction torque using the expression (16) Itis implemented by making a regularly updated correctionT(ta) to the registered motor temperature Tm(t)

T (t) = Tm (t) + T (ta) (17)

where T(t) is temperature parameter in the expression (16) tis time and ta is correction calculation time

To calculate the correction T(ta) let us turn to theexpression (16) and express the temperature parameter fromit

T (ta) = minus1120575 sdot ln(120591 (ta) minus 120591c minus 120572L2 (ta)120574radic1003816100381610038161003816 (ta)1003816100381610038161003816 minus 120573120574) (18)

where 120591(ta) L(ta) and (ta) are friction torque load torqueand velocity in the joint at the time ta Then

T (ta) = T (ta) minus Tm (ta) (19)

where Tm(ta) is motor temperature registered at the correc-tion calculation time

Friction torque 120591(ta) and load torque L(ta) included in(18) can be obtained using the method (8) which involvesperforming two counter-directed motions at constant veloc-ity The need for performing special motions may necessitateinterrupting the robotrsquos control program which is generallyundesirable so the interruption procedure should have rea-sonable periodicity

Our study has shown that the more intensive thermalprocesses in the joint currently are (see Figure 4) the larger achange of the temperature correction T turns out to be andthe more often its adjustment is required This observationhas been used to develop a heuristic periodicity criterion

t minus ta gt min (k sdot t tmax) (20)

where k is scale coefficient regulating the periodicity ofcorrections and tmax is parameter setting the time uponexpiration of which correction is made forcibly

The temperature correction adjustment mechanism Tis launched when the condition (20) is true As a measure


Inputs120593 joint angular position degs120593 joint angular velocity degs120593(ta) joint angular velocity value for temperature adjustment degsTm motor temperature ∘Ct time of robotrsquos control program executing stmax time parameter upon expiration of which temperature correction ismade forcibly s120591c 120572 120573 120574 120575 identified parameters of friction function eq (16)k periodicity criterion parameterOutputst time which it takes the motor temperature Tm to change by 1∘C sta adaptation time parameter sTm(ta) motor temperature at time ta

∘C120591(ta) friction torque at time ta N sdotmL(ta) load torque at time ta N sdotmT(ta) temperature parameter at time ta ∘CT(ta) temperature correction at time ta

∘C120591(t) friction torque N sdotm ROBOTrsquoS CONTROL PROGRAM CYCLEwhile robotrsquos control program executing Temperature Tm change monitoring cycle

if |Tm minus Tm(ta)| gt= 1 if motor temperature Tm changed by 1∘Ct = |t minus ta| time for temperature Tm change

end if Temperature parameter T adjustment cycle

if t ==0 or t minus ta gt min(k sdot t tmax) compliance of heuristic periodicity criterion eq (20)Pause of robotrsquos control program executingTm(ta) = Tm rewriting temporary variableta = t ascending time value to adaptation time parameter120593 = (ta) assigning the velocity value (ta) for the robot joint velocityMoving to 120593 = 1800 joint clockwise motion


u+(120593) = u assigning uminus until mowing counterclockwise120591(ta) = 120591(120593) and L(ta) = lg(120593) calculating by eq (8)T(ta) = by eq (18) temperature parameter adjustmentT(ta) = T(ta) minus Tm(ta) temperature correction calculation by eq (19)end ifT(t) = Tm(t) + T(ta) temperature parameter calculation online by eq (17)120591(t) = by eq (16) friction torque estimation online using T(t) with temperature

correction calculated after adjustment cycleMotions of robotrsquos control program executing

end while

Algorithm 3 Pseudocode demonstrating proposed friction torque estimation technique making temperature parameter adjustment

of temperature change intensity we used the time t whichit takes the motor temperature Tm to change by 1∘C Thischoice was made in view of the peculiar features of temper-ature diagnostics in the control systems of KUKA robotswhere registration step is 1∘C Parameter tmax provides extraoperational security in established temperature conditionsAlgorithm 3 presenting pseudocode for realising describedtechnique

Letrsquos demonstrate the proposed friction torque estimationtechnique using the experimental data shown earlier inFigure 5These data present a pairs sequence of axle counter-directed motions performed at four velocities at different

temperatures in heat-up and cool-down modes Each pairof motions allows using the method (8) getting a frictiontorque which can be used both for calculating the tempera-ture correction T and for comparison purposes in analyzingthe accuracy of friction torque estimates

The experiment in Figure 5 showed a substantial differ-ence in the dependences of friction torque 120591 uponmotor tem-perature Tm in heat-up and cool-down modes In Figures 9and 10 these modes are presented separately and labelled withfriction torque estimatesThe estimates are obtained using theexpression (16) with temperature correction in accordancewith (17) at times determined by the condition (20) for k = 5


40

60

80

100

120

25 30 35 40 45 50 55 60

0

0 10025 50 75

evaluation datadata measuring (correction motion)

experiment dataexperiment data (polynomial fitted)

Time t (min)


Fric

tion

(Nm

)

minus20

minus10

ocorrection temperature ΔT ( C)

oΔT

(C)

4=1012 degs

3=675 degs

2=338 degs

1=191 degs

Figure 9 The result of the friction function temperature parameteradjustment for heat-up data

tmax = 60 minutes In the presented experiments to calculatetemperature corrections T a velocity (ta) = 675 degs isused The labels show moments of time when adjustmentswere made and corresponding values of the temperaturecorrection T

The statistical study showed that when choosing thevelocity for calculating the temperature correction T it isadvisable to use the observations below for guidance Turningto (16) it can be noted that the higher the velocity themore pronounced the contribution of temperature influenceon the friction torque compared to its other componentsHowever in practice as velocity grows it is becoming moredifficult to stabilize it And as a result the registered torque(1) has a growing component related to acceleration whichaccording to the method (8) should be absent Consequentlyit is recommended that in the general case mid-rangevelocities should be used for calculating the temperaturecorrection

The accuracy of friction torque estimates in the proposedtechnique is linked with how frequently the temperaturecorrection is adjusted T In the experiments in Figures 9and 10 six adjustments were made in each of them for thevalues of the parameters k = 5 and tmax = 60 minutesUsing the same experimental data and changing the valuesof the parameters k and tmax let us consider the effect oftemperature adjustment frequency on the accuracy of friction

Time t (min)

30 35 40 45 50 55 60

30

40

50

60

70

80

90

100

20

010

300 0100 50150200


Fric

tion

(Nm

)evaluation datadata measuring (correction motion)



oΔT

(C)

4=1012 degs

2=338 degs

1=191 degs

3=675 degs

Figure 10The result of the friction function temperatureparameteradjustment for cool-down data

torque estimates Let us use standard deviation taken for thewhole range of velocities under study as an accuracy test

120576 = radic 1nm

nsumi=1

msumj=1(120591(j)i minus 120591(j)i )2 (21)

where n is number of experiments for a given velocity m = 4is number of velocities under study 120591(j)i is friction torqueobtained with the use of the method (8) and smoothed outusing polynomial approximation (as shown in Figures 9 and10) and 120591(j)i is friction torque estimate obtained using theproposed technique

Thefindings of a comparative experiment are presented inFigure 11 in the form of a deviation histogram 120576 depending onthe number of corrections for heat-up and cool-downmodesIt is noteworthy that in both modes the jointrsquos temperaturevaries quite widelymdasharound 40∘C (see Figure 4) Also thecooling process lasting for 300 minutes runs three times asslow as the heating process with the adjustment procedurecarried out three times as rare with comparable frictiontorque estimation accuracy This confirms the universalityof the proposed criterion (20) The presented histogramillustrates that even in the modes of intensive natural heat-up and cool-down of the robot about seven correctionsare required for a temperature variation range of 40∘C Thecorresponding standard deviation does not exceed 2 Nsdotm

On the whole it should be noted that the correctionT can be updated in relatively long time periods owingto high inertia of thermal processes and in view of the


1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14Number of corrections

warmingcooling

0

RMS

corr

ectio

n er

ror

(Nm

)

Figure 11 Friction torque estimate accuracy depending on thenumber of temperature corrections

temperature estimation in (17) which indirectly takes intoaccount its change rate This periodicity does not affect therobotrsquos performance to any significant degree and ensureshigh accuracy of friction torque estimation in its joints in realtime without using any auxiliary equipment

4 Conclusions

The study has explored the friction function in the secondjoint of the KUKA KR10 R900 sixx robot depending on loadvelocity and temperature To separate friction torque froman axial control torque signal an experimental techniquenot requiring knowledge of the robotrsquos mass-inertial char-acteristics has been used Nonlinear nature of the frictiontorque dependence upon load torque has been confirmedexperimentally and its description proposed The combinedinfluence of velocity motor winding temperature and jointhousing temperature on friction torque during heat-up andcool-down has been explored A function that approximatesmean experimental data describing the dependence has beenproposed The study has demonstrated the absence of theStribeck effect in the joint

The experimental data do not allow building a definitefriction torque dependence on the friction area temperaturesince the robotrsquos design does not allow measuring temper-ature that determines friction in the joint To solve thisproblem a technique to estimate temperature in friction areahas been proposed In a nutshell the technique consists inindirect estimation of temperature using the axial torque ofthe robot in the process of its performing special motions andadjusting the friction function temperature correction Anadjustment frequency criterion independent of temperaturemode has been proposedThe effect of adjustment frequencyon the accuracy of torque estimation has been studiedThe conducted experiments have shown that the proposedsolution allows estimating friction torque with a standard

deviation not exceeding 2NsdotmRecommendations for choos-ing special motion velocities and adjustment periodicity havebeen given

The friction function estimation technique is univer-sal and allows developing a friction model depending onthe jointrsquos angular velocity load and temperature in anyserial industrial robot with a serial kinematic structure Thefriction-area temperature estimation technique can be testedin robot operation modes when substantial changes in thejoint temperature occur

Precise knowledge of axial friction forces will allowestimating the external forces acting on the robot usingdata on its axial torques The availability of mass-inertialcharacteristics and the robotrsquos kinematic model along withaxial friction models will help estimate the external forcesacting on the process tools as well as implement force-torquecontrol algorithms Within the latter task some identificationmethods described in articles [6 8] potentially can be com-bined with described nonlinear friction model Neverthelessthe temperature factor compensation is important in themodes of significant temperature changes

Data Availability

The plots graphics and estimation results data used tosupport the findings of this study are included within thearticle

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was financially supported by the Ministry ofEducation and Science of the Russian Federation [statecontract 02G25310173]

References

[1] R Bittencourt Friction change detection in industrial robotarms [Master thesis] KTH Electrical Engineering StockholmSweden 2007

[2] R Waiboer Dynamic modelling identification and simulationof industrial robots ndash for off-line programming of robotised laserwelding University of Twente 2007

[3] W Susanto R Babuska F Liefhebber and T van der WeidenldquoAdaptive Friction Compensation Application to a RoboticManipulatorrdquo IFAC Proceedings Volumes vol 41 no 2 pp2020ndash2024 2008

[4] B Bona and A Curatella ldquoIdentification of industrial robotparameters for advanced model-based controllers designrdquo inProceedings of the 2005 IEEE International Conference onRobotics and Automation pp 1681ndash1686 Spain April 2005

[5] L Marton and F Van Der Linden ldquoTemperature dependentfriction estimation Application to lubricant health monitor-ingrdquoMechatronics vol 22 no 8 pp 1078ndash1084 2012

[6] M Gautier A Janot and P-O Vandanjon ldquoA new closed-loop output error method for parameter identification of robot


dynamicsrdquo IEEE Transactions on Control Systems Technologyvol 21 no 2 pp 428ndash444 2013

[7] P Hamon M Gautier and P Garrec ldquoNew dry friction modelwith load- and velocity-dependence and dynamic identificationof multi-dof robotsrdquo in Proceedings of the 2011 IEEE Interna-tional Conference on Robotics and Automation ICRA 2011 pp1077ndash1084 China May 2011

[8] A Janot P-O Vandanjon and M Gautier ldquoA generic instru-mental variable approach for industrial robot identificationrdquoIEEE Transactions on Control Systems Technology vol 22 no1 pp 132ndash145 2014

[9] A Janot P C Young and M Gautier ldquoIdentification andcontrol of electro-mechanical systems using state-dependentparameter estimationrdquo International Journal of Control vol 90no 4 pp 643ndash660 2017

[10] A C Bittencourt E Wernholt S Sander-Tavallaey and TBrogardh ldquoAn extended friction model to capture load andtemperature effects in robot jointsrdquo in Proceedings of the 23rdIEEERSJ 2010 International Conference on Intelligent Robotsand Systems IROS 2010 pp 6161ndash6167 Taiwan October 2010

[11] M Ruderman F Hoffmann and T Bertram ldquoModeling andidentification of elastic robot joints with hysteresis and back-lashrdquo IEEE Transactions on Industrial Electronics vol 56 no 10pp 3840ndash3847 2009

[12] L Simoni M Beschi G Legnani and A Visioli ldquoFrictionmodeling with temperature effects for industrial robot manipu-latorsrdquo in Proceedings of the IEEERSJ International Conferenceon Intelligent Robots and Systems IROS 2015 pp 3524ndash3529Germany October 2015

[13] J Swevers W Verdonck and J De Schutter ldquoDynamic modelidentification for industrial robotsrdquo IEEE Control Systems Mag-azine vol 27 no 5 pp 58ndash71 2007

[14] B Bhushan Introduction to Tribology John Wiley amp Sons LtdTheAtrium SouthernGate ChichesterWest Sussex PO19 8SQUK 2013

[15] F B Carlson A Robertsson and R Johansson ldquoModeling andidentification of position and temperature dependent frictionphenomena without temperature sensingrdquo in Proceedings ofthe IEEERSJ International Conference on Intelligent Robots andSystems IROS 2015 pp 3045ndash3051 Germany October 2015

[16] KR AGILUS sixx WP With W and C Variants SpecificationKUKA Roboter GmbH p 133 2015

[17] Z Qin Q Ren L Baron M Balazinski and L Birglen ldquoJointfriction identification for robots using TSK fuzzy system basedon subtractive clusteringrdquo in Proceedings of the 2008 AnnualMeeting of the North American Fuzzy Information ProcessingSociety NAFIPS 2008 USA May 2008

[18] M Indri I Lazzero A Antoniazza andAM Bottero ldquoFrictionmodeling and identification for industrial manipulatorsrdquo inProceedings of the 2013 IEEE 18th International Conference onEmerging Technologies and Factory Automation ETFA 2013Italy September 2013

[19] E-J Kim K Seki M Iwasaki and S-H Lee ldquoModeling andidentification of serial two-link manipulator considering jointnonlinearities for industrial robots controlrdquo in Proceedings ofthe 25th IEEERSJ International Conference on Robotics andIntelligent Systems IROS 2012 pp 2718ndash2723 Portugal October2012

[20] V Sathish S Ramaswamy and S Butail ldquoA simulation basedapproach to detect wear in industrial robotsrdquo in Proceedings ofthe 11th IEEE International Conference on Automation Science

and Engineering CASE 2015 pp 1570ndash1575 Sweden August2015

[21] L Le Tien A Albu-Schaffer A de Luca and G HirzingerldquoFriction observer and compensation for control of robots withjoint torquemeasurementrdquo in Proceedings of the IEEERSJ Inter-national Conference on Intelligent Robots and Systems (IROSrsquo08) pp 3789ndash3795 Nice France September 2008

[22] E D Markus ldquoFlatness based feed-forward control of a flexiblerobot arm under gravity and joint frictionrdquo in Proceedings ofthe 12th International Conference on Informatics in ControlAutomation and Robotics ICINCO 2015 pp 174ndash180 FranceJuly 2015

[23] C Lehmann B Olofsson K Nilsson et al ldquoRobot JointModeling and Parameter Identification Using the ClampingMethodrdquo IFAC Proceedings Volumes vol 46 no 9 pp 813ndash8182013

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018


Active and Passive Electronic Components

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of



Journal ofEngineeringVolume 2018

SensorsJournal of



RotatingMachinery


Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation


Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom


onmodelsmay have limited accuracywhich deteriorates withtime due to various unmeasured factors [4]

This study explores friction in one of the joints of theKUKA KR10 R900 sixx robot-manipulator [16] Its controlsystem allows measuring the motor temperature only Toaccount for the thermal factor influence on friction in therobotrsquos joint a technique is proposed based on introducinga motor temperature correction which is regularly calcu-lated using the friction torque registered when special axialmotions are performed by the robot

2 Overview

21 Friction Function and Its Components Axial controltorque u of the industrial robot can be described by

u = M (120593) + C (120593 ) + L minus sign () 120591 (1)

where M(120593) is inertia matrix C(120593 ) is vector of centrifugaland Coriolis forces 120593 and are angular positionsvelocities and accelerations in the robotrsquos axles respectivelyL is axial load torque 120591 is friction torque counter-directed tothe axial velocity

Friction is a very complex process and to some degreeunique to every mechanism [14] The studies dedicated totribology use a wide range of friction function formalizations[2 17] One of the fullest descriptions of the friction torque 120591in a robot-manipulator joint is presented in the study [10] Itestablishes a dependence of friction torque 120591 on load torqueL angular velocity and the jointrsquos temperature T and canbe presented as follows

120591 (L T) = 120591L (L) + 120591s () + 120591v (T) + 120591c (2)

where 120591L is friction component dependent on load 120591s isStribeck friction 120591v is viscous friction 120591c is Coulomb frictionLet us consider these components in greater detail

Coulomb friction 120591c is related to the force of intermolec-ular interaction of the mating surfaces and is a constant [14]

Theway load L has an effect on the friction component 120591L is linked with the pressing force of the mating surfaces theirroughness the presence of lubricant between them and thematerial properties of the parts [14] Axial load L of the robotconsists of holding torque lg(120593) (caused by the gravitationalforce and dependent on the robotrsquos kinematics) and loadtorque l(t) (caused by the external forces applied to the robotat the current time t)

L = lg (120593) + l (t) (3)

To describe the dependence of friction on load both linear[5 7 10] and nonlinear functions [2] can be used Most oftenthis dependence is described as follows

120591L (L) = 120572 |L| 120572 gt 0 (4)

where 120572 is a friction coefficientThe viscous friction component 120591v arises when the jointrsquos

mechanical elements move in viscous fluid This dependenceis often expressed linearly in terms of velocity [6 9 15

17] in some cases a nonlinear function approximation isused power [12] trigonometrical [18] and exponential [11]functions The value and characteristics of viscous frictionare substantially dependent on the viscosity of lubricant used[14] which in turn depends on temperature For this reasona more general description of viscous friction involves thejointrsquos both velocity and temperature T In the studies[2 5 10] this dependence is formalized as follows

120591v (T) = (120573 + 120574eminusT120575+1205750) 10038161003816100381610038161003816100381610038161003816 120573 120574 120575 1205750 gt 0 (5)

where 120573 120574 are coefficients to characterize viscosity 120575 1205750 arecoefficients to take temperature into account

The nature of the Stribeck effect 120591s is related to thecatching of peaks on rough surfaces at themicrolevel througha layer of liquid lubricant The effect depends on the finishof the surfaces lubricant viscosity and quantity and it isregistered at low velocities and canmanifest itself to a varyingdegree The Stribeck effect is identified in the mechanismsexplored in the studies [3 18 19] where it is described asfollows

120591s () = 120582eminus|s|120583 10038161003816100381610038161003816100381610038161003816 (6)

where s ndash Stribeckrsquos velocity 120582 and 120583 ndash coefficients tocharacterize the amplitude and slope of the nonlinearity ofStribeck friction respectively The expression (6) reflects thedependence of Stribeck friction on velocity only howeverthere are studies which to describe this component furtherhave taken into account the influence of the jointrsquos tempera-ture and load [10] There are also a number of joint designsin which the Stribeck effect has been either not detected orneglected by the researchers [2 4 5 12 13 20]

22 FrictionMeasurement Methods Themost common solu-tion for estimating friction torque 120591 in robot-manipulatorjoints is separating it using measurable control torque uwhere the contribution of the remaining components of theequation (1) is calculated with due regard to the mass-inertialcharacteristics of the robotrsquos links and then eliminated[12 19ndash22] If these characteristics are unknown variousadditional computational approaches andmethodologies canbe used to identify them [2ndash4 6ndash9 13]

The study [1] proposes an alternative method that doesnot require identifying mass-inertial characteristics of therobotrsquos links In a nutshell it consists in paired counter-directed motions performed by one robot axle at constantvelocity || = const without external load l = 0 At constantspeed there is no influence of inertia forces Centrifugal andCoriolis forces are directed radially from considered jointand create torque acting only on subsequent joints and linksThen according to (1) allow the components of the inertiaforcesM(120593) = 0 and those of centrifugal and Coriolis forcesC(120593 ) = 0 while the axial control torque u counteractsonly the friction forces 120591 and the load torque L(120593) = lg(120593)caused by the gravitational forces Both components dependon the angle 120593 Notably the gravitational component of theload torque lg(120593) does not depend on the motion directionsign( 120593) whilst the friction torque component 120591 changes signas the direction reverses





end for








x

yz

Axis 1

Axis 2

Axis 3Axis 4

Axis 5

Axis 6

+-170deg

+45deg-190deg

+156 deg-120deg

+-185deg

+-120deg

+-350deg







+ -



100

20

30

40

50

60

70

80

90

Fric

tion

(Nm

)



L(L) + C( T)

=90 degs T=31 Co


=338 degs T=32 Co

=191 degs T=32 Co




120591 (L T) = 120591L (L) + G (T) (9)


0 100 200 300 400

20

30

40

50

60

70Warming Cooling

Time t (min)

Tem

pera

ture

(C)o

Tm

T＝



120591L (L) = 120572L2 (10)
















end while







20 25 30 35 40 45 50 55 60 6520

30

40

50

60

70

80

90

100

110

warmingcooling


Fric

tion

(Nm

)

1=191 degs

4=1012 degs

3=675 degs

2=338 degs


20

30

40

50

60

70

80

90

100

110

Fric

tion

(Nm

)

warmingcooling


4=1012 degs

3=675 degs






25 30 35 40 45 502020

30

40

50

60

70

80

90

100

110


Aver

age f

rictio

nlowast

(Nm

)


1=191 degs

2=338 degs

3=675 degs

4=1012 degs






120591v (T0T) = 120591v (T0) minus 120591v (T) (14)




20 40 60 80 100 120 140 160 180 200

20

40

60

80

100

120

0

140

o


Fric

tion

(Nm

)

T0=20 C





120591 (L T) = 120591c + 120572L2 + (120573 + 120574eminusT120575) 1003816100381610038161003816100381610038161003816100381612 (16)





T (t) = Tm (t) + T (ta) (17)






















end while







40

60

80

100

120

25 30 35 40 45 50 55 60

0

0 10025 50 75



Time t (min)


Fric

tion

(Nm

)

minus20

minus10


oΔT

(C)

4=1012 degs

3=675 degs

2=338 degs

1=191 degs





Time t (min)

30 35 40 45 50 55 60

30

40

50

60

70

80

90

100

20

010

300 0100 50150200


Fric

tion

(Nm




oΔT

(C)

4=1012 degs

2=338 degs

1=191 degs

3=675 degs



120576 = radic 1nm

nsumi=1

msumj=1(120591(j)i minus 120591(j)i )2 (21)





1

2

3

4

5

6


warmingcooling

0

RMS

corr

ectio

n er

ror

(Nm

)



4 Conclusions






Data Availability




Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






end for








x

yz

Axis 1

Axis 2

Axis 3Axis 4

Axis 5

Axis 6

+-170deg

+45deg-190deg

+156 deg-120deg

+-185deg

+-120deg

+-350deg







+ -



100

20

30

40

50

60

70

80

90

Fric

tion

(Nm

)



L(L) + C( T)

=90 degs T=31 Co


=338 degs T=32 Co

=191 degs T=32 Co




120591 (L T) = 120591L (L) + G (T) (9)


0 100 200 300 400

20

30

40

50

60

70Warming Cooling

Time t (min)

Tem

pera

ture

(C)o

Tm

T＝



120591L (L) = 120572L2 (10)
















end while







20 25 30 35 40 45 50 55 60 6520

30

40

50

60

70

80

90

100

110

warmingcooling


Fric

tion

(Nm

)

1=191 degs

4=1012 degs

3=675 degs

2=338 degs


20

30

40

50

60

70

80

90

100

110

Fric

tion

(Nm

)

warmingcooling


4=1012 degs

3=675 degs






25 30 35 40 45 502020

30

40

50

60

70

80

90

100

110


Aver

age f

rictio

nlowast

(Nm

)


1=191 degs

2=338 degs

3=675 degs

4=1012 degs






120591v (T0T) = 120591v (T0) minus 120591v (T) (14)




20 40 60 80 100 120 140 160 180 200

20

40

60

80

100

120

0

140

o


Fric

tion

(Nm

)

T0=20 C





120591 (L T) = 120591c + 120572L2 + (120573 + 120574eminusT120575) 1003816100381610038161003816100381610038161003816100381612 (16)





T (t) = Tm (t) + T (ta) (17)






















end while







40

60

80

100

120

25 30 35 40 45 50 55 60

0

0 10025 50 75



Time t (min)


Fric

tion

(Nm

)

minus20

minus10


oΔT

(C)

4=1012 degs

3=675 degs

2=338 degs

1=191 degs





Time t (min)

30 35 40 45 50 55 60

30

40

50

60

70

80

90

100

20

010

300 0100 50150200


Fric

tion

(Nm




oΔT

(C)

4=1012 degs

2=338 degs

1=191 degs

3=675 degs



120576 = radic 1nm

nsumi=1

msumj=1(120591(j)i minus 120591(j)i )2 (21)





1

2

3

4

5

6


warmingcooling

0

RMS

corr

ectio

n er

ror

(Nm

)



4 Conclusions






Data Availability




Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



+ -



100

20

30

40

50

60

70

80

90

Fric

tion

(Nm

)



L(L) + C( T)

=90 degs T=31 Co


=338 degs T=32 Co

=191 degs T=32 Co




120591 (L T) = 120591L (L) + G (T) (9)


0 100 200 300 400

20

30

40

50

60

70Warming Cooling

Time t (min)

Tem

pera

ture

(C)o

Tm

T＝



120591L (L) = 120572L2 (10)
















end while







20 25 30 35 40 45 50 55 60 6520

30

40

50

60

70

80

90

100

110

warmingcooling


Fric

tion

(Nm

)

1=191 degs

4=1012 degs

3=675 degs

2=338 degs


20

30

40

50

60

70

80

90

100

110

Fric

tion

(Nm

)

warmingcooling


4=1012 degs

3=675 degs






25 30 35 40 45 502020

30

40

50

60

70

80

90

100

110


Aver

age f

rictio

nlowast

(Nm

)


1=191 degs

2=338 degs

3=675 degs

4=1012 degs






120591v (T0T) = 120591v (T0) minus 120591v (T) (14)




20 40 60 80 100 120 140 160 180 200

20

40

60

80

100

120

0

140

o


Fric

tion

(Nm

)

T0=20 C





120591 (L T) = 120591c + 120572L2 + (120573 + 120574eminusT120575) 1003816100381610038161003816100381610038161003816100381612 (16)





T (t) = Tm (t) + T (ta) (17)






















end while







40

60

80

100

120

25 30 35 40 45 50 55 60

0

0 10025 50 75



Time t (min)


Fric

tion

(Nm

)

minus20

minus10


oΔT

(C)

4=1012 degs

3=675 degs

2=338 degs

1=191 degs





Time t (min)

30 35 40 45 50 55 60

30

40

50

60

70

80

90

100

20

010

300 0100 50150200


Fric

tion

(Nm




oΔT

(C)

4=1012 degs

2=338 degs

1=191 degs

3=675 degs



120576 = radic 1nm

nsumi=1

msumj=1(120591(j)i minus 120591(j)i )2 (21)





1

2

3

4

5

6


warmingcooling

0

RMS

corr

ectio

n er

ror

(Nm

)



4 Conclusions






Data Availability




Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia













end while







20 25 30 35 40 45 50 55 60 6520

30

40

50

60

70

80

90

100

110

warmingcooling


Fric

tion

(Nm

)

1=191 degs

4=1012 degs

3=675 degs

2=338 degs


20

30

40

50

60

70

80

90

100

110

Fric

tion

(Nm

)

warmingcooling


4=1012 degs

3=675 degs






25 30 35 40 45 502020

30

40

50

60

70

80

90

100

110


Aver

age f

rictio

nlowast

(Nm

)


1=191 degs

2=338 degs

3=675 degs

4=1012 degs






120591v (T0T) = 120591v (T0) minus 120591v (T) (14)




20 40 60 80 100 120 140 160 180 200

20

40

60

80

100

120

0

140

o


Fric

tion

(Nm

)

T0=20 C





120591 (L T) = 120591c + 120572L2 + (120573 + 120574eminusT120575) 1003816100381610038161003816100381610038161003816100381612 (16)





T (t) = Tm (t) + T (ta) (17)






















end while







40

60

80

100

120

25 30 35 40 45 50 55 60

0

0 10025 50 75



Time t (min)


Fric

tion

(Nm

)

minus20

minus10


oΔT

(C)

4=1012 degs

3=675 degs

2=338 degs

1=191 degs





Time t (min)

30 35 40 45 50 55 60

30

40

50

60

70

80

90

100

20

010

300 0100 50150200


Fric

tion

(Nm




oΔT

(C)

4=1012 degs

2=338 degs

1=191 degs

3=675 degs



120576 = radic 1nm

nsumi=1

msumj=1(120591(j)i minus 120591(j)i )2 (21)





1

2

3

4

5

6


warmingcooling

0

RMS

corr

ectio

n er

ror

(Nm

)



4 Conclusions






Data Availability




Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



20 25 30 35 40 45 50 55 60 6520

30

40

50

60

70

80

90

100

110

warmingcooling


Fric

tion

(Nm

)

1=191 degs

4=1012 degs

3=675 degs

2=338 degs


20

30

40

50

60

70

80

90

100

110

Fric

tion

(Nm

)

warmingcooling


4=1012 degs

3=675 degs






25 30 35 40 45 502020

30

40

50

60

70

80

90

100

110


Aver

age f

rictio

nlowast

(Nm

)


1=191 degs

2=338 degs

3=675 degs

4=1012 degs






120591v (T0T) = 120591v (T0) minus 120591v (T) (14)




20 40 60 80 100 120 140 160 180 200

20

40

60

80

100

120

0

140

o


Fric

tion

(Nm

)

T0=20 C





120591 (L T) = 120591c + 120572L2 + (120573 + 120574eminusT120575) 1003816100381610038161003816100381610038161003816100381612 (16)





T (t) = Tm (t) + T (ta) (17)






















end while







40

60

80

100

120

25 30 35 40 45 50 55 60

0

0 10025 50 75



Time t (min)


Fric

tion

(Nm

)

minus20

minus10


oΔT

(C)

4=1012 degs

3=675 degs

2=338 degs

1=191 degs





Time t (min)

30 35 40 45 50 55 60

30

40

50

60

70

80

90

100

20

010

300 0100 50150200


Fric

tion

(Nm




oΔT

(C)

4=1012 degs

2=338 degs

1=191 degs

3=675 degs



120576 = radic 1nm

nsumi=1

msumj=1(120591(j)i minus 120591(j)i )2 (21)





1

2

3

4

5

6


warmingcooling

0

RMS

corr

ectio

n er

ror

(Nm

)



4 Conclusions






Data Availability




Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



20 40 60 80 100 120 140 160 180 200

20

40

60

80

100

120

0

140

o


Fric

tion

(Nm

)

T0=20 C





120591 (L T) = 120591c + 120572L2 + (120573 + 120574eminusT120575) 1003816100381610038161003816100381610038161003816100381612 (16)





T (t) = Tm (t) + T (ta) (17)






















end while







40

60

80

100

120

25 30 35 40 45 50 55 60

0

0 10025 50 75



Time t (min)


Fric

tion

(Nm

)

minus20

minus10


oΔT

(C)

4=1012 degs

3=675 degs

2=338 degs

1=191 degs





Time t (min)

30 35 40 45 50 55 60

30

40

50

60

70

80

90

100

20

010

300 0100 50150200


Fric

tion

(Nm




oΔT

(C)

4=1012 degs

2=338 degs

1=191 degs

3=675 degs



120576 = radic 1nm

nsumi=1

msumj=1(120591(j)i minus 120591(j)i )2 (21)





1

2

3

4

5

6


warmingcooling

0

RMS

corr

ectio

n er

ror

(Nm

)



4 Conclusions






Data Availability




Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia












end while







40

60

80

100

120

25 30 35 40 45 50 55 60

0

0 10025 50 75



Time t (min)


Fric

tion

(Nm

)

minus20

minus10


oΔT

(C)

4=1012 degs

3=675 degs

2=338 degs

1=191 degs





Time t (min)

30 35 40 45 50 55 60

30

40

50

60

70

80

90

100

20

010

300 0100 50150200


Fric

tion

(Nm




oΔT

(C)

4=1012 degs

2=338 degs

1=191 degs

3=675 degs



120576 = radic 1nm

nsumi=1

msumj=1(120591(j)i minus 120591(j)i )2 (21)





1

2

3

4

5

6


warmingcooling

0

RMS

corr

ectio

n er

ror

(Nm

)



4 Conclusions






Data Availability




Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



40

60

80

100

120

25 30 35 40 45 50 55 60

0

0 10025 50 75



Time t (min)


Fric

tion

(Nm

)

minus20

minus10


oΔT

(C)

4=1012 degs

3=675 degs

2=338 degs

1=191 degs





Time t (min)

30 35 40 45 50 55 60

30

40

50

60

70

80

90

100

20

010

300 0100 50150200


Fric

tion

(Nm




oΔT

(C)

4=1012 degs

2=338 degs

1=191 degs

3=675 degs



120576 = radic 1nm

nsumi=1

msumj=1(120591(j)i minus 120591(j)i )2 (21)





1

2

3

4

5

6


warmingcooling

0

RMS

corr

ectio

n er

ror

(Nm

)



4 Conclusions






Data Availability




Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



1

2

3

4

5

6


warmingcooling

0

RMS

corr

ectio

n er

ror

(Nm

)



4 Conclusions






Data Availability




Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia
























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


friction model of industrial robot joint with temperature...

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