Research Article A Novel Time-Varying Friction...

Research ArticleA Novel Time-Varying Friction CompensationMethod for Servomechanism

Bin Feng1 Dongsheng Zhang12 Jun Yang1 and Shijie Guo1

1School of Mechanical Engineering Xirsquoan Jiaotong University No 28 Xianning West Road Xirsquoan Shaanxi 710049 China2State Key Laboratory for Manufacturing Systems Engineering Xirsquoan Jiaotong University Xirsquoan Shaanxi 710049 China

Correspondence should be addressed to Dongsheng Zhang zdsxjtu1974gmailcom

Received 29 May 2014 Revised 10 September 2014 Accepted 12 September 2014

Academic Editor Xingsheng Gu

Copyright copy 2015 Bin Feng et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Friction is an inevitable nonlinear phenomenon existing in servomechanisms Friction errors often affect their motion and contouraccuracies during the reverse motion To reduce friction errors a novel time-varying friction compensation method is proposed tosolve the problem that the traditional friction compensation methods hardly deal with This problem leads to an unsatisfactoryfriction compensation performance and the motion and contour accuracies cannot be maintained effectively In this methoda trapezoidal compensation pulse is adopted to compensate for the friction errors A generalized regression neural networkalgorithm is used to generate the optimal pulse amplitude function The optimal pulse duration function and the pulse amplitudefunction can be established by the pulse characteristic parameter learning and then the optimal friction compensation pulse canbe generated The feasibility of friction compensation method was verified on a high-precision X-Y worktable The experimentalresults indicated that the motion and contour accuracies were improved greatly with reduction of the friction errors in differentworking conditions Moreover the overall friction compensation performance indicators were decreased by more than 54 andthis friction compensation method can be implemented easily on most of servomechanisms in industry

1 Introduction

Friction is a very complex time-varying phenomenon andexists in the sliding surfaces of servomechanism extensivelysuch as bearings and guide-ways Friction errors occurduring the reverse motion and the servo motion controllershardly deal with them effectively Friction errors affectmotion accuracy significantly and degrade contour accuracyof computer numerical control (CNC) machine tools [1ndash3] More and more researchers have suggested that thefriction should be taken into consideration seriously for high-precision servomechanisms [4] Friction phenomenon canbe alleviated by modifying servomechanism design or usinglubricants However even with the proper lubrication anddesign friction cannot be completely eliminated [5]

To reduce friction errors the following two ways are usu-ally adopted One is to improve the bandwidth of servomech-anism such as using a disturbance observer employing

a neural network controller and optimizing control param-eters [6ndash8] The other is to develop a friction compensationstrategy with small modifications to the servo motion con-troller such as adopting a frictionmodel applying a repetitivecontrol strategy and using a friction compensation pulse [9]The disturbance observer is often used to estimate the exter-nal perturbations and reduce the adverse effects caused by thenonlinear uncertainties [10] Using the disturbance observeror optimizing the control parameters friction errors can besuppressed effectively to a certain extent but the desiredperformance can hardly be achieved [11] Even if the neuralnetwork controller can effectively reduce the friction errorsdesigning such kind of controller is very complicated and itis helpless for most of servomechanisms adopting the con-ventional proportion-integration-differentiation (PID) con-trollers [12] To solve these problems in-depth study on thefriction compensation strategy has been carried out Frictionmodels are commonly used to predict friction force and to

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 269391 16 pageshttpdxdoiorg1011552015269391

2 Mathematical Problems in Engineering

realize friction compensation [13] Several friction modelshave been applied in describing the friction characteristicsof sliding surfaces such as Stribeck model Dahl model andLuGre model [14] However friction is time-varying andclosely related to humidity temperature ageing inevitablemisalignment of axis unevendistributed lubricant and so onSo it is very difficult to obtain an accurate frictionmodel [15]Meanwhile the identification results are unrepeatable evenfor well-prepared experiments and can hardly be applied inthe servomechanisms in industry [16] The repetitive controlstrategy proposed by Tung et al [17] compensated the frictionerrors by adding the compensation command based on theprevious measured friction errors to the position commandObviously this approach cannot be applied in the workingconditions where the feed rate and trajectory vary frequentlyMei et al adopted a rectangular compensation pulse to reducefriction errors [18] The rectangular compensation pulse isadded to the position commandThe amplitude and durationof optimal rectangular compensation pulse can be obtainedby the friction compensation experiments Similarly Chenet al used a double compensation pulse to compensate thefriction errors [19]However it iswell known that the positionloop bandwidth is much smaller than the velocity loopbandwidth and the current loop bandwidth in servomech-anisms [20] Therefore adding the compensation pulse tothe position command limits friction errors to be furtherreduced Moreover one of themost important characteristicsof friction is the time-varying characteristic So far this wayneglects the time-varying characteristic It is considered thatthe friction is a time-invariant physical phenomenon How-ever specifically for servomechanism in industry frictionmay vary with time greatly and the friction compensationperformance becomes worse As a result motion and contouraccuracies will degrade and cannot be maintained effectivelyTherefore it is very urgent to develop a new friction compen-sation method against the time-varying friction

In this paper a novel time-varying friction compensationmethod is proposed to solve this problem When the fric-tion existing in servomechanism has changed greatly as aresult the friction compensation performance cannot satisfythe required friction compensation performance A noveltrapezoidal compensation pulse was adopted to compensatethe friction errors and a friction compensation performanceevaluation function was designed Then an optimal pulseduration function and a pulse amplitude function wereestablished by the pulse characteristic parameter learningAccording to these functions the optimal pulse amplitudeand the pulse duration can be calculated under differentworking conditions Hence the required friction compen-sation performance can be obtained and the motion andcontour accuracies of servomechanism can be guaranteedeffectively In addition according to different working con-ditions the friction errors can be compensated adaptively

The paper is organized as follows The generation mech-anism of friction errors and compensation strategy areintroduced briefly in Section 2 Then the novel time-varyingfriction compensation method is mainly described in Sec-tions 3 and 4 Experimental results are presented in Section 5Finally conclusions are given in Section 6

2 Analysis of Friction Errorsand Friction Compensation

To reveal the generation mechanism of friction errors and tovalidate the friction compensation method proposed by thispaper a high-precision X-Y worktable was set up Figure 1shows the high-precision X-Y worktable Each axis is drivenby a ball screw coupled to a servo motor and its amplifierThe X-Y worktable motion is controlled by an open CNCsystem composed of a host computer and a slave computer[21 22] The host computer is responsible for the operationdata sampling motion monitoring and so on A digitalPID motion controller is developed on the slave computerand generates the control signals which are sent to thecorresponding amplifiers Some logic control operations canbe implemented by the logic signals such as emergency stopThe position feedback signals are produced by the gratingscales installed on the X-Y worktable The velocity feedbacksignals can be obtained by the encoders directly coupled tothe servo motors These feedback signals are sampled by thedigital PID motion controller In addition the related dataof each axis such as position feedback and velocity feedbackare transmitted to the host computer in real time The mainspecifications of this high-precisionX-Y worktable are shownin Table 1

As shown in Figure 2 the controller structure of 119909-axisincludes a cascade PID feedback controller a velocity feedfor-ward controller and an acceleration feedforward controllerThe controller structure of119910-axis is similarwith the controllerstructure of 119909-axis The cascade PID feedback controller iscomposed of a position controller a velocity controller anda current controller to reject both the steady-state errorsand the external disturbances In addition these feedforwardcontrollers are used to further improve the dynamics of ser-vomechanism The position controller and the velocity con-troller adopt the proportion controller and the proportion-integration controller respectively The current controller iscontained in the amplifier and can be equivalent as a torqueconstant Meanwhile to show the superiority of the proposedmethod a disturbance observer was developed to suppressthe friction errors The current differences between thenormal model output and physical plant output are inducedby the external nonlinear friction force These differences areconsidered as equivalent perturbationswhich can be added tocurrent commandThus the friction errors can be suppressedto a certain extent

To make the proposed method clearly the friction com-pensation method is illustrated with the worktable in the 119909-direction as an example in this paper Figure 3 shows theactual velocity and the tracking errors during the reversemotion of the worktable in the 119909-direction with the feedrate 119865 = 1000mmsdotminminus1 and the circular radius 119877 = 25mmAs shown in Figure 3 the prominent errors caused by thefriction force are friction errors which degrade the motionaccuracy of the worktable The whole process of reversemotion will be briefly introduced as follows

When the worktable arrives at a reverse position at themoment 119905stick its motion is stopped From then on thefriction errors begin to generate and to increase until they

Mathematical Problems in Engineering 3

X-axisamplifier

Host computer

Slave computer

Open CNC system

Y-axis

Y-axis

amplifier

Data bus

X-axis control signal

Y-axis control signal

Logic control signals

Feedback signals

Limit switch

X-axis grating scale

Y-axis grating scale

X-Y worktable

Ballscrew

Cable

X-axis

Figure 1 High-precision X-Y worktable

Positioncontroller

Position feedback Velocity feedback

Positioncommand

Velocitycommand

Velocityfeedforward

controller

Accelerationfeedforward

controller

Velocitycontroller Amplifier Servomotor Mechanical

subsystem

Plant

Inverse ofnormal model

Lowpass filter Disturbance observer

Currentcommand

Friction force

+

+ + +

+

+

+

+

++

minus minus

minus

minus

Trapezoidalcompensation

pulse Vb

Figure 2 Controller structure of 119909-axis

40 4005 401 4015 402 4025 403

0

0005

001

0015

Trac

king

erro

rs (m

m)

0

1

2

3

Tracking errorsActual velocity

minus0005

t (s)

minus1

Actu

al v

eloc

ity (m

mmiddotsminus

1)

tstick

Tmax

tslip terr tn

TzTdTb

Figure 3 Dynamic reverse motion process of the worktable in the119909-direction

Table 1 Main specifications of high-precision119883-119884 worktable

Parameter ValueWorktable area (mmtimesmm) 560 times 420119883119884 stroke (mmtimesmm) 250 times 150Maximum allowed acceleration 119886

119897

(mmsdotsminus2) 1500Inertia of the plant for 119909-axis (kgsdotm2) 000298Inertia of the plant for 119910-axis (kgsdotm2) 000358Torque constant (NsdotmsdotVminus1) 268Rated torque of servo motor (Nsdotm) 716Sampling period 119879 (s) 0001Screw lead (mm) 16Resolution of encoder (120583m) 05Resolution of scale (120583m) 05

reach a peak at the moment 119905err Then when the drivingforce overcomes the break-away force the worktable startsto slip at the moment 119905slip However it is essential to note


that the friction affects the position control system in theform of external force or torque and takes time for the effectsof friction to be transformed into the position output Inaddition there exists a backlash during the reverse motionthis contributes to an additional delay Let 119879

119889

be the sum ofthese delays and a transition time 119879

119887

from the moment 119905stickto the moment of 119905slip can be expressed as

119879119887

= 119905slip minus 119905stick (1)

The moment 119905err can be described as

119905err = 119879max + 119905stick (2)

where 119879max is a time interval from the moment 119905stick to themoment 119905err and it can be expressed as

119879max = 119879119889 + 119879119887 (3)

The elastic junctions appear in the sliding surfaces andbehave like springs during the reverse motion There is apresliding displacement 119863

119887

which is an approximately linearfunction of the driving force till the driving force reachesthe break-away force Meanwhile there is an additional mea-sured error Δ119864 caused by the external noise and grating scaleresolution The presliding displacement 119863

119887

and measurederror Δ119864 are very small and can be neglected The delay time119879119889

is inevitable and hard to be further decreased Howeverits value is much smaller than the transition time 119879

119887

andits effects on the tracking errors can be ignored Moreoverthe position command is not influenced during the reversemotion Thus a great peak error 119864

119901

is produced and can beexpressed as

119864119901

= 119875119888119909

(119905stick + 119879max) minus (119875119891119909 (119905stick) + 119863119887 + Δ119864)

= 119875119888119909

(119905stick + 119879119889 + 119879119887) minus 119875119891119909 (119905stick) minus 119863119887 minus Δ119864

asymp 119875119888119909

(119905stick + 119879119887) minus 119875119891119909 (119905stick)

(4)

where 119875119888119909

and 119875119891119909

are the position command and positionfeedback of the worktable respectively Considering theexisting delay 119879

119889

the worktable actually starts to slip at themoment 119905err At this moment the position command andposition feedback are 119875

119888119909

(119905stick + 119879max) and 119875119891119909(119905stick) + 119863119887 +Δ119864 respectivelyThe peak error119864

119901

rises as the transition time119879119887

and it can be further declined by decreasing the transitiontime 119879

119887

However the transition time 119879119887

is determined by thebreak-away force which varies with the time-varying frictionThus the time-varying characteristic should be consideredseriously in designing a friction compensator The frictionerrors begin to attenuate after the moment 119905err and disappearat the moment 119905

119899

A time interval 119879119911

from the moment 119905stickto the moment 119905

119899

can be expressed as

119879119911

= 119905119899

minus 119905stick (5)

To reduce the friction errors an external friction com-pensation pulse is commonly employed to decrease the tran-sition time 119879

119887

Compared with the position loop the velocityloop has a much higher bandwidth Moreover comparedwith being added to the current command the additionalexternal friction compensation pulse is added to velocitycommand and has a smaller impact on the servomechanismTherefore it is reasonable to add the pulse to velocitycommand Meanwhile as shown in Figure 2 a trapezoidalcompensation pulse119881

119887

is proposed to compensate the frictionerrors Compared with the conventional rectangular frictioncompensation pulse the trapezoidal compensation pulsehas some advantages such as better friction compensationperformance smaller impact on servomechanism and betterflexibility

When the worktable arrives at a reverse position at themoment 119905stick that is 119894119879 the trapezoidal compensation pulse119881119887

compensates the friction errors at the moment 119905stick119879 thatis (119894 + 1)119879 and can be expressed as

119881119887

(119905) =

119860119901

sdot sgn (119889119903

) 119905 isin (119905stick119879 (119879119898 + 119905stick119879))

0

119860119875

(119905) =

(119905 minus 119905stick) 119865119901

119879119903

119905 isin (119905stick119879 (119905stick119879 + 119879119903))

119865119901

119905 isin ((119879119903

+ 119905stick) (119905stick + 119879119898 minus 119879119903))

((119905stick + 119879119898) minus 119905) 119865119901

119879119903

119905 isin ((119905stick + 119879119898 minus 119879119903) (119905stick + 119879119898))

119889119903

= 119875119888119909

((119894 + 1) 119879) minus 119875119888119909

(119894119879)

sgn (119889119903

) =

minus1 119889119903

isin (minusinfin 0)

0 119889119903

= 0

1 119889119903

isin (0 +infin)

(6)


where 119865119901

119879119898

and 119879119903

are the amplitude duration and risetime of the pulse respectively 119879 is the sampling period and119889119903

is the difference between the position command of twoconsecutive sampling instants 119860

119901

is the value of pulse and119879119903

is generally set as a constant The pulse characteristicparameters are the pulse amplitude 119865

119901

and the pulse duration119879119898

With 119889119903

gt 0 the generated trapezoidal compensationpulse 119881

119887

is shown in Figure 4A reasonable friction compensation pulse is essential to

achieve the desired friction compensation performance Evenwith a smaller friction compensation pulse amplitude or ashort friction compensation pulse duration the great frictionerrors still appear On the contrary with a high frictioncompensation pulse amplitude or a long friction compensa-tion pulse duration the great oscillations of tracking errorsappear andmotion accuracy degrades greatly To evaluate thefriction compensation performance a friction compensationperformance evaluation function 119864

119886

can be given as follows

119864119886

=

sum(119894+119873

119886)

119896=119894

10038161003816100381610038161003816119875119888119909

(119896119879) minus 119875119891119909

(119896119879)

10038161003816100381610038161003816

119873119886

119873119886

=

119879119872

119879

(7)

where 119879119872

is the monitoring time and 119873119886

is the numberof sampling points per 119879

119872

The friction compensation per-formance is better as this function value decreases In thispaper a pulse characteristic parameter learning is proposedto search the optimal pulse duration and the pulse amplitudeThe pulse characteristic parameter learning is an automaticoptimization process composed of friction compensationpulse amplitude learning and friction compensation pulseduration learning The friction compensation pulse durationlearning is used to search the optimal pulse duration and toestablish the optimal pulse duration function The frictioncompensation pulse amplitude learning is used to search theoptimal pulse amplitude and to establish the optimal pulseamplitude function

To compensate the friction errors in different trajectoriesit is necessary to establish the relationships between the fric-tion compensation pulse characteristic parameters and thecharacteristic parameters of motion trajectory On one handacceleration is one of the main characteristic parameters ofmotion trajectory and can be easily obtained On the otherhand the acceleration at the moment 119905stick that is reverseacceleration is closely related to the transition time119879

119887

and thebreak-away force [23] Therefore relationships between thereverse acceleration and the pulse characteristic parametersneed to be established to realize friction compensation indifferent trajectories When the high-precision X-Y work-table performs a circular motion the reverse accelerationis equal to the centripetal acceleration [24] Furthermorethe centripetal acceleration can be obtained by the positioncommand of circular motion trajectory and then the reverseacceleration can be calculated as

119886 = (

119865

60

)

2

1

119877

= 1205962

119877 (8)

0

tstickTtstick

Fp

Tr Tr

iT (i + 1)T (i + N)TTm

Ap

(mmmiddotsminus

1)

t (s)

Figure 4 Trapezoidal compensation pulse

where 120596 is the angular velocity of circular motion trajectoryand 119886 is the reverse acceleration According to this equationthe reverse acceleration can be obtained and modified easilyThe position command of a circular motion trajectory for thehigh-precision X-Y worktable can be written as

119875119888119909

= 119877 sin (120596119905)

119875119888119910

= 119877 sin (120596119905)

120596 = radic

119886

119877

(9)

where 119875119888119910

is the position command of the worktable in the119910-direction

The relationships can be built by the friction compensa-tion pulse duration learning and the friction compensationpulse amplitude learning When the friction existing in theservomechanism has changed greatly the values of pulsecharacteristic parameters need to be learned to solve theproblems caused by the time-varying friction Thus themotion and contour accuracies of servomechanism can beguaranteed effectively

3 Friction Compensation PulseDuration Learning

To obtain the optimal pulse duration and to establish theoptimal pulse duration function the friction compensationpulse duration learning is proposed in this paper Consider-ing the practical working conditions the learning efficiencyand the required friction compensation performance differ-ent reverse acceleration intervals and their increments areadopted to satisfy different requirements of friction compen-sation To make this friction compensation method simplethree different reverse acceleration intervals are adopted inthis paper The related parameters are set as follows

119886119894

minimum reverse acceleration119886119898

maximum reverse acceleration1198861

reverse acceleration 1


1198862


pulse amplitude increment1198731

number of steps in the reverse acceleration inter-val 11198732



iteration number of coarse learning119873119865

iteration number of fine learning119879119890

pulse duration increment

The reverse acceleration configuration is shown inFigure 5

Themaximum amplitude of friction compensation pulse119865119898

can be expressed as

119865119898

= 119879119903

119886119897

(10)

where 119886119897

is the maximum allowed acceleration The initialamplitude of the friction compensation pulse 119865

119894

can beobtained as

119865119894

= 120578119865119898

(11)

where 120578 is the friction compensation coefficient Generallythe value of 120578 is between 01 and 015 Without the frictioncompensation the reverse acceleration 119886 updates automati-cally in the order of 119886

119894

1198861

1198862

and 119886119898

At each reverse accel-eration 119886 the worktable performs a sinusoidal movement Asa result the time interval 119879

119911

can be automatically calculatedas 119879

119911119894

1198791199111

1198791199112

and 119879119911119898

respectively based on the trackingerrors The initial pulse duration 119879

119898

can be expressed as

119879119898

= 2119879119903

(12)

The initial value of pulse amplitude 119865119901

can be obtained as

119865119901

= 119865119894

(13)

At the reverse acceleration 119886119894

the worktable performs asinusoidal movement The induced friction errors are com-pensated by the generated friction compensation pulse Inthis paper a pulse duration learning evaluation function119864119905

was designed The pulse duration learning performancebecomes better as this function value decreases The pulseduration learning evaluation function 119864

119905

can be given asfollows

119864119905

=

sum(119894+119873

119905)

119896=119894

10038161003816100381610038161003816119875119888119909

(119896119879) minus 119875119891119909

(119896119879)

10038161003816100381610038161003816

119873119905

119873119905

=

2119879119911

119879

(14)

where 119873119905

is the number of sampling points per 2119879119911

Whenthe sinusoidal movement terminates the pulse amplitude 119865

119901

can be updated as

119865119901

= 119865119901

+ 119865119905119904

(15)

0

Reverseacceleration

interval 1

Reverseacceleration

interval 2

Reverseacceleration

interval 3

ama2a1ai a

Figure 5 Reverse acceleration configuration

and 119865119901

keeps updating until 119865119901

gt 119865119898

Then the pulse dura-tion 119879

119898

can be updated as

119879119898

= 119879119898

+ 119879119890

(16)

The aforementioned process is repeated until 119879119898

gt 119879119911119894

The value of pulse duration 119879

119898

is the 119879119898119894

which is consideredas the optimal pulse duration 119879

119900119898

at the reverse acceleration119886119894

and it corresponds to theminimumof pulse duration learn-ing evaluation function 119864

119905

Similarly the reverse acceleration119886 can be updated automatically in the order of 119886

1

1198862

and119886119898

The corresponding optimal pulse duration 119879119900119898

can beobtained as 119879

1198981

1198791198982

and 119879119898119898

respectively According to theresults of pulse duration learning an optimal pulse durationfunction 119879

119900119898

can be expressed as

119879119900119898

=

119879119898119894

119886 isin (0 119886119894

)

119879119898119894

+

1198791198981

minus 119879119898119894

1198861

minus 119886119894

(119886 minus 119886119894

) 119886 isin [119886119894

1198861

)

1198791198981

+

1198791198982

minus 1198791198981

1198862

minus 1198861

(119886 minus 1198861

) 119886 isin [1198861

1198862

)

1198791198982

+

119879119898119898

minus 1198791198982

119886119898

minus 1198862

(119886 minus 1198862

) 119886 isin [1198862

119886119898

)

119879119898119898

119886 isin [119886119898

119886119897

]

(17)

Thus the corresponding optimal pulse duration 119879119900119898

can becalculated at different reverse accelerations

4 Friction Compensation PulseAmplitude Learning

To obtain the optimal pulse amplitude an optimal pulseamplitude function can be established by a coarse learningstage a fine learning stage and the generation of optimalpulse amplitude function in this paper

41 Coarse Learning Stage An initial pulse amplitude arraycan be obtained in the coarse learning stage The reverseacceleration increment of coarse learning stage Δ119886

119888

in differ-ent reverse acceleration intervals can be calculated as

Δ119886119888

=

1198861

minus 119886119894

1198731

119886 isin [119886119894

1198861

)

Δ119886119888

=

1198862

minus 1198861

1198732

119886 isin [1198861

1198862

)

Δ119886119888

=

119886119898

minus 1198862

1198733

119886 isin [1198862

119886119898

]

(18)


The amplitude increment of the friction compensation pulsein the coarse learning stage 119865

119888119904

can be calculated as

119865119888119904

=

119865119898

minus 119865119894

119873119862

(19)

At the beginning of the coarse learning stage the initial valueof reverse acceleration 119886 can be expressed as

119886 = 119886119888119896

(20)

where 119896 = 1 and 119886 = 119886119888119896

= 119886119894

The initial value of pulseamplitude 119865

119901

can be expressed as

119865119901

= 119865119894

(21)

According to (17) the pulse duration 119879119898


119879119898

= 119879119900119898

(119886) (22)

The monitoring time 119879119872

can be expressed as

119879119872

= 2119879119900119898

(119886) (23)

At the reverse acceleration 119886 the worktable performs asinusoidal movement The induced friction errors can becompensated by the generated friction compensation pulseIn this paper the evaluation function 119864

119886

is employed toevaluate the friction compensation performanceThe frictioncompensation performance becomes better as the functionvalue decreases When the sinusoidal movement terminatesthe pulse amplitude 119865

119901

can be updated as

119865119901

= 119865119901

+ 119865119888119904

(24)

And 119865119901


gt 119865119898

The value of pulseamplitude 119865

119901

is 119865119888119896

which corresponds to the minimum ofevaluation function119864

119886

at the reverse acceleration 119886Then thereverse acceleration 119886 can be updated as

119896 = 119896 + 1

119886 = 119886119888119896

= 119886 + Δ119886119888

(25)

The aforementioned process is repeated until 119886 gt 119886119898

and thecoarse learning stage is finished At different reverse acceler-ations the pulse amplitude array in the coarse learning stagecan be expressed as [119865

1198881

1198651198882

119865119888119896

119865119888119899

] 119896 = 1 2 (119873

1

+1198732

+1198733

+2) 119896 le 119899The corresponding reverse accelera-tion array can be expressed as [119886

1198881

1198861198882

119886119888119896

119886119888119899

] where119886119898

= 119886119888119899

42 Fine Learning Stage To further improve the frictioncompensation performance on the basis of the resultsobtained in the coarse learning stage a fine learning stage isadopted The reverse acceleration array can be expanded as[1198861198881

11988611988811989112

1198861198882

11988611988811989123

1198861198883

119886119888119891(119894)(119894+1)

119886119888(ℎminus1)

119886119888119891(ℎminus1)

119886119888ℎ

]119894 = 1 2 (ℎ minus 1) where ℎ = 2119899 minus 1 119886

119888119899

= 119886119888ℎ

The elementof this array 119886

119888119891(119894)(119894+1)

can be expressed as

Δ119886119891

=

Δ119886119888

2

119886119888119891(119894)(119894+1)

= 119886119888(119894)

+ Δ119886119891

119894 = 1 2 ℎ minus 1

(26)

where Δ119886119891

is the reverse acceleration increment in the finelearning stage The subscript indexes of these elements arerenumbered and can be recorded as

[1198861198911

1198861198912

119886119891119895

119886119891ℎ

]

119895 = 1 2 ℎ ℎ = 2119899 minus 1

(27)

Similarly the generated pulse amplitude array in the coarselearning stage can be expanded as [119865

1198881

11986511988811989112

1198651198882

11986511988811989123

1198651198883

119865119888119891(119894)(119894+1)

119865119888(ℎminus1)

119865119888119891(ℎminus1)

119865119888ℎ

] where ℎ = 2119899minus1 119865119888119899

=

119865119888ℎ

and the element of this pulse amplitude array 119865119888119891(119894)(119894+1)

can be expressed as

119865119888119891(119894)(119894+1)

=

119865119888(119894)

+ 119865119888(119894+1)

2

119894 = 1 2 ℎ minus 1 (28)

The subscript indexes of the elements in the expanded pulseamplitude array are renumbered and can be recorded as

[1198651198871198911

1198651198871198912

1198651198871198913

119865119887119891119895

119865119887119891(ℎminus1)

119865119887119891ℎ

]

119895 = 1 2 ℎ ℎ = 2119899 minus 1

(29)

where 119865119887119891119895

is the element of the expanded pulse amplitudearray The pulse amplitude increment in the fine learningstage 119865

119891119904


119865119891119904

=

119865cs119873119865

(30)

At the beginning of the fine learning stage the initial value ofreverse acceleration 119886 can be expressed as

119886 = 119886119891119895

(31)

where 119895 = 1 and 119886 = 1198861198911

= 119886119894


119901

can be expressed as

119865119901

= 119865119887119891119895

minus

119865119888119904

2

(32)



119879119898

= 119879119900119898

(119886) (33)

The worktable performs a sinusoidal movement at thereverse acceleration 119886 The induced friction errors are com-pensated by the generated friction compensation pulseMeanwhile the evaluation function119864

119886

is also adoptedWhenthe sinusoidal movement terminates the pulse amplitude 119865

119901

can be updated as

119865119901

= 119865119901

+ 119865119891119904

(34)

And 119865119901


gt 119865119887119891119895

+ (119865119888119904

2) Theoptimal value of pulse amplitude119865

119901

is119865119887119891119895

which correspondsto the minimum of evaluation function 119864

119886

at the reverseacceleration 119886 Then the reverse acceleration 119886 can beupdated as

119895 = 119895 + 1

119886 = 119886119891119895

= 119886 + Δ119886119891

(35)


The aforementioned process is repeated until 119886 gt 119886119891ℎ

and the fine learning stage is finished At different reverseaccelerations the optimal pulse amplitude array can beexpressed as

[1198651198911

1198651198912

1198651198913

119865119891119895

119865119891(ℎminus1)

119865119891ℎ

]

119895 = 1 2 ℎ ℎ = 2119899 minus 1

(36)

43 Generation of Optimal Pulse Amplitude Function Dueto the fact that there is a complicated nonlinear relationshipbetween the optimal pulse amplitude array and the corre-sponding reverse acceleration array it is very difficult toachieve the satisfactory fitting performance with the approx-imate equation or the conventional least square methodHowever the neural network has a strong ability in nonlinearfitting and can map the arbitrary nonlinear relationships[25] Meanwhile its learning rule is easy to implement ona computer Therefore the GRNN algorithm is proposed totrain the complicated nonlinear relationship in this paper

An optimal pulse amplitude function (OFGN) is gener-ated by the generalized regression neural network (GRNN)algorithm due to its advantages such as simple structurehigh training efficiency and global convergence Moreoverthe high accurate fitting can be obtained by the OFGN TheGRNN algorithm includes an input layer a mode layer asummation layer and an output layerThenumber of neuronsin the input layer is equal to the dimension of the optimalpulse amplitude array and each neuron is a simply distributedunit which transfers the input variables to the mode layerdirectly The summation layer sums these neurons and theoutput layer exports the value of the optimal pulse amplitude119865119900119901

which can be expressed as

119865119900119901

=

1198651198911

119886 isin (0 119886119894

)

OFGN 119886 isin [119886119894

119886119898

)

119865119891ℎ

119886 isin [119886119898

119886119897

]

(37)

Thus the corresponding value of optimal pulse amplitude119865119900119901

can be calculated at different reverse accelerations 119886

5 Experimental Investigation

To verify the effectiveness of this friction compensationmethod a friction compensation module was developed andembedded into the open CNC system The flowchart ofthe module is shown in Figure 6 When the actual valueof friction compensation performance evaluation functioncannot satisfy the required friction compensation perfor-mance that is 119864

119886

gt 119864119903

it indicates that the pulsecharacteristic parameter learning is necessary Exiting themodule or performing the pulse characteristic parameterlearning is optional If the pulse characteristic parameterlearning is required according to working conditions andthe required friction compensation performance the relatedparameters of friction compensation module are set Thefriction compensation pulse duration learning is performedautomatically and an optimal pulse duration function is

established Then the friction compensation pulse ampli-tude learning is performed automatically and an optimalpulse amplitude function is generated The process of pulsecharacteristic parameter learning is finished When thefriction compensation is enabled the optimal characteristicparameters can be calculated by the optimal pulse amplitudefunction and the pulse duration function The friction errorsare compensated by the generated friction compensationpulse during the reverse motion In addition the frictioncompensation performance can be evaluated and monitoredonline In this paper the setting values of this module areshown in Table 2

With the circular radius119877= 50mm the high-precisionX-Y worktable carries out the friction compensation pulse char-acteristic parameter learning automatically Figure 7 showsthe results of friction compensation pulse characteristicparameter learning at the reverse acceleration 119886 isin [119886

119894

119886119898

]As shown in Figure 7(a) the optimal friction compensationpulse durations of high-precision X-Y worktable decreasecontinuously with the reverse acceleration 119886 The optimalfriction compensation pulse duration of 119910-axis is longer thanthat of 119909-axis The optimal pulse amplitude arrays of high-precision X-Y worktable and the corresponding optimalpulse amplitude function curves are shown in Figure 7(b)The optimal pulse amplitude of 119909-axis is larger than that of119910-axis Moreover it shows that these optimal pulse amplitudefunctions generated by the GRNN algorithm can achieve theaccurate fitting of optimal pulse amplitude arrays

Figure 8 shows the circular contour errors with thefeed rate 119865 = 500mmsdotminminus1 and the circular radius 119877 =25mm It is noted that the prominent contour errors occur infour quadrants The prominent contour errors in quadrantsA and C are mainly induced by the nonlinear frictionduring the reverse motion of 119909-axis Similarly the prominentcontour errors in quadrants B and D are mainly inducedby the nonlinear friction during the reverse motion of 119910-axis Generally the prominent contour errors in quadrantsA and C are similar The same is true with the prominentcontour errors in quadrants B and D Therefore the frictioncompensation performance can be studied by these trackingerrors and contour errors in quadrants A and B and can bemonitored during the monitoring time 119879

119872

To verify the feasibility of this friction compensation

method friction compensation experiments were carriedout on the high-precision X-Y worktable with the radius119877 = 25mm and the feed rates 119865 = 500mmsdotminminus11000mmsdotminminus1 2000mmsdotminminus1 and 3000mmsdotminminus1Moreover to show the superiority of the proposed methoda disturbance observer was designed to suppress the frictionerrors under the aforementioned working conditions Inthis paper the friction compensation performances for thehigh-precisionX-Y worktable are comprehensively evaluatedby a set of friction compensation performance indicators asfollows

119862119883119864

peak value of the contour errors during thereverse motion of 119909-axis119862119884119864

peak value of the contour errors during thereverse motion of 119910-axis


Start

Yes

NoReverse acceleration

No

Friction compensation pulse amplitude learning

Command position

Arrive at the reverse position

End

Friction compensation implementation

Set related parameters offriction compensation module

No

Yes

Yes

No

Yes

Evaluate friction compensation performance and monitor

friction errors (7)

Generation of friction compensation pulse (6)

Generation of optimal pulse amplitude function

(37)

Friction compensation

Calculate optimal pulse duration (17) and amplitude (37)

Pulse characteristic parameter learning

Friction compensation pulse duration learning

Generation of optimal pulse duration function

(17)

Satisfy the required friction compensation

performance

Exit the module

(18)ndash(36)

(10)ndash(16)

Figure 6 Flowchart of friction compensation module

119875119883119864

absolute peak value of the tracking errors duringthe reverse motion of 119909-axis119875119884119864


root mean square of the tracking errors duringthe reverse motion of 119909-axis119864119884119877


absolute mean of the tracking errors during thereverse motion of 119909-axis119864119884119872

absolute mean of the tracking errors during thereverse motion of 119910-axis

These indicators can be calculated by sampling the posi-tion command and the position feedback during the moni-toring time119879

119872

and are comparedwith friction compensation(WFC) disturbance observer (WDOB) and without frictioncompensation (WTFC)

With the circular radius 119877 = 25mm and the feedrates 119865 = 500mmsdotminminus1 and 3000mmsdotminminus1 as shown inFigures 9 and 10 the tracking errors and the contour errors inquadrants A and B are compared during the monitoring time

Table 2 Setting values of friction compensation module

Parameter 119909-axis 119910-axis119886119894

(mmsdotsminus2) 5 5119886119898

(mmsdotsminus2) 150 1501198861



10 101198732

8 81198733

8 8119873119862

10 10119873119865

4 4119864119903

(mm) 001 001119865ts (mmsdotsminus1) 05 05119865119894

(mmsdotsminus1) 05 05119879119890

(s) 0005 0005119879119903

(s) 0003 0003

119879119872

under three situations with friction compensation withdisturbance observer and without friction compensation Asshown in Figures 9 and 10 the friction errors can be decreased


20 40 60 80 100 120 140003

004

005

006

007

008

a (mmmiddotsminus2)

Tom

(s)

X-axisY-axis

(a)

20 40 60 80 100 120 140a (mmmiddotsminus2)

Optimal pulse amplitude array ofOptimal pulse amplitude function curve ofOptimal pulse amplitude array ofOptimal pulse amplitude function curve of

0

05

1

15

2

25

3

35

4

45

Fop

(mmmiddotsminus

1)

X-axisX-axis

Y-axisY-axis

(b)

Figure 7 Results of friction compensation pulse characteristic parameter learning (a) optimal friction compensation pulse duration curves(b) optimal pulse amplitude arrays and the corresponding optimal pulse amplitude function curves

Table 3 Friction compensation performance indicators during the reverse motion of 119909-axis

Trajectory WTFC (120583m) WFC (120583m) WDOB (120583m)119862119883119864

119864119883119872

119875119883119864

119864119883119877

119862119883119864

119864119883119872

119875119883119864

119864119883119877

119862119883119864

119864119883119872

119875119883119864

119864119883119877

119865 = 500mmsdotminminus1119877 = 25mm 119886 = 278mmsdotsminus2 689 333 693 384 250 154 253 161 496 224 473 262




WTFC without friction compensation WFC with friction compensation WDOB with disturbance observer

with friction compensation anddisturbance observer and thecontour errors as well as the tracking errors were decreasedwith the reduction of friction errors So it can be seen that thebetter friction compensation performance can be achieved bythe proposed friction compensation method With differentfeed rates and reverse accelerations the friction compensa-tion performance indicators are shown in Tables 3 and 4As shown in Tables 3 and 4 these friction compensationperformance indicators with the friction compensation aresmaller than those with the disturbance observer Further-more as shown in Table 3 during the reverse motion of119909-axis the friction compensation performance indicatorswere decreased greatly with the friction compensation The119862119883119864

119864119883119872

119875119883119864

and 119864119883119877

with different feed rates weredecreased more than 63 54 63 and 58 respectivelyMeanwhile as shown in Table 4 during the reverse motionof 119910-axis the friction compensation performance indicatorswere decreased greatly as well The 119862

119884119864

119864119884119872

119875119884119864

and 119864119884119877

with different feed rates were decreased more than 71 6271 and 69 respectively Compared with the disturbanceobserver the friction compensationmethod proposed by thispaper ismore effective and feasible to compensate the frictionerrors

To verify the conclusion that this friction compensationmethod can be applied to compensating friction errorswith different motion trajectories different S-shaped motiontrajectories S

1

S2

S3

and S4

based on trapezoidal velocityprofile are adoptedThe parameters ofmotion trajectories S

1

S2

S3

and S4

are as follows the accelerations in the accel-eration and deceleration sections are 10mmsdotsminus2 20mmsdotsminus220mmsdotsminus2 and 30mmsdotsminus2 respectivelyThe velocities in con-stant velocity sections are 10mmsdotsminus1 20mmsdotsminus1 30mmsdotsminus1and 30mmsdotsminus1 respectivelyThemotiondistances are 30mm30mm 50mm and 80mm respectively Moreover thesemotion trajectories have different accelerations velocitiesand distances which can be used to test the limitations of this




119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877






C

B

D

minus10120583m

10120583m

30∘

210∘

60∘

240∘

90∘

270∘

120∘

300∘

150∘

330∘

180∘ 0∘

0120583m

A

Figure 8 Circular contour errors with 119865 = 500mmsdotminminus1 119877 =25mm

proposed method This friction compensation experiment iscarried out on the worktable in the 119910-direction

Figure 11 shows the experimental results with the motiontrajectories of S

1

S2

S3

and S4

Figures 11(b) and 11(d)indicate that the friction errors were decreased greatly insame motion distances at different accelerations Figures11(d) and 11(f) show that the friction errors were decreasedgreatly in different motion distances at same accelerationsFigures 11(b) 11(f) and 11(h) illustrate the friction errors weredecreased greatly in different motion distances at differentaccelerations Meanwhile the different accelerations implydifferent velocities As shown in Figure 11 the motion andcontour accuracies were improved with the reduction of thefriction errors Table 5 shows that the friction compensationperformance indicators were decreased significantly withdifferent motion trajectories Moreover the 119862

119884119864

119864119884119872

119875119884119864

and 119864

119884119877

with the motion trajectories of S1

S2

S3

and S4

were decreased by more than 76 71 76 and 75respectively Tables 4 and 5 together indicate that this frictioncompensation method proposed by this paper can be appliedto different motion trajectories

6 Conclusion and Discussion

Friction is a complex physical phenomenon and varies withtime It exerts some adverse effects on precision motionThe conventional friction compensation methods neglectthe time-varying characteristic and cannot cope with theproblems caused by the time-varying friction effectivelyMeanwhile these methods can hardly compensate the fric-tion errors under different working conditions In this papera novel time-varying friction compensation method is pro-posed The main contributions are as follows

(1) A novel trapezoidal compensation pulse is adoptedin this paper The friction errors can be compensatedby adding the friction compensation pulse to thevelocity command A reasonable friction compensa-tion pulse is essential to achieve the desired frictioncompensation performance To evaluate the frictioncompensation performance a friction compensationperformance evaluation function was designed Thepulse characteristic parameter learning is an auto-matic optimization process and can be employedto search the optimal pulse characteristic parameterand to establish the optimal pulse duration functionand pulse amplitude function Then the optimalpulse duration and the pulse amplitude under dif-ferent working conditions can be obtained Whenthe friction has changed greatly these functions canbe established by the pulse characteristic parameterlearning Thus the required friction compensationperformance can be achieved even if the frictionvaries with time Moreover this method has someadvantages such as automation intelligence flexibil-ity and practicality It can be implemented easily onmost of servomechanisms in industry

(2) A generalized regression neural network algorithm isemployed to train the nonlinear relationship betweenthe optimal pulse amplitude array and the corre-sponding reverse acceleration array An optimal pulseamplitude function is generated by this algorithmThe experimental results show that the accurate fittingof optimal pulse amplitude arrays can be achieved bythe generated optimal pulse amplitude function


0 005 01 015 020

0001

0002

0003

0004

0005

0006

0007

Con

tour

erro

rs (m

m)

TM (s)

(a)

0 005 01 015 020

0001

0002

0003

0004

0005

0006

0007

Con

tour

erro

rs (m

m)

TM (s)

(b)

0 005 01 015 02

0

Trac

king

erro

rs (m

m)

TM (s)

minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

WTFCWFCWDOB

(c)

0

Trac

king

erro

rs (m

m)

0 005 01 015 02TM (s)

minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

WTFCWFCWDOB

(d)

Figure 9 119865 = 500mmsdotminminus1 119877 = 25mm (a) contour errors in quadrant A (b) contour errors in quadrant B (c) corresponding trackingerrors of 119909-axis in quadrant A and (d) corresponding tracking errors of 119910-axis in quadrant B WTFC without friction compensation WFCwith friction compensation WDOB with disturbance observer

Table 5 Friction compensation performance indicators with different S-shaped motion trajectories

Trajectory WTFC (120583m) WFC (120583m)119862119884119864

119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877

S1119871 = 30mm 119886 = 10mmsdotsminus2 718 225 718 302 100 038 100 047

S2

119871 = 30mm 119886 = 20mmsdotsminus2 789 232 789 329 184 069 184 077

S3

119871 = 50mm 119886 = 20mmsdotsminus2 846 245 846 332 192 071 192 082

S4

119871 = 80mm 119886 = 30mmsdotsminus2 928 366 928 455 215 073 215 085

WTFC without friction compensation WFC with friction compensation


0 002 004 006 008 01 012

0

0005

001

0015

002

0025

Con

tour

erro

rs (m

m)

minus0005

TM (s)

(a)

0 002 004 006 008 01 012

0

0002

0004

0006

0008

001

0012

0014

0016

Con

tour

erro

rs (m

m)

minus0002

TM (s)

(b)

0 002 004 006 008 01 012

0

0005

Trac

king

erro

rs (m

m)

TM (s)

minus0025

minus002

minus0015

minus001

minus0005

WTFCWFCWDOB

(c)

0 002 004 006 008 01 012

0

0002

Trac

king

erro

rs (m

m)

TM (s)

minus0016

minus0014

minus0012

minus001

minus0008

minus0006

minus0004

minus0002

WTFCWFCWDOB

(d)


(3)Thenovel time-varying friction compensationmethodwas verified on a high-precision X-Y worktable withdifferent feed rates in different trajectories The fric-tion errors were compensated adaptively and thefriction compensation performance was evaluatedcomprehensively by the friction compensation indi-cators Meanwhile to show the superiority of theproposed friction compensation method the distur-bance observer was developed to suppress the frictionerrors The experiment results show that the pro-posed friction compensationmethod ismore effectiveand feasible to compensate the friction errors Thefriction compensation performance indicators were

decreased by more than 54 and this friction com-pensation method can be used in different workingconditions

Notation

119886 Reverse acceleration (mmsdotsminus2)1198861

Reverse acceleration 1 (mmsdotsminus2)1198862

Reverse acceleration 2 (mmsdotsminus2)119886119888119891(119894)(119894+1)

Element of reverse acceleration array(mmsdotsminus2)

119886119894

Minimum reverse acceleration (mmsdotsminus2)119886119897

Maximum allowed acceleration (mmsdotsminus2)


0 1 2 3 4 5 6 7 80

5

10

15

20

25

30Po

sitio

n co

mm

and

(mm

)

t (s)

S1

(a)

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)

WTFC WFC

minus0008minus0007minus0006minus0005minus0004minus0003minus0002minus0001

TM (s)

(b)

0 1 2 3 4 50

5

10

15

20

25

30

Posit

ion

com

man

d (m

m)

t (s)

S2

(c)

WTFC WFC

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)


TM (s)

(d)

Posit

ion

com

man

d (m

m)

0 1 2 3 4 5 605

101520253035404550

t (s)

S3

(e)

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)

TM (s)

minus001minus0009minus0008minus0007minus0006minus0005minus0004minus0003minus0002minus0001

WTFC WFC

(f)

Posit

ion

com

man

d (m

m)

0 1 2 3 4 5 6 70

1020304050607080

t (s)

S4

(g)

0 005 01 015 02

Trac

king

erro

rs (m

m)

TM (s)

00001


WTFC WFC

(h)

Figure 11 Experimental results with the motion trajectories of S1

S2

and S3

as well as S4

(a) S-shaped motion trajectory S1

(b) trackingerrors at the reverse position of motion trajectory S

1

(c) S-shaped motion trajectory S2

(d) tracking errors at the reverse position of motiontrajectory S

2

(e) S-shaped motion trajectory S3

(f) tracking errors at the reverse position of motion trajectory S3

(g) S-shaped curve motiontrajectory S

4

and (h) tracking errors at the reverse position of motion trajectory S4

WTFC without friction compensation WFC withfriction compensation

119886119898

Maximum reverse acceleration (mmsdotsminus2)119860119901

Value of pulse (mmsdotsminus1)119889119903

Difference of position command (mm)119863119887

Presliding displacement (mm)119864119886

Friction compensation performance eval-uation function (mm)

119864119901

Peak error (mm)119864119903

Required friction compensation perform-ance (mm)

119864119905

Pulse duration learning evaluation func-tion (mm)

119865119887119891119895

Element of expanded pulse amplitudearray (mmsdotsminus1)

119865119888119904

Amplitude increment of friction compen-sation pulse in the coarse learning stage(mmsdotsminus1)

119865119891119904

Pulse amplitude increment in the finelearning stage (mmsdotsminus1)

119865119894

Initial amplitude of the friction compen-sation pulse (mmsdotsminus1)

119865119898

Maximum amplitude of friction compen-sation pulse (mmsdotsminus1)


119865119901

Friction compensation pulse amplitude(mmsdotsminus1)

119865119905119904

Pulse amplitude increment (mmsdotsminus1)1198731

Number of steps in the reverse accelera-tion interval 1

1198732


1198733


119873119886

Number of sampling points per 2119905119898

119873119862

Iteration number of coarse learning119873119865

Iteration number of fine learning119875119888119909

119875119891119909

Position command and position feedbackof the 119909-axis (mm)

119905err Moment at the peak error (s)119905119899

Moment of friction errors that disap-peared (s)

119905slip Moment when the worktable starts tomotion (s)

119905stick Moment at the reverse position (s)119905stick119879 Moment at the moment of (119894 + 1)119879 (s)119879 Sampling period (s)119879119887

Transition time (s)119879119889

Sum of delays (s)119879119890

Pulse duration increment (s)119879119898

Friction compensation pulse duration (s)119879max Time interval from the moment 119905stick to

the moment 119905err (s)1198791198981

1198791198982

and 119879

119898119898

optimal duration at the reverse accelera-tions 119886

1

1198862

and 119886119898

(s)119879119900119898

Optimal pulse duration (s)119879119903

Pulse rise time (s)119879119911

Time interval from the moment 119905stick tothe moment 119905

119899

(s)119879119911119894

1198791199111

1198791199112

and 119879119911119898

Value of time interval 119879119911

at the reverseaccelerations 119886

119894

1198861

1198862

and 119886119898

(s)119879119872

Monitoring time (s)119881119887

Trapezoidal compensation pulseΔ119886

119888

Reverse acceleration increment in thecoarse learning stage (mmsdotsminus2)

Δ119886119891

Reverse acceleration increment in the finelearning stage (mmsdotsminus2)

Δ119864 Additional measured error (mm)120578 Friction compensation coefficient120596 Angular velocity of circular motion tra-

jectory (radsdotsminus1)

Conflict of Interests

The authors declare that they have no financial and personalrelationships with other people or organizations that caninappropriately influence their work there is no professionalor other personal interest of any nature or kind in anyproduct or company that could be construed as influencingthe position presented in or the review of the paper

Acknowledgment

This work was supported by the National Hi-tech ResearchandDevelopment Program of China [Grant 2012AA040701]

References

[1] K Erkorkmaz and Y Altintas ldquoHigh speed CNC system designPart IImodeling and identification of feed drivesrdquo InternationalJournal ofMachineToolsampManufacture vol 41 no 10 pp 1487ndash1509 2001

[2] S-S Yeh Z-H Tsai and P-L Hsu ldquoApplications of integratedmotion controllers for precise CNC machinesrdquo The Interna-tional Journal of Advanced Manufacturing Technology vol 44no 9-10 pp 906ndash920 2009

[3] X-C Xi A-N Poo and G-S Hong ldquoTracking error-basedstatic friction compensation for a bi-axial CNC machinerdquoPrecision Engineering vol 34 no 3 pp 480ndash488 2010

[4] B Armstrong-Helouvry ldquoStick slip and control in low-speedmotionrdquo IEEE Transactions on Automatic Control vol 38 no10 pp 1483ndash1496 1993

[5] C Hsieh and Y-C Pan ldquoDynamic behavior and modelling ofthe pre-sliding static frictionrdquo Wear vol 242 no 1-2 pp 1ndash172000

[6] E Schrijver and J van Dijk ldquoDisturbance observers for rigidmechanical systems equivalence stability and designrdquo Journalof Dynamic Systems Measurement and Control vol 124 no 4pp 539ndash548 2002

[7] XMeiMTsutsumi T Yamazaki andN Sun ldquoStudy of the fric-tion error for a high-speed high precision tablerdquo InternationalJournal of Machine Tools and Manufacture vol 41 no 10 pp1405ndash1415 2001

[8] R R Selmic and F L Lewis ldquoNeural-network approximationof piecewise continuous functions application to friction com-pensationrdquo IEEE Transactions on Neural Networks vol 13 no3 pp 745ndash751 2002

[9] S-S Yeh and J-T Sun ldquoFriction modeling and compensationfor feed drive motions of CNCmilling machinesrdquo Journal of theChinese Society ofMechanical Engineers vol 33 no 1 pp 39ndash492012

[10] W-S Huang C-W Liu P-L Hsu and S-S Yeh ldquoPrecisioncontrol and compensation of servomotors and machine toolsvia the disturbance observerrdquo IEEE Transactions on IndustrialElectronics vol 57 no 1 pp 420ndash429 2010

[11] M Iwasaki T Shibata and N Matsui ldquoDisturbance-observer-based nonlinear friction compensation in table drive systemrdquoIEEEASME Transactions on Mechatronics vol 4 no 1 pp 3ndash81999

[12] S N Huang and K K Tan ldquoIntelligent friction modelingand compensation using neural network approximationsrdquo IEEETransactions on Industrial Electronics vol 59 no 8 pp 3342ndash3349 2012

[13] S-S Yeh and H-C Su ldquoDevelopment of friction identificationmethods for feed drives of CNC machine toolsrdquo The Interna-tional Journal of Advanced Manufacturing Technology vol 52no 1ndash4 pp 263ndash278 2011

[14] H Olsson K J Astrom C C de Wit M Gafvert andP Lischinsky ldquoFriction models and friction compensationrdquoEuropean Journal of Control vol 4 no 3 pp 176ndash195 1998

[15] B Armstrong-Helouvry P Dupont and C C de Wit ldquoAsurvey of models analysis tools and compensation methods for


the control of machines with frictionrdquo Automatica vol 30 no7 pp 1083ndash1138 1994

[16] M-W Sun Z-HWang Y-KWang and Z-Q Chen ldquoOn low-velocity compensation of brushless DC servo in the absence offrictionmodelrdquo IEEE Transactions on Industrial Electronics vol60 no 9 pp 3897ndash3905 2013

[17] E Tung G Anwar and M Tomizuka ldquoLow velocity frictioncompensation and feedforward solution based on repetitivecontrolrdquo in Proceedings of the American Control Conference pp2615ndash2620 IEEE Boston Ma USA June 1991

[18] X Mei M Tsutsumi T Tao and N Sun ldquoStudy on thecompensation of error by stick-slip for high-precision tablerdquoInternational Journal of Machine Tools amp Manufacture vol 44no 5 pp 503ndash510 2004

[19] G S Chen X S Mei and T Tao ldquoFriction compensation usinga double pulse method for a high-speed high-precision tablerdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 225 no 5 pp1263ndash1272 2011

[20] Y Altintas A Verl C Brecher L Uriarte and G PritschowldquoMachine tool feed drivesrdquo CIRP AnnalsmdashManufacturing Tech-nology vol 60 no 2 pp 779ndash796 2011

[21] KK TanK Z TangH FDou and SNHuang ldquoDevelopmentof an integrated and open-architecture precisionmotion controlsystemrdquo Control Engineering Practice vol 10 no 7 pp 757ndash7722002

[22] G Pritschow Y Altintas F Jovane et al ldquoOpen controllerarchitecturemdashpast present and futurerdquo CIRP AnnalsmdashManu-facturing Technology vol 50 no 2 pp 463ndash470 2001

[23] E-C Park H Lim and C-H Choi ldquoPosition control of X-Ytable at velocity reversal using presliding friction characteris-ticsrdquo IEEE Transactions on Control Systems Technology vol 11no 1 pp 24ndash31 2003

[24] D Liu Research on Precision Control and Dynamic Characteris-tics for Servo Driven System in NC Machine Tool Xirsquoan JiaotongUniversity Xirsquoan China 2010

[25] J M Fines and A Agah ldquoMachine tool positioning errorcompensation using artificial neural networksrdquo EngineeringApplications of Artificial Intelligence vol 21 no 7 pp 1013ndash10262008

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


realize friction compensation [13] Several friction modelshave been applied in describing the friction characteristicsof sliding surfaces such as Stribeck model Dahl model andLuGre model [14] However friction is time-varying andclosely related to humidity temperature ageing inevitablemisalignment of axis unevendistributed lubricant and so onSo it is very difficult to obtain an accurate frictionmodel [15]Meanwhile the identification results are unrepeatable evenfor well-prepared experiments and can hardly be applied inthe servomechanisms in industry [16] The repetitive controlstrategy proposed by Tung et al [17] compensated the frictionerrors by adding the compensation command based on theprevious measured friction errors to the position commandObviously this approach cannot be applied in the workingconditions where the feed rate and trajectory vary frequentlyMei et al adopted a rectangular compensation pulse to reducefriction errors [18] The rectangular compensation pulse isadded to the position commandThe amplitude and durationof optimal rectangular compensation pulse can be obtainedby the friction compensation experiments Similarly Chenet al used a double compensation pulse to compensate thefriction errors [19]However it iswell known that the positionloop bandwidth is much smaller than the velocity loopbandwidth and the current loop bandwidth in servomech-anisms [20] Therefore adding the compensation pulse tothe position command limits friction errors to be furtherreduced Moreover one of themost important characteristicsof friction is the time-varying characteristic So far this wayneglects the time-varying characteristic It is considered thatthe friction is a time-invariant physical phenomenon How-ever specifically for servomechanism in industry frictionmay vary with time greatly and the friction compensationperformance becomes worse As a result motion and contouraccuracies will degrade and cannot be maintained effectivelyTherefore it is very urgent to develop a new friction compen-sation method against the time-varying friction

In this paper a novel time-varying friction compensationmethod is proposed to solve this problem When the fric-tion existing in servomechanism has changed greatly as aresult the friction compensation performance cannot satisfythe required friction compensation performance A noveltrapezoidal compensation pulse was adopted to compensatethe friction errors and a friction compensation performanceevaluation function was designed Then an optimal pulseduration function and a pulse amplitude function wereestablished by the pulse characteristic parameter learningAccording to these functions the optimal pulse amplitudeand the pulse duration can be calculated under differentworking conditions Hence the required friction compen-sation performance can be obtained and the motion andcontour accuracies of servomechanism can be guaranteedeffectively In addition according to different working con-ditions the friction errors can be compensated adaptively

The paper is organized as follows The generation mech-anism of friction errors and compensation strategy areintroduced briefly in Section 2 Then the novel time-varyingfriction compensation method is mainly described in Sec-tions 3 and 4 Experimental results are presented in Section 5Finally conclusions are given in Section 6

2 Analysis of Friction Errorsand Friction Compensation

To reveal the generation mechanism of friction errors and tovalidate the friction compensation method proposed by thispaper a high-precision X-Y worktable was set up Figure 1shows the high-precision X-Y worktable Each axis is drivenby a ball screw coupled to a servo motor and its amplifierThe X-Y worktable motion is controlled by an open CNCsystem composed of a host computer and a slave computer[21 22] The host computer is responsible for the operationdata sampling motion monitoring and so on A digitalPID motion controller is developed on the slave computerand generates the control signals which are sent to thecorresponding amplifiers Some logic control operations canbe implemented by the logic signals such as emergency stopThe position feedback signals are produced by the gratingscales installed on the X-Y worktable The velocity feedbacksignals can be obtained by the encoders directly coupled tothe servo motors These feedback signals are sampled by thedigital PID motion controller In addition the related dataof each axis such as position feedback and velocity feedbackare transmitted to the host computer in real time The mainspecifications of this high-precisionX-Y worktable are shownin Table 1

As shown in Figure 2 the controller structure of 119909-axisincludes a cascade PID feedback controller a velocity feedfor-ward controller and an acceleration feedforward controllerThe controller structure of119910-axis is similarwith the controllerstructure of 119909-axis The cascade PID feedback controller iscomposed of a position controller a velocity controller anda current controller to reject both the steady-state errorsand the external disturbances In addition these feedforwardcontrollers are used to further improve the dynamics of ser-vomechanism The position controller and the velocity con-troller adopt the proportion controller and the proportion-integration controller respectively The current controller iscontained in the amplifier and can be equivalent as a torqueconstant Meanwhile to show the superiority of the proposedmethod a disturbance observer was developed to suppressthe friction errors The current differences between thenormal model output and physical plant output are inducedby the external nonlinear friction force These differences areconsidered as equivalent perturbationswhich can be added tocurrent commandThus the friction errors can be suppressedto a certain extent

To make the proposed method clearly the friction com-pensation method is illustrated with the worktable in the 119909-direction as an example in this paper Figure 3 shows theactual velocity and the tracking errors during the reversemotion of the worktable in the 119909-direction with the feedrate 119865 = 1000mmsdotminminus1 and the circular radius 119877 = 25mmAs shown in Figure 3 the prominent errors caused by thefriction force are friction errors which degrade the motionaccuracy of the worktable The whole process of reversemotion will be briefly introduced as follows

When the worktable arrives at a reverse position at themoment 119905stick its motion is stopped From then on thefriction errors begin to generate and to increase until they


X-axisamplifier

Host computer

Slave computer

Open CNC system

Y-axis

Y-axis

amplifier

Data bus




Feedback signals

Limit switch



X-Y worktable

Ballscrew

Cable

X-axis


Positioncontroller


Positioncommand

Velocitycommand

Velocityfeedforward

controller


controller


subsystem

Plant



Currentcommand

Friction force

+

+ + +

+

+

+

+

++

minus minus

minus

minus


pulse Vb


40 4005 401 4015 402 4025 403

0

0005

001

0015

Trac

king

erro

rs (m

m)

0

1

2

3


minus0005

t (s)

minus1

Actu

al v

eloc

ity (m

mmiddotsminus

1)

tstick

Tmax

tslip terr tn

TzTdTb




119897





119889


119887


119879119887



119905err = 119879max + 119905stick (2)


119879max = 119879119889 + 119879119887 (3)


119887


119887



119887


119901


119864119901

= 119875119888119909


= 119875119888119909


asymp 119875119888119909


(4)

where 119875119888119909

and 119875119891119909


119889


119888119909


119901



119887



119899



119899

can be expressed as

119879119911

= 119905119899



119887


119887




119881119887

(119905) =

119860119901

sdot sgn (119889119903

) 119905 isin (119905stick119879 (119879119898 + 119905stick119879))

0

119860119875

(119905) =

(119905 minus 119905stick) 119865119901

119879119903

119905 isin (119905stick119879 (119905stick119879 + 119879119903))

119865119901

119905 isin ((119879119903


((119905stick + 119879119898) minus 119905) 119865119901

119879119903


119889119903

= 119875119888119909

((119894 + 1) 119879) minus 119875119888119909

(119894119879)

sgn (119889119903

) =

minus1 119889119903

isin (minusinfin 0)

0 119889119903

= 0

1 119889119903

isin (0 +infin)

(6)


where 119865119901

119879119898

and 119879119903



119901



119901


With 119889119903


119887



119886


119864119886

=

sum(119894+119873

119886)

119896=119894

10038161003816100381610038161003816119875119888119909

(119896119879) minus 119875119891119909

(119896119879)

10038161003816100381610038161003816

119873119886

119873119886

=

119879119872

119879

(7)

where 119879119872



119872



119887


119886 = (

119865

60

)

2

1

119877

= 1205962

119877 (8)

0

tstickTtstick

Fp

Tr Tr


Ap

(mmmiddotsminus

1)

t (s)



119875119888119909

= 119877 sin (120596119905)

119875119888119910

= 119877 sin (120596119905)

120596 = radic

119886

119877

(9)

where 119875119888119910





119886119894





1198862











can be expressed as

119865119898

= 119879119903

119886119897

(10)

where 119886119897


119894

can beobtained as

119865119894

= 120578119865119898

(11)


119894

1198861

1198862

and 119886119898


119911


119911119894

1198791199111

1198791199112

and 119879119911119898


119898

can be expressed as

119879119898

= 2119879119903

(12)


can be obtained as

119865119901

= 119865119894

(13)




119905


119864119905

=

sum(119894+119873

119905)

119896=119894

10038161003816100381610038161003816119875119888119909

(119896119879) minus 119875119891119909

(119896119879)

10038161003816100381610038161003816

119873119905

119873119905

=

2119879119911

119879

(14)

where 119873119905



119901

can be updated as

119865119901

= 119865119901

+ 119865119905119904

(15)

0

Reverseacceleration

interval 1

Reverseacceleration

interval 2

Reverseacceleration

interval 3

ama2a1ai a


and 119865119901


gt 119865119898


119898

can be updated as

119879119898

= 119879119898

+ 119879119890

(16)


gt 119879119911119894


119898

is the 119879119898119894


119900119898



119905


1

1198862

and119886119898



1198981

1198791198982

and 119879119898119898


119900119898

can be expressed as

119879119900119898

=

119879119898119894

119886 isin (0 119886119894

)

119879119898119894

+

1198791198981

minus 119879119898119894

1198861

minus 119886119894

(119886 minus 119886119894

) 119886 isin [119886119894

1198861

)

1198791198981

+

1198791198982

minus 1198791198981

1198862

minus 1198861

(119886 minus 1198861

) 119886 isin [1198861

1198862

)

1198791198982

+

119879119898119898

minus 1198791198982

119886119898

minus 1198862

(119886 minus 1198862

) 119886 isin [1198862

119886119898

)

119879119898119898

119886 isin [119886119898

119886119897

]

(17)






119888


Δ119886119888

=

1198861

minus 119886119894

1198731

119886 isin [119886119894

1198861

)

Δ119886119888

=

1198862

minus 1198861

1198732

119886 isin [1198861

1198862

)

Δ119886119888

=

119886119898

minus 1198862

1198733

119886 isin [1198862

119886119898

]

(18)



119888119904


119865119888119904

=

119865119898

minus 119865119894

119873119862

(19)


119886 = 119886119888119896

(20)

where 119896 = 1 and 119886 = 119886119888119896

= 119886119894


119901

can be expressed as

119865119901

= 119865119894

(21)



119879119898

= 119879119900119898

(119886) (22)


can be expressed as

119879119872

= 2119879119900119898

(119886) (23)


119886


119901

can be updated as

119865119901

= 119865119901

+ 119865119888119904

(24)

And 119865119901


gt 119865119898


119901

is 119865119888119896


119886


119896 = 119896 + 1

119886 = 119886119888119896

= 119886 + Δ119886119888

(25)



1198881

1198651198882

119865119888119896

119865119888119899

] 119896 = 1 2 (119873

1

+1198732

+1198733


1198881

1198861198882

119886119888119896

119886119888119899

] where119886119898

= 119886119888119899


11988611988811989112

1198861198882

11988611988811989123

1198861198883

119886119888119891(119894)(119894+1)

119886119888(ℎminus1)

119886119888119891(ℎminus1)

119886119888ℎ


119888119899

= 119886119888ℎ


119888119891(119894)(119894+1)

can be expressed as

Δ119886119891

=

Δ119886119888

2

119886119888119891(119894)(119894+1)

= 119886119888(119894)

+ Δ119886119891

119894 = 1 2 ℎ minus 1

(26)

where Δ119886119891


[1198861198911

1198861198912

119886119891119895

119886119891ℎ

]

119895 = 1 2 ℎ ℎ = 2119899 minus 1

(27)


1198881

11986511988811989112

1198651198882

11986511988811989123

1198651198883

119865119888119891(119894)(119894+1)

119865119888(ℎminus1)

119865119888119891(ℎminus1)

119865119888ℎ

] where ℎ = 2119899minus1 119865119888119899

=

119865119888ℎ


can be expressed as

119865119888119891(119894)(119894+1)

=

119865119888(119894)

+ 119865119888(119894+1)

2

119894 = 1 2 ℎ minus 1 (28)


[1198651198871198911

1198651198871198912

1198651198871198913

119865119887119891119895

119865119887119891(ℎminus1)

119865119887119891ℎ

]

119895 = 1 2 ℎ ℎ = 2119899 minus 1

(29)

where 119865119887119891119895


119891119904


119865119891119904

=

119865cs119873119865

(30)


119886 = 119886119891119895

(31)

where 119895 = 1 and 119886 = 1198861198911

= 119886119894


119901

can be expressed as

119865119901

= 119865119887119891119895

minus

119865119888119904

2

(32)



119879119898

= 119879119900119898

(119886) (33)


119886


119901

can be updated as

119865119901

= 119865119901

+ 119865119891119904

(34)

And 119865119901


gt 119865119887119891119895

+ (119865119888119904


119901

is119865119887119891119895


119886


119895 = 119895 + 1

119886 = 119886119891119895

= 119886 + Δ119886119891

(35)




[1198651198911

1198651198912

1198651198913

119865119891119895

119865119891(ℎminus1)

119865119891ℎ

]

119895 = 1 2 ℎ ℎ = 2119899 minus 1

(36)




119865119900119901

=

1198651198911

119886 isin (0 119886119894

)

OFGN 119886 isin [119886119894

119886119898

)

119865119891ℎ

119886 isin [119886119898

119886119897

]

(37)





119886

gt 119864119903




119894

119886119898



119872



119862119883119864




Start

Yes


No


Command position


End



No

Yes

Yes

No

Yes


friction errors (7)



(37)






(17)


performance

Exit the module

(18)ndash(36)

(10)ndash(16)


119875119883119864








119872





(mmsdotsminus2) 5 5119886119898




10 101198732

8 81198733

8 8119873119862

10 10119873119865

4 4119864119903


(mmsdotsminus1) 05 05119879119890

(s) 0005 0005119879119903

(s) 0003 0003

119879119872



20 40 60 80 100 120 140003

004

005

006

007

008

a (mmmiddotsminus2)

Tom

(s)

X-axisY-axis

(a)



0

05

1

15

2

25

3

35

4

45

Fop

(mmmiddotsminus

1)

X-axisX-axis

Y-axisY-axis

(b)




119864119883119872

119875119883119864

119864119883119877

119862119883119864

119864119883119872

119875119883119864

119864119883119877

119862119883119864

119864119883119872

119875119883119864

119864119883119877







119864119883119872

119875119883119864

and 119864119883119877


119884119864

119864119884119872

119875119884119864

and 119864119884119877



1

S2

S3

and S4


1

S2

S3

and S4





119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877






C

B

D

minus10120583m

10120583m

30∘

210∘

60∘

240∘

90∘

270∘

120∘

300∘

150∘

330∘

180∘ 0∘

0120583m

A




1

S2

S3

and S4


119884119864

119864119884119872

119875119884119864

and 119864

119884119877


S2

S3

and S4







0 005 01 015 020

0001

0002

0003

0004

0005

0006

0007

Con

tour

erro

rs (m

m)

TM (s)

(a)

0 005 01 015 020

0001

0002

0003

0004

0005

0006

0007

Con

tour

erro

rs (m

m)

TM (s)

(b)

0 005 01 015 02

0

Trac

king

erro

rs (m

m)

TM (s)

minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

WTFCWFCWDOB

(c)

0

Trac

king

erro

rs (m

m)

0 005 01 015 02TM (s)

minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

WTFCWFCWDOB

(d)




119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877


S2

119871 = 30mm 119886 = 20mmsdotsminus2 789 232 789 329 184 069 184 077

S3

119871 = 50mm 119886 = 20mmsdotsminus2 846 245 846 332 192 071 192 082

S4

119871 = 80mm 119886 = 30mmsdotsminus2 928 366 928 455 215 073 215 085



0 002 004 006 008 01 012

0

0005

001

0015

002

0025

Con

tour

erro

rs (m

m)

minus0005

TM (s)

(a)

0 002 004 006 008 01 012

0

0002

0004

0006

0008

001

0012

0014

0016

Con

tour

erro

rs (m

m)

minus0002

TM (s)

(b)

0 002 004 006 008 01 012

0

0005

Trac

king

erro

rs (m

m)

TM (s)

minus0025

minus002

minus0015

minus001

minus0005

WTFCWFCWDOB

(c)

0 002 004 006 008 01 012

0

0002

Trac

king

erro

rs (m

m)

TM (s)

minus0016

minus0014

minus0012

minus001

minus0008

minus0006

minus0004

minus0002

WTFCWFCWDOB

(d)




Notation





119886119894




0 1 2 3 4 5 6 7 80

5

10

15

20

25

30Po

sitio

n co

mm

and

(mm

)

t (s)

S1

(a)

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)

WTFC WFC


TM (s)

(b)

0 1 2 3 4 50

5

10

15

20

25

30

Posit

ion

com

man

d (m

m)

t (s)

S2

(c)

WTFC WFC

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)


TM (s)

(d)

Posit

ion

com

man

d (m

m)

0 1 2 3 4 5 605

101520253035404550

t (s)

S3

(e)

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)

TM (s)


WTFC WFC

(f)

Posit

ion

com

man

d (m

m)

0 1 2 3 4 5 6 70

1020304050607080

t (s)

S4

(g)

0 005 01 015 02

Trac

king

erro

rs (m

m)

TM (s)

00001


WTFC WFC

(h)


S2

and S3

as well as S4



1



2




4



119886119898






119864119901



119864119905


119865119887119891119895


119865119888119904


119865119891119904


119865119894


119865119898



119865119901


119865119905119904



1198732


1198733


119873119886


119873119862



119875119891119909










the moment 119905err (s)1198791198981

1198791198982

and 119879

119898119898


1

1198862

and 119886119898

(s)119879119900119898




119899

(s)119879119911119894

1198791199111

1198791199112

and 119879119911119898



119894

1198861

1198862

and 119886119898

(s)119879119872



119888


Δ119886119891






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










X-axisamplifier

Host computer

Slave computer

Open CNC system

Y-axis

Y-axis

amplifier

Data bus




Feedback signals

Limit switch



X-Y worktable

Ballscrew

Cable

X-axis


Positioncontroller


Positioncommand

Velocitycommand

Velocityfeedforward

controller


controller


subsystem

Plant



Currentcommand

Friction force

+

+ + +

+

+

+

+

++

minus minus

minus

minus


pulse Vb


40 4005 401 4015 402 4025 403

0

0005

001

0015

Trac

king

erro

rs (m

m)

0

1

2

3


minus0005

t (s)

minus1

Actu

al v

eloc

ity (m

mmiddotsminus

1)

tstick

Tmax

tslip terr tn

TzTdTb




119897





119889


119887


119879119887



119905err = 119879max + 119905stick (2)


119879max = 119879119889 + 119879119887 (3)


119887


119887



119887


119901


119864119901

= 119875119888119909


= 119875119888119909


asymp 119875119888119909


(4)

where 119875119888119909

and 119875119891119909


119889


119888119909


119901



119887



119899



119899

can be expressed as

119879119911

= 119905119899



119887


119887




119881119887

(119905) =

119860119901

sdot sgn (119889119903

) 119905 isin (119905stick119879 (119879119898 + 119905stick119879))

0

119860119875

(119905) =

(119905 minus 119905stick) 119865119901

119879119903

119905 isin (119905stick119879 (119905stick119879 + 119879119903))

119865119901

119905 isin ((119879119903


((119905stick + 119879119898) minus 119905) 119865119901

119879119903


119889119903

= 119875119888119909

((119894 + 1) 119879) minus 119875119888119909

(119894119879)

sgn (119889119903

) =

minus1 119889119903

isin (minusinfin 0)

0 119889119903

= 0

1 119889119903

isin (0 +infin)

(6)


where 119865119901

119879119898

and 119879119903



119901



119901


With 119889119903


119887



119886


119864119886

=

sum(119894+119873

119886)

119896=119894

10038161003816100381610038161003816119875119888119909

(119896119879) minus 119875119891119909

(119896119879)

10038161003816100381610038161003816

119873119886

119873119886

=

119879119872

119879

(7)

where 119879119872



119872



119887


119886 = (

119865

60

)

2

1

119877

= 1205962

119877 (8)

0

tstickTtstick

Fp

Tr Tr


Ap

(mmmiddotsminus

1)

t (s)



119875119888119909

= 119877 sin (120596119905)

119875119888119910

= 119877 sin (120596119905)

120596 = radic

119886

119877

(9)

where 119875119888119910





119886119894





1198862











can be expressed as

119865119898

= 119879119903

119886119897

(10)

where 119886119897


119894

can beobtained as

119865119894

= 120578119865119898

(11)


119894

1198861

1198862

and 119886119898


119911


119911119894

1198791199111

1198791199112

and 119879119911119898


119898

can be expressed as

119879119898

= 2119879119903

(12)


can be obtained as

119865119901

= 119865119894

(13)




119905


119864119905

=

sum(119894+119873

119905)

119896=119894

10038161003816100381610038161003816119875119888119909

(119896119879) minus 119875119891119909

(119896119879)

10038161003816100381610038161003816

119873119905

119873119905

=

2119879119911

119879

(14)

where 119873119905



119901

can be updated as

119865119901

= 119865119901

+ 119865119905119904

(15)

0

Reverseacceleration

interval 1

Reverseacceleration

interval 2

Reverseacceleration

interval 3

ama2a1ai a


and 119865119901


gt 119865119898


119898

can be updated as

119879119898

= 119879119898

+ 119879119890

(16)


gt 119879119911119894


119898

is the 119879119898119894


119900119898



119905


1

1198862

and119886119898



1198981

1198791198982

and 119879119898119898


119900119898

can be expressed as

119879119900119898

=

119879119898119894

119886 isin (0 119886119894

)

119879119898119894

+

1198791198981

minus 119879119898119894

1198861

minus 119886119894

(119886 minus 119886119894

) 119886 isin [119886119894

1198861

)

1198791198981

+

1198791198982

minus 1198791198981

1198862

minus 1198861

(119886 minus 1198861

) 119886 isin [1198861

1198862

)

1198791198982

+

119879119898119898

minus 1198791198982

119886119898

minus 1198862

(119886 minus 1198862

) 119886 isin [1198862

119886119898

)

119879119898119898

119886 isin [119886119898

119886119897

]

(17)






119888


Δ119886119888

=

1198861

minus 119886119894

1198731

119886 isin [119886119894

1198861

)

Δ119886119888

=

1198862

minus 1198861

1198732

119886 isin [1198861

1198862

)

Δ119886119888

=

119886119898

minus 1198862

1198733

119886 isin [1198862

119886119898

]

(18)



119888119904


119865119888119904

=

119865119898

minus 119865119894

119873119862

(19)


119886 = 119886119888119896

(20)

where 119896 = 1 and 119886 = 119886119888119896

= 119886119894


119901

can be expressed as

119865119901

= 119865119894

(21)



119879119898

= 119879119900119898

(119886) (22)


can be expressed as

119879119872

= 2119879119900119898

(119886) (23)


119886


119901

can be updated as

119865119901

= 119865119901

+ 119865119888119904

(24)

And 119865119901


gt 119865119898


119901

is 119865119888119896


119886


119896 = 119896 + 1

119886 = 119886119888119896

= 119886 + Δ119886119888

(25)



1198881

1198651198882

119865119888119896

119865119888119899

] 119896 = 1 2 (119873

1

+1198732

+1198733


1198881

1198861198882

119886119888119896

119886119888119899

] where119886119898

= 119886119888119899


11988611988811989112

1198861198882

11988611988811989123

1198861198883

119886119888119891(119894)(119894+1)

119886119888(ℎminus1)

119886119888119891(ℎminus1)

119886119888ℎ


119888119899

= 119886119888ℎ


119888119891(119894)(119894+1)

can be expressed as

Δ119886119891

=

Δ119886119888

2

119886119888119891(119894)(119894+1)

= 119886119888(119894)

+ Δ119886119891

119894 = 1 2 ℎ minus 1

(26)

where Δ119886119891


[1198861198911

1198861198912

119886119891119895

119886119891ℎ

]

119895 = 1 2 ℎ ℎ = 2119899 minus 1

(27)


1198881

11986511988811989112

1198651198882

11986511988811989123

1198651198883

119865119888119891(119894)(119894+1)

119865119888(ℎminus1)

119865119888119891(ℎminus1)

119865119888ℎ

] where ℎ = 2119899minus1 119865119888119899

=

119865119888ℎ


can be expressed as

119865119888119891(119894)(119894+1)

=

119865119888(119894)

+ 119865119888(119894+1)

2

119894 = 1 2 ℎ minus 1 (28)


[1198651198871198911

1198651198871198912

1198651198871198913

119865119887119891119895

119865119887119891(ℎminus1)

119865119887119891ℎ

]

119895 = 1 2 ℎ ℎ = 2119899 minus 1

(29)

where 119865119887119891119895


119891119904


119865119891119904

=

119865cs119873119865

(30)


119886 = 119886119891119895

(31)

where 119895 = 1 and 119886 = 1198861198911

= 119886119894


119901

can be expressed as

119865119901

= 119865119887119891119895

minus

119865119888119904

2

(32)



119879119898

= 119879119900119898

(119886) (33)


119886


119901

can be updated as

119865119901

= 119865119901

+ 119865119891119904

(34)

And 119865119901


gt 119865119887119891119895

+ (119865119888119904


119901

is119865119887119891119895


119886


119895 = 119895 + 1

119886 = 119886119891119895

= 119886 + Δ119886119891

(35)




[1198651198911

1198651198912

1198651198913

119865119891119895

119865119891(ℎminus1)

119865119891ℎ

]

119895 = 1 2 ℎ ℎ = 2119899 minus 1

(36)




119865119900119901

=

1198651198911

119886 isin (0 119886119894

)

OFGN 119886 isin [119886119894

119886119898

)

119865119891ℎ

119886 isin [119886119898

119886119897

]

(37)





119886

gt 119864119903




119894

119886119898



119872



119862119883119864




Start

Yes


No


Command position


End



No

Yes

Yes

No

Yes


friction errors (7)



(37)






(17)


performance

Exit the module

(18)ndash(36)

(10)ndash(16)


119875119883119864








119872





(mmsdotsminus2) 5 5119886119898




10 101198732

8 81198733

8 8119873119862

10 10119873119865

4 4119864119903


(mmsdotsminus1) 05 05119879119890

(s) 0005 0005119879119903

(s) 0003 0003

119879119872



20 40 60 80 100 120 140003

004

005

006

007

008

a (mmmiddotsminus2)

Tom

(s)

X-axisY-axis

(a)



0

05

1

15

2

25

3

35

4

45

Fop

(mmmiddotsminus

1)

X-axisX-axis

Y-axisY-axis

(b)




119864119883119872

119875119883119864

119864119883119877

119862119883119864

119864119883119872

119875119883119864

119864119883119877

119862119883119864

119864119883119872

119875119883119864

119864119883119877







119864119883119872

119875119883119864

and 119864119883119877


119884119864

119864119884119872

119875119884119864

and 119864119884119877



1

S2

S3

and S4


1

S2

S3

and S4





119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877






C

B

D

minus10120583m

10120583m

30∘

210∘

60∘

240∘

90∘

270∘

120∘

300∘

150∘

330∘

180∘ 0∘

0120583m

A




1

S2

S3

and S4


119884119864

119864119884119872

119875119884119864

and 119864

119884119877


S2

S3

and S4







0 005 01 015 020

0001

0002

0003

0004

0005

0006

0007

Con

tour

erro

rs (m

m)

TM (s)

(a)

0 005 01 015 020

0001

0002

0003

0004

0005

0006

0007

Con

tour

erro

rs (m

m)

TM (s)

(b)

0 005 01 015 02

0

Trac

king

erro

rs (m

m)

TM (s)

minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

WTFCWFCWDOB

(c)

0

Trac

king

erro

rs (m

m)

0 005 01 015 02TM (s)

minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

WTFCWFCWDOB

(d)




119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877


S2

119871 = 30mm 119886 = 20mmsdotsminus2 789 232 789 329 184 069 184 077

S3

119871 = 50mm 119886 = 20mmsdotsminus2 846 245 846 332 192 071 192 082

S4

119871 = 80mm 119886 = 30mmsdotsminus2 928 366 928 455 215 073 215 085



0 002 004 006 008 01 012

0

0005

001

0015

002

0025

Con

tour

erro

rs (m

m)

minus0005

TM (s)

(a)

0 002 004 006 008 01 012

0

0002

0004

0006

0008

001

0012

0014

0016

Con

tour

erro

rs (m

m)

minus0002

TM (s)

(b)

0 002 004 006 008 01 012

0

0005

Trac

king

erro

rs (m

m)

TM (s)

minus0025

minus002

minus0015

minus001

minus0005

WTFCWFCWDOB

(c)

0 002 004 006 008 01 012

0

0002

Trac

king

erro

rs (m

m)

TM (s)

minus0016

minus0014

minus0012

minus001

minus0008

minus0006

minus0004

minus0002

WTFCWFCWDOB

(d)




Notation





119886119894




0 1 2 3 4 5 6 7 80

5

10

15

20

25

30Po

sitio

n co

mm

and

(mm

)

t (s)

S1

(a)

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)

WTFC WFC


TM (s)

(b)

0 1 2 3 4 50

5

10

15

20

25

30

Posit

ion

com

man

d (m

m)

t (s)

S2

(c)

WTFC WFC

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)


TM (s)

(d)

Posit

ion

com

man

d (m

m)

0 1 2 3 4 5 605

101520253035404550

t (s)

S3

(e)

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)

TM (s)


WTFC WFC

(f)

Posit

ion

com

man

d (m

m)

0 1 2 3 4 5 6 70

1020304050607080

t (s)

S4

(g)

0 005 01 015 02

Trac

king

erro

rs (m

m)

TM (s)

00001


WTFC WFC

(h)


S2

and S3

as well as S4



1



2




4



119886119898






119864119901



119864119905


119865119887119891119895


119865119888119904


119865119891119904


119865119894


119865119898



119865119901


119865119905119904



1198732


1198733


119873119886


119873119862



119875119891119909










the moment 119905err (s)1198791198981

1198791198982

and 119879

119898119898


1

1198862

and 119886119898

(s)119879119900119898




119899

(s)119879119911119894

1198791199111

1198791199112

and 119879119911119898



119894

1198861

1198862

and 119886119898

(s)119879119872



119888


Δ119886119891






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119889


119887


119879119887



119905err = 119879max + 119905stick (2)


119879max = 119879119889 + 119879119887 (3)


119887


119887



119887


119901


119864119901

= 119875119888119909


= 119875119888119909


asymp 119875119888119909


(4)

where 119875119888119909

and 119875119891119909


119889


119888119909


119901



119887



119899



119899

can be expressed as

119879119911

= 119905119899



119887


119887




119881119887

(119905) =

119860119901

sdot sgn (119889119903

) 119905 isin (119905stick119879 (119879119898 + 119905stick119879))

0

119860119875

(119905) =

(119905 minus 119905stick) 119865119901

119879119903

119905 isin (119905stick119879 (119905stick119879 + 119879119903))

119865119901

119905 isin ((119879119903


((119905stick + 119879119898) minus 119905) 119865119901

119879119903


119889119903

= 119875119888119909

((119894 + 1) 119879) minus 119875119888119909

(119894119879)

sgn (119889119903

) =

minus1 119889119903

isin (minusinfin 0)

0 119889119903

= 0

1 119889119903

isin (0 +infin)

(6)


where 119865119901

119879119898

and 119879119903



119901



119901


With 119889119903


119887



119886


119864119886

=

sum(119894+119873

119886)

119896=119894

10038161003816100381610038161003816119875119888119909

(119896119879) minus 119875119891119909

(119896119879)

10038161003816100381610038161003816

119873119886

119873119886

=

119879119872

119879

(7)

where 119879119872



119872



119887


119886 = (

119865

60

)

2

1

119877

= 1205962

119877 (8)

0

tstickTtstick

Fp

Tr Tr


Ap

(mmmiddotsminus

1)

t (s)



119875119888119909

= 119877 sin (120596119905)

119875119888119910

= 119877 sin (120596119905)

120596 = radic

119886

119877

(9)

where 119875119888119910





119886119894





1198862











can be expressed as

119865119898

= 119879119903

119886119897

(10)

where 119886119897


119894

can beobtained as

119865119894

= 120578119865119898

(11)


119894

1198861

1198862

and 119886119898


119911


119911119894

1198791199111

1198791199112

and 119879119911119898


119898

can be expressed as

119879119898

= 2119879119903

(12)


can be obtained as

119865119901

= 119865119894

(13)




119905


119864119905

=

sum(119894+119873

119905)

119896=119894

10038161003816100381610038161003816119875119888119909

(119896119879) minus 119875119891119909

(119896119879)

10038161003816100381610038161003816

119873119905

119873119905

=

2119879119911

119879

(14)

where 119873119905



119901

can be updated as

119865119901

= 119865119901

+ 119865119905119904

(15)

0

Reverseacceleration

interval 1

Reverseacceleration

interval 2

Reverseacceleration

interval 3

ama2a1ai a


and 119865119901


gt 119865119898


119898

can be updated as

119879119898

= 119879119898

+ 119879119890

(16)


gt 119879119911119894


119898

is the 119879119898119894


119900119898



119905


1

1198862

and119886119898



1198981

1198791198982

and 119879119898119898


119900119898

can be expressed as

119879119900119898

=

119879119898119894

119886 isin (0 119886119894

)

119879119898119894

+

1198791198981

minus 119879119898119894

1198861

minus 119886119894

(119886 minus 119886119894

) 119886 isin [119886119894

1198861

)

1198791198981

+

1198791198982

minus 1198791198981

1198862

minus 1198861

(119886 minus 1198861

) 119886 isin [1198861

1198862

)

1198791198982

+

119879119898119898

minus 1198791198982

119886119898

minus 1198862

(119886 minus 1198862

) 119886 isin [1198862

119886119898

)

119879119898119898

119886 isin [119886119898

119886119897

]

(17)






119888


Δ119886119888

=

1198861

minus 119886119894

1198731

119886 isin [119886119894

1198861

)

Δ119886119888

=

1198862

minus 1198861

1198732

119886 isin [1198861

1198862

)

Δ119886119888

=

119886119898

minus 1198862

1198733

119886 isin [1198862

119886119898

]

(18)



119888119904


119865119888119904

=

119865119898

minus 119865119894

119873119862

(19)


119886 = 119886119888119896

(20)

where 119896 = 1 and 119886 = 119886119888119896

= 119886119894


119901

can be expressed as

119865119901

= 119865119894

(21)



119879119898

= 119879119900119898

(119886) (22)


can be expressed as

119879119872

= 2119879119900119898

(119886) (23)


119886


119901

can be updated as

119865119901

= 119865119901

+ 119865119888119904

(24)

And 119865119901


gt 119865119898


119901

is 119865119888119896


119886


119896 = 119896 + 1

119886 = 119886119888119896

= 119886 + Δ119886119888

(25)



1198881

1198651198882

119865119888119896

119865119888119899

] 119896 = 1 2 (119873

1

+1198732

+1198733


1198881

1198861198882

119886119888119896

119886119888119899

] where119886119898

= 119886119888119899


11988611988811989112

1198861198882

11988611988811989123

1198861198883

119886119888119891(119894)(119894+1)

119886119888(ℎminus1)

119886119888119891(ℎminus1)

119886119888ℎ


119888119899

= 119886119888ℎ


119888119891(119894)(119894+1)

can be expressed as

Δ119886119891

=

Δ119886119888

2

119886119888119891(119894)(119894+1)

= 119886119888(119894)

+ Δ119886119891

119894 = 1 2 ℎ minus 1

(26)

where Δ119886119891


[1198861198911

1198861198912

119886119891119895

119886119891ℎ

]

119895 = 1 2 ℎ ℎ = 2119899 minus 1

(27)


1198881

11986511988811989112

1198651198882

11986511988811989123

1198651198883

119865119888119891(119894)(119894+1)

119865119888(ℎminus1)

119865119888119891(ℎminus1)

119865119888ℎ

] where ℎ = 2119899minus1 119865119888119899

=

119865119888ℎ


can be expressed as

119865119888119891(119894)(119894+1)

=

119865119888(119894)

+ 119865119888(119894+1)

2

119894 = 1 2 ℎ minus 1 (28)


[1198651198871198911

1198651198871198912

1198651198871198913

119865119887119891119895

119865119887119891(ℎminus1)

119865119887119891ℎ

]

119895 = 1 2 ℎ ℎ = 2119899 minus 1

(29)

where 119865119887119891119895


119891119904


119865119891119904

=

119865cs119873119865

(30)


119886 = 119886119891119895

(31)

where 119895 = 1 and 119886 = 1198861198911

= 119886119894


119901

can be expressed as

119865119901

= 119865119887119891119895

minus

119865119888119904

2

(32)



119879119898

= 119879119900119898

(119886) (33)


119886


119901

can be updated as

119865119901

= 119865119901

+ 119865119891119904

(34)

And 119865119901


gt 119865119887119891119895

+ (119865119888119904


119901

is119865119887119891119895


119886


119895 = 119895 + 1

119886 = 119886119891119895

= 119886 + Δ119886119891

(35)




[1198651198911

1198651198912

1198651198913

119865119891119895

119865119891(ℎminus1)

119865119891ℎ

]

119895 = 1 2 ℎ ℎ = 2119899 minus 1

(36)




119865119900119901

=

1198651198911

119886 isin (0 119886119894

)

OFGN 119886 isin [119886119894

119886119898

)

119865119891ℎ

119886 isin [119886119898

119886119897

]

(37)





119886

gt 119864119903




119894

119886119898



119872



119862119883119864




Start

Yes


No


Command position


End



No

Yes

Yes

No

Yes


friction errors (7)



(37)






(17)


performance

Exit the module

(18)ndash(36)

(10)ndash(16)


119875119883119864








119872





(mmsdotsminus2) 5 5119886119898




10 101198732

8 81198733

8 8119873119862

10 10119873119865

4 4119864119903


(mmsdotsminus1) 05 05119879119890

(s) 0005 0005119879119903

(s) 0003 0003

119879119872



20 40 60 80 100 120 140003

004

005

006

007

008

a (mmmiddotsminus2)

Tom

(s)

X-axisY-axis

(a)



0

05

1

15

2

25

3

35

4

45

Fop

(mmmiddotsminus

1)

X-axisX-axis

Y-axisY-axis

(b)




119864119883119872

119875119883119864

119864119883119877

119862119883119864

119864119883119872

119875119883119864

119864119883119877

119862119883119864

119864119883119872

119875119883119864

119864119883119877







119864119883119872

119875119883119864

and 119864119883119877


119884119864

119864119884119872

119875119884119864

and 119864119884119877



1

S2

S3

and S4


1

S2

S3

and S4





119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877






C

B

D

minus10120583m

10120583m

30∘

210∘

60∘

240∘

90∘

270∘

120∘

300∘

150∘

330∘

180∘ 0∘

0120583m

A




1

S2

S3

and S4


119884119864

119864119884119872

119875119884119864

and 119864

119884119877


S2

S3

and S4







0 005 01 015 020

0001

0002

0003

0004

0005

0006

0007

Con

tour

erro

rs (m

m)

TM (s)

(a)

0 005 01 015 020

0001

0002

0003

0004

0005

0006

0007

Con

tour

erro

rs (m

m)

TM (s)

(b)

0 005 01 015 02

0

Trac

king

erro

rs (m

m)

TM (s)

minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

WTFCWFCWDOB

(c)

0

Trac

king

erro

rs (m

m)

0 005 01 015 02TM (s)

minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

WTFCWFCWDOB

(d)




119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877


S2

119871 = 30mm 119886 = 20mmsdotsminus2 789 232 789 329 184 069 184 077

S3

119871 = 50mm 119886 = 20mmsdotsminus2 846 245 846 332 192 071 192 082

S4

119871 = 80mm 119886 = 30mmsdotsminus2 928 366 928 455 215 073 215 085



0 002 004 006 008 01 012

0

0005

001

0015

002

0025

Con

tour

erro

rs (m

m)

minus0005

TM (s)

(a)

0 002 004 006 008 01 012

0

0002

0004

0006

0008

001

0012

0014

0016

Con

tour

erro

rs (m

m)

minus0002

TM (s)

(b)

0 002 004 006 008 01 012

0

0005

Trac

king

erro

rs (m

m)

TM (s)

minus0025

minus002

minus0015

minus001

minus0005

WTFCWFCWDOB

(c)

0 002 004 006 008 01 012

0

0002

Trac

king

erro

rs (m

m)

TM (s)

minus0016

minus0014

minus0012

minus001

minus0008

minus0006

minus0004

minus0002

WTFCWFCWDOB

(d)




Notation





119886119894




0 1 2 3 4 5 6 7 80

5

10

15

20

25

30Po

sitio

n co

mm

and

(mm

)

t (s)

S1

(a)

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)

WTFC WFC


TM (s)

(b)

0 1 2 3 4 50

5

10

15

20

25

30

Posit

ion

com

man

d (m

m)

t (s)

S2

(c)

WTFC WFC

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)


TM (s)

(d)

Posit

ion

com

man

d (m

m)

0 1 2 3 4 5 605

101520253035404550

t (s)

S3

(e)

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)

TM (s)


WTFC WFC

(f)

Posit

ion

com

man

d (m

m)

0 1 2 3 4 5 6 70

1020304050607080

t (s)

S4

(g)

0 005 01 015 02

Trac

king

erro

rs (m

m)

TM (s)

00001


WTFC WFC

(h)


S2

and S3

as well as S4



1



2




4



119886119898






119864119901



119864119905


119865119887119891119895


119865119888119904


119865119891119904


119865119894


119865119898



119865119901


119865119905119904



1198732


1198733


119873119886


119873119862



119875119891119909










the moment 119905err (s)1198791198981

1198791198982

and 119879

119898119898


1

1198862

and 119886119898

(s)119879119900119898




119899

(s)119879119911119894

1198791199111

1198791199112

and 119879119911119898



119894

1198861

1198862

and 119886119898

(s)119879119872



119888


Δ119886119891






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where 119865119901

119879119898

and 119879119903



119901



119901


With 119889119903


119887



119886


119864119886

=

sum(119894+119873

119886)

119896=119894

10038161003816100381610038161003816119875119888119909

(119896119879) minus 119875119891119909

(119896119879)

10038161003816100381610038161003816

119873119886

119873119886

=

119879119872

119879

(7)

where 119879119872



119872



119887


119886 = (

119865

60

)

2

1

119877

= 1205962

119877 (8)

0

tstickTtstick

Fp

Tr Tr


Ap

(mmmiddotsminus

1)

t (s)



119875119888119909

= 119877 sin (120596119905)

119875119888119910

= 119877 sin (120596119905)

120596 = radic

119886

119877

(9)

where 119875119888119910





119886119894





1198862











can be expressed as

119865119898

= 119879119903

119886119897

(10)

where 119886119897


119894

can beobtained as

119865119894

= 120578119865119898

(11)


119894

1198861

1198862

and 119886119898


119911


119911119894

1198791199111

1198791199112

and 119879119911119898


119898

can be expressed as

119879119898

= 2119879119903

(12)


can be obtained as

119865119901

= 119865119894

(13)




119905


119864119905

=

sum(119894+119873

119905)

119896=119894

10038161003816100381610038161003816119875119888119909

(119896119879) minus 119875119891119909

(119896119879)

10038161003816100381610038161003816

119873119905

119873119905

=

2119879119911

119879

(14)

where 119873119905



119901

can be updated as

119865119901

= 119865119901

+ 119865119905119904

(15)

0

Reverseacceleration

interval 1

Reverseacceleration

interval 2

Reverseacceleration

interval 3

ama2a1ai a


and 119865119901


gt 119865119898


119898

can be updated as

119879119898

= 119879119898

+ 119879119890

(16)


gt 119879119911119894


119898

is the 119879119898119894


119900119898



119905


1

1198862

and119886119898



1198981

1198791198982

and 119879119898119898


119900119898

can be expressed as

119879119900119898

=

119879119898119894

119886 isin (0 119886119894

)

119879119898119894

+

1198791198981

minus 119879119898119894

1198861

minus 119886119894

(119886 minus 119886119894

) 119886 isin [119886119894

1198861

)

1198791198981

+

1198791198982

minus 1198791198981

1198862

minus 1198861

(119886 minus 1198861

) 119886 isin [1198861

1198862

)

1198791198982

+

119879119898119898

minus 1198791198982

119886119898

minus 1198862

(119886 minus 1198862

) 119886 isin [1198862

119886119898

)

119879119898119898

119886 isin [119886119898

119886119897

]

(17)






119888


Δ119886119888

=

1198861

minus 119886119894

1198731

119886 isin [119886119894

1198861

)

Δ119886119888

=

1198862

minus 1198861

1198732

119886 isin [1198861

1198862

)

Δ119886119888

=

119886119898

minus 1198862

1198733

119886 isin [1198862

119886119898

]

(18)



119888119904


119865119888119904

=

119865119898

minus 119865119894

119873119862

(19)


119886 = 119886119888119896

(20)

where 119896 = 1 and 119886 = 119886119888119896

= 119886119894


119901

can be expressed as

119865119901

= 119865119894

(21)



119879119898

= 119879119900119898

(119886) (22)


can be expressed as

119879119872

= 2119879119900119898

(119886) (23)


119886


119901

can be updated as

119865119901

= 119865119901

+ 119865119888119904

(24)

And 119865119901


gt 119865119898


119901

is 119865119888119896


119886


119896 = 119896 + 1

119886 = 119886119888119896

= 119886 + Δ119886119888

(25)



1198881

1198651198882

119865119888119896

119865119888119899

] 119896 = 1 2 (119873

1

+1198732

+1198733


1198881

1198861198882

119886119888119896

119886119888119899

] where119886119898

= 119886119888119899


11988611988811989112

1198861198882

11988611988811989123

1198861198883

119886119888119891(119894)(119894+1)

119886119888(ℎminus1)

119886119888119891(ℎminus1)

119886119888ℎ


119888119899

= 119886119888ℎ


119888119891(119894)(119894+1)

can be expressed as

Δ119886119891

=

Δ119886119888

2

119886119888119891(119894)(119894+1)

= 119886119888(119894)

+ Δ119886119891

119894 = 1 2 ℎ minus 1

(26)

where Δ119886119891


[1198861198911

1198861198912

119886119891119895

119886119891ℎ

]

119895 = 1 2 ℎ ℎ = 2119899 minus 1

(27)


1198881

11986511988811989112

1198651198882

11986511988811989123

1198651198883

119865119888119891(119894)(119894+1)

119865119888(ℎminus1)

119865119888119891(ℎminus1)

119865119888ℎ

] where ℎ = 2119899minus1 119865119888119899

=

119865119888ℎ


can be expressed as

119865119888119891(119894)(119894+1)

=

119865119888(119894)

+ 119865119888(119894+1)

2

119894 = 1 2 ℎ minus 1 (28)


[1198651198871198911

1198651198871198912

1198651198871198913

119865119887119891119895

119865119887119891(ℎminus1)

119865119887119891ℎ

]

119895 = 1 2 ℎ ℎ = 2119899 minus 1

(29)

where 119865119887119891119895


119891119904


119865119891119904

=

119865cs119873119865

(30)


119886 = 119886119891119895

(31)

where 119895 = 1 and 119886 = 1198861198911

= 119886119894


119901

can be expressed as

119865119901

= 119865119887119891119895

minus

119865119888119904

2

(32)



119879119898

= 119879119900119898

(119886) (33)


119886


119901

can be updated as

119865119901

= 119865119901

+ 119865119891119904

(34)

And 119865119901


gt 119865119887119891119895

+ (119865119888119904


119901

is119865119887119891119895


119886


119895 = 119895 + 1

119886 = 119886119891119895

= 119886 + Δ119886119891

(35)




[1198651198911

1198651198912

1198651198913

119865119891119895

119865119891(ℎminus1)

119865119891ℎ

]

119895 = 1 2 ℎ ℎ = 2119899 minus 1

(36)




119865119900119901

=

1198651198911

119886 isin (0 119886119894

)

OFGN 119886 isin [119886119894

119886119898

)

119865119891ℎ

119886 isin [119886119898

119886119897

]

(37)





119886

gt 119864119903




119894

119886119898



119872



119862119883119864




Start

Yes


No


Command position


End



No

Yes

Yes

No

Yes


friction errors (7)



(37)






(17)


performance

Exit the module

(18)ndash(36)

(10)ndash(16)


119875119883119864








119872





(mmsdotsminus2) 5 5119886119898




10 101198732

8 81198733

8 8119873119862

10 10119873119865

4 4119864119903


(mmsdotsminus1) 05 05119879119890

(s) 0005 0005119879119903

(s) 0003 0003

119879119872



20 40 60 80 100 120 140003

004

005

006

007

008

a (mmmiddotsminus2)

Tom

(s)

X-axisY-axis

(a)



0

05

1

15

2

25

3

35

4

45

Fop

(mmmiddotsminus

1)

X-axisX-axis

Y-axisY-axis

(b)




119864119883119872

119875119883119864

119864119883119877

119862119883119864

119864119883119872

119875119883119864

119864119883119877

119862119883119864

119864119883119872

119875119883119864

119864119883119877







119864119883119872

119875119883119864

and 119864119883119877


119884119864

119864119884119872

119875119884119864

and 119864119884119877



1

S2

S3

and S4


1

S2

S3

and S4





119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877






C

B

D

minus10120583m

10120583m

30∘

210∘

60∘

240∘

90∘

270∘

120∘

300∘

150∘

330∘

180∘ 0∘

0120583m

A




1

S2

S3

and S4


119884119864

119864119884119872

119875119884119864

and 119864

119884119877


S2

S3

and S4







0 005 01 015 020

0001

0002

0003

0004

0005

0006

0007

Con

tour

erro

rs (m

m)

TM (s)

(a)

0 005 01 015 020

0001

0002

0003

0004

0005

0006

0007

Con

tour

erro

rs (m

m)

TM (s)

(b)

0 005 01 015 02

0

Trac

king

erro

rs (m

m)

TM (s)

minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

WTFCWFCWDOB

(c)

0

Trac

king

erro

rs (m

m)

0 005 01 015 02TM (s)

minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

WTFCWFCWDOB

(d)




119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877


S2

119871 = 30mm 119886 = 20mmsdotsminus2 789 232 789 329 184 069 184 077

S3

119871 = 50mm 119886 = 20mmsdotsminus2 846 245 846 332 192 071 192 082

S4

119871 = 80mm 119886 = 30mmsdotsminus2 928 366 928 455 215 073 215 085



0 002 004 006 008 01 012

0

0005

001

0015

002

0025

Con

tour

erro

rs (m

m)

minus0005

TM (s)

(a)

0 002 004 006 008 01 012

0

0002

0004

0006

0008

001

0012

0014

0016

Con

tour

erro

rs (m

m)

minus0002

TM (s)

(b)

0 002 004 006 008 01 012

0

0005

Trac

king

erro

rs (m

m)

TM (s)

minus0025

minus002

minus0015

minus001

minus0005

WTFCWFCWDOB

(c)

0 002 004 006 008 01 012

0

0002

Trac

king

erro

rs (m

m)

TM (s)

minus0016

minus0014

minus0012

minus001

minus0008

minus0006

minus0004

minus0002

WTFCWFCWDOB

(d)




Notation





119886119894




0 1 2 3 4 5 6 7 80

5

10

15

20

25

30Po

sitio

n co

mm

and

(mm

)

t (s)

S1

(a)

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)

WTFC WFC


TM (s)

(b)

0 1 2 3 4 50

5

10

15

20

25

30

Posit

ion

com

man

d (m

m)

t (s)

S2

(c)

WTFC WFC

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)


TM (s)

(d)

Posit

ion

com

man

d (m

m)

0 1 2 3 4 5 605

101520253035404550

t (s)

S3

(e)

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)

TM (s)


WTFC WFC

(f)

Posit

ion

com

man

d (m

m)

0 1 2 3 4 5 6 70

1020304050607080

t (s)

S4

(g)

0 005 01 015 02

Trac

king

erro

rs (m

m)

TM (s)

00001


WTFC WFC

(h)


S2

and S3

as well as S4



1



2




4



119886119898






119864119901



119864119905


119865119887119891119895


119865119888119904


119865119891119904


119865119894


119865119898



119865119901


119865119905119904



1198732


1198733


119873119886


119873119862



119875119891119909










the moment 119905err (s)1198791198981

1198791198982

and 119879

119898119898


1

1198862

and 119886119898

(s)119879119900119898




119899

(s)119879119911119894

1198791199111

1198791199112

and 119879119911119898



119894

1198861

1198862

and 119886119898

(s)119879119872



119888


Δ119886119891






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1198862











can be expressed as

119865119898

= 119879119903

119886119897

(10)

where 119886119897


119894

can beobtained as

119865119894

= 120578119865119898

(11)


119894

1198861

1198862

and 119886119898


119911


119911119894

1198791199111

1198791199112

and 119879119911119898


119898

can be expressed as

119879119898

= 2119879119903

(12)


can be obtained as

119865119901

= 119865119894

(13)




119905


119864119905

=

sum(119894+119873

119905)

119896=119894

10038161003816100381610038161003816119875119888119909

(119896119879) minus 119875119891119909

(119896119879)

10038161003816100381610038161003816

119873119905

119873119905

=

2119879119911

119879

(14)

where 119873119905



119901

can be updated as

119865119901

= 119865119901

+ 119865119905119904

(15)

0

Reverseacceleration

interval 1

Reverseacceleration

interval 2

Reverseacceleration

interval 3

ama2a1ai a


and 119865119901


gt 119865119898


119898

can be updated as

119879119898

= 119879119898

+ 119879119890

(16)


gt 119879119911119894


119898

is the 119879119898119894


119900119898



119905


1

1198862

and119886119898



1198981

1198791198982

and 119879119898119898


119900119898

can be expressed as

119879119900119898

=

119879119898119894

119886 isin (0 119886119894

)

119879119898119894

+

1198791198981

minus 119879119898119894

1198861

minus 119886119894

(119886 minus 119886119894

) 119886 isin [119886119894

1198861

)

1198791198981

+

1198791198982

minus 1198791198981

1198862

minus 1198861

(119886 minus 1198861

) 119886 isin [1198861

1198862

)

1198791198982

+

119879119898119898

minus 1198791198982

119886119898

minus 1198862

(119886 minus 1198862

) 119886 isin [1198862

119886119898

)

119879119898119898

119886 isin [119886119898

119886119897

]

(17)






119888


Δ119886119888

=

1198861

minus 119886119894

1198731

119886 isin [119886119894

1198861

)

Δ119886119888

=

1198862

minus 1198861

1198732

119886 isin [1198861

1198862

)

Δ119886119888

=

119886119898

minus 1198862

1198733

119886 isin [1198862

119886119898

]

(18)



119888119904


119865119888119904

=

119865119898

minus 119865119894

119873119862

(19)


119886 = 119886119888119896

(20)

where 119896 = 1 and 119886 = 119886119888119896

= 119886119894


119901

can be expressed as

119865119901

= 119865119894

(21)



119879119898

= 119879119900119898

(119886) (22)


can be expressed as

119879119872

= 2119879119900119898

(119886) (23)


119886


119901

can be updated as

119865119901

= 119865119901

+ 119865119888119904

(24)

And 119865119901


gt 119865119898


119901

is 119865119888119896


119886


119896 = 119896 + 1

119886 = 119886119888119896

= 119886 + Δ119886119888

(25)



1198881

1198651198882

119865119888119896

119865119888119899

] 119896 = 1 2 (119873

1

+1198732

+1198733


1198881

1198861198882

119886119888119896

119886119888119899

] where119886119898

= 119886119888119899


11988611988811989112

1198861198882

11988611988811989123

1198861198883

119886119888119891(119894)(119894+1)

119886119888(ℎminus1)

119886119888119891(ℎminus1)

119886119888ℎ


119888119899

= 119886119888ℎ


119888119891(119894)(119894+1)

can be expressed as

Δ119886119891

=

Δ119886119888

2

119886119888119891(119894)(119894+1)

= 119886119888(119894)

+ Δ119886119891

119894 = 1 2 ℎ minus 1

(26)

where Δ119886119891


[1198861198911

1198861198912

119886119891119895

119886119891ℎ

]

119895 = 1 2 ℎ ℎ = 2119899 minus 1

(27)


1198881

11986511988811989112

1198651198882

11986511988811989123

1198651198883

119865119888119891(119894)(119894+1)

119865119888(ℎminus1)

119865119888119891(ℎminus1)

119865119888ℎ

] where ℎ = 2119899minus1 119865119888119899

=

119865119888ℎ


can be expressed as

119865119888119891(119894)(119894+1)

=

119865119888(119894)

+ 119865119888(119894+1)

2

119894 = 1 2 ℎ minus 1 (28)


[1198651198871198911

1198651198871198912

1198651198871198913

119865119887119891119895

119865119887119891(ℎminus1)

119865119887119891ℎ

]

119895 = 1 2 ℎ ℎ = 2119899 minus 1

(29)

where 119865119887119891119895


119891119904


119865119891119904

=

119865cs119873119865

(30)


119886 = 119886119891119895

(31)

where 119895 = 1 and 119886 = 1198861198911

= 119886119894


119901

can be expressed as

119865119901

= 119865119887119891119895

minus

119865119888119904

2

(32)



119879119898

= 119879119900119898

(119886) (33)


119886


119901

can be updated as

119865119901

= 119865119901

+ 119865119891119904

(34)

And 119865119901


gt 119865119887119891119895

+ (119865119888119904


119901

is119865119887119891119895


119886


119895 = 119895 + 1

119886 = 119886119891119895

= 119886 + Δ119886119891

(35)




[1198651198911

1198651198912

1198651198913

119865119891119895

119865119891(ℎminus1)

119865119891ℎ

]

119895 = 1 2 ℎ ℎ = 2119899 minus 1

(36)




119865119900119901

=

1198651198911

119886 isin (0 119886119894

)

OFGN 119886 isin [119886119894

119886119898

)

119865119891ℎ

119886 isin [119886119898

119886119897

]

(37)





119886

gt 119864119903




119894

119886119898



119872



119862119883119864




Start

Yes


No


Command position


End



No

Yes

Yes

No

Yes


friction errors (7)



(37)






(17)


performance

Exit the module

(18)ndash(36)

(10)ndash(16)


119875119883119864








119872





(mmsdotsminus2) 5 5119886119898




10 101198732

8 81198733

8 8119873119862

10 10119873119865

4 4119864119903


(mmsdotsminus1) 05 05119879119890

(s) 0005 0005119879119903

(s) 0003 0003

119879119872



20 40 60 80 100 120 140003

004

005

006

007

008

a (mmmiddotsminus2)

Tom

(s)

X-axisY-axis

(a)



0

05

1

15

2

25

3

35

4

45

Fop

(mmmiddotsminus

1)

X-axisX-axis

Y-axisY-axis

(b)




119864119883119872

119875119883119864

119864119883119877

119862119883119864

119864119883119872

119875119883119864

119864119883119877

119862119883119864

119864119883119872

119875119883119864

119864119883119877







119864119883119872

119875119883119864

and 119864119883119877


119884119864

119864119884119872

119875119884119864

and 119864119884119877



1

S2

S3

and S4


1

S2

S3

and S4





119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877






C

B

D

minus10120583m

10120583m

30∘

210∘

60∘

240∘

90∘

270∘

120∘

300∘

150∘

330∘

180∘ 0∘

0120583m

A




1

S2

S3

and S4


119884119864

119864119884119872

119875119884119864

and 119864

119884119877


S2

S3

and S4







0 005 01 015 020

0001

0002

0003

0004

0005

0006

0007

Con

tour

erro

rs (m

m)

TM (s)

(a)

0 005 01 015 020

0001

0002

0003

0004

0005

0006

0007

Con

tour

erro

rs (m

m)

TM (s)

(b)

0 005 01 015 02

0

Trac

king

erro

rs (m

m)

TM (s)

minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

WTFCWFCWDOB

(c)

0

Trac

king

erro

rs (m

m)

0 005 01 015 02TM (s)

minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

WTFCWFCWDOB

(d)




119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877


S2

119871 = 30mm 119886 = 20mmsdotsminus2 789 232 789 329 184 069 184 077

S3

119871 = 50mm 119886 = 20mmsdotsminus2 846 245 846 332 192 071 192 082

S4

119871 = 80mm 119886 = 30mmsdotsminus2 928 366 928 455 215 073 215 085



0 002 004 006 008 01 012

0

0005

001

0015

002

0025

Con

tour

erro

rs (m

m)

minus0005

TM (s)

(a)

0 002 004 006 008 01 012

0

0002

0004

0006

0008

001

0012

0014

0016

Con

tour

erro

rs (m

m)

minus0002

TM (s)

(b)

0 002 004 006 008 01 012

0

0005

Trac

king

erro

rs (m

m)

TM (s)

minus0025

minus002

minus0015

minus001

minus0005

WTFCWFCWDOB

(c)

0 002 004 006 008 01 012

0

0002

Trac

king

erro

rs (m

m)

TM (s)

minus0016

minus0014

minus0012

minus001

minus0008

minus0006

minus0004

minus0002

WTFCWFCWDOB

(d)




Notation





119886119894




0 1 2 3 4 5 6 7 80

5

10

15

20

25

30Po

sitio

n co

mm

and

(mm

)

t (s)

S1

(a)

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)

WTFC WFC


TM (s)

(b)

0 1 2 3 4 50

5

10

15

20

25

30

Posit

ion

com

man

d (m

m)

t (s)

S2

(c)

WTFC WFC

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)


TM (s)

(d)

Posit

ion

com

man

d (m

m)

0 1 2 3 4 5 605

101520253035404550

t (s)

S3

(e)

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)

TM (s)


WTFC WFC

(f)

Posit

ion

com

man

d (m

m)

0 1 2 3 4 5 6 70

1020304050607080

t (s)

S4

(g)

0 005 01 015 02

Trac

king

erro

rs (m

m)

TM (s)

00001


WTFC WFC

(h)


S2

and S3

as well as S4



1



2




4



119886119898






119864119901



119864119905


119865119887119891119895


119865119888119904


119865119891119904


119865119894


119865119898



119865119901


119865119905119904



1198732


1198733


119873119886


119873119862



119875119891119909










the moment 119905err (s)1198791198981

1198791198982

and 119879

119898119898


1

1198862

and 119886119898

(s)119879119900119898




119899

(s)119879119911119894

1198791199111

1198791199112

and 119879119911119898



119894

1198861

1198862

and 119886119898

(s)119879119872



119888


Δ119886119891






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119888119904


119865119888119904

=

119865119898

minus 119865119894

119873119862

(19)


119886 = 119886119888119896

(20)

where 119896 = 1 and 119886 = 119886119888119896

= 119886119894


119901

can be expressed as

119865119901

= 119865119894

(21)



119879119898

= 119879119900119898

(119886) (22)


can be expressed as

119879119872

= 2119879119900119898

(119886) (23)


119886


119901

can be updated as

119865119901

= 119865119901

+ 119865119888119904

(24)

And 119865119901


gt 119865119898


119901

is 119865119888119896


119886


119896 = 119896 + 1

119886 = 119886119888119896

= 119886 + Δ119886119888

(25)



1198881

1198651198882

119865119888119896

119865119888119899

] 119896 = 1 2 (119873

1

+1198732

+1198733


1198881

1198861198882

119886119888119896

119886119888119899

] where119886119898

= 119886119888119899


11988611988811989112

1198861198882

11988611988811989123

1198861198883

119886119888119891(119894)(119894+1)

119886119888(ℎminus1)

119886119888119891(ℎminus1)

119886119888ℎ


119888119899

= 119886119888ℎ


119888119891(119894)(119894+1)

can be expressed as

Δ119886119891

=

Δ119886119888

2

119886119888119891(119894)(119894+1)

= 119886119888(119894)

+ Δ119886119891

119894 = 1 2 ℎ minus 1

(26)

where Δ119886119891


[1198861198911

1198861198912

119886119891119895

119886119891ℎ

]

119895 = 1 2 ℎ ℎ = 2119899 minus 1

(27)


1198881

11986511988811989112

1198651198882

11986511988811989123

1198651198883

119865119888119891(119894)(119894+1)

119865119888(ℎminus1)

119865119888119891(ℎminus1)

119865119888ℎ

] where ℎ = 2119899minus1 119865119888119899

=

119865119888ℎ


can be expressed as

119865119888119891(119894)(119894+1)

=

119865119888(119894)

+ 119865119888(119894+1)

2

119894 = 1 2 ℎ minus 1 (28)


[1198651198871198911

1198651198871198912

1198651198871198913

119865119887119891119895

119865119887119891(ℎminus1)

119865119887119891ℎ

]

119895 = 1 2 ℎ ℎ = 2119899 minus 1

(29)

where 119865119887119891119895


119891119904


119865119891119904

=

119865cs119873119865

(30)


119886 = 119886119891119895

(31)

where 119895 = 1 and 119886 = 1198861198911

= 119886119894


119901

can be expressed as

119865119901

= 119865119887119891119895

minus

119865119888119904

2

(32)



119879119898

= 119879119900119898

(119886) (33)


119886


119901

can be updated as

119865119901

= 119865119901

+ 119865119891119904

(34)

And 119865119901


gt 119865119887119891119895

+ (119865119888119904


119901

is119865119887119891119895


119886


119895 = 119895 + 1

119886 = 119886119891119895

= 119886 + Δ119886119891

(35)




[1198651198911

1198651198912

1198651198913

119865119891119895

119865119891(ℎminus1)

119865119891ℎ

]

119895 = 1 2 ℎ ℎ = 2119899 minus 1

(36)




119865119900119901

=

1198651198911

119886 isin (0 119886119894

)

OFGN 119886 isin [119886119894

119886119898

)

119865119891ℎ

119886 isin [119886119898

119886119897

]

(37)





119886

gt 119864119903




119894

119886119898



119872



119862119883119864




Start

Yes


No


Command position


End



No

Yes

Yes

No

Yes


friction errors (7)



(37)






(17)


performance

Exit the module

(18)ndash(36)

(10)ndash(16)


119875119883119864








119872





(mmsdotsminus2) 5 5119886119898




10 101198732

8 81198733

8 8119873119862

10 10119873119865

4 4119864119903


(mmsdotsminus1) 05 05119879119890

(s) 0005 0005119879119903

(s) 0003 0003

119879119872



20 40 60 80 100 120 140003

004

005

006

007

008

a (mmmiddotsminus2)

Tom

(s)

X-axisY-axis

(a)



0

05

1

15

2

25

3

35

4

45

Fop

(mmmiddotsminus

1)

X-axisX-axis

Y-axisY-axis

(b)




119864119883119872

119875119883119864

119864119883119877

119862119883119864

119864119883119872

119875119883119864

119864119883119877

119862119883119864

119864119883119872

119875119883119864

119864119883119877







119864119883119872

119875119883119864

and 119864119883119877


119884119864

119864119884119872

119875119884119864

and 119864119884119877



1

S2

S3

and S4


1

S2

S3

and S4





119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877






C

B

D

minus10120583m

10120583m

30∘

210∘

60∘

240∘

90∘

270∘

120∘

300∘

150∘

330∘

180∘ 0∘

0120583m

A




1

S2

S3

and S4


119884119864

119864119884119872

119875119884119864

and 119864

119884119877


S2

S3

and S4







0 005 01 015 020

0001

0002

0003

0004

0005

0006

0007

Con

tour

erro

rs (m

m)

TM (s)

(a)

0 005 01 015 020

0001

0002

0003

0004

0005

0006

0007

Con

tour

erro

rs (m

m)

TM (s)

(b)

0 005 01 015 02

0

Trac

king

erro

rs (m

m)

TM (s)

minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

WTFCWFCWDOB

(c)

0

Trac

king

erro

rs (m

m)

0 005 01 015 02TM (s)

minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

WTFCWFCWDOB

(d)




119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877


S2

119871 = 30mm 119886 = 20mmsdotsminus2 789 232 789 329 184 069 184 077

S3

119871 = 50mm 119886 = 20mmsdotsminus2 846 245 846 332 192 071 192 082

S4

119871 = 80mm 119886 = 30mmsdotsminus2 928 366 928 455 215 073 215 085



0 002 004 006 008 01 012

0

0005

001

0015

002

0025

Con

tour

erro

rs (m

m)

minus0005

TM (s)

(a)

0 002 004 006 008 01 012

0

0002

0004

0006

0008

001

0012

0014

0016

Con

tour

erro

rs (m

m)

minus0002

TM (s)

(b)

0 002 004 006 008 01 012

0

0005

Trac

king

erro

rs (m

m)

TM (s)

minus0025

minus002

minus0015

minus001

minus0005

WTFCWFCWDOB

(c)

0 002 004 006 008 01 012

0

0002

Trac

king

erro

rs (m

m)

TM (s)

minus0016

minus0014

minus0012

minus001

minus0008

minus0006

minus0004

minus0002

WTFCWFCWDOB

(d)




Notation





119886119894




0 1 2 3 4 5 6 7 80

5

10

15

20

25

30Po

sitio

n co

mm

and

(mm

)

t (s)

S1

(a)

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)

WTFC WFC


TM (s)

(b)

0 1 2 3 4 50

5

10

15

20

25

30

Posit

ion

com

man

d (m

m)

t (s)

S2

(c)

WTFC WFC

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)


TM (s)

(d)

Posit

ion

com

man

d (m

m)

0 1 2 3 4 5 605

101520253035404550

t (s)

S3

(e)

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)

TM (s)


WTFC WFC

(f)

Posit

ion

com

man

d (m

m)

0 1 2 3 4 5 6 70

1020304050607080

t (s)

S4

(g)

0 005 01 015 02

Trac

king

erro

rs (m

m)

TM (s)

00001


WTFC WFC

(h)


S2

and S3

as well as S4



1



2




4



119886119898






119864119901



119864119905


119865119887119891119895


119865119888119904


119865119891119904


119865119894


119865119898



119865119901


119865119905119904



1198732


1198733


119873119886


119873119862



119875119891119909










the moment 119905err (s)1198791198981

1198791198982

and 119879

119898119898


1

1198862

and 119886119898

(s)119879119900119898




119899

(s)119879119911119894

1198791199111

1198791199112

and 119879119911119898



119894

1198861

1198862

and 119886119898

(s)119879119872



119888


Δ119886119891






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












[1198651198911

1198651198912

1198651198913

119865119891119895

119865119891(ℎminus1)

119865119891ℎ

]

119895 = 1 2 ℎ ℎ = 2119899 minus 1

(36)




119865119900119901

=

1198651198911

119886 isin (0 119886119894

)

OFGN 119886 isin [119886119894

119886119898

)

119865119891ℎ

119886 isin [119886119898

119886119897

]

(37)





119886

gt 119864119903




119894

119886119898



119872



119862119883119864




Start

Yes


No


Command position


End



No

Yes

Yes

No

Yes


friction errors (7)



(37)






(17)


performance

Exit the module

(18)ndash(36)

(10)ndash(16)


119875119883119864








119872





(mmsdotsminus2) 5 5119886119898




10 101198732

8 81198733

8 8119873119862

10 10119873119865

4 4119864119903


(mmsdotsminus1) 05 05119879119890

(s) 0005 0005119879119903

(s) 0003 0003

119879119872



20 40 60 80 100 120 140003

004

005

006

007

008

a (mmmiddotsminus2)

Tom

(s)

X-axisY-axis

(a)



0

05

1

15

2

25

3

35

4

45

Fop

(mmmiddotsminus

1)

X-axisX-axis

Y-axisY-axis

(b)




119864119883119872

119875119883119864

119864119883119877

119862119883119864

119864119883119872

119875119883119864

119864119883119877

119862119883119864

119864119883119872

119875119883119864

119864119883119877







119864119883119872

119875119883119864

and 119864119883119877


119884119864

119864119884119872

119875119884119864

and 119864119884119877



1

S2

S3

and S4


1

S2

S3

and S4





119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877






C

B

D

minus10120583m

10120583m

30∘

210∘

60∘

240∘

90∘

270∘

120∘

300∘

150∘

330∘

180∘ 0∘

0120583m

A




1

S2

S3

and S4


119884119864

119864119884119872

119875119884119864

and 119864

119884119877


S2

S3

and S4







0 005 01 015 020

0001

0002

0003

0004

0005

0006

0007

Con

tour

erro

rs (m

m)

TM (s)

(a)

0 005 01 015 020

0001

0002

0003

0004

0005

0006

0007

Con

tour

erro

rs (m

m)

TM (s)

(b)

0 005 01 015 02

0

Trac

king

erro

rs (m

m)

TM (s)

minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

WTFCWFCWDOB

(c)

0

Trac

king

erro

rs (m

m)

0 005 01 015 02TM (s)

minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

WTFCWFCWDOB

(d)




119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877


S2

119871 = 30mm 119886 = 20mmsdotsminus2 789 232 789 329 184 069 184 077

S3

119871 = 50mm 119886 = 20mmsdotsminus2 846 245 846 332 192 071 192 082

S4

119871 = 80mm 119886 = 30mmsdotsminus2 928 366 928 455 215 073 215 085



0 002 004 006 008 01 012

0

0005

001

0015

002

0025

Con

tour

erro

rs (m

m)

minus0005

TM (s)

(a)

0 002 004 006 008 01 012

0

0002

0004

0006

0008

001

0012

0014

0016

Con

tour

erro

rs (m

m)

minus0002

TM (s)

(b)

0 002 004 006 008 01 012

0

0005

Trac

king

erro

rs (m

m)

TM (s)

minus0025

minus002

minus0015

minus001

minus0005

WTFCWFCWDOB

(c)

0 002 004 006 008 01 012

0

0002

Trac

king

erro

rs (m

m)

TM (s)

minus0016

minus0014

minus0012

minus001

minus0008

minus0006

minus0004

minus0002

WTFCWFCWDOB

(d)




Notation





119886119894




0 1 2 3 4 5 6 7 80

5

10

15

20

25

30Po

sitio

n co

mm

and

(mm

)

t (s)

S1

(a)

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)

WTFC WFC


TM (s)

(b)

0 1 2 3 4 50

5

10

15

20

25

30

Posit

ion

com

man

d (m

m)

t (s)

S2

(c)

WTFC WFC

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)


TM (s)

(d)

Posit

ion

com

man

d (m

m)

0 1 2 3 4 5 605

101520253035404550

t (s)

S3

(e)

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)

TM (s)


WTFC WFC

(f)

Posit

ion

com

man

d (m

m)

0 1 2 3 4 5 6 70

1020304050607080

t (s)

S4

(g)

0 005 01 015 02

Trac

king

erro

rs (m

m)

TM (s)

00001


WTFC WFC

(h)


S2

and S3

as well as S4



1



2




4



119886119898






119864119901



119864119905


119865119887119891119895


119865119888119904


119865119891119904


119865119894


119865119898



119865119901


119865119905119904



1198732


1198733


119873119886


119873119862



119875119891119909










the moment 119905err (s)1198791198981

1198791198982

and 119879

119898119898


1

1198862

and 119886119898

(s)119879119900119898




119899

(s)119879119911119894

1198791199111

1198791199112

and 119879119911119898



119894

1198861

1198862

and 119886119898

(s)119879119872



119888


Δ119886119891






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start

Yes


No


Command position


End



No

Yes

Yes

No

Yes


friction errors (7)



(37)






(17)


performance

Exit the module

(18)ndash(36)

(10)ndash(16)


119875119883119864








119872





(mmsdotsminus2) 5 5119886119898




10 101198732

8 81198733

8 8119873119862

10 10119873119865

4 4119864119903


(mmsdotsminus1) 05 05119879119890

(s) 0005 0005119879119903

(s) 0003 0003

119879119872



20 40 60 80 100 120 140003

004

005

006

007

008

a (mmmiddotsminus2)

Tom

(s)

X-axisY-axis

(a)



0

05

1

15

2

25

3

35

4

45

Fop

(mmmiddotsminus

1)

X-axisX-axis

Y-axisY-axis

(b)




119864119883119872

119875119883119864

119864119883119877

119862119883119864

119864119883119872

119875119883119864

119864119883119877

119862119883119864

119864119883119872

119875119883119864

119864119883119877







119864119883119872

119875119883119864

and 119864119883119877


119884119864

119864119884119872

119875119884119864

and 119864119884119877



1

S2

S3

and S4


1

S2

S3

and S4





119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877






C

B

D

minus10120583m

10120583m

30∘

210∘

60∘

240∘

90∘

270∘

120∘

300∘

150∘

330∘

180∘ 0∘

0120583m

A




1

S2

S3

and S4


119884119864

119864119884119872

119875119884119864

and 119864

119884119877


S2

S3

and S4







0 005 01 015 020

0001

0002

0003

0004

0005

0006

0007

Con

tour

erro

rs (m

m)

TM (s)

(a)

0 005 01 015 020

0001

0002

0003

0004

0005

0006

0007

Con

tour

erro

rs (m

m)

TM (s)

(b)

0 005 01 015 02

0

Trac

king

erro

rs (m

m)

TM (s)

minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

WTFCWFCWDOB

(c)

0

Trac

king

erro

rs (m

m)

0 005 01 015 02TM (s)

minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

WTFCWFCWDOB

(d)




119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877


S2

119871 = 30mm 119886 = 20mmsdotsminus2 789 232 789 329 184 069 184 077

S3

119871 = 50mm 119886 = 20mmsdotsminus2 846 245 846 332 192 071 192 082

S4

119871 = 80mm 119886 = 30mmsdotsminus2 928 366 928 455 215 073 215 085



0 002 004 006 008 01 012

0

0005

001

0015

002

0025

Con

tour

erro

rs (m

m)

minus0005

TM (s)

(a)

0 002 004 006 008 01 012

0

0002

0004

0006

0008

001

0012

0014

0016

Con

tour

erro

rs (m

m)

minus0002

TM (s)

(b)

0 002 004 006 008 01 012

0

0005

Trac

king

erro

rs (m

m)

TM (s)

minus0025

minus002

minus0015

minus001

minus0005

WTFCWFCWDOB

(c)

0 002 004 006 008 01 012

0

0002

Trac

king

erro

rs (m

m)

TM (s)

minus0016

minus0014

minus0012

minus001

minus0008

minus0006

minus0004

minus0002

WTFCWFCWDOB

(d)




Notation





119886119894




0 1 2 3 4 5 6 7 80

5

10

15

20

25

30Po

sitio

n co

mm

and

(mm

)

t (s)

S1

(a)

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)

WTFC WFC


TM (s)

(b)

0 1 2 3 4 50

5

10

15

20

25

30

Posit

ion

com

man

d (m

m)

t (s)

S2

(c)

WTFC WFC

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)


TM (s)

(d)

Posit

ion

com

man

d (m

m)

0 1 2 3 4 5 605

101520253035404550

t (s)

S3

(e)

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)

TM (s)


WTFC WFC

(f)

Posit

ion

com

man

d (m

m)

0 1 2 3 4 5 6 70

1020304050607080

t (s)

S4

(g)

0 005 01 015 02

Trac

king

erro

rs (m

m)

TM (s)

00001


WTFC WFC

(h)


S2

and S3

as well as S4



1



2




4



119886119898






119864119901



119864119905


119865119887119891119895


119865119888119904


119865119891119904


119865119894


119865119898



119865119901


119865119905119904



1198732


1198733


119873119886


119873119862



119875119891119909










the moment 119905err (s)1198791198981

1198791198982

and 119879

119898119898


1

1198862

and 119886119898

(s)119879119900119898




119899

(s)119879119911119894

1198791199111

1198791199112

and 119879119911119898



119894

1198861

1198862

and 119886119898

(s)119879119872



119888


Δ119886119891






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










20 40 60 80 100 120 140003

004

005

006

007

008

a (mmmiddotsminus2)

Tom

(s)

X-axisY-axis

(a)



0

05

1

15

2

25

3

35

4

45

Fop

(mmmiddotsminus

1)

X-axisX-axis

Y-axisY-axis

(b)




119864119883119872

119875119883119864

119864119883119877

119862119883119864

119864119883119872

119875119883119864

119864119883119877

119862119883119864

119864119883119872

119875119883119864

119864119883119877







119864119883119872

119875119883119864

and 119864119883119877


119884119864

119864119884119872

119875119884119864

and 119864119884119877



1

S2

S3

and S4


1

S2

S3

and S4





119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877






C

B

D

minus10120583m

10120583m

30∘

210∘

60∘

240∘

90∘

270∘

120∘

300∘

150∘

330∘

180∘ 0∘

0120583m

A




1

S2

S3

and S4


119884119864

119864119884119872

119875119884119864

and 119864

119884119877


S2

S3

and S4







0 005 01 015 020

0001

0002

0003

0004

0005

0006

0007

Con

tour

erro

rs (m

m)

TM (s)

(a)

0 005 01 015 020

0001

0002

0003

0004

0005

0006

0007

Con

tour

erro

rs (m

m)

TM (s)

(b)

0 005 01 015 02

0

Trac

king

erro

rs (m

m)

TM (s)

minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

WTFCWFCWDOB

(c)

0

Trac

king

erro

rs (m

m)

0 005 01 015 02TM (s)

minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

WTFCWFCWDOB

(d)




119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877


S2

119871 = 30mm 119886 = 20mmsdotsminus2 789 232 789 329 184 069 184 077

S3

119871 = 50mm 119886 = 20mmsdotsminus2 846 245 846 332 192 071 192 082

S4

119871 = 80mm 119886 = 30mmsdotsminus2 928 366 928 455 215 073 215 085



0 002 004 006 008 01 012

0

0005

001

0015

002

0025

Con

tour

erro

rs (m

m)

minus0005

TM (s)

(a)

0 002 004 006 008 01 012

0

0002

0004

0006

0008

001

0012

0014

0016

Con

tour

erro

rs (m

m)

minus0002

TM (s)

(b)

0 002 004 006 008 01 012

0

0005

Trac

king

erro

rs (m

m)

TM (s)

minus0025

minus002

minus0015

minus001

minus0005

WTFCWFCWDOB

(c)

0 002 004 006 008 01 012

0

0002

Trac

king

erro

rs (m

m)

TM (s)

minus0016

minus0014

minus0012

minus001

minus0008

minus0006

minus0004

minus0002

WTFCWFCWDOB

(d)




Notation





119886119894




0 1 2 3 4 5 6 7 80

5

10

15

20

25

30Po

sitio

n co

mm

and

(mm

)

t (s)

S1

(a)

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)

WTFC WFC


TM (s)

(b)

0 1 2 3 4 50

5

10

15

20

25

30

Posit

ion

com

man

d (m

m)

t (s)

S2

(c)

WTFC WFC

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)


TM (s)

(d)

Posit

ion

com

man

d (m

m)

0 1 2 3 4 5 605

101520253035404550

t (s)

S3

(e)

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)

TM (s)


WTFC WFC

(f)

Posit

ion

com

man

d (m

m)

0 1 2 3 4 5 6 70

1020304050607080

t (s)

S4

(g)

0 005 01 015 02

Trac

king

erro

rs (m

m)

TM (s)

00001


WTFC WFC

(h)


S2

and S3

as well as S4



1



2




4



119886119898






119864119901



119864119905


119865119887119891119895


119865119888119904


119865119891119904


119865119894


119865119898



119865119901


119865119905119904



1198732


1198733


119873119886


119873119862



119875119891119909










the moment 119905err (s)1198791198981

1198791198982

and 119879

119898119898


1

1198862

and 119886119898

(s)119879119900119898




119899

(s)119879119911119894

1198791199111

1198791199112

and 119879119911119898



119894

1198861

1198862

and 119886119898

(s)119879119872



119888


Δ119886119891






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877






C

B

D

minus10120583m

10120583m

30∘

210∘

60∘

240∘

90∘

270∘

120∘

300∘

150∘

330∘

180∘ 0∘

0120583m

A




1

S2

S3

and S4


119884119864

119864119884119872

119875119884119864

and 119864

119884119877


S2

S3

and S4







0 005 01 015 020

0001

0002

0003

0004

0005

0006

0007

Con

tour

erro

rs (m

m)

TM (s)

(a)

0 005 01 015 020

0001

0002

0003

0004

0005

0006

0007

Con

tour

erro

rs (m

m)

TM (s)

(b)

0 005 01 015 02

0

Trac

king

erro

rs (m

m)

TM (s)

minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

WTFCWFCWDOB

(c)

0

Trac

king

erro

rs (m

m)

0 005 01 015 02TM (s)

minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

WTFCWFCWDOB

(d)




119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877


S2

119871 = 30mm 119886 = 20mmsdotsminus2 789 232 789 329 184 069 184 077

S3

119871 = 50mm 119886 = 20mmsdotsminus2 846 245 846 332 192 071 192 082

S4

119871 = 80mm 119886 = 30mmsdotsminus2 928 366 928 455 215 073 215 085



0 002 004 006 008 01 012

0

0005

001

0015

002

0025

Con

tour

erro

rs (m

m)

minus0005

TM (s)

(a)

0 002 004 006 008 01 012

0

0002

0004

0006

0008

001

0012

0014

0016

Con

tour

erro

rs (m

m)

minus0002

TM (s)

(b)

0 002 004 006 008 01 012

0

0005

Trac

king

erro

rs (m

m)

TM (s)

minus0025

minus002

minus0015

minus001

minus0005

WTFCWFCWDOB

(c)

0 002 004 006 008 01 012

0

0002

Trac

king

erro

rs (m

m)

TM (s)

minus0016

minus0014

minus0012

minus001

minus0008

minus0006

minus0004

minus0002

WTFCWFCWDOB

(d)




Notation





119886119894




0 1 2 3 4 5 6 7 80

5

10

15

20

25

30Po

sitio

n co

mm

and

(mm

)

t (s)

S1

(a)

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)

WTFC WFC


TM (s)

(b)

0 1 2 3 4 50

5

10

15

20

25

30

Posit

ion

com

man

d (m

m)

t (s)

S2

(c)

WTFC WFC

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)


TM (s)

(d)

Posit

ion

com

man

d (m

m)

0 1 2 3 4 5 605

101520253035404550

t (s)

S3

(e)

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)

TM (s)


WTFC WFC

(f)

Posit

ion

com

man

d (m

m)

0 1 2 3 4 5 6 70

1020304050607080

t (s)

S4

(g)

0 005 01 015 02

Trac

king

erro

rs (m

m)

TM (s)

00001


WTFC WFC

(h)


S2

and S3

as well as S4



1



2




4



119886119898






119864119901



119864119905


119865119887119891119895


119865119888119904


119865119891119904


119865119894


119865119898



119865119901


119865119905119904



1198732


1198733


119873119886


119873119862



119875119891119909










the moment 119905err (s)1198791198981

1198791198982

and 119879

119898119898


1

1198862

and 119886119898

(s)119879119900119898




119899

(s)119879119911119894

1198791199111

1198791199112

and 119879119911119898



119894

1198861

1198862

and 119886119898

(s)119879119872



119888


Δ119886119891






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 005 01 015 020

0001

0002

0003

0004

0005

0006

0007

Con

tour

erro

rs (m

m)

TM (s)

(a)

0 005 01 015 020

0001

0002

0003

0004

0005

0006

0007

Con

tour

erro

rs (m

m)

TM (s)

(b)

0 005 01 015 02

0

Trac

king

erro

rs (m

m)

TM (s)

minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

WTFCWFCWDOB

(c)

0

Trac

king

erro

rs (m

m)

0 005 01 015 02TM (s)

minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

WTFCWFCWDOB

(d)




119864119884119872

119875119884119864

119864119884119877

119862119884119864

119864119884119872

119875119884119864

119864119884119877


S2

119871 = 30mm 119886 = 20mmsdotsminus2 789 232 789 329 184 069 184 077

S3

119871 = 50mm 119886 = 20mmsdotsminus2 846 245 846 332 192 071 192 082

S4

119871 = 80mm 119886 = 30mmsdotsminus2 928 366 928 455 215 073 215 085



0 002 004 006 008 01 012

0

0005

001

0015

002

0025

Con

tour

erro

rs (m

m)

minus0005

TM (s)

(a)

0 002 004 006 008 01 012

0

0002

0004

0006

0008

001

0012

0014

0016

Con

tour

erro

rs (m

m)

minus0002

TM (s)

(b)

0 002 004 006 008 01 012

0

0005

Trac

king

erro

rs (m

m)

TM (s)

minus0025

minus002

minus0015

minus001

minus0005

WTFCWFCWDOB

(c)

0 002 004 006 008 01 012

0

0002

Trac

king

erro

rs (m

m)

TM (s)

minus0016

minus0014

minus0012

minus001

minus0008

minus0006

minus0004

minus0002

WTFCWFCWDOB

(d)




Notation





119886119894




0 1 2 3 4 5 6 7 80

5

10

15

20

25

30Po

sitio

n co

mm

and

(mm

)

t (s)

S1

(a)

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)

WTFC WFC


TM (s)

(b)

0 1 2 3 4 50

5

10

15

20

25

30

Posit

ion

com

man

d (m

m)

t (s)

S2

(c)

WTFC WFC

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)


TM (s)

(d)

Posit

ion

com

man

d (m

m)

0 1 2 3 4 5 605

101520253035404550

t (s)

S3

(e)

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)

TM (s)


WTFC WFC

(f)

Posit

ion

com

man

d (m

m)

0 1 2 3 4 5 6 70

1020304050607080

t (s)

S4

(g)

0 005 01 015 02

Trac

king

erro

rs (m

m)

TM (s)

00001


WTFC WFC

(h)


S2

and S3

as well as S4



1



2




4



119886119898






119864119901



119864119905


119865119887119891119895


119865119888119904


119865119891119904


119865119894


119865119898



119865119901


119865119905119904



1198732


1198733


119873119886


119873119862



119875119891119909










the moment 119905err (s)1198791198981

1198791198982

and 119879

119898119898


1

1198862

and 119886119898

(s)119879119900119898




119899

(s)119879119911119894

1198791199111

1198791199112

and 119879119911119898



119894

1198861

1198862

and 119886119898

(s)119879119872



119888


Δ119886119891






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 002 004 006 008 01 012

0

0005

001

0015

002

0025

Con

tour

erro

rs (m

m)

minus0005

TM (s)

(a)

0 002 004 006 008 01 012

0

0002

0004

0006

0008

001

0012

0014

0016

Con

tour

erro

rs (m

m)

minus0002

TM (s)

(b)

0 002 004 006 008 01 012

0

0005

Trac

king

erro

rs (m

m)

TM (s)

minus0025

minus002

minus0015

minus001

minus0005

WTFCWFCWDOB

(c)

0 002 004 006 008 01 012

0

0002

Trac

king

erro

rs (m

m)

TM (s)

minus0016

minus0014

minus0012

minus001

minus0008

minus0006

minus0004

minus0002

WTFCWFCWDOB

(d)




Notation





119886119894




0 1 2 3 4 5 6 7 80

5

10

15

20

25

30Po

sitio

n co

mm

and

(mm

)

t (s)

S1

(a)

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)

WTFC WFC


TM (s)

(b)

0 1 2 3 4 50

5

10

15

20

25

30

Posit

ion

com

man

d (m

m)

t (s)

S2

(c)

WTFC WFC

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)


TM (s)

(d)

Posit

ion

com

man

d (m

m)

0 1 2 3 4 5 605

101520253035404550

t (s)

S3

(e)

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)

TM (s)


WTFC WFC

(f)

Posit

ion

com

man

d (m

m)

0 1 2 3 4 5 6 70

1020304050607080

t (s)

S4

(g)

0 005 01 015 02

Trac

king

erro

rs (m

m)

TM (s)

00001


WTFC WFC

(h)


S2

and S3

as well as S4



1



2




4



119886119898






119864119901



119864119905


119865119887119891119895


119865119888119904


119865119891119904


119865119894


119865119898



119865119901


119865119905119904



1198732


1198733


119873119886


119873119862



119875119891119909










the moment 119905err (s)1198791198981

1198791198982

and 119879

119898119898


1

1198862

and 119886119898

(s)119879119900119898




119899

(s)119879119911119894

1198791199111

1198791199112

and 119879119911119898



119894

1198861

1198862

and 119886119898

(s)119879119872



119888


Δ119886119891






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6 7 80

5

10

15

20

25

30Po

sitio

n co

mm

and

(mm

)

t (s)

S1

(a)

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)

WTFC WFC


TM (s)

(b)

0 1 2 3 4 50

5

10

15

20

25

30

Posit

ion

com

man

d (m

m)

t (s)

S2

(c)

WTFC WFC

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)


TM (s)

(d)

Posit

ion

com

man

d (m

m)

0 1 2 3 4 5 605

101520253035404550

t (s)

S3

(e)

0 005 01 015 02

00001

Trac

king

erro

rs (m

m)

TM (s)


WTFC WFC

(f)

Posit

ion

com

man

d (m

m)

0 1 2 3 4 5 6 70

1020304050607080

t (s)

S4

(g)

0 005 01 015 02

Trac

king

erro

rs (m

m)

TM (s)

00001


WTFC WFC

(h)


S2

and S3

as well as S4



1



2




4



119886119898






119864119901



119864119905


119865119887119891119895


119865119888119904


119865119891119904


119865119894


119865119898



119865119901


119865119905119904



1198732


1198733


119873119886


119873119862



119875119891119909










the moment 119905err (s)1198791198981

1198791198982

and 119879

119898119898


1

1198862

and 119886119898

(s)119879119900119898




119899

(s)119879119911119894

1198791199111

1198791199112

and 119879119911119898



119894

1198861

1198862

and 119886119898

(s)119879119872



119888


Δ119886119891






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119865119901


119865119905119904



1198732


1198733


119873119886


119873119862



119875119891119909










the moment 119905err (s)1198791198981

1198791198982

and 119879

119898119898


1

1198862

and 119886119898

(s)119879119900119898




119899

(s)119879119911119894

1198791199111

1198791199112

and 119879119911119898



119894

1198861

1198862

and 119886119898

(s)119879119872



119888


Δ119886119891






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article A Novel Time-Varying Friction...

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Transcript of Research Article A Novel Time-Varying Friction...