a novel approach for nurbs interpolation with minimal feed … · 2019. 7. 30. · a novel approach...

Research ArticleA Novel Approach for NURBS Interpolation with Minimal FeedRate Fluctuation Based on Improved Adams-Moulton Method

Junquan Peng1 Xinhua Liu1 Lei Si12 and Jingjing Liu1

1School of Mechanical and Electrical Engineering China University of Mining amp Technology Xuzhou China2School of Information and Electrical Engineering China University of Mining amp Technology Xuzhou China

Correspondence should be addressed to Xinhua Liu l_xinhua_2006126com

Received 14 November 2016 Accepted 8 March 2017 Published 22 March 2017

Academic Editor Alessandro Gasparetto

Copyright copy 2017 Junquan Peng et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In order to reduce the feed rate fluctuation of interpolation a novel approach for NURBS interpolation with minimal feed ratefluctuation based on improved Adams-Moulton (IAM) method is proposed At first the representation and calculation of NURBScurve interpolation are described Then the constraints of feeding step length are firstly given out to calculate the minimal hopingfeeding step length and the detailed IAMmethod of NURBS curve interpolation is presented Finally simulations and experimentsare carried out to verify the feasibility and applicability of proposed IAMmethod

1 Introduction

Along with rapid progress and development of sciencetechnology and social economy the machining accuracyof CNC machine tools is increasingly demanding [1] Justbecause of this condition conventional linear and circularinterpolation cannot fit the requirement of machining inmodern CNC systems However NURBS method offers acommon mathematical form for the precise representationof standard analytical shapes as well as free-form curves andsurfaces [2ndash4] Since NURBS curve has lots of advantages indesigning complicated shapes compared with conventionallinear and circular its interpolation methods have beendeveloped in CADCAM applications [5 6]

Although many approximation methods connected withNURBS interpolation have been proposed to obtain efficientmachining feed rate fluctuation is an important challengeTo the best of our knowledge the problem of NURBSinterpolation for CNC machines has almost not been dealtwith

Recent publications relevant to this paper are mainlysummarized Farouki and Tsai [7] proposed PH curveinterpolation which could give rational analytical relationsbetween arc length and parameters as well as effectivelyreducing the interpolation output feed rate fluctuation Tsai

and Chung [8] reduced the interpolation output feed ratefluctuation through iterative methods especially in curvecurvature at the output of the feed rate fluctuation Yongand Narayanaswami [9] proposed a spline curve interpola-tion algorithm based on the second-order Taylor expansionmethod Yeh and Hsu [10] proposed a speed-controlledinterpolator by incorporating a compensatory parameter tominimize the feed rate fluctuation Erkorkmaz and Altintas[11 12] adopted quintic polynomial for interpolation withminimal feed rate fluctuation Zhiming et al [13] proposeda real-time parametric interpolation technique Cheng etal [14] compared different NURBS interpolation algorithmson the basis of computing time interpolation accuracyand feed rate fluctuation Tikhon et al [15] proposed aNURBS curve interpolation device based on the adaptivefeed rate control Lei et al [16] proposed an inverse lengthfunction method which cubicquintic Hermite spline wasused to achieve very low feed rate fluctuation in non-real-time processes Lin et al [17] proposed a dynamics-basedinterpolator with real-time look-ahead algorithm to generatea smooth and jerk-limited accelerationdeceleration feedrate profile Jia et al [18] presented a NURBS interpolationmethod based on Adams algorithm which ensured the real-time of the algorithm through reasonable simplification andapproximation Wu et al [19 20] presented a nonuniform

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 8718760 10 pageshttpsdoiorg10115520178718760

2 Mathematical Problems in Engineering

rational B-spline interpolation algorithm with compensatoryparameter to minimize feed rate fluctuation and contourerror by improving the inverse length function method withthe usage of biarc guide curve Sun et al [21] proposed a noveladaptive federate interpolation method for NURBS tool pathwith drive constraints which can acquire smooth feed rateprofile Liu et al [22] proposed a double secant interpolationalgorithm and a fast calculation and derivation algorithm ofNURBS to solve the problem of feed rate fluctuation Liuet al [23] proposed a new NURBS interpolation methodwith minimal feed rate fluctuation for CNCmachine tools toobtain the sampling curve parameter

Although many approaches for NURBS interpolationhave been proposed in above literature they have somecommon disadvantages summarized as follows Firstly mostof researchers presented their methods for NURBS interpola-tion with the second-order Taylor expansion method How-ever the second-order Taylor expansion method referred tosecond derivative of NURBS curve and it was complicatedSecondly some researchers considered the problem of feedrate fluctuation Nevertheless results of feed rate fluctuationwere very high and unsatisfactory

In order to solve the above problems a novel approachfor NURBS interpolation with minimal feed rate fluctuationbased on improved Adams-Moulton method is applied andthe rest of this paper is organized as follows In Section 2representation and calculation ofNURBS curve interpolationare presented The improved predictor-corrector formula ofNURBS curve interpolation is proposed in Section 3 InSection 4 simulations with the trident and butterfly NURBScurves are conducted to verify the feasibility of the proposedmethod Experimental verification is conducted in Section 5Our conclusions are summarized in Section 6

2 Representation and Calculation ofNURBS Curve Interpolation

Suppose that 119862(119906) represents a NURBS curve and is given by[2 24 25]

119862 (119906) = sum119899119894=0119873119894119901 (119906) 119908119894119875119894sum119899119894=0119873119894119901 (119906) 119908119894 (1)

where 119875119894 are the control points 119908119894 are the weights of 119875119894(119899+1) are the number of the control points and119901 is the degree

of NURBS curve 119873119894119901(119906) is the 119901th-degree B-spline basisfunction and can be calculated using the recursive formulasgiven as follows

1198731198940 (119906) = 1 if 119906119894 le 119906 le 119906119894+10 otherwise

119873119894119901 (119906) = 119906 minus 119906119894119906119894+119901 minus 119906119894119873119894119901minus1 (119906)+ 119906119894+119901+1 minus 119906119906119894+119901+1 minus 119906119894+1119873119894+1119901minus1 (119906)

(2)

where 1199060 1199061 119906119899+119901+1 represents the knot vectors and 119906 isthe interpolation parameter

The node vector in parameter coordinates is not uni-formly distributed for NURBS curve interpolation in generalso the value of the B-spline basis function is not the sameand it is necessary to obtain all the B-spline basis functionsrespectively In addition the weight factor value of NURBScurve interpolation needs to be obtained and this makesthe computation more time-consuming However becauseof the simple expression and high derivative evaluationefficiency the matrix expression of the NURBS curve canbe used to solve the complicated calculation problem in thepretreatment process

Due to the fact that the NURBS curve of degree 3 isused widely the following example is used to be illustratedIf NURBS curve control polygon [1198750 1198751 119875119899] weights[1199080 1199081 119908119899] and node vectors [1199060 1199061 119906119899+119901+1] aregiven then NURBS curve of degree 3 can be expressed asfollows [18]

119862 (119905) = sum119894119895=119894minus3 1199081198951198751198951198731198953 (119906)sum119894119895=119894minus3 1199081198951198731198953 (119906)

= [1 119905 1199052 1199053]119872119894 [ 119908119894minus3119875119894minus3119908119894minus2119875119894minus2119908119894minus1119875119894minus1119908119894119875119894

][1 119905 1199052 1199053]119872119894 [ 119908119894minus3119908119894minus2119908119894minus1

119908119894

] (3)

where 119894 = 3 4 119899 119905 isin [0 1] 119905 = (119906 minus 119906119894)(119906119894+1 minus 119906119894) =(119906minus119906119894)nabla1119894 nabla1119894 = 119906119894+1 minus119906119894 nabla2119894 = 119906119894+2 minus119906119894+1 nabla3119894 = 119906119894+3 minus119906119894+2especially nabla0119894 = 0 and the expression of119872119894 is as follows

M119894 = [[[[[

11989811 11989812 11989813 1198981411989821 11989822 11989823 1198982411989831 11989832 11989833 1198983411989841 11989842 11989843 11989844]]]]]

=

[[[[[[[[[[[[[[[[[[[

(nabla1119894 )2nabla2119894minus1nabla3119894minus2 1 minus 11989811 minus 11989813 (nabla1119894minus1)2nabla2119894minus1nabla3119894minus1 0minus311989811 311989811 minus 11989823 3nabla1119894minus1nabla1119894nabla2119894minus1nabla3119894minus1 0311989811 minus311989811 minus 11989833 3 (nabla1119894 )2nabla2119894minus1nabla3119894minus1 0minus11989811 11989811 minus 11989843 minus 11989844 minus(1311989833 + 11989844 + (nabla1119894 )2nabla2119894 nabla3119894minus1)

(nabla1119894 )2nabla2119894 nabla3119894

]]]]]]]]]]]]]]]]]]]

(4)

Mathematical Problems in Engineering 3

Here the expression of parameters is as follows

1198610 = 11989811119908119894minus3119875119894minus3 + 11989812119908119894minus2119875119894minus2 + 11989813119908119894minus1119875119894minus1+ 11989814119908119894119875119894




11986110158400 = 11989811119908119894minus3 + 11989812119908119894minus2 + 11989813119908119894minus1 + 1198981411990811989411986110158401 = 11989821119908119894minus3 + 11989822119908119894minus2 + 11989823119908119894minus1 + 1198982411990811989411986110158402 = 11989831119908119894minus3 + 11989832119908119894minus2 + 11989833119908119894minus1 + 1198983411990811989411986110158403 = 11989841119908119894minus3 + 11989842119908119894minus2 + 11989843119908119894minus1 + 11989844119908119894

(5)

When (4) and (5) are substituted into (3) the newexpression of NURBS curve is as follows

119862 (119905) = sum119894119895=119894minus3 1199081198951198751198951198731198953 (119906)sum119894119895=119894minus3 1199081198951198731198953 (119906) = 1198610 + 1198611119905 + 11986121199052 + 1198613119905311986110158400 + 11986110158401119905 + 119861101584021199052 + 119861101584031199053 119905 isin [0 1] 119894 = 3 4 119899

(6)

According to (6) molecular coefficients 1198610 1198611 1198612 1198613and denominator coefficients 11986110158400 11986110158401 11986110158402 11986110158403 have little todo with parameter 119905 Therefore it is able to avoid therepeated calculation of the real-time interpolation processwith condition that the numerical value of these coefficientsis obtained That is improvement of interpolation speed ispromoted

3 Improved Adams-Moulton Method ofNURBS Curve Interpolation

31 Constraints of Feeding Step Length In this subsection thesystem architecture of proposed IAM method is establishedin Figure 1 which is utilized to implement CNC control tasks

32 Constraints of Feeding Step Length Normally estimatingfeed step length of trajectory space is calculated as (7)and based on constraints of the maximum feed rate chorderror normal acceleration and jerk Therefore constraintequations of the maximum feed rate chord error normalacceleration and jerk are given as formulas (8) (9) (10) and(11) respectively

Δ 119894 = 1003817100381710038171003817119862 (119898+1) minus 119862 (119898)1003817100381710038171003817 asymp radic8 sdot 120575 sdot 120588119894 (7)

Δ119871 1198941 = 119865max sdot 119879 (8)

Δ119871 1198942 = radic8 sdot 120588119894 sdot 120575max sdot 119879 (9)

Δ119871 1198943 = radic119860max sdot 120588119894 sdot 119879 (10)

Δ119871 1198944 = 3radic119869max sdot 120588119894 sdot 119879 (11)

where 119865max 120575max 119860max 119869max are the maximum feed ratechord error normal acceleration and jerk respectively 119879 120588119894120575 are sample time the radius of curvature at the parametervalue 119906119894 and the chord error respectively

However the chord error involves the curvature calcula-tion so (12) is adopted to calculate the curvature

120575 = 100381610038161003816100381610038161003816100381610038161003816119862 (119906119898 + 119906119898+12 ) minus 119862 (119906119898) + 119862 (119906119898+1)2100381610038161003816100381610038161003816100381610038161003816 (12)

In addition to avoid the curvature calculation formulas(9) (10) and (11) can be expressed as formulas (13) (14) and(15) respectively

Δ119871 1198942 = radic120575max120575 sdot Δ 119894 (13)

Δ119871 1198943 = 1198792 sdot radic 119886119873max2 sdot 120575maxsdot Δ119871 1198942 (14)

Δ119871 1198944 = 1198794 sdot 3radic119869max sdot Δ119871 11989421205752maxsdot Δ119871 1198942 (15)

In summary hoping feeding step length should be satis-fied with the following formula

Δ119871 119894 = min (Δ119871 1198941 Δ119871 1198942 Δ119871 1198943 Δ119871 1198944) (16)

33 Improved Adams-MoultonMethod To implement NURBSinterpolation the Adams-Moulton (AM) method is adoptedhere Using the AM method the curve approximation up isgiven as follows [18]

119906119898+1 = 119906119898 + 11987924 (9119891119896+1 + 19119891119896 minus 5119891119896minus1 + 119891119896minus2)119891119896 = 119889119906119889119905

10038161003816100381610038161003816100381610038161003816119905=119905119898 = 119881 (119906119898)10038171003817100381710038171198621015840 (119906119898)1003817100381710038171003817 (17)

where 119881(119906119898) 119879 and 1198621015840(119906119898) denote the feed rate thesampling time and the first derivatives of the NURBS curverespectively

Although the second-order derivative operation isavoided by AM method an implicit is introduced and it isalso needed to solve first-order derivative Therefore theforward-center-backward difference is used to replace thedifferential to improve AM method so that the operationtime of interpolation algorithm can be shortened and thereal-time performance is able to be ensured The three


G062 K-U-X-Y-Z-W-FU-X-Y-Z-W-U-X-Y-Z-W-

helliphelliphelliphelliphelliphellipU-U-

hellip

NC le NC interpretor

Date buer

Reading NUBRScurve date

Calculation of NURBScurve interpolation

Calculation of feedingstep length constraints

Improved predictor-corrector formula ofNURBS interpolation

Matrix transform evaluation

Maximum feed rate chord error normal acceleration and jerkconstraints

Inverse quadratic interpolation

IAM method for NURBSinterpolation

NC interpreting

IAM interpolation

Interpolationexecuting

Generating curve pointsXi Yi Zi

Degree p control point Pi Weights wi knotvector u0 un+p+1

Figure 1 The system architecture of proposed NURBS interpolation method

different forward-center-backward differences are as informulas (18) (19) and (20) respectively

1199061015840119898+1 = 119906119898+1 minus 119906119898119879 (Backward difference)1199061015840119898 = 119906119898+1 minus 119906119898119879 (Forward difference)

1199061015840119898minus1 = 119906119898 minus 119906119898minus22119879 (Central difference)1199061015840119898minus2 = 119906119898minus1 minus 119906119898minus2119879 (Forward difference)

(18)

1199061015840119898+1 = 119906119898+1 minus 119906119898119879 (Backward difference)1199061015840119898 = 119906119898 minus 119906119898minus1119879 (Backward difference)


(19)

1199061015840119898+1 = 119906119898+1 minus 119906119898119879 (Backward difference)1199061015840119898 = 119906119898+1 minus 119906119898minus12119879 (Central difference)

1199061015840119898minus1 = 119906119898 minus 119906119898minus1119879 (Forward difference)1199061015840119898minus2 = 119906119898minus1 minus 119906119898minus2119879 (Forward difference)

(20)

When (18) (19) and (20) are substituted into (17) theimproved AM method is as in formulas (21) (22) and (23)respectively

1199060119898+1 = 18 sdot (13119906119898 minus 2119906119898minus1 minus 3119906119898minus2) (21)

1199061119898+1 = 110 sdot (21119906119898 minus 12119906119898minus1 + 119906119898minus2) (22)



45

40

35

30

25

20

15

10

5

0

NURBS curveControl polygonControl points

y-a

xis (

mm

)

x-axis (mm)454035302520151050

(a) The trident contour


80

70

60

50

40

30

20

10

0

y-a

xis (

mm

)

x-axis (mm)100806040200

(b) The butterfly contour

Figure 2 NURBS curves for simulations

Due to the slow convergence rate of the traditionalcorrection formula the InverseQuadratic Interpolation (IQI)is adopted to calculate the estimated parameter 119898+1 Whenparameters 1199060119898+1 1199061119898+1 1199062119898+1 are substituted into (7) esti-mated feed step lengths Δ1198710119898+1 Δ1198711119898+1 Δ1198712119898+1 are obtainedrespectively So the estimated parameter 119898+1 calculationformula is as follows [26]

119898+1 = 1199062119898+1minus 119903 sdot (119903 minus 119902) sdot (1199062119898+1 minus 1199061119898+1) + (1 minus 119903) sdot 119904 sdot (1199062119898+1 minus 1199060119898+1)(119902 minus 1) sdot (119903 minus 1) sdot (119904 minus 1) (24)

where 119902 = Δ1198710119898+1Δ1198711119898+1 119903 = Δ1198712119898+1Δ1198711119898+1 119904 = Δ1198712119898+1Δ1198710119898+1Theoretically sampling step size should satisfy the follow-

ing equation 1003817100381710038171003817119862 (119898+1) minus 119862 (119898)1003817100381710038171003817 = 119881 (119898) sdot 119879 (25)

However truncation errors and the discrepancy errorsbetween machining trajectory and the demand curve willmake (25) dissatisfied that is feed rate fluctuation existedSo feed rate fluctuation 120576 should be considered and it can bedefined as follows

120576 = 119881 (119898) sdot 119879 minus 1003817100381710038171003817119862 (119898+1) minus 119862 (119898)1003817100381710038171003817119881 (119898) sdot 119879 sdot 100 (26)

4 Simulation Study

In this section two NURBS curves of degree 3 named tridentand butterfly shown in Figure 2 are used as examples to testthe feasibility of the developed interpolation scheme and thecommand generator programs are written by Matlab and are

Table 1 The interpolation conditions for two third-degree NURBScurves

Parameters Symbols UnitsNamely feed rate 119865 80mmsMaximum acceleration 119860max 1200mms2

Maximum jerk 119869max 25000mms3

Maximum chord error 120575max 0001mmSampling time 119879 0001 s

executed on a personal computer with Intel(R) Core(TM) i5-2450M 250GHz CPU According to Figure 2 the trident andbutterfly curves parameters are given in Appendix and theinterpolation conditions for two NURBS curves of degree 3are listed in Table 1

In this section feed rate precision is only considered andfeed rate scheduling [27] is applied as shown in Figures 3 and4The AMmethod and IAMmethod are compared based onfeed rate precision Finally simulation results of the tridentand butterfly NURBS curve are shown in Figures 5 and 6respectively In addition important data of simulations aregiven in Table 2

From Figure 5 it can be seen that the maximum feedrate fluctuation of the AM method is about 00017 Themaximum feed rate fluctuation of the IAM method is about00008 and it is almost close to zero Overall the proposedIAMmethod is suitable for other simple third-degreeNURBScurves In Figure 6 it can be seen that the maximum feedrate fluctuation of the AM method is about 0023 Themaximum feed rate fluctuation of the IAM method is about00033 Feed rate fluctuation is reduced about 14 times withthe IAMmethod From above all the proposed IAMmethodis suitable for other complex third-degree NURBS curves and


0

10

20

30

40

50

60

70

80Fe

edra

te (m

ms

)

05 1 150Time (s)

(a) The feed rate profile

minus1500

minus1000

minus500

0

500

1000

1500

Acce

lera

tion

(mm

s2)

05 1 150Time (s)

(b) The acceleration profile

times104

05 1 150Time (s)

minus3

minus2

minus1

0

1

2

3

Jerk

(mm

s3)

(c) The jerk profile

Figure 3 Feed rate scheduling results of the trident NURBS curve

Table 2 Simulation results of two interpolation methods

Interpolation algorithm The maximum federate fluctuation ratio () Interpolation time (s)Trident Butterfly Trident Butterfly

AMmethod 00017 0023 1800 5808IAMmethod 00008 00033 1908 5910

it is satisfied formodernCNCmachining FromTable 2 it canbe seen that the interpolation time of trident and butterflyNURBS curve based on the AM method is lower than theproposed IAMmethod but gap is very small In summer theproposed IAM interpolation method in this paper has betterflexibility than the AMmethod

5 Experimental Verification

The CNC platform is developed for verifying the proposedIAM method experimentally The hardware and softwarestructures are as follows

(i) CPU Intel(R) Core(TM) i5-2450M 250GHz


0

10

20

30

40

50

60

70

80Fe

edra

te (m

ms

)

1 2 3 4 50Time (s)


minus1500

minus1000

minus500

0

500

1000

1500

1 2 3 4 50Time (s)

Acce

lera

tion

(mm

s2)


minus3

minus2

minus1

0

1

2

3

1 2 3 4 50Time (s)

Jerk

(mm

s3)

times104


Figure 4 Feed rate scheduling results of the butterfly NURBS curve

0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

Parameter (u)

Feed

rate

uc

tuat

ion

()

times10minus3

(a) The AMmethod

0 02 04 06 08 10

1

07

06

05

04

03

02

08

Parameter (u)

Feed

rate

uc

tuat

ion

()

times10minus3

(b) The IAMmethod

Figure 5 Simulation results of the trident NURBS curve


0 02 04 06 08 10

5

10

15

20

25

Parameter (u)

Feed

rate

uc

tuat

ion

()

times10minus3

(a) The AMmethod

0 02 04 06 08 1Parameter (u)

0

05

1

15

2

25

3

35

Feed

rate

uc

tuat

ion

()

times10minus3

(b) The IAMmethod

Figure 6 Simulation results of the butterfly NURBS curve

(ii) Servo system

(a) 119909-119910-axis Panasonic MBDKT2510E (driver) +Panasonic MHMJ042G1U (motor)

(iii) Drive system ball screws 5mm screw pitch

(iv) Software

(a) OS 32-bit Windows 7(b) Real-time core motion control card based

on DSP+FPGA including linear circular andNURBS interpolator and other basic modulesthe sampling time being 1ms

The butterfly NURBS curve is represented and ma-chined by the CNC platform The maximum feed rateis 4800mmmin The machining result is shown in Figure 7In Figure 7(b) there are bulges at positions 1 and 3 Howeverinterpolation lines are continuous at positions 1 and 3 fromFigure 7(c) In Figure 7(b) the break point is sunken andunsmooth at position 2 Nevertheless transition of the breakpoint is smooth at position 2 from Figure 7(c) From aboveall the final experimental results show the proposed IAMmethod is more feasible than AMmethod

6 Conclusions

In this paper an improved Adams-Moulton method forNURBS interpolation with minimal feed rate fluctuation hasbeen proposed The main conclusions of the study are asfollows

(i) The theoreticalmodel of the proposed IAMmethod isbuilt based on the derived analytic relations betweenthe kinematic characteristics and the parameters ofpath curve as well as feed rate profile

(ii) The proposed IAM method has been validated bysimulations and experimental tests It is effective andreliable for the feed rate interpolation of geometricallysimple and complex NURBS curves of degree 3 andsmaller feed rate fluctuation based on the proposedIAMmethod can be obtained compared with the AMmethod

Appendix

Parameters of Trident and ButterflyNURBS Curves

The Simple Trident NURBS Curve

Control points (mm) (20 0) (40 40) (24 16) (2040) (16 16) (0 40) (20 0)Knot vector [0 0 0 0 025 05 075 1 1 1 1]Weight vector [1 1 1 1 1 1 1]

The Complex Butterfly NURBS Curve

Control points (mm) (54493 52139) (5550752139) (56082 49615) (56780 44971) (6957551358) (77786 58573) (90526 67081) (10597363801) (100400 47326) (94567 39913) (9236930485) (83440 33757) (91892 28509) (8944420393) (83218 15446) (87621 4830) (809459267) (79834 14535) (76074 8522) (7018312550) (64171 16865) (59993 22122) (5568036359) (56925 24995) (59765 19828) (5449314940) (49220 19828) (52060 24994) (5330536359) (48992 22122) (44814 16865) (3880212551) (32911 8521) (29152 14535) (280409267) (21364 4830) (25768 15447) (1953920391) (17097 28512) (25537 33750) (1660230496) (14199 39803) (8668 47408) (3000


Host

Servodriver

PC soware

Servomotore ball screw

sliding table

Motion control card

(a) Hardware and software of the host computer

1

2 3

(b) The machining result of AMmethod

1

23

(c) The machining result of IAMmethod

Figure 7 The machining of the butterfly curve

63794) (18465 67084) (31197 58572) (3941151358) (52204 44971) (52904 49614) (5347852139) (54492 52139)Knot vector [0 0 0 0 00083 00150 00361 0085501293 01509 01931 02273 02435 02561 0269202889 03170 03316 03482 03553 03649 0383704005 04269 04510 04660 04891 05000 0510905340 05489 05731 05994 06163 06351 0644706518 06683 06830 07111 07307 07439 0756507729 08069 08491 08707 09145 09639 0985009917 1 1 1 1]Weight vector [10000 10000 10000 10000 1000010000 10000 10000 10000 10000 10000 2000010000 10000 50000 30000 10000 11000 1000010000 10000 10000 10000 10000 10000 1000010000 10000 10000 10000 10000 10000 1000010000 10000 30000 50000 10000 10000 2000010000 10000 10000 10000 10000 10000 1000010000 10000 10000 10000]

Conflicts of Interest

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

Acknowledgments

The support of National Natural Science Foundation(no 51605477) Discipline Frontier Research Project (no2015XKQY10) and Priority Academic ProgramDevelopmentof Jiangsu Higher Education Institutions (PAPD) in carryingout this research is gratefully acknowledged

References

[1] Q Cheng C Wu P Gu W Chang and D Xuan ldquoAn analysismethodology for stochastic characteristic of volumetric errorin multiaxis CNC machine toolrdquo Mathematical Problems inEngineering vol 2013 Article ID 863283 12 pages 2013

[2] L Piegl and W Tiller The NURBS Books Springer Berlingermany 2nd edition 1997

[3] M M Emami and B Arezoo ldquoA look-ahead command gener-ator with control over trajectory and chord error for NURBScurve with unknown arc lengthrdquo CAD Computer Aided Designvol 42 no 7 pp 625ndash632 2010

[4] A Safari H G Lemu S Jafari and M Assadi ldquoA com-parative analysis of nature-inspired optimization approachesto 2D geometric modelling for turbomachinery applicationsrdquoMathematical Problems in Engineering vol 2013 Article ID716237 15 pages 2013


[5] J Jahanpour and M R Alizadeh ldquoA novel acc-jerk-limitedNURBS interpolation enhanced with an optimized S-shapedquintic feedrate scheduling schemerdquo The International Journalof Advanced Manufacturing Technology vol 77 no 9ndash12 pp1889ndash1905 2015

[6] Y Wang D Yang R Gai S Wang and S Sun ldquoDesignof trigonometric velocity scheduling algorithm based on pre-interpolation and look-ahead interpolationrdquo International Jour-nal of Machine Tools andManufacture vol 96 pp 94ndash105 2015

[7] R T Farouki and Y-F Tsai ldquoExact Taylor series coefficients forvariable-feedrate CNC curve interpolatorsrdquo Computer-AidedDesign vol 33 no 2 pp 155ndash165 2001

[8] M-C Tsai and C-W Chung ldquoA real-time predictor-correctorinterpolator for CNC machiningrdquo Journal of ManufacturingScience and Engineering Transactions of the ASME vol 125 no3 pp 449ndash460 2003

[9] T Yong and R Narayanaswami ldquoA parametric interpolator withconfined chord errors acceleration and deceleration for NCmachiningrdquo Computer-Aided Design vol 35 no 13 pp 1249ndash1259 2003

[10] S-S Yeh and P-L Hsu ldquoSpeed-controlled interpolator formachining parametric curvesrdquo CAD Computer Aided Designvol 31 no 5 pp 349ndash357 1999

[11] K Erkorkmaz and Y Altintas ldquoHigh speed CNC systemdesign Part I jerk limited trajectory generation and quinticspline interpolationrdquo International Journal ofMachine Tools andManufacture vol 41 no 9 pp 1323ndash1345 2001

[12] K Erkorkmaz and Y Altintas ldquoHigh speed CNC system designPart IImodeling and identification of feed drivesrdquo InternationalJournal of Machine Tools and Manufacture vol 41 no 10 pp1487ndash1509 2001

[13] X Zhiming C Jincheng and F Zhengjin ldquoPerformance evalu-ation of a real-time interpolation algorithm for NURBS curvesrdquoInternational Journal of Advanced Manufacturing Technologyvol 20 no 4 pp 270ndash276 2002

[14] M-Y Cheng M-C Tsai and J-C Kuo ldquoReal-time NURBScommand generators for CNC servo controllersrdquo InternationalJournal ofMachine Tools andManufacture vol 42 no 7 pp 801ndash813 2002

[15] M Tikhon T J Ko S H Lee and H Sool Kim ldquoNURBSinterpolator for constant material removal rate in open NCmachine toolsrdquo International Journal of Machine Tools andManufacture vol 44 no 2-3 pp 237ndash245 2004

[16] W T Lei M P Sung L Y Lin and J J Huang ldquoFastreal-time NURBS path interpolation for CNC machine toolsrdquoInternational Journal of Machine Tools amp Manufacture vol 47no 10 pp 1530ndash1541 2007

[17] M-T Lin M-S Tsai and H-T Yau ldquoDevelopment of adynamics-based NURBS interpolator with real-time look-ahead algorithmrdquo International Journal of Machine Tools andManufacture vol 47 no 15 pp 2246ndash2262 2007

[18] Q-X Jia Z-X Xu and X-S Liu ldquoNURBS interpolation basedon Adams algorithmrdquo Journal of Jilin University (Engineeringand Technology Edition) vol 39 no 1 pp 215ndash218 2009

[19] J-C Wu X-Q Tang J-H Chen and H-C Zhou ldquoReal timefast non-uniform rational B-spline interpolation algorithmrdquoComputer Integrated Manufacturing Systems vol 17 no 6 pp1224ndash1227 2011

[20] J Wu H Zhou X Tang and J Chen ldquoA NURBS interpolationalgorithm with continuous feedraterdquo The International Journalof Advanced Manufacturing Technology vol 59 no 5ndash8 pp623ndash632 2012

[21] Y Sun Y Zhao Y Bao and D Guo ldquoA novel adaptive-feedrate interpolation method for NURBS tool path withdrive constraintsrdquo International Journal of Machine Tools andManufacture vol 77 pp 74ndash81 2014

[22] Q Liu H Liu S Zhou C Li and S Yuan ldquoPrinciple and devel-opment of NURBS interpolation algorithm with zero-feedratefluctuationrdquo Computer Integrated Manufacturing Systems vol21 no 10 pp 2659ndash2667 2015

[23] H Liu Q Liu S Zhou C Li and S Yuan ldquoA NURBSinterpolation method with minimal feedrate fluctuation forCNC machine toolsrdquo The International Journal of AdvancedManufacturing Technology vol 78 no 5ndash8 pp 1241ndash1250 2015

[24] X Li J Zhang and Y Zheng ldquoNURBS-based isogeometricanalysis of beams and plates using high order shear deformationtheoryrdquoMathematical Problems in Engineering vol 2013 ArticleID 159027 9 pages 2013

[25] MAnnoni A Bardine S Campanelli P Foglia andC A PreteldquoA real-time configurable NURBS interpolator with boundedacceleration jerk and chord errorrdquo CAD Computer AidedDesign vol 44 no 6 pp 509ndash521 2012

[26] T Sauer Numerical Analysis Addison-Wesley Longman Pub-lishing Co Inc Boston Mass USA 2005

[27] L Xinhua P Junquan S Lei and W Zhongbin ldquoA novelapproach for NURBS interpolation through the integrationof acc-jerk-continuous-based control method and look-aheadalgorithmrdquo International Journal of Advanced ManufacturingTechnology vol 88 no 1 pp 961ndash969 2017

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


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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


rational B-spline interpolation algorithm with compensatoryparameter to minimize feed rate fluctuation and contourerror by improving the inverse length function method withthe usage of biarc guide curve Sun et al [21] proposed a noveladaptive federate interpolation method for NURBS tool pathwith drive constraints which can acquire smooth feed rateprofile Liu et al [22] proposed a double secant interpolationalgorithm and a fast calculation and derivation algorithm ofNURBS to solve the problem of feed rate fluctuation Liuet al [23] proposed a new NURBS interpolation methodwith minimal feed rate fluctuation for CNCmachine tools toobtain the sampling curve parameter

Although many approaches for NURBS interpolationhave been proposed in above literature they have somecommon disadvantages summarized as follows Firstly mostof researchers presented their methods for NURBS interpola-tion with the second-order Taylor expansion method How-ever the second-order Taylor expansion method referred tosecond derivative of NURBS curve and it was complicatedSecondly some researchers considered the problem of feedrate fluctuation Nevertheless results of feed rate fluctuationwere very high and unsatisfactory

In order to solve the above problems a novel approachfor NURBS interpolation with minimal feed rate fluctuationbased on improved Adams-Moulton method is applied andthe rest of this paper is organized as follows In Section 2representation and calculation ofNURBS curve interpolationare presented The improved predictor-corrector formula ofNURBS curve interpolation is proposed in Section 3 InSection 4 simulations with the trident and butterfly NURBScurves are conducted to verify the feasibility of the proposedmethod Experimental verification is conducted in Section 5Our conclusions are summarized in Section 6

2 Representation and Calculation ofNURBS Curve Interpolation

Suppose that 119862(119906) represents a NURBS curve and is given by[2 24 25]

119862 (119906) = sum119899119894=0119873119894119901 (119906) 119908119894119875119894sum119899119894=0119873119894119901 (119906) 119908119894 (1)

where 119875119894 are the control points 119908119894 are the weights of 119875119894(119899+1) are the number of the control points and119901 is the degree

of NURBS curve 119873119894119901(119906) is the 119901th-degree B-spline basisfunction and can be calculated using the recursive formulasgiven as follows

1198731198940 (119906) = 1 if 119906119894 le 119906 le 119906119894+10 otherwise

119873119894119901 (119906) = 119906 minus 119906119894119906119894+119901 minus 119906119894119873119894119901minus1 (119906)+ 119906119894+119901+1 minus 119906119906119894+119901+1 minus 119906119894+1119873119894+1119901minus1 (119906)

(2)

where 1199060 1199061 119906119899+119901+1 represents the knot vectors and 119906 isthe interpolation parameter

The node vector in parameter coordinates is not uni-formly distributed for NURBS curve interpolation in generalso the value of the B-spline basis function is not the sameand it is necessary to obtain all the B-spline basis functionsrespectively In addition the weight factor value of NURBScurve interpolation needs to be obtained and this makesthe computation more time-consuming However becauseof the simple expression and high derivative evaluationefficiency the matrix expression of the NURBS curve canbe used to solve the complicated calculation problem in thepretreatment process

Due to the fact that the NURBS curve of degree 3 isused widely the following example is used to be illustratedIf NURBS curve control polygon [1198750 1198751 119875119899] weights[1199080 1199081 119908119899] and node vectors [1199060 1199061 119906119899+119901+1] aregiven then NURBS curve of degree 3 can be expressed asfollows [18]

119862 (119905) = sum119894119895=119894minus3 1199081198951198751198951198731198953 (119906)sum119894119895=119894minus3 1199081198951198731198953 (119906)

= [1 119905 1199052 1199053]119872119894 [ 119908119894minus3119875119894minus3119908119894minus2119875119894minus2119908119894minus1119875119894minus1119908119894119875119894

][1 119905 1199052 1199053]119872119894 [ 119908119894minus3119908119894minus2119908119894minus1

119908119894

] (3)

where 119894 = 3 4 119899 119905 isin [0 1] 119905 = (119906 minus 119906119894)(119906119894+1 minus 119906119894) =(119906minus119906119894)nabla1119894 nabla1119894 = 119906119894+1 minus119906119894 nabla2119894 = 119906119894+2 minus119906119894+1 nabla3119894 = 119906119894+3 minus119906119894+2especially nabla0119894 = 0 and the expression of119872119894 is as follows

M119894 = [[[[[

11989811 11989812 11989813 1198981411989821 11989822 11989823 1198982411989831 11989832 11989833 1198983411989841 11989842 11989843 11989844]]]]]

=

[[[[[[[[[[[[[[[[[[[

(nabla1119894 )2nabla2119894minus1nabla3119894minus2 1 minus 11989811 minus 11989813 (nabla1119894minus1)2nabla2119894minus1nabla3119894minus1 0minus311989811 311989811 minus 11989823 3nabla1119894minus1nabla1119894nabla2119894minus1nabla3119894minus1 0311989811 minus311989811 minus 11989833 3 (nabla1119894 )2nabla2119894minus1nabla3119894minus1 0minus11989811 11989811 minus 11989843 minus 11989844 minus(1311989833 + 11989844 + (nabla1119894 )2nabla2119894 nabla3119894minus1)

(nabla1119894 )2nabla2119894 nabla3119894

]]]]]]]]]]]]]]]]]]]

(4)








(5)



(6)






Δ119871 1198941 = 119865max sdot 119879 (8)






120575 = 100381610038161003816100381610038161003816100381610038161003816119862 (119906119898 + 119906119898+12 ) minus 119862 (119906119898) + 119862 (119906119898+1)2100381610038161003816100381610038161003816100381610038161003816 (12)


Δ119871 1198942 = radic120575max120575 sdot Δ 119894 (13)




Δ119871 119894 = min (Δ119871 1198941 Δ119871 1198942 Δ119871 1198943 Δ119871 1198944) (16)



10038161003816100381610038161003816100381610038161003816119905=119905119898 = 119881 (119906119898)10038171003817100381710038171198621015840 (119906119898)1003817100381710038171003817 (17)






hellip


Date buer









NC interpreting

IAM interpolation








(18)



(19)



(20)






45

40

35

30

25

20

15

10

5

0


y-a

xis (

mm

)

x-axis (mm)454035302520151050



80

70

60

50

40

30

20

10

0

y-a

xis (

mm

)

x-axis (mm)100806040200









4 Simulation Study










0

10

20

30

40

50

60

70

80Fe

edra

te (m

ms

)

05 1 150Time (s)


minus1500

minus1000

minus500

0

500

1000

1500

Acce

lera

tion

(mm

s2)

05 1 150Time (s)


times104

05 1 150Time (s)

minus3

minus2

minus1

0

1

2

3

Jerk

(mm

s3)











0

10

20

30

40

50

60

70

80Fe

edra

te (m

ms

)

1 2 3 4 50Time (s)


minus1500

minus1000

minus500

0

500

1000

1500

1 2 3 4 50Time (s)

Acce

lera

tion

(mm

s2)


minus3

minus2

minus1

0

1

2

3

1 2 3 4 50Time (s)

Jerk

(mm

s3)

times104



0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

Parameter (u)

Feed

rate

uc

tuat

ion

()

times10minus3

(a) The AMmethod

0 02 04 06 08 10

1

07

06

05

04

03

02

08

Parameter (u)

Feed

rate

uc

tuat

ion

()

times10minus3

(b) The IAMmethod



0 02 04 06 08 10

5

10

15

20

25

Parameter (u)

Feed

rate

uc

tuat

ion

()

times10minus3

(a) The AMmethod

0 02 04 06 08 1Parameter (u)

0

05

1

15

2

25

3

35

Feed

rate

uc

tuat

ion

()

times10minus3

(b) The IAMmethod


(ii) Servo system



(iv) Software




6 Conclusions




Appendix







Host

Servodriver

PC soware


sliding table

Motion control card


1

2 3


1

23






Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















(5)



(6)






Δ119871 1198941 = 119865max sdot 119879 (8)






120575 = 100381610038161003816100381610038161003816100381610038161003816119862 (119906119898 + 119906119898+12 ) minus 119862 (119906119898) + 119862 (119906119898+1)2100381610038161003816100381610038161003816100381610038161003816 (12)


Δ119871 1198942 = radic120575max120575 sdot Δ 119894 (13)




Δ119871 119894 = min (Δ119871 1198941 Δ119871 1198942 Δ119871 1198943 Δ119871 1198944) (16)



10038161003816100381610038161003816100381610038161003816119905=119905119898 = 119881 (119906119898)10038171003817100381710038171198621015840 (119906119898)1003817100381710038171003817 (17)






hellip


Date buer









NC interpreting

IAM interpolation








(18)



(19)



(20)






45

40

35

30

25

20

15

10

5

0


y-a

xis (

mm

)

x-axis (mm)454035302520151050



80

70

60

50

40

30

20

10

0

y-a

xis (

mm

)

x-axis (mm)100806040200









4 Simulation Study










0

10

20

30

40

50

60

70

80Fe

edra

te (m

ms

)

05 1 150Time (s)


minus1500

minus1000

minus500

0

500

1000

1500

Acce

lera

tion

(mm

s2)

05 1 150Time (s)


times104

05 1 150Time (s)

minus3

minus2

minus1

0

1

2

3

Jerk

(mm

s3)











0

10

20

30

40

50

60

70

80Fe

edra

te (m

ms

)

1 2 3 4 50Time (s)


minus1500

minus1000

minus500

0

500

1000

1500

1 2 3 4 50Time (s)

Acce

lera

tion

(mm

s2)


minus3

minus2

minus1

0

1

2

3

1 2 3 4 50Time (s)

Jerk

(mm

s3)

times104



0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

Parameter (u)

Feed

rate

uc

tuat

ion

()

times10minus3

(a) The AMmethod

0 02 04 06 08 10

1

07

06

05

04

03

02

08

Parameter (u)

Feed

rate

uc

tuat

ion

()

times10minus3

(b) The IAMmethod



0 02 04 06 08 10

5

10

15

20

25

Parameter (u)

Feed

rate

uc

tuat

ion

()

times10minus3

(a) The AMmethod

0 02 04 06 08 1Parameter (u)

0

05

1

15

2

25

3

35

Feed

rate

uc

tuat

ion

()

times10minus3

(b) The IAMmethod


(ii) Servo system



(iv) Software




6 Conclusions




Appendix







Host

Servodriver

PC soware


sliding table

Motion control card


1

2 3


1

23






Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












hellip


Date buer









NC interpreting

IAM interpolation








(18)



(19)



(20)






45

40

35

30

25

20

15

10

5

0


y-a

xis (

mm

)

x-axis (mm)454035302520151050



80

70

60

50

40

30

20

10

0

y-a

xis (

mm

)

x-axis (mm)100806040200









4 Simulation Study










0

10

20

30

40

50

60

70

80Fe

edra

te (m

ms

)

05 1 150Time (s)


minus1500

minus1000

minus500

0

500

1000

1500

Acce

lera

tion

(mm

s2)

05 1 150Time (s)


times104

05 1 150Time (s)

minus3

minus2

minus1

0

1

2

3

Jerk

(mm

s3)











0

10

20

30

40

50

60

70

80Fe

edra

te (m

ms

)

1 2 3 4 50Time (s)


minus1500

minus1000

minus500

0

500

1000

1500

1 2 3 4 50Time (s)

Acce

lera

tion

(mm

s2)


minus3

minus2

minus1

0

1

2

3

1 2 3 4 50Time (s)

Jerk

(mm

s3)

times104



0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

Parameter (u)

Feed

rate

uc

tuat

ion

()

times10minus3

(a) The AMmethod

0 02 04 06 08 10

1

07

06

05

04

03

02

08

Parameter (u)

Feed

rate

uc

tuat

ion

()

times10minus3

(b) The IAMmethod



0 02 04 06 08 10

5

10

15

20

25

Parameter (u)

Feed

rate

uc

tuat

ion

()

times10minus3

(a) The AMmethod

0 02 04 06 08 1Parameter (u)

0

05

1

15

2

25

3

35

Feed

rate

uc

tuat

ion

()

times10minus3

(b) The IAMmethod


(ii) Servo system



(iv) Software




6 Conclusions




Appendix







Host

Servodriver

PC soware


sliding table

Motion control card


1

2 3


1

23






Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










45

40

35

30

25

20

15

10

5

0


y-a

xis (

mm

)

x-axis (mm)454035302520151050



80

70

60

50

40

30

20

10

0

y-a

xis (

mm

)

x-axis (mm)100806040200









4 Simulation Study










0

10

20

30

40

50

60

70

80Fe

edra

te (m

ms

)

05 1 150Time (s)


minus1500

minus1000

minus500

0

500

1000

1500

Acce

lera

tion

(mm

s2)

05 1 150Time (s)


times104

05 1 150Time (s)

minus3

minus2

minus1

0

1

2

3

Jerk

(mm

s3)











0

10

20

30

40

50

60

70

80Fe

edra

te (m

ms

)

1 2 3 4 50Time (s)


minus1500

minus1000

minus500

0

500

1000

1500

1 2 3 4 50Time (s)

Acce

lera

tion

(mm

s2)


minus3

minus2

minus1

0

1

2

3

1 2 3 4 50Time (s)

Jerk

(mm

s3)

times104



0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

Parameter (u)

Feed

rate

uc

tuat

ion

()

times10minus3

(a) The AMmethod

0 02 04 06 08 10

1

07

06

05

04

03

02

08

Parameter (u)

Feed

rate

uc

tuat

ion

()

times10minus3

(b) The IAMmethod



0 02 04 06 08 10

5

10

15

20

25

Parameter (u)

Feed

rate

uc

tuat

ion

()

times10minus3

(a) The AMmethod

0 02 04 06 08 1Parameter (u)

0

05

1

15

2

25

3

35

Feed

rate

uc

tuat

ion

()

times10minus3

(b) The IAMmethod


(ii) Servo system



(iv) Software




6 Conclusions




Appendix







Host

Servodriver

PC soware


sliding table

Motion control card


1

2 3


1

23






Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

10

20

30

40

50

60

70

80Fe

edra

te (m

ms

)

05 1 150Time (s)


minus1500

minus1000

minus500

0

500

1000

1500

Acce

lera

tion

(mm

s2)

05 1 150Time (s)


times104

05 1 150Time (s)

minus3

minus2

minus1

0

1

2

3

Jerk

(mm

s3)











0

10

20

30

40

50

60

70

80Fe

edra

te (m

ms

)

1 2 3 4 50Time (s)


minus1500

minus1000

minus500

0

500

1000

1500

1 2 3 4 50Time (s)

Acce

lera

tion

(mm

s2)


minus3

minus2

minus1

0

1

2

3

1 2 3 4 50Time (s)

Jerk

(mm

s3)

times104



0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

Parameter (u)

Feed

rate

uc

tuat

ion

()

times10minus3

(a) The AMmethod

0 02 04 06 08 10

1

07

06

05

04

03

02

08

Parameter (u)

Feed

rate

uc

tuat

ion

()

times10minus3

(b) The IAMmethod



0 02 04 06 08 10

5

10

15

20

25

Parameter (u)

Feed

rate

uc

tuat

ion

()

times10minus3

(a) The AMmethod

0 02 04 06 08 1Parameter (u)

0

05

1

15

2

25

3

35

Feed

rate

uc

tuat

ion

()

times10minus3

(b) The IAMmethod


(ii) Servo system



(iv) Software




6 Conclusions




Appendix







Host

Servodriver

PC soware


sliding table

Motion control card


1

2 3


1

23






Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

10

20

30

40

50

60

70

80Fe

edra

te (m

ms

)

1 2 3 4 50Time (s)


minus1500

minus1000

minus500

0

500

1000

1500

1 2 3 4 50Time (s)

Acce

lera

tion

(mm

s2)


minus3

minus2

minus1

0

1

2

3

1 2 3 4 50Time (s)

Jerk

(mm

s3)

times104



0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

Parameter (u)

Feed

rate

uc

tuat

ion

()

times10minus3

(a) The AMmethod

0 02 04 06 08 10

1

07

06

05

04

03

02

08

Parameter (u)

Feed

rate

uc

tuat

ion

()

times10minus3

(b) The IAMmethod



0 02 04 06 08 10

5

10

15

20

25

Parameter (u)

Feed

rate

uc

tuat

ion

()

times10minus3

(a) The AMmethod

0 02 04 06 08 1Parameter (u)

0

05

1

15

2

25

3

35

Feed

rate

uc

tuat

ion

()

times10minus3

(b) The IAMmethod


(ii) Servo system



(iv) Software




6 Conclusions




Appendix







Host

Servodriver

PC soware


sliding table

Motion control card


1

2 3


1

23






Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 02 04 06 08 10

5

10

15

20

25

Parameter (u)

Feed

rate

uc

tuat

ion

()

times10minus3

(a) The AMmethod

0 02 04 06 08 1Parameter (u)

0

05

1

15

2

25

3

35

Feed

rate

uc

tuat

ion

()

times10minus3

(b) The IAMmethod


(ii) Servo system



(iv) Software




6 Conclusions




Appendix







Host

Servodriver

PC soware


sliding table

Motion control card


1

2 3


1

23






Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Host

Servodriver

PC soware


sliding table

Motion control card


1

2 3


1

23






Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









a novel approach for nurbs interpolation with minimal feed … · 2019. 7. 30. · a novel approach...

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