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Development of a real-time look-ahead interpolation methodology with spline-fitting technique

ORIGINAL ARTICLE

Development of a real-time look-ahead

interpolation methodology with spline-fitting technique for high-speed machining

Meng-Shiun Tsai &Hao-Wei Nien &Hong-Tzong Yau

Received:18May 2009/Accepted:17July 2009/Published online:9August 2009#Springer-Verlag London Limited 2009

Abstract Methodologies for converting short line seg-ments into parametric curves were proposed in the past.However,most of the algorithms only consider the position continuity at the junctions of parametric curves.The discontinuity of the slope and curvature at the junctions of the parametric curve might cause feedrate fluctuation and velocity discontinuous.This paper proposes a look-ahead interpolation scheme for short line segments.The proposed interpolation method consists of two modules:spline-fitting and acceleration/deceleration (acc/dec)feedrate-planning modules.The spline-fitting module first looks ahead several short line segments and converts them into parametric curves.The continuities of the slope and curvature at each junctions of the spline curve are ensured.Then the acc/dec feedrate-planning module proposes a new algorithm to determine the feedrate at the junction of the fitting curve and unfitted short segments,and the corner feedrate within the fitting curve.The chord error and acceleration of the trajectory are bounded with the proposed algorithm.Simulations are performed to validate the tracking and contour accuracies of the proposed method.The computational efforts between the proposed algorithm and the non-uniform rational B-spline (NURBS)-fitting technique are compared to demonstrate the efficiency of the proposed method.Finally,experiments on a PC-based control system are conducted to demonstrate that the proposed interpolation method can achieve better accuracy

and reduce machining time as compared to the approximation optimal feedrate interpolation algorithm.Keywords Short line segments .Spline fitting .Look-ahead algorithm .High-speed machining

1Introduction

In modern computer-aided design (CAD)systems,free-form or contoured geometric shapes are usually adopted in design of complex parts.However,linear (G01)or circular (G02,G03)interpolations are still widely used in traditional computer numerical control (CNC)system.Therefore,computer-aided manufacturing (CAM)systems need to generate short line segments to approximate the contoured geometric curve under the given tolerance.If the requirement of the part accuracy becomes strict,the segmentation approach could suffer from the following problems:(1)high-speed machining cannot be achieved for the NC codes with very short line segments;(2)at the junction of adjacent segments,the feedrate fluctuation and velocity discontinuity could be severe;(3)high acceleration/deceleration (acc/dec)might cause system vibration and reduce machining quality.

To plan the feedrate profile for the consecutive short line segments,Han et al.[1]presented an interpolation scheme for two blocks of short line segments.Hu et al.[2]developed an approximate optimal feedrate algorithm to deal with a large amount of short line segments.Ye et al.[3]proposed a look-ahead algorithm which can predict the variation of the curvature and realize adaptive control of the feedrate based on the characteristics of the trajectory.The above methods might solve the problems such as feedrate fluctuations.However,high-speed machining

M.-S.Tsai :H.-W.Nien (*):H.-T.Yau Department of Mechanical Engineering,National Chung Cheng University,

168University Road,Minhsiung Township,

Chiayi County 62102Taiwan,Republic of China e-mail:hwnien@https://www.sodocs.net/doc/172432621.html, M.-S.Tsai

e-mail:imetsai@https://www.sodocs.net/doc/172432621.html,.tw

Int J Adv Manuf Technol (2010)47:621–638DOI 10.1007/s00170-009-2220-7

could not be achieved because of the length constraints for the short line segments.

Another approach to solve the above problems is to convert short line segments into parametric curves and then apply parametric interpolation for the fitted curve.There were many studies [4,5]showing that parametric interpolation could have the merits of achieving high-speed and high-precision machinery.In general,there are two methods for converting line segments into parametric curves as shown in Fig.1.One is the approximation method [6,7]and the other is the interpolation method [8–11].Based on the given data points,the approximation method generates the fitted curve by using the least-square technique.The advantages of the approxi-mation method are that the fitted curves can be described with less data number.Yau and Kuo [6]applied offline non-uniform rational B-spline (NURBS)curve-fitting technique to convert the conventional NC codes from CAD/CAM systems to the curves with a NURBS format.Yeh and Su [7]implemented the NURBS curve-fitting technique on real-time machining.

Although the approximation method can reduce the data number,the problem is that the fitted curves might not pass through the original data points as shown in Fig.1a .On the other hand,the interpolation method shown in Fig.1b can ensure that the data points are all located in the fitted curves.Yau and Wang [8]developed the fast Bezier interpolator with a look-ahead function to deal with segmentation problems.The slope discontinuity might still occur at the junction of the fitted line segments.Li et al.[9]presented a real-time interpolation algorithm using the spline-fitting technique.It was shown that within the fitting curve,only a simple acceleration/deceleration (acc/dec)was applied to plan the feedrate profile.Furthermore,the spline-fitting technique only considers the position continuity at the junction of adjacent segments.Li et al.[10]proposed a NURBS pre-interpolator for a five-axis machining which was validated by simulation.Wang and Yau [11]developed a real-time NURBS interpolator with a look-ahead function to handle numerous short line segments.

However,the maximum number of the fitted segments is only 20because on-line fitting of the NURBS curve is very time-consuming.

Although previous works have proposed several methods in the interpolation of short line segments,they only considered the position continuity at the junctions of the parametric curves.The discontinuity of the slope and curvature at the junctions of the parametric curves might cause feedrate fluctuation and the velocity to be discontinuous.Furthermore,the on-line curve-fitting method for the NURBS curve requires large computational efforts.In this paper,a look-ahead interpolation with spline-curve fitting technique for short line segments is proposed.The fitting algorithm can ensure that the position,slope,and curvature at the junctions of the parametric curves,and unfitted line segments are all continuous.Furthermore,the acc/dec feedrate-planning module develops an algorithm which can look-ahead 30blocks to ensure that the chord error and the acceleration are all bounded in the motion trajectory.One of the merits of the proposed method is that the fitting time can be reduced significantly as compared to the NURBS-fitting technique [11],and thus real-time implemen-tation of the curve fitting for large number of short line segments becomes feasible.Finally,simulations and experi-ments are conducted to demonstrate that the proposed algorithm can achieve better accuracy and reduce machining time as compared to the approximate optimal feedrate interpolation (AOF)[2].

2Design of spline-fitting technique [12]

In this study,cubic spline curve is used to curve-fit the short line segments such that the computation efforts can be reduced significantly.The spline curve is made of different Bezier curves connected together.The conditions of maintaining the continuity of the position,slope,and curvature at the junction of each Bezier curve are first derived.Then,the spline-curve fitting method is proposed and the process is explained subsequently.

Data point Trajectory path Fitted curve

Fig.1The methods for

converting short line segments into parametric curves;a approximation method;b interpolation method

2.1Introduction to cubic spline curve

Suppose C (u )represents an n th-degree Bezier curve and is defined as follows [13]:C eu T?

X n i ?0

B i ;n eu TP i ;for 0 u 1e1T

where P i represents the control points,and B i,n (u )is the

n th-degree Bernstein polynomial given by B i ;n eu T?

n !

i !n ài eT!

u i 1àu eTn ài

e2T

The cubic Bezier curve is adopted here such that fast computation of the interpolation points can be achieved in real-time.As shown in Fig.2,several Bezier curves can be connected into a single spline curve.The general form of a spline curve is defined as follows:S eu T?C i u ài à1eTeT;

for i à1 u i

e3T

where i=1,?,N .It is shown that the slope and curvature at the junction point P 3as shown in Fig.2might be discontinuous if the control points are not properly selected.In order to maintain the continuity of the slope and curvature for the two curves,the conditions for connecting two Bezier curves should be developed.2.2The continuity conditions for connecting two Bezier curves

Several geometric characteristics in connecting two Bezier curve are illustrated by Fig.3.It is shown that the first

Bezier curve C 1(u )has four control points P 10

,P 11,P 12,P 13and the second Bezier curve C 2(u )has the other four

control points P 20,P 21,P 22,P 2

3.The point Q is the junction of the C 1(u )and C 2(u ).The first derivatives at the junction point Q can be expressed as follows:

C 0

1eu T u ?1?3Q àP 1

2àáe4T

C 0

2eu T u ?0?3P 21

àQ àáe5T

If the first derivatives at the point Q are different,it could cause the slope at the point Q become discontinuous as shown in Fig.3a .To obtain a smooth transition between the two curves,the first derivative at the point Q should be set to be equivalent and the condition is given as:

Q ?

P 12tP 212

e6T

Geometrically,the point Q is the midpoint of the line

P 12P 21

as shown in Fig.3b .However,this condition set by Eq.6is not sufficient for the curve interpolation where the curvature at the junction point Q should also remain continuous for the two curves.The importance of the curvature continuity can be visualized by imaging that a tool is moving around the curves as shown in Fig.3b .As the tool is moving on the first curve C 1(u ),it is pushed

against the left side of the line P 12P 21

.When the tool passes the junction point Q ,the tool is moving on the second curve

and it is pushed against the right side of the line P 12P 21

.The jerk could occur at the junction Q because the curvature changes sign when passing the transition region.The curvature of a curve at each point can be obtained as follows:

k eu T?

C 0eu T?C 00eu Tj j

C 0eu Tj j 3

e7T

It can be seen from Eq.7that the continuity of the curvature at the point Q can be achieved when the first and second derivatives at the point Q between the two curves are equivalent.The second derivatives at the point Q for C 1(u )and C 2(u )curves are represented as the following:C 00

1eu Tu ?1?6P 11

à2P 1

2tQ àá e8T

C 00

2eu Tu ?0?6Q à2P 21

tP 22àá e9T

)

(1u C )

(2u C )

(u C N 0

P 1

P 2

P 3

P C i (u ): i -th Bezier Curve

Fig.2Cubic spline curve

By equating Eq.8with Eq.9,the following condition

can be obtained as:

2P 12

àP 11?2P 21àP 22e10T

Geometrically,the left-hand side of Eq.10corresponds

to a particular point S 1on the line through P 11and P 1

2as shown in Fig.3c .The point S 1is given as follows:S 1?

P 12

P 12àP 11

e11T

Similarly,the point S 2which corresponds to the right-hand side of Eq.10is given as:

S 2?P 21tP 21àP 2

2àáe12TThe second derivatives are continuous at the junction

point Q if and only if two points S 1and S 2are equal.The

coincidence point of the S 1and S 2is assumed to be the

point S which corresponds to Fig.3d where the second derivatives at the Q are equivalent.Based on Eqs.11and

12,the control points P 12and P 2

1can be represented as follows:

P 12

P 11tS 2

e13T

P 21

P 22tS 2

e14T

These conditions only ensure that the continuity of the junction point Q .However,the point within the Bezier curve,such as the point A shown in Fig.3d ,could occur.Similar to the point Q ,the curvature of the point A changes

(a)

P 11

P 1

P 11

P 1

0P 1

10P 1

2P 1

P P ==20

13Q

21P 2

P 2

2P 2

2P 22P 2

2P 2

3P 2

3P 1S 2

S S

)

(1u C )

(2u C )

(2u C )(2u C )

(2u C (e)

2P 1

0P 11

P 2

2P 2

3P S

P )

(1u C )

(2u C (b)

(d)

Fig.3Connecting two Bezier curves;a the C 0gluing Bezier curves at the point Q ;b the C 1connecting Bezier curves at the point Q ,c the almost C 2con-necting Bezier curves at the point Q ,d the C 2connecting Bezier curves at the point Q ;e the C 2connecting Bezier curves with zero second derivative at the end points

sign and a large jerk could occur at this point.To eliminate such a condition,one can set the second derivatives at the

end points P 10and P 2

3to be zero as shown in Fig.3e .By setting this condition,the geometric corner within the block can be eliminated and the control points are given as:

P 11?

P 10tP 122

e15T

P 22

P 21tP 232

e16T

Based on Eqs.6,13,14,15,and 16,one can obtain a cubic spline curve with the continuous slope and the zero curvature at the junction point Q .2.3Spline-curve fitting method

In this section,a spline-curve fitting method is presented for converting small line segments into a parametric curve.The spline curve is composed of N segment Bezier curves.The data points Q 0,?,Q N as shown in Fig.4are obtained from

NC code.The points S 0,?,S N and P i 0,?P i

3are the spline control points and i th Bezier control points,respectively.To ensure that the fitting curve passes all of the data points

Q 0,?,Q N ,the control points P i 0and P i

3are assigned to be the data points Q i ?1and Q i ,respectively.The end points of

the fitting curve P 10and P N

3are set to be S 0and S N .To maintain the continuity of the position,slope,and curvature

at the junction of each Bezier curve,the conditions mentioned in Section 2.2can be represented as follows:

Q i ?

P i

2tP i t11

e17T

P i 2

S i tP i

1e18T

P i t11

S i tP i t1

e19T

P i 1

P i 2tS i à12

e20T

P i t12

P i t11tS i t1e21T

By substituting Eqs.18,19,20,and 21into Eq.17,the linear equations for the data points and spline control points can be represented as the following:

Q i ?

16S i à1t23S i t1

S i t1;for i ?1;ááá;N à1e22T

000P S Q ==20

31P P Q ==3

0232P P Q ==i

Q N

N N P S Q 3==1

S 2

S i

S 1

2P 11

P 2

2P Fig.4The illustration of the spline-curve fitting method

The linear equations can be expressed as the matrix form and given as:

1 6

6000ááá000 1410ááá000 0141ááá000 ........................ 0000ááá141 0000ááá006 2

|??????????????????????????????{z??????????????????????????????}

...

S Nà1

S N

|????{z????}

...

Q Nà1

Q N

|??????{z??????}

e23T

The spline control points S0,?,S N can be solved iteratively by using the Gauss elimination method.To be more efficient in the computation,one can calculate the inverse of matrix A in advance and store the results in the memory.After the spline control points S0,?,S N are determined,the next step is to derive the Bezier control points P i0,?P i3.The i th Bezier control points P i1and P i2can be computed by solving Eqs.18and20simultaneously and are given as follows:

P i1?2

S ià1t

S i;for i?1;...;Ne24T

P i2?1

S ià1t

S i;for i?1;ááá;Ne25T

With the obtained Bezier control points P i0,?P i3,the spline curve defined by Eq.3can be determined.

The above method is then applied to fit short line segments into a cubic spline curve in real-time interpolation. The process of the spline-curve fitting method is summarized as follows:

(a)Given the data points from the NC codes,one can

determine the points of Q0,Q1,?Q N.

(b)Set the i th Bezier control point P i0and P i3to be Q i?1

and Q i,respectively.

(c)Calculate the spline control points S0,S1,?S N by using

Eq.23.

(d)Calculate the i th Bezier control points P i1and P i2by

using Eqs.24and25,respectively.

After obtaining the control points P i0,P i1,P i2,and P i3for i=1,2,?,N,the cubic spline curve is determined by the Bezier curves by using Eq.3.

It is noticed that the maximum number of the short line segments depends upon not only the computational capability but also the fitting accuracy.It is because even the data points can be fitted into a spline curve,the fitted curve path and the original line command path might not be coincided even though the fitted curve passes the command points.The fitting criterion,which is based on the bi-chord error formulation,will be derived in the next section such that the fitting accuracy of the path is ensured fit the given tolerance.

3System architecture and look-ahead algorithm

3.1System architecture

An X–Y table using a PC-based control system is developed in the laboratory as shown in Fig.5.The controller consists of three main programs:CNC interpreter,look-ahead algorithm,and motion controller.The CNC interpreter reads NC codes to generate and store NC blocks.The look-ahead algorithm includes two different modules.The first module is the spline-fitting module for the NC blocks. It is designed to fit the short line segments by using the bi-chord error formulation.The second acc/dec feedrate-planning module plans the feedrate profile based on the chord errors,curvatures,and acceleration limits.Then,the look-ahead algorithm outputs the feedrate profile to the interpolator and the position commands for X and Y-axes are generated. Finally,the position commands are sent to the servo controller to perform real-time motion control.

To verify the proposed algorithm,the experiments are performed on an X–Y table with YASKAWA SGDL-04AF servo motors and SGDL-04AS servo drivers as show in Fig.5.The CNC interpreter,look-ahead function,and motion controller were implemented on a PC platform using a Pentium IV1.4GHz CPU with a1024MB memory under Microsoft Windows XP operating system.The real-time extensions(RTX)[14]are used to ensure the operating system with real-time performance.The real-time con-straints of the look-ahead algorithm and motion controller are set to10and0.5ms,respectively.The PC interface sends voltage commands and receives the feedback signals through Advantech PCI-1716D/A card and PCI-1784 encoder card.The built-in incremental encoders and the linear scales mounted on the X–Y table were used for velocity and position feedbacks.The resolution of the encoder and linear scale are1,000pulse/rev and1μm, respectively.

3.2Look-ahead algorithm

3.2.1Spline-interpolation fitting module

The first task of the spline-fitting module is to determine whether the data points could be used in the curve fitting. To achieve this task,the bi-chord error formulation is taken

into consideration.The bi-chord error criterion is illustrated in Fig.6where Q i ?1,Q i ,and Q i +1are the data points and the L 1and L 2are the block lengths between the data points.The chord errors δ1and δ2can be calculated as follows [8]:d 1?R 1àcos f 1eT

e26Td 2?R 1àcos f 2eT?R 1àcos p àq àf 1eT?

e27T

R ?

L 12sin f 1

e28T

f 1?tan à1

L 1sin p àq eTL 2tL 1cos p àq eT

e29T

If the chord errors δ1and δ2are both smaller than the given fitted tolerance δtol ,it means that the two NC blocks (Q i Q i à1and Q i t1Q i )can be fitted into a spline curve.By applying the bi-chord error formulation for the NC blocks,the fitted block number (N t )can be acquired.

After the fitted data points are obtained,the second task of the spline-fitting module is to fit the data points into a spline curve by using the technique derived in the previous section.The flowchart for the spline-fitting module is shown in Fig.7.The maximum fitted blocks (N max )can be determined which is dependent upon the computational capability.

3.2.2Acceleration/deceleration feedrate-planning module

After obtaining the fitted spline curve,the acc/dec feedrate-planning module with a look-ahead algorithm is applied.

The feedrates at some critical points are determined first.The first step is to determine the feedrate at the junction of adjacent segments which include the linear segment and spline curve as shown in Fig.8.The design criterion is that the feedrate should be decreased when the curvature of the junction points exceeds the threshold value κth defined as:k th ?

A max V 2max

e30T

?i Q

+i Q

Fig.6Schematic diagram of the bi-chord error formulation

Fig.5System architecture of a PC-based control system

where A max is the maximum acceleration limits and V max is the desired feedrate in the NC block.Based on the tangent

vectors T !1and T !

2shown in Fig.8,the curvature κc at the junction can be calculated by using Eqs.28and 29.Figure 8

describes the four possible conditions of connecting the linear segment and spline curve.The feedrate at the junction under the four conditions is determined consistently based on the curvature at the junction.

After obtaining the curvature of the junction points,the feedrate V c at the junction of adjacent segments are given as:

V c ?

????????????

A max =p k c ;k c >k th

V max

;k c k th 8<:e31T

The concept can be illustrated by an example shown in Fig.9where the Q 0,Q 1,Q 2,Q 3,Q 4,and Q 5are the data points from NC codes.If the five segments are fitted into the spline curve,it is found that the chord error of segment Q 3Q 4shown in Fig.9(a)exceeds the fitted tolerance.By applying the bi-chord error formulation,only four segments should be combined into a spline curve and the segment Q 4Q 5should perform the linear interpolation such that the chord error of segment Q 3Q 4can be confined.After the fitting process,the feedrate at junction point Q 4can be calculated by applying Eq.31.

After obtaining the feedrate at the junction of adjacent segments,the second step is to determine the feedrates at the sharp corners within the fitted curve.As shown in Fig.9b ,the curvature of the points of Q 1,Q 2,and Q 3might be large.The contour accuracy at these points might exceed the allowable tolerance as the feedrate increases.The idea of the acc/dec planning algorithm is to treat the motion trajectory as a whole and thus the same criteria in determining the feedrate at the junction is also applied to the so-called sharp corners within the fitted curve.That is,

Fig.7The flowchart of the spline-fitting module

(d)

Spline

)

(1u S )

(2u S 0

2)(==

u u S T 1Spline

Linear

(b)

)

S 0

2)(==u u S 1

T Linear

Linear

(a)

Spline

Linear

(c)

T 1)

(u S Fig.8The four connection types;a the connection with two linear segments;b the connection between linear

segment and spline curve;c the connection between spline curve and linear segment;d the connection with spline curves

if the curvature of the points Q i within the fitting curve exceeds the threshold value κth defined as Eq.30,these points can be regarded as the sharp corners.The feedrate V m sp at the sharp corners can be calculated and given as:V m

sp ?

??????????

A max sp

s e32T

where k m sp is the curvature at the sharp corners and m is the

index of the sharp corners.

The points Q i within the fitting curve need further examination on chord errors due to the interpolation process.It was shown that the chord error of the interpolation accuracy at the sharp corners depends upon the feedrate by the following approximation formulation [15]:

d m

sp ?1k sp à?????????????????????????????????????????

?1k sp

!2à

V m sp T s 2 2v u u t e33T

where T s is the sampling time of the motion controller.

When the feedrate at sharp corners increases to a higher value,the chord errors at the sharp corners might exceed the given tolerance δmax .To confine the chord errors at the sharp corners,the feedrates at the sharp corners can be obtained by integrating Eq.33with Eq.32and rewritten as:V m

sp ?min ??????????A max k m sp s ;2T s ??????????????????????????2d max k m sp

àd 2max s ()

e34T

After obtaining the feedrates at the junction point and sharp corner,the third step is to plan the feedrate profile such that the acceleration is limited.The planning technique is to first divide the fitted curve into small curve segments.For example,if the curvature of the point Q 2as shown in Fig.9b is greater than curvature threshold κth ,the point Q 2is regarded as the sharp corner.Then the fitted curve is divided into two segments,such as Q 0Q 1Q 2and Q 2Q 3Q 4.

The length of each segment L seg is then calculated by

applying adaptive Simpson ’s method [16].

To illustrate the procedure of planning the acc/dec feedrate profile in the look-ahead algorithm,an example shown in Fig.10is demonstrated.Here,the feedrates at the

junction points or the sharp corners denoted as b V

seg are first obtained by using Eqs.31and 34.The feedrate at the junction between the N i and N i +1segment can be expressed

as b V

seg ;N i .At the N k segment,the acc/dec planning technique looks ahead N la segments as shown in Fig.10a .Here,the N la is selected to be 30.Given the feedrates for each of the N la segments,the following inequality equations should be satisfied in order to generate the feedrate profile under the limit of the maximum acc/dec A max .L seg ;N i !

b V 2seg ;N i àb V 2seg ;N i à1

2sgn b V

seg ;N i àb V seg ;N i à1 A max ;for N k i N k tN la e35T

To ensure that the length constraint defined in Eq.35is

satisfied for each of the N la segments,the lengths of each segment are first calculated and stored into the memory during the look-ahead process.The procedure starts from the last segment N k tN la of the look-ahead buffer and examine the length constraint sequentially from the last segment to the first look-ahead segment.If the length

constraint for the N i is not satisfied,either of the b V

seg ;N i à1and b V

seg ;N i should be adjusted.As shown in Fig.10b ,two conditions could occurr dependent upon the values of b V

seg ;N i à1and b V seg ;N i .Under the acceleration condition,the feedrate b V

seg ;N i of the N i segment is adjusted by using the following equation:e V

seg ;N i ?min b V

seg ;N i ;????????????????????????????????????????????b V seg ;N i à1t2A max L seg ;N i q

e36T

where e V

seg ;N i is the adjusted feedrate after considering the maximum acc/dec limit A max .Equation 36ensures that the

Q 1

Q 2

Q 3

Q 4

Q 0

Q 1

Q 2

Q 3

Q 4

Q 5

Q (a)

(b)

chord error

Fig.9Example of the proposed interpolation algorithm;a spline-fitting method without bi-chord error test;b Spline-fitting method with bi-chord error test

criteria set by Eqs.31and 34and the length constraint are

satisfied.However,if the N i segment is under the

deceleration condition,the feedrate b V

seg ;N i à1is adjusted such that the above criteria are also satisfied and it is given as the following:

e V

seg ;N i à1?min b V

seg ;N i à1;?????????????????????????????????????????b V seg ;N i t2A max L seg ;N i q

e37T

where e V

seg ;N i à1is the adjusted feedrate under the deceleration condition.By considering the length constraints for each of the N la segments,the feedrate at the N k segment can be obtained.After the interpolation for the N k segment,the look-ahead algorithm is then applied to the next segment.

The above look-ahead algorithm is better illustrated by the example shown in Fig.10.Here the N i segment does not satisfy the length constraint due to the short length.

Equation 37is applied and the feedrate of the b V

seg ;N i à1is reduced such that the limit of the maximum acc/dec is satisfied.Consequentially,the length constraint set by Eq.35for the N i ?1segment should be calculated using

the adjusted feedrate e V

seg ;N i à1.If the N i ?1segment does not either satisfy the length constraint,the feedrate b V

seg ;N i à2should also be adjusted by using Eq.37.Subsequently,the adjusted feedrate profile with consideration of the corner,junction,and A max effects is shown in Fig.10c .The flowchart of the acc/dec feedrate-planning module is shown in Fig.11which describes the details of the process.

3.3Real-time interpolator

After obtaining the feedrate profile by applying the look-ahead algorithm,the interpolator performs linear or spline interpolation to generate the position commands.The spline curves can be interpolated by representing the parameter u

i-1

(a)

Look-ahead Segments

(b)

N seg ,?,??i N seg V (c)

?V Fig.10Illustration for the acc/dec planning technique of look-ahead algorithm;a trajectory path;b adjusted feedrate under different acc/dec conditions;c Feedrate profiles (solid the original feedrate,dashed the adjusted feedrate)

as a function of time.The second-order approximation interpolation is utilized to implement the spline interpolation in this paper,and it is given as [17]:

u k t1

?u k tV u k eTT s S 0u k t

T 2

s 2A u k eTS 0u k àV 2u k eTS 0u k eTáS 00u k eT?

S 0u k eTj j (

)

e38T

where V (u k )and A (u k )are the feedrate and acceleration planned in the look-ahead algorithm,respectively.

4Numerical simulation and experimental verification To evaluate the proposed interpolation algorithm,the shark

and crab contours shown in Fig.12a and b are used as working examples.The number of NC blocks for the shark and crab contours are 1,005and 1,348,respectively.By applying the spline-fitting technique,the numbers of blocks of the shark and crab contours are reduced to 67and 214,respectively.

The block diagram of the servo controller used in simulations and experiments is shown in Fig.13,where

Fig.11The flowchart of the acceleration/deceleration feedrate-planning module

Fig.12The NC codes for working examples;a shark contour;b crab contour

R j ,U j ,and Y j are the position commands,voltage commands and actual positions,respectively.The parameter j is the index of each axis.The parameters a j and b j were identified by measuring the frequency response from the voltage command to the velocity feedback [18].The parameters K vp ,j and T i,j in the velocity loop were designed by setting the damping ratio 1.0and bandwidth 80Hz of the closed-loop transfer function.The proportional gain K pp ,j in the position loop was determined by the desired bandwidth 40Hz of the position loop.The velocity feedforward gain

K vff,j is used to reduce the tracking errors.The closed-loop transfer function G c,j (s )is given as the following:

G c ;j es T?

Y j es TR j es T

?b j K vff ;j K vp ;j s 2

tb j K vp ;j K pp ;j tK vff ;j

T i ;j

s tb j K pp

;j K

vp ;j

T i ;j

s 3ta j tb j K vp ;j àás 2tb j K vp ;j K pp ;j t1T i ;j

s tb j K pp ;j K vp ;j

T i ;j e39T

Fig.13Block diagram of the servo controller

Parameters

Symbol

Value

Interpolator

Maximum fitted block

N max 100

Maximum acceleration limit A max 2,450mm/sec 2Fitted tolerance δtol 10μm Chord tolerance

δmax

1μm

Maximum look-ahead segments N la 30

Servo controller

X axis

System dynamics a x 73.229s

b x 7.374×103mm/V Velocity controller

K vp,x 5.766×10?2V s/mm T i,x 6.845×10?3s ?1Position controller

K pp,x 156.063s ?1Velocity feedforward controller K vff,x 0.95

Y axis

System dynamics a y 70.415s

b y 6.937×103mm/volt Velocity controller

K vp,y 6.118×10?2V s/mm T i,y 6.933×10?3s ?1Position controller

K pp,y 155.517s ?1Velocity feedforward controller

K vff,y

0.95

Table 1Parameters of the interpolator and the servo controller for numerical simulations and experiments

profiles;b X axis tracking errors;c Y axis tracking errors;d contour errors

All parameters of the interpolator and the servo controller for numerical simulations and experiments are listed in Table1unless state otherwise.

4.1Numerical simulation

In this section,numerical simulations are performed using the shark contour.The performance between the approxi-mate optimal feedrate interpolation(AOF)[2]and the look-ahead algorithm with spline-fitting method(LASF)are compared.The AOF developed an algorithm to find the approximate optimal feedrate and then the feedrate profile for the short line segments is applied based on the optimal value of the feedback.

The comparisons between the AOF and the LASF interpolation algorithms are shown in Fig.14.The feedrate profiles are shown in Fig.14a.When the short line segments are fitted into a spline curve,such as OA,AB, etc.,the LASF algorithm within the curve can achieve the maximum feedrate equal to3,000mm/min.The statistical data are summarized in Table2which shows that the LASF algorithm can reduce24.06%,44.01%,and47.75%of the machining time as compared to the AOF under the given maximum feedrate of1,000,2,000,and3,000mm/min, respectively.The maximum contour error for the LASF algorithm can be reduced by55.97%,35.66%,and57.99% as compared to the AOF under the feedrate of1,000,2,000, and3,000mm/min,respectively.

Figure14b,c,and d show that the tracking and the contour errors of the LASF algorithm are smaller than those of the AOF.It is because the continuity of the slope and curvature at each junction point of the spline curves are ensured when the short line segments are fitted into a spline curve.Based on Eqs.20and21,the slope and curvature at the junction points of adjacent segments are also continu-ous.Therefore,the feedrate fluctuation and velocity discontinuity problem can be eliminated.Furthermore,the velocity for the X and Y-axes are smoother than the AOF algorithm.

It is mentioned that the computing time of the spline-fitting algorithm can be much reduced as compared to the NURBS approach[11].From Eq.23,the calculation of the inverse matrix of A could be very time-consuming especially when the dimension of A becomes large.The advantage of the proposed method is that the A in Eq.23is a constant matrix which is not dependent upon the NC codes.Therefore,the inverse of the A matrix can be computed in advance and stored in the memory.On the other hand,the coefficient matrix using the NURBS method depends upon the NC points and the inversion of the matrix cannot be obtained beforehand in the interpolation and should be performed in real-time.The comparisons on the computation time between the NURBS and the proposed method are shown in Fig.15.It is shown that the computing time for the spline-fitting algorithm is much less than that of the NURBS method.

4.2Experimental verification

The experiments are performed using the setup shown in Fig.5.A larger number of the look-ahead segments increase the computational time of the look-ahead algo-rithm.To ensure real-time implementation,the maximum number of the look-ahead segment N la is set to be30.The computational time of the look-ahead algorithm is shown in Table3.It can be seen that most of the computational power is consumed in the acc/dec feedrate-planning module.The real-time constraint of10ms can be satisfied. Note that the average computational time of the spline-fitting module is given as11.8and9.3μs for the shark and

Table2Numerical Simulation performance comparisons between the AOF and LASF interpolation algorithms for the shark contour

Maximum feedrate (mm/min)Interpolation

algorithm

Tracking error(μm)Contour error(μm)Time(s)

X axis Y axis Max RMS Mean IAE

Max RMS Max RMS

1,000AOF21.989 4.05121.645 3.65816.248 1.0280.2748.40130.526 LASF19.153 1.91918.211 1.9317.1530.8930.48511.249323.1805 2,000AOF57.0587.88871.863 6.87122.168 1.9540.61813.46922.022 LASF38.078 4.55635.836 4.44314.261 1.989 1.42617.53212.328 3,000AOF102.77710.95367.9939.34845.547 3.233 1.14719.19616.4085 LASF57.9397.12151.7817.15219.131 2.685 2.50221.3368.5725

Max the maximum value of the contour error,RMS root mean square of the contour error,Mean the average contour error,IAE integral absolute error

crab contours,respectively.The time for the shark contour is higher because more short blocks can be fitted into the spline curve as shown in Fig.12a .

The shark contour is tested under the feedrate commands equal to 1,000,2,000,and 3,000mm/min,respectively.The tracking and contour error comparisons between the AOF and LASF interpolation algorithms are shown in Fig.16.The statistical data are summarized in Table 4.It is clear that the proposed interpolation can achieve better tracking/contour performances and reduce the machining time.The AOF algorithm which only maintains the continuity of position for the consecutive segments could cause large contour errors at the area of C,E,and J as shown in Fig.16c .The LASF algorithm guarantees that the continu-ity of the position,slope,and curvature for the consecutive segments and junction points are continuous.Table 4shows that the LASF algorithm reduces the maximum contour error and the machining time by 46.93%and 47.75%as

compared to the AOF algorithm when the maximum feedrate equals to 3,000mm/min.

To further verify the LASF interpolation,the crab shown in Fig.12b is tested.The tracking and contour error comparisons between the AOF and LASF interpolation algorithms are shown in Fig.17.The statistical data are summarized in Table 5.It shows that the contour errors using the AOF algorithm are large.The LASF algorithm improves contouring accuracy significantly,especially for the short line segments region.As shown in Table 5,maximum feedrate is 3,000mm/min,the maximum contour errors and machining time can be reduced by 38.24%and 55.53%as compared to the AOF algorithm.

5Conclusion

A new look-ahead algorithm with spline-fitting interpola-tion scheme which consists of the spline-fitting and acc/dec feedrate-planning modules is proposed in this paper.The conditions to ensure the continuity of the position,slope,and curvature at each junction point of the spline curves and line segments are derived.Then,the bi-chord error formulation is derived to determine whether the short line segments can be combined into a spline curve.The sharp corners of the fitted curve are identified by applying the curvature threshold criterion.The feedrates at the junction of adjacent segments and the feedrates at the sharp corners within the fitted curve can be calculated by utilizing the curvature threshold criterion.Based on the chord errors at the sharp corners,curvatures and acceleration limits,the linear acc/dec feedrate profile is adopted to generate the feedrate profile.Simulations were performed to validate the LASF interpolation algorithm and the computing efficiency is demonstrated.Finally,experiments are conducted to show that the LASF interpolation can achieve better accuracy and reduce machining time as compared to the AOF algorithm.

Algorithm Computational time (μs)Shark contour

Crab contour Look-ahead algorithm 2239.54392.0Spline-fitting module

11.89.3Acc/dec feedrate-planning module

2227.74382.7Feedrate at the junction of adjacent segments

35.714.7Feedrate at the sharp corner within the fitted curve 23.216.1Acc/Dec planning technique 2168.84351.9Motion controller 19.119.1Interpolator 12.912.9Servo controller 6.2

6.2

Table 3Real-time performance evaluation for PC-based motion controller with the LASF interpolation algorithm

Fitted Blocks

C o m p u t i n g T i m e (m s )

Fig.15Computing time between the NURBS and the spline-fitting techniques

Table 4Performance comparisons between the AOF and LASF interpolation algorithms for the shark contour Maximum feedrate (mm/min)

Interpolation algorithm

Tracking error (μm)Contour error (μm)Time (s)

X axis Y axis Max

RMS

Mean

IAE

Max

RMS Max RMS 1,000AOF 23.634 2.94415.097 2.27317.398 1.318 1.39942.71530.526LASF 14.131 2.72712.452 2.31413.165 1.224 1.36931.75423.18052,000AOF 41.377 5.02660.875 3.7720.461 1.881 1.79339.48622.022LASF 23.704 4.87820.394 4.09115.862 1.851 2.01524.84412.3283,000

AOF 54.7377.11237.134 4.85931.195 2.639 2.28137.42116.4085LASF

37.599

6.732

28.552

5.731

16.555

2.275

2.497

21.414

8.5725

Fig.16Experimental error comparisons of different interpolation algorithms for the shark contour (maximum feedrate equal to 3,000mm/min);a X axis tracking errors;b Y axis tracking errors;c contour errors

Table 5Performance comparisons between the AOF and LASF interpolation algorithms for the crab contour Maximum feedrate (mm/min)

Interpolation algorithm

Tracking error (μm)Contour error (μm)Time (s)

X axis Y axis Max

RMS

Mean

IAE

Max

RMS Max RMS 1000AOF 27.591 2.81516.142 2.3126.241 1.1420.92176.83583.439LASF 18.497 2.49418.335 2.52415.412 1.1950.82145.42855.29152000AOF 62.417 4.8374.001 3.28267.092 1.673 1.23175.08761.0135LASF 25.496 4.62125.564 4.70724.235 2.083 1.41740.31328.4373000

AOF 69.601 6.66280.685 4.67470.032 2.777 1.59469.14143.3685LASF

37.399

6.823

43.671

7.241

43.247

3.496

2.274

43.865

19.283

Fig.17Experimental error comparisons of different interpolation algorithms for the crab contour (maximum feedrate equal to 3,000mm/min);a X axis tracking errors;b Y axis tracking errors;c contour errors

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Development of a real-time look-ahead interpolation methodology with spline-fitting technique

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