Accuracy improvement of three-axis CNC machining centers by quasi-static error compensation

Journal of Manufacturing Systems Vol. 16/No. 5 1997

Accuracy Improvement of Three-Axis CNC Machining Centers by Quasi-Static Error Compensation X.B. Chen and A. Geddam, City University of Hong Kong, Hong Kong Z.J. Yuan, Harbin Institute of Technology, Harbin, China

Abstract

center through a single-board microcomputer. Kiridena and Ferreira3 used the D-H method and developed a model that showed the effects of the positioning error of machine axes on the accuracy (position and orientation) of the cutting tool in its workspace. Lin and Ehmann4 used a generalized methodology for the evaluation of volumetric errors in the workspace of a multiaxis machine of arbitrary configuration. They used a modified coordinate frame assignment rule and introduced the concept of fixed and moving error reference frames for kinematic error motions. Their approach offers the ability to assign the coordinate origins of the joint frames at the physical joint. Based on a rigid body model, Zhang et al. s calculated the geometric errors of coordinate measuring machines, successfully applied sot~,vare error compensation to machines, and obtained "a factor of 10 improvement in machine accuracy." Elshennawy and Ham n also used rigid body kinematics to develop a model for geometric positioning errors in coordinate measuring machines and applied an error compensation technique for improving the performance of machines by an average factor of four. Assuming linear elemental error characteristics, Ferreira and Liu7 combined them to form an overall model of the volumetric errors in the machine using rigid body kinematics. The model parameters were expressed as functions of 15 error measurements made at nine points on the edges of a cubical workspace. Based on the concepts developed by Ferreira and Liu, 7 Kiridena and Ferreiras~° presented an approach for the development of a general n-th order quasi-static error model for CNC machining centers. They also proposed a method for estimating the parameters in the model and experimentally demonstrated that the accuracy of parameter estimation and the first-order model were very high. Under the assumption of the availability of an error model, Kiridena and Ferreira developed a computa-

This paper discusses the development of a general quasistatic error model for multiaxis CNC machining centers using rigid body kinematics. To predict the quasi-static errors at any point in the workspace, a new method was proposed using the meshing concepts developed in the finite element method literature. Taking the nonlinearity of quasi-static errors into account, computational approaches to compensating errors of the basic motions of a CNC machine are developed. The strategies developed are tested on a CNC machine, and the results show that a 80-90% reduction of quasi-static errors is gained after introducing compensation.

Keywords: Quasi-Static Errors, Model, Error Compensation, Nonlinearity

Introduction The geometric errors of machined parts are due to the machine tool structure and machining process. Error sources attributable to the machine tool structure can be classified as quasi-static errors, thermally induced errors, and dynamic errors. Quasi-static errors are defined as "errors of relative position between the tool and workpiece that are varying slowly in time and are related to the structure of the machine tool itself.' '~ Quasi-static error sources include the geometric and kinematic errors of the machine and errors due to static and slowly varying forces such as the dead weight of the machine's components, overconstrained slides, and so on. It was reported that quasi-static errors account for about 70% of all errors attributable to a machine tool. ~ So the ability to compensate for such errors could significantly enhance the accuracy of a machine tool and consequently reduce the error observed on parts machined on it. Many investigators have addressed this problem from different perspectives. Donmez et al.2 used rigid body kinematics to propose a general methodology of error compensation, and they applied it on a turning

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Journal of Manufacturing Systems Vol. 16/No. 5 1997

2. Elemental~component approaches: These ap-

tional approach to compensate quasi-static errors of three-axis machining centers. Using rigid body kinematics, Srivastava, Veldhuis, and Elbestawitn established a model to obtain volumetric (position and orientation) errors at the tool throughout the workspace due to geometric and thermal errors of individual components on a five-axis CNC machine tool. They assumed that the individual joint (or axis) error ,caries linearly with movement along the axis and exponentially with time to represent the thermal deformation, and they solved the compensation motion of each axis using a special symbolic mathematical package. Zhang, Ouyang, and Lu 12 proposed a displacement method for machine geometry calibration. By measuring the displacement error along 22 lines in the machine work zone, the volumetric errors of a threeaxis machine tool were determined. Kreng, Liu, and Chu 13 developed an error model to express the observed error in a three-a~ds machining center workspace as a function of its elemental errors. Using a specially designed pallet, the error model was updated periodically. The experimental results indicated that the error can be predicted within 25 txm when the method is applied. Shin, Chin, and Brink14established a comprehensive characterization procedure for CNC machining centers and illustrated that static and dynamic characteristics as well as transient and steadystate features can be extracted from seven complementary characterization tests. Duffie and Yang~s also developed a kinematic model for a machine. By using reference parts as the metrology artifacts, Mou and Liu l~d7 developed an error correction method. After that, they proposed an adaptive methodology by adopting current machine tool conditions obtained from various sensors and then modifying the error model coefficients for more accurate error estimation and compensation,ls2° The modeling approaches described above, with the exception of a few, might be categorized into the following three classes:

The advantage of the synthetic approaches is that it is possible to find out the sensitivity of the machine's accuracy to different elemental errors, and to estimate the elemental errors from the three-dimensional errors measured at a few points in the workspace. Because of this advantage, more and more researchers are applying the synthetic approaches to their modeling of quasi-static errors of machine tools.7-1°,~'~° Assuming linear or quadratic elemental error characteristics and using rigid body kinematics, these authors combine them to form an overall model of the volumetric errors in the machine workspace. The model parameters were expressed as functions of the measurements of some points on various kinematic reference standards and artifact standard parts. Because the nature of the error components associated with each link is complex, sometimes it has to be expressed by higher order polynomial functions. Consequently, the derivation of the error model and estimation of the parameters of the model will become very complicated. Thus, the error predication based on these models may not be precise enough. To alleviate the above problem, a new error modeling and prediction method will be developed in this paper. The methodology has two major steps, as follows:

1. Workspace approaches: These approaches measure and analyze errors in the workspace without attempting to resolve them to their actual sources. Typically, these approaches require large amounts of data (and hence measurements) and their validity for machine states that have not been encountered in the model calibration is always in doubt.

1. Development of a general mathematical model to represent the quasi-static errors of multiaxis CNC machine tools, and 2. Construction of an quasi-static error map and development of a practical method for predicting quasi-static errors at any point in the workspace of a CNC machine tool based on the measurements of error components.

proaches address the problem of computing deformation of machine members (elemental errors). They usually do not analyze the propagation of these errors through the kinematic system of the machine to influence its three-dimensional accuracy. 3. Synthetic approaches: These approaches, which bridge the gap between the two approaches above, analyze emphatically how the elemental errors are magnified/suppressed during their propagation through the kinematic system of the machine to influence the three-dimensional accuracy in its workspace.

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Journal of Manufacturing Systems Vol. 16/No. 5 1997

For improving the performance of CNC machining tools by compensating quasi-static errors, hardware and software approaches have been used to acquire the expected results. Hardware approaches developed by Dufour and Groppetti n would be adequate without modification. However, in most production environments, it is difficult to make hardware modifications on existing machine tools due to technical limitations and warranty problems. In such cases, a complete software compensation scheme that performs as a preprocessor is an attractive alternative. Once an adequate and accurate model is available for characterizing the quasi-static errors in the workspace of a machine, the compensation scheme can be implemented by using the uncompensated NC code to generate a compensated NC program to achieve anticipated improvement in machining accuracy. One important aspect of the error model that results from the analysis given in Geddam and Chen z2 is that the variation of quasi-static errors in the workspace is nonlinear. Consequently, the linear motion of the tool, under the influence of errors, will actually be a curve in the workspace. Hence, compensations given at the starting and ending points of a line will be inadequate in correcting the errors at intermediate points. Considering basic motions, the majority of CNC machines can perform three kinds of basic motions: rapid positioning motion, linear interpolation motion, and helical interpolation motion? s This paper, taking the nonlinearity of quasi-static errors into account, will develop computational approaches to compensating quasi-static errors of these three motions of a CNC machine.

[~

-~'+

--a,13' a,+Aa,]

i

bi+Abi[

(1)

where Aai, Abi, and Aci are the translational errors, and txi, 13i,and 7i are extremely small angular errors of the inaccurate link due to the rotation about the X, Y, and Z axes, respectively. All error terms in this link transformation matrix are assumed to be independent of position and are treated as constants. The joint constraint matrix, J~, is modified and used to describe the relative motion of two coordinate systems along an imperfect joint. For an imperfect prismatic joint, the modified joint constraint matrix can be written as follows:

1 - e z ( M ) ev(M) 41rn + & ( M ) ez(M) 1 -e~(M) 42m + ~ ( M ) 4~rn + &( M) J(M)=[-eyT) e~(M) 1 0

0

1

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when M = X

4,=0 42--1 43=0, 4,=0 42--0 43=1,

when M = Y

(2)

when M = Z

where e~(M), er(M), and e~(M) are small angular errors, and ~(M), By(M), and 8~(M) are the translational errors of the inaccurate prismatic joint. For an imperfect revolute joint, the modified joint constraint matrix for revolving 0x about the Z axis of a reference frame can be written as follows:

c(o~ +tz (o~))

Modeling of Quasi-Static Errors

s(o~ +e~(Oz))

Any multiaxis machine tool can be considered to consist of a series of links connected together by prismatic and revolute joints. These components comprise a closed chain between the tool edge and the workpiece. For a multiaxis machine, each link and joint can be modeled as a link transformation matrix and a joint constraint matrix, respectively,u,2s For an inaccurate link, suppose the link has three rotational error components about and three translational error components along a reference coordinate system's axes. Under the assumptions of small errors, the resultant link transformation matrix, L~, across the actual link can be given as follows:

J(Oz) =

ex(o~).s(oz +e~(Oz))G(o, ).C(Oz +e~(o~)) 0

325

-S(6 +eAo~))

G(Oz)

C(Oz+e~(o~))

-eAo~) 6x(O~)

G(o=).s(oz +e=(0=))+ ex(oz).c(o, +eA0z))

1

¢5~(0z)

0

0

1

(3)

Journal of Manufacturing Systems Vol. 16/No. 5 1997

Joint

1(2)

Z

w\

/

Y

~ k Machinebase

X

.

Link2 X

Link4

,'e )ase

Joint3(X)

Figure 1

Figure 2

Schematic Diagram of Typical Vertical Machine Tool

Kinematic Equivalent for Machine Tool in Figure 1

E = Ww- Wr=(R-I) • Wr+P

where 8z(0z) is the error of rotation 0z and ex(0D and ey(0D are the angular errors about the X and Y axes, respectively. 8x(0=), 8y(0z), and 8z(0D are the translational errors. All error terms in the joint constraint matrix are treated as functions of position in translation or rotation when quasi-static errors are only considered. Given any n-axis CNC machine tool with an arbitrary combination of prismatic and revolute joints, the homogeneous transformation matrix (HTM) can be used to build a mathematical model describing the resultant positioning errors. The relationship between the position vector of a point in the tool coordinate system and that in the workpiece coordinate system can be expressed as follows:

(5)

To demonstrate the general procedure for error model derivation, a typical vertical CNC machine center is considered in this paper. It has three translational axes, as shown in The whole machine can be modeled as a kinematic chain with four links connected in series by three prismatic joints. One end of the chain is attached to the workpiece, while the other end is attached to the tool on the spindle, as shown in The error is then

Figure 1.

Figure2.

E

= R-IL4(R-1j3(R-1L3(Rj2(RL2(RjI(RL1 W T -1- PL1) -fPJ, q- eL2) Jc P J2 -Jr-p-1L3 ) -1- P - l J3 q'- P-IL4 -- WT (6)

where

[WlWl=Hm<-" N [WrlI = 7IR1 TI p ] "IWrlI

(4)

-1 I I

m=l

where Wr and Ww represent the position vector of a point in the tool and workpiece coordinate system, respectively. =A,._I is the HTM of the i-th solid link and joint; it can be expressed as =A,,_, = arm "L=. R and P represent a 3 × 3 orientation submatrix and a 3 × 1 position vector of the total HTM, respectively. Let the tool coordinate system coincide with the workpiece system; then the positioning error E can be derived from the difference between

<

-1 = Li

3 i = 3,4;

- -

= J(X) -1

Prediction of Quasi-Static Errors Before predicting the errors, it is necessary to measure geometric and kinematic errors at some positions in the workspace of the machine tool. In this paper, a laser interferometer measurement system and an electronic differential level were used to measure machine linkage errors. These errors

Wrand Wee:

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Journal of Manufacturing Systems Vol.

16/No.

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1997

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Figure 4

Measured Linear Errors of XAxis

Measured Straightness Errors of XAxis

include linear errors of feed mechanisms, straightness errors of guideways, angular errors of machine slides, and squareness errors among machine axes. The measurement results show that the linear errors of the X axis are about 32 tzm in a 700-mm travel (Figure 3), of the Yaxis are 6 Ixm in a 250-mm travel, and of the Z axis are 5.5 I~m in a 200-mm travel. The maximum straightness error was found in the X axis, which is 2 Ixm along the Z axis (Figure 4). The maximum angular error with a magnitude of 6 arcsec was the pitch angular error of the X axis (Figure 5). Most previous work related to error model and compensation either on machine tools or CMMs adopted the assumption that error components change linearly with position. However, the test result shows that most of the geometric error components, especially the straightness and angular errors, are nonlinear with position, which will make the error prediction and compensation more complex. Based on Eq. (6), the position errors in three dimensions at the measured points of the CNC machine can be calculated. These position errors at a limited number of points in the workspace of the CNC machine can be used to construct the quasistatic error map of the machine that can be stored in the control computer of the machine and will be a basis of error prediction. For error prediction, the meshing concepts developed in the finite element method (FEM) literature are used to subdivide the workspace into a finite number of smaller three-dimensional elements. 26,27 The points at which the position errors have been calculated are chosen as nodal points. By connecting all the nodal points, the workspace is meshed into many elements. Once the elements have been constructed and the errors at the nodal points are deter-

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Figure 5 Measured Pitch Angular Errors of XAxis

mined, the errors at any position in every element can be interpolated by introducing suitable interpolation functions. n e = mle 1 + mze 2 + m3e3 +... + m,e, = E miei i=1

(7)

where e =interpolated value of the position errors for the desired point, [AX, AY, AZ]T e/=known position errors at nodal point Pi, [AXe, AY. AZJ ~ mi=interpolation functions at nodal point Pi n =total number of nodal points for each element In three-dimensional elements, tetrahedral and hexahedral elements are the two families most widely used. 27Although it is conceivable that many types of functions could serve as interpolation functions in these elements, only polynomials have been used in error prediction. Polynomials are relatively easy to manipulate mathematically. For the same type of element, polynomials of various orders, such as linear, quadratic, cubic, and so on, could be used to

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Journal o f Manu/acturing Systems Vol. t6/No. 5 1997

characterize the nature of error distribution in the element. Usually, if the variation of errors is complex, higher order polynomials are recommended for obtaining more precisely interpolated values. However, for higher order polynomials, more nodal points for each element are required, and consequently, larger computer memory, higher computational burden, and an additional cost associated with measurements will also needed. In this paper, only hexahedral elements will be considered. Figure 6 shows the eight-nodal brick element and a local normalized coordinate system (6, "q, 0 with its origin at the centroid (xc, yc, Zc) of the brick. The coordinates of the centroid are:

|

/

Z

~ ,

2a

),

/Z ~

3

Figure 6 Eight-Nodal Brick Element

m, = (1 - ~z) (1 + "q n~) (1 + ~ ~i)/4

(12)

3. Cubic element a. C o r n e r nodes

(8)

m, = (1 + ~ 6) (1 + n ~q,) (1 + ~ 5) [9(62 + ,q2 + ~2) -19]/64

z~ = (zl + z2)/2 = (zs + z6)/2, and so on The local normalized coordinates are defined in terms of global coordinates by the following: = (x - x~)/a, Xl = (y - y~)/b, ~ = (z - z~)/c

2b~

O(c. c zc) l /" 6

X

xc = (xl + xs)/2 = (x2 + x6)/2, and so on

y~ = (vl + y4)/2 = (Y2 + y3)/2, and so on

8

5

(13)

b. Typical midside node, ~ = +1 / 3, "qi = +1, ~i = ±1

(9)

m, = 9(1-62) (1+9~ ~i) (l+~q nO (1+~ ~)/64 (14)

Error Compensation at a Point

where x, y, and z are coordinates of any point in the elements; a, b, and c are the element half lengths in the x, y, and z directions, respectively. It follows that each serendipity coordinate has a value between -1 and +1. The interpolation functions for hexahedral elements are incomplete polynomials and are derived by inspection. The results are as follows:

Consider the error compensation at a point first, which will be a basis of error compensation of the basic motions of a CNC machine tool. Suppose Xd Xc

= desired tool position = error-compensated position e(Xd) = error at desired position

Let Xc be defined by the following: 1. L i n e a r element

m, = (1 + ~ Fa) (1 + Xl Xll) (1 + gg~)/8

Xc = X,t - e(X,)

When the tool is commanded to move to Arc, the actual position reached by the tool is given by the following:

(lO)

2. Quadratic element

X . =X~ + e(X¢) = X d - e(X,3 + e ( X d - e(Xd))

a.

Comer nodes

(16)

where X, = actual position that the tool will reach after compensation e(X~) = error at error-compensated position

m,=(1 + ~/~) (1 + n n~) (1 + [ ~i) (~ ~.+ -q -qi + ~[~ - 2 ) / 8

(15)

(ll)

Consequently, the residual error of the tool after compensation, which is the difference between the

b. Typical midside node, ~. = O, "qi = +1, ~ = +1

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Journal of Manufacturing Systems Vol. 16/No. 5 1997

actual position and the desired position, can be obtained by the following:

e(0) = e ( X a) n

Xc(n ) = Xc(n -

1) - e ( n - 1) = X a - ~ e(i - 1)

(17)

e = Xo - Xd = e(Xd -- e(Xd)) -- e(Xd)

i=1

(21)

X a ( n ) = X c ( n ) + e( X ~ ( n ) )

Based on the first error-compensated position X~ and the first residual error e, the second error-compensated position can be established as follows: X~(2) : X~ - e : Xa - e(Xa - e(Xd))

e(n) = Xa(n)-

Xc(2): second error-compensated position The actual position reached by the tool after introducing the second error compensation is given by the following:

Compensation Along the Rapid Positioning Motion

X,,(2) = Xc(2) + e(Xc(2)) = Xd -- e(X,t - e(Xd)) (19)

The purpose of a rapid positioning motion is to move the tool rapidly from its current position to a destination position with no requirement on the cutter path. Only the final position is of importance to the machining operation. So the scheme for calculating the error-compensated position at a point can be directly used to calculate the error-compensated position of the destination in the rapid positioning motion.

The residual error can be given by the following: e(2) = e(Xd -- e(Xd -- e(Xd - e(Xd) ) ) -(20)

Similarly, it is possible to establish the third errorcompensated position. Thus, a recursive scheme of compensation can be developed and used to iteratively calculate the error-compensated position until the difference between the actual position and the desired position is smaller than a predefined tolerance value. F i g u r e 7 is a flowchart of the recursive scheme of compensation. By referring to F i g u r e 7, the recursive equations can be expressed as follows:

x~

Compensation Along the Linear Interpolation Motion A linear interpolation motion means to move the tool from its current position to a destination position with a linear requirement on the cutter path. Considering the motion of the tool along a straight

/

Desired position

X a - ~e(i -

1)

1,2,3,...

The previous section presented the scheme for computing the error-compensated position in order to move the tool to a desired position in the presence of errors. Now the results of compensation at a point will be generalized to compensate errors along the three kinds of basic motions (rapid positioning motion, linear interpolation motion, and helical interpolation motion) of a CNC machine tool.

where

e ( X d - e(Xd))

n =

Error Compensation Along the Basic Motions of CNC Machine Tools

(18)

+ e ( X d - e ( X d - e(Xd)))

X d

Predict error by

interpolation

i=1

Figure 7 Flowchart of Recursive Scheme of Compensation

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Journal of Manufacturing Systems Vol. 16/No. 5 1997

line between points X~ and X2, the trajectory X can be parameterized as follows: X = )(1 + k(X2 - X 0

k E [0,1]

(22) X~c

First, the compensation can be given at the starting (k = 0) and ending (k = 1) of the line. Using the compensation scheme at a point, the error-compensated positions at the starting and ending points can be computed, respectively, satisfying that the residual error is smaller than a predefined tolerance A. With the two new error-compensated positions, a new line X~ is given by the following: X ~ = X I ~ + k(X~c-X~c)

kE

[0,1]

Figure 8 Determination of Point at Which Original Straight Line is to Be Broken in the Compensation Scheme Along a Line

errors for the motion along a straight-line trajectory was simulated on a three-axis machine. The purpose of the simulation was to verify that the compensation strategy succeeded in producing a corrected trajectory that stayed within a predefined tolerance value. First, suppose the trajectory is a diagonal line across the workspace from (10.0, 10.0, 10.0) to (680.0, 240.0, 190.0), and the predefined tolerance value is 1 Ixm. Using the above compensation approach, the original straight line has to be broken up into eight parts and the decomposition of the trajectory is given as the list of coordinates in Figure 9. The position errors along the line with and without error compensation are shown in Figure 10. It is observable from the figures that the trajectory followed by the tool when commanded to move along the compensated command trajectory was well within the predefined tolerance value from the desired trajectory.

(23)

The same parameter k can be used for the modified trajectory X~ because it is merely a scaled and displaced image of X (that is, it remains a straight line), and consequently, a one-to-one correspondence remains between the original and modified trajectories. The residual errors that remain uncompensated by only introducing corrections at the starting and ending points are given by the following: e = X c + e(Xc) - X = e(Xlc + k(X2c - Xlc)) + (XI~ - XO + k[(X2c - Xa) - (X~c - X1)] k ~ [0,1 ]

(24)

If the square of the magnitudes of the residual errors at all points of the trajectory are maintained less than or equal to the square of the predefined tolerance value (that is, A2), the positions of the tool will be kept within a sphere of radius A from each point along the trajectory. But due to the nonlinear variation of quasi-static errors in the workspace, they are often greater than the square of the predefined tolerance value (Figure 8). In such a case, the original straight line has to be broken up into two parts. The broken position is determined by the position at which the square of the residual error has its maximum. Suppose the coordinate of this position is Xb; X~ and Xb, Xb and X~ can be taken as the starting and ending points of the two new lines, and the error-compensated trajectories can be established using the scheme mentioned above until the square of the magnitude of the residual error is less than or equal to As at each point. The approach to compensate the quasi-static

Compensation Along the Helical Interpolation Motion A helical interpolation allows an arc interpolation on a specified plane, and at the same time, a linear interpolation on a third axis that is perpendicular to the specified plane. An arc interpolation can be considered as a special case of a helical interpolation, when the linear interpolation is zero. The helical interpolation is one of the most common interpolations used in NC programs of CNC machine tools. In the following discussion, it is assumed that the arc interpolation is performed parallel to the X1X2 plane and the linear interpolation is mounted in the X3 direction, as shown in Figure 11. If the tool has to move, it moves along the helical arc SME whose center axis is the C-C axis. The helical arc is parameterized as follows:

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Journal of Manufacturing Systems Vol. 16INo. 5 1997

>>Line

x10.000 X60.25 1 x144.005 X261.266 X445.527 X495.778 X546.032 X629.786 X680.039

210.000; 223.500; 246.001; 271.503; 2127.009; 2140.510; 2154.012; 2176.512; 2190.013;

Y 10.000 Y27.25 1 Y56.001 Y96.253 Y159.506 Y176.758 Y194.010 Y222.764 Y240.015

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

k

Figure 9 Command Trajectory

x1 = Xlc + Y cos x2 = x2c + Y

[e,

+ I@,

-

0 Error without compensation

Segments

8 Error with compensation

(a) Errors along Xaxis

OS)]

sin [O,+ k(8, - f3,)]

k E uul (25)

x3 = X3S + YX3e - X3S)

where xle, x2eare the coordinates in the center axis in the X, and X2 direction, r is the radius of projection of the helical arc on the XIX2 plane, and 0,, 0, are the angles between CiSi, CjEj, and axis XI, respectively. The approach to compensation along the helical interpolation motion is similar to the compensation for the motion along linear interpolation. But in this case, instead of two points, three points, including two end points (k = 0 and 1) and the midpoint M (k = 0.5), of the helical interpolation are required. Applying the compensation scheme at a point, three corrected positions, S’, M’, E’, are obtained. The whole error-compensation trajectory should pass through the three corrected positions and should satisfy the following equation: (x’, - x’rJ2 + (x’2 - x’2J2 = (I”‘)2

= 6 .-5 E zI

2 0 -2 -4 -6

2. p

-8 -10

$

-12 -14 -16

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

l Error without compensation

2 zs ii ..E .e z

0 -2 -4 -6

(26)

k

0 Error without compensation

&I Error with compensation

(a) Errors along Zaxis Figure 10 Position Errors Along a Line With and Without Error Compensation

X’l = X’l, + Y’ cos [cl’, + k(w, - WJ]

x'3 = X'ls + k(x'3, - x'3J

;m

rl] .[i;;;zJ

1

8 Error with compensation

x‘~ = x’2c + Y’sin [O’, + k(0’, - OIs)] _.::_[;;

0.9

(b) Errors along Y axis

where x’re, x’2care the x1 and x2direction coordinates of the error-compensation trajectory center axis, respectively, and r’ is the radius of projection of the error-compensation trajectory on the XIX2 plane. For points S’, M’, E’, this can be rewritten in a matrix form as:

I”,;;

0.8

k

(27)

k E P,ll (28)

where 0 Is, 8’, are the angles between C’jS’r, C’jE’j and the X, axis, respectively.

By solving the above equation, the coordinate xic, x’2e and radius Y’ of the error compensation can be found. So the error-compensation trajectory can be established and parameterized as

Because the variation of quasi-static errors is nonlinear in the workspace, one must now check if the actual positions of intermediate points on the original trajectory lie within the predefined tolerance

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Journal of Manufacturing Systems Vol. 16/No. 5 1997

>>Helical arc X500.027 X448.517 X294.295 Xt54.419 X100.006

C

Error-compensated trajectory ~

Originaltrajectory

Y10.005 Y154.357 Y220.175 Y157.497 Y30.002

Z50.014; Z75.012 Z102.503 Z127.499 Zt49.997

R19.998; Rt9.989; R19.993; R20.004;

Figure 12 Command Trajectory Segments of Helical Arc Compensation

E,

;

I/

nal helical arc has to be broken up into two parts. The broken position is also determined by the position at which the square of the residual errors has its maximum. On each part, the error compensation scheme established above is applied interactively until the square of the magnitude of the residual errors is less than or equal to the predefined tolerance value. Simulations have been conducted to observe the working of this compensation scheme. Suppose a helical interpolation starts at [500.0 10.0 50.0] and ends at [100.0 30.0 150.0], with a center axis whose coordinates are 30.0 and 20.0 in the X and Y directions, respectively. The arc interpolation of the helical arc is performed on the X-Y plane. The predefined tolerance value is 1 ~m. Using the above compensation scheme, the original helical arc had to be broken up into four parts. Figure 12 lists the command trajectory segments of the helical arc compensation. Figure 13 shows the errors along the X,, Y, and Z axes with and without error compensation. Once again, the residual errors are well within the predefined tolerance value from the desired trajectory.

/

I I

oforiginal Vaject(xy

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(a)Ahelical arc

.,,..

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(b)ProjectionofhelicalarconX~X=plane Figure 11 Error Compensation Along Helical Interpolation

Compensation Results and Evaluation To evaluate the compensation schemes, typical motion trajectories have been checked with and without error compensation. The chosen trajectories are three lines along X, Y, and Z axes, respectively, as shown in Figure 14, and the predefined tolerance values are set as 5 Ixm and 1 txm, respectively. Figure 15 shows the quasi-static errors along these three lines before compensation. Using the compensation scheme for a line, the decomposition of these trajectories is given as the list of coordinates in Figure 16. The simulation residual errors and the actual residual errors measured by the interferometer measurement system are shown in Figure 17 and Figure 18 when the predefined tolerance values are 5 Ixm and 1 ~m, respectively. Because the measurements by the interferometer measurement

value from the original trajectory. For every k in the range [0,1], the residual errors that remain uncompensated can be expressed by the following: e = [(X'l + el(X') X1]2 + [(X'2 + e2(X') - x2]2 + [(x'3 + e3(X') -x3] 2 (29) -

where X" = [x'l, x'2, x'3], the error-compensated position: el(X'), e2(X'), and e3(X') are quasi-static errors at the error-compensated position X" in the X1, X2, and X3 axes, respectively. Similar to the compensation scheme for a straight line, if the square of the magnitudes of the residual errors cannot be maintained less than or equal to the square of the predefined tolerance value, the origi-

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16/No.

5

1997

.~.. ............. r . . . . . . .

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machine tool modeling, the errors of measurements, and the existence of thermal effects, the actual trajectory is not the same as the simulation trajectory. Further, in some part of the line the actual residual errors have gone beyond the predefined tolerance. However, it is observable that a very significant improvement in the accuracy of the CNC machine tool has been achieved. When the predefined tolerance is 5 mm, the maximum actual linear error of line 1 is reduced from 30.2 Ixm to 4.4 ixm, of line 2 from 5.1 txm to 0.9 ixm, and of line 3 from 4.6 Ixm to 3.0 ixm. Similarly, when the predefined tolerance is 1 Ixm, the maximum of line 1 is reduced to 2.5 Ixm, of line 2 to

0 Error with compensation

(a) Errors along Zaxis

Figure 13 Position

Errors

Along

Helical

Error

Compensation

Arc

With

and

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system can be made only in a single direction, only the linear errors for the three trajectories are measured. It is observable from these figures that the simulation trajectories followed by the tool when commanded to move along the compensated command trajectories were well within the predefined tolerance values from the desired trajectory. Due to the limitations of the

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Journal of Manufacturing Systems Vol. 1 6 / N o . 5 1997

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ric accuracy in its workspace. By the model, the quasi-static errors in the workspace can be calculated based on the measurement o f error components. Then the quasi-static error map o f CNC machine tools can be constructed. 2. Error prediction at any point in the workspace o f CNC machine tools is essential for developing a successful compensation scheme. For this purpose, the meshing concepts developed in the FEM literature were used to subdivide the workspace into smaller three-dimensional elements. By introducing a suitable interpolation scheme, the errors at an arbitrary position in any element can be estimated. 3. An error compensation scheme has been presented based on the availability o f an adequate and accurate model o f quasi-static errors. At

Quasi-Static Errors Before Compensation

1.2 txm, and o f line 3 to 1.6 txm. It can be seen that when the predefined tolerance value is set as a smaller value, the actual linear errors can be controlled within a smaller range. Further, the improvement o f the accuracy o f CNC machine tools is more obvious, while the nonlinearity o f the quasi-static error variation is more serious.

Conclusions 1. A general quasi-static error model for multiaxis CNC machine tools was developed using rigid b o d y kinematics. This model compiled the effects o f inaccuracies in the geometry and motion o f the machine members on its volumet-

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Vol. 16/No. 5 1997

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(c) Errors along Line 3

Figure 17 Simulation a n d Actual Errors After C o m p e n s a t i o n ( w h e n A = 5

Figure 18

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Simulation and Actual Errors After C o m p e n s a t i o n (when A = 1 ~ m )

first, for compensating errors when locating the tool at a point, a recursive scheme o f compensation has been addressed. Then, taking into account the nonlinearity o f quasi-static errors, computational approaches are developed for compensating errors o f three basic motions o f a CNC machine (rapid positioning motion, linear interpolation motion, and helical arc interpolation motion). Finally, the experimental results on

a CNC machine tool show that a 80-90% reduction o f quasi-static errors has been gained after compensation.

Acknowledgment This paper is based on an interim report o f Research Project No. 9040090 funded by UGC. The authors are grateful for the support received from the City University o f Hong Kong.

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17. J. Mou and C.R. Liu, "A Method for Enhancing the Accuracy of CNC Machine Tools for On-Machine Inspection" Journal of Manufacturing Systems (vl 1, n4, 1992), pp229-237. 18. J. Mou and C.R. Liu, "An Adaptive Methodology for Machine Tool Error Correction" ASME Transactions Journal of Engineeringfor Industry (v117, 1995), pp389-399. 19. J. Mou, M.A. Donmez, and S. Cetinkunt, "An Adaptive Error Correction Method Using Feature-Based Analysis Techniques for Machine Performance Improvement, Part 1: Theory Derivation," ASME Transactions Journal of Engineeringfor Industry (v117, 1995), pp584-590. 20. J. Mou, M.A. Donmez, and S. Cetinkunt, "An Adaptive Error Correction Method Using Feature-Based Analysis Techniques for Machine Performance Improvement, Part 2: Experimental Verification," ASME Transactions Journalof Engineeringfor Industry (v I 17, 1995), pp591-600. 21. E Dufour and R. Groppetti, "Computer Aided Accuracy Improvement in Large NC Machine-Tools," MTDR Conference Proceedings 22 (1981), pp611-618. 22. A. Geddam, and X.B. Chen, "Modeling, Prediction and Compensation of Quasistatic errors in Multi-Axis CNC Machining Centers" 3rd CIRP Workshop on Design & Implementation of intelligent Manufacturing Systems (v3, 1996), pp193-198. 23. C.H. Chang and M.A. Melkanoff, NC Machine Programming and Software Design (Englewood Cliffs, NJ: Prentice-Hall, 1989). 24. P. Sheth and J.Jr. Uicker, "IMP (Integrated Mechanisms Program), A Computer-Aided Design Analysis System for Mechanisms and Linkages;' ASME Transactions Journal of Engineering for Industry (v94, 1972), pp454-464. 25. P. Paul, Robot Manipulators: Mathematics, Programming, and Control (Cambridge, MA: MIT Press, 1982). 26. EL. Stasa, Applied Finite Element Analysis for Engineers (CBS Publishing Japan Ltd., 1985). 27. K.H. Huebner, E.A. Thornton, and T.G. Byrom, The Finite Element Methodfor Engineers (New York: John Wiley & Sons, 1995).

References 1. J. Hocken andthe Machine-ToolTask Force, Teclmology of MachineTools (v5, I980), Machine-Tool Accuracy, UCRL-52960-5 (Lawrence Livermore Laboratory, University of California). 2. M.A. Donmez, D.S. Blomquist, R.J. Hocken, C.R. Liu, and M.M. Barash, "A General Methodology for Machine Tool Accuracy Enhancement by Error Compensation" Precision Engineering (v8, n4, Oct. 1986), pp187-196. 3. V. Kil"idena and P.M. Ferreira, "Mapping the Effects of Positioning Errors on the Volumetric Accuracy of Five-Axis CNC Machine Tools" International Journal of Machine Tools & Manufacture (v33, n3, 1993), pp 417-437. 4. D. Lin and K.E Ehmann, "Direct Volumetric Error Evaluation for Multi-Axis Machines," International Journal of Machine Tools & Manufacture (v33, n5, 1993), pp675-693. 5. G. Zhang, R. Veale, T. Charlton, B. Borchardt, and R. Hocken, "Error Compensation of Coordinate Measuring Machines," Annals of CIRP (v34, 1985), pp445-448. 6. Ahmad K. Elshennawy and Inyong Ham, "Performance Improvement in Coordinate Measuring Machines by Error Compensation," Journal of Manufacturing Systems (v9, n2, 1990), pp151-158. 7. P.M. Ferreira and C.R. Liu, "A Method for Estimating and Compensating Quasistatic Errors of Machine Tools" ASME Transactions Journal of Engineeringfor Industry (v115, 1993), pp149-159. 8. S.B. Kiridena and P.M. Ferreira, "Kinematic Modeling of Quasistatic Errors of Three-Axis Machining Centers" International Journal of Machine Tools & Manufacture (v34, nl, 1994), pp85-100. 9. S.B. Kiridena and P.M. Ferreira, "Parameter Estimation and Model Verification of First Order Quasistatic Error Model for Three-Axis Machining Centers," International Journal of Machine Tools & Manufacture (v34, nl, 1994), pp101-125. 10. S.B. Kiridena and P.M. Ferreira, "Computational Approaches to Compensating Quasistatie Errors of Three-Axis Machining Centers," International Journal of Machine Tools & Manufacture (v34, hi, 1994), ppi27-145. 11. A.K. Srivastava, S.C. Veldhuis, and M.A. Elbestawit, "Modelling Geometric and Thermal Errors in a Five-Axis CNC Machine Tool," International Journal of Machine Tools & Manufacture (v35, n9, 1994), pp1321-1337. 12. G. Zhang, R. Ouyang, and B. Lu, "A Displacement Method for Machine Geometry Calibration" Annals of CIRP (v37, 1988), pp515-518. 13. V.B. Kreng, C.R. Liu, and C.N. Chu, "A Kinematic Model for Machine Tool Accuracy Characterization," International Journal of Advanced Manufacturing Technology (n9, 1994), pp79-86. 14. YungC. Shin, Henry Chin, and Michael J. Brink, "Characterization of CNC Machining Centers," Journal of Manufacturing Systems (vl0, n5, 1991), pp407-421. 15. N.A. Duffie and S.M. Yang, "Generation of Parametric Kinematic Error-Correction Functions from Volumetric Error Measurements," Annals of CIRP (v34, 1985), pp435-438. 16. J. Mou and C.R. Liu, "An Error Correction Method for CNC Machine Tools Using Reference Parts," Transactions of the North American Manufacturing Research Institution of SME (v12, 1994), pp275-282.

Authors' Biographies X.B. Chen is a research assistant in the Department of Manufacturing Engineering at the City University of Hong Kong. He received his MS in mechanical engineering and PhD in mechanical manufacturing from Harbin Institute of Technology (China) in 1991 and 1996, respectively. His research interests are in the areas of precision engineering, intelligent sensing of manufacturing processes, compensation control of manufacturing systems, and robot accuracy. A. Geddam is an associate professor in the Department of Manufacturing Engineering at the City University of Hong Kong. His research interests are in precision engineering, machine tools, and machine dynamics and control. He received his PbD in machine tools and manufacturing engineering from the University of Manchester Institute of Science and Technology in 1971. Z.J. Yuan is a professor in the Department of Manufacturing Engineering at Harbin Institute of Technology (China). His research interests are in precision and ultraprecision engineering and automation of manufacturing processes. Professor Yuan is a active member of CIRP.

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