A new technique for volumetric error assessment of CNC machine tools incorporating ball bar measurement and 3D volumetric error model

~ Pergamon

Int. J. Mach. Tools Manufact. Vol. 37, No. I1, pp. 1583-1596, 1997 © 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0890-6955/97517.00 + .00

PII: S0890---6955(97)00029-1

A NEW TECHNIQUE FOR VOLUMETRIC ERROR ASSESSMENT OF CNC MACHINE TOOLS INCORPORATING BALL BAR MEASUREMENT

A N D 3D V O L U M E T R I C E R R O R M O D E L

HEUI JAE PAHK,?:I: YOUNG SAM KIMt and JOON HEE MOON'i" (Received 9 September 1996; in final form 13 March 1997) Abstract--This paper presents a useful technique for assessing the volumetric errors in multiaxis machine tools using a kinematic double ball bar. This system has been developed based on a volumetric error model which describes the three-diraensional errors of machine tools. The developed system requires input of the measured radial data performed on the three orthogonal planes, and analyzes parametric errors such as positional, straightness, angular, squareness, and backlash errors. The developed system can also assess the dynamic performance of machine tools such as errors due to the servo gain mismatch. The developed system has been tested using an actual machine tool, showing high potential for error assessment of multiaxis machine tools. © 1997 Elsevier Science Ltd

1. INTRODUCTION

Development of efficient techniques for performance verification of machine tools has been considered as an important task for accuracy enhancement and quality assurance for users and manufacturers of machine tools and coordinate measuring machines. In order to assess the accuracy of a machine tool, the double ball bar or kinematic ball bar has been frequently used. Bryan [1] proposed a ball bar consisting of precision balls and LVDT for checking the accuracy of machine tools, and Knapp [2] proposed a similar method using a two-dimensional probe and master disc. Kunzmann et al. [3] used the kinematic bail bar for the parametric error calibration of machine tools, and Kakino et al. [4] demonstrated the relationship between the ball bar measurement and the various parametric errors for machine tools. Burdekin and Park [5] proposed a similar type of ball bar using two LVDTs. This paper presents a new technique for analyzing the three-dimensionai parametric errors of CNC machine tools, in the volumetric sense, using ball bar measurement. This system has been developed based on a volumetric error model which equates the threedimensional errors of machine tools. The parametric errors such as positional, straightness, angular, and backlash errors are modeled as polynomial functions of the position along each axis. The squareness error and the errors due to servo gain mismatch are also modeled between the neighboring axes. Then the three-dimensional volumetric error model is constructed accordinl, to the kinematic chain of the considered CNC machine tools. The constructed volumetric error model is then applied to the ball bar measurement data. The developed system inputs the measured radial data from the ball bar measurement in three orthogonal planes, such as XY, YZ, and ZX. The least-squares technique is used to evaluate the coefficients of the respective parametric errors, and thus the parametric errors can be successfully analyzed. The developed system has been applied to practical CNC machine tools, and the vo]lumetric error assessment has been efficiently performed. 2. ERROR MEASUREMENT USING THE KINEMATIC BALL BAR

There are several types of commercially available ball bars (e.g. see Ref. [6]). A ball bar usually consists of a kinematic artifact linking two precision bails, where precision

tDepartment of Mechanical Design and Production Engineering, Seoui National University, Seoul, South Korea. :~Author to whom correspondence should be addressed. 1583

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LVDTs are located between the two balls. Fig. 1 shows a typical set-up of a commercially available ball bar, where one ball is fixed on the table of a machine tool and the other is attached to the spindle of the machine tool. In Fig. 1, let O(0,0,0) be the center point of the ball on the table, and P(X,Y,Z) be the nominal coordinates of the ball center attached to the spindle. When the machine is commanded to move to P(X,Y,Z) position, the actual position of the machine is assumed to be P ' ( X ' , Y ' , Z ' ) . Thus, the machine geometric error can be defined as the difference between the two coordinates. That is AX=X' - X AY=Y'-

Y

(1)

A Z = Z' - Z

where AX, Ay, AZ are the X, Y, Z error components at the nominal P(X,Y,Z) position with respect to the O(0,0,0) point. When the error components AX, AY, AZ are present, the error in the distance between the two points, AR, can be evaluated as (R + AR) 2 - X '2 + y,2 + Z,2

=

(X + ~tX)2 + (Y + AY)2 + (Z + AZ) 2

(2)

where R is the nominal distance between the two points, O and P. When ignoring the second order terms of error components and remembering that R 2 = X 2 + y2 + Z 2, Eqn (2) gives

(3)

AR = (XAX + YAY + ZAZ)/R

Eqn (3) gives the error in the length direction of the ball bar when the error components (&X, AY,AZ) are present during the machine movement from O to P, and can be used for error diagnosis of the machine tool. Spindle

(X,Y,Z)

Ball Bar

(o,o,o)

Fig. 1. Error measurement using the kinematic ball bar.

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3. PARAMETRICERROR MODELING FOR MACHINE TOOLS

It is widely acknowledged that there are six geometric error components when a machine element moves along a guideway, as in Fig. 2. They are three translational errors (positional error, horizontal/vertical straightness errors) and three rotational errors (pitch, yaw, and roll errors). When multiaxis movement is considered, each axis has the above six geometric error components, and three squareness errors are present between the XY axis, Y-Z axis, and Z - X axis. There are also dynamic errors due to servo gain mismatch between the X-Y axis, Y-Z axis, and Z - X axis. In this paper, polynomial based error modeling has been performed for the above geometric/parametric error components of machine tools, prior to the error analysis. 3.1. Positional error components

Positional error components, ~x(X), By(Y), &(Z), along the X, Y, Z axes, which are mainly due to scale errors and lead screw pitch errors, are usually defined as the difference between actual coordinates and nominal coordinates along the X, Y, Z axes. Thus it is possible to model the positional error as a polynomial function of position along each axis. Therefore, the positional error components are N

(4)

~x(X) = dXxl(X[R ) + dxx2(X/R) 2 + ... = E d3fxi(X/R) i

i=1 N

(5)

~y(Y) = dyy,(YIR) + dyy2(YIR) 2 + ... = E dyyi(¥1R)' i=l

N

az(Z) = azz,(Zm) + azz2(Z/R) ~ + . . . . ~ azz,(Z/R)'

(6)

i=l

where X/R, Y/R, ZIR are the dimensionless coordinates of the position along each axis, and dxxi, dyy, d:x~ are the coefficients of the polynomial. 3.2. Straightness error components Straightness error is mainly due to non-straightness of a guideway and/or due to the beating interfaces in machine tools. As the straightness error is defined as the perpendicular deviation along ,each axis, it can be modeled as a second order polynomial function of position, while the first order polynomial function is considered for modeling the Vertical Straightness Error

~

Z

x.~ ~ L ~ ~

~'l

~ Y

~ ~ J '

Horizo,t,l ~

Straighmess

Error

Translatory Machine axis

Fig. 2. Six DOF parametric errors along a machine axis.

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Heui Jae Pahk et al.

squareness error between the neighboring axis. The squareness term will be discussed in a later section. N

6y(X) = dyx2(XIR) 2 + d y x 3 ( X [ g ) 3 + . . . .

Z dyxi(X/R) i

(7)

i=2

N

8z(X) = dZx2(X/R) 2 + dZx3(X/R) 3 + . . . .

Z dzxi(X/R) i

(8)

i=2

N

8x(Y) = dxy2(Y/R) 2 + dxy3(Y/R) 3 + . . . .

Z dxYi(Y/R)i

(9)

i=2

N

~ . dzyi(YIR) i

¢$z(Y) = dzy2(Y/R) 2 + dzy3(Y/R) 3 + . . . .

(lO)

i=2

N

6x(Z) = dxz2(Z/R) 2 + dxz3(Z/R) 3 +

dxz~(ZIR) ~

(11)

¢~y(Z) = dyz2(ZlR) 2 + Hyz3(ZIR) 3 +... = Z dyzi(Z/R)i

(12)

...

=

i=2

N

i=2

where the 8y(X), 8z(X); 8x(Y), 8z(Y); 8x(Z), By(Z) are the straightness error components along the X, Y, and Z axes, respectively. Constants dyxi, dzxi, dxyi, dzyi, dxzi, dyz~ are the coefficients of the polynomial function, which is to be determined. 3.3. Angular error components 3.3.1. Pitch and y a w error. Pitch and yaw errors are the angular error components in the perpendicular direction along each axis; they are influenced by guideway geometry and beating interfaces of machine tools, and thus can be related to the straightness profile along each axis. For example, the pitch error, Ex(Y), along the Y axis can be defined as the derivative of the vertical straightness error along the Y axis: N

Ex( Y) = ~( Sz( Y) )/~y = ~. idzy,(Y/R)' - t/R

(13)

i=2

The yaw error component, Ez(Y), along the Y axis can be derived as the derivative of the horizontal straightness error along the Y axis, that is N

Ez(Y) = -- ~(~x(Y))/~y = -- Z idA'yi(Y/R)i- I/R

(14)

i=2

where the minus sign is considered for the sign convention of the angular error components. Similarly, the rest of the pitch and yaw errors are modeled as the derivatives of the respective straightness error profiles, that is N

Ey(X) = - O(Sz(X))lOx = -

~'. idzx,(XIR)'- llR i=2

(15)

Volumetricerror assessmentin machinetools

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N

Ez(X) = a(&¢(X))/~x = E idyxi(X/R) i - n[R

(16)

i=2

N

Ex(Z) = - ~)(6y(Z))l~z = -

~ idyzi(ZIRy - '/R

(17)

i=2

N

Ey(Z) = 3(&:(Z))IOZ = ~ idxzi(ZlR) i - 'IR

(18)

i=2

3.3.2. Roll error. The roll error components, Ex(X), Ey(Y), Ez(Z), are defined as the angular error in the axial direction. Thus the roll error components can be modeled as the polynomial function of the position along each axis: N

Ex(X) = E exxi(XIR) i

(19)

i=l

N

Ey(Y) = E eyyi( Y[R)i

(20)

i=l

N

Ez(Z) = ~ , ezzi(Z/R) i

(21)

i=l

where Ex(X), Ey(Y), Ez(Z) are the roll error components along the X, Y and Z axes, and exxi, eyyi, ezzi are the coefficients for the polynomial model. 3.4. Backlash error components Backlash error is mainly caused by backlash in the screw/gear assembly during the motion reversal. When the amount of backlash in the X axis is Bx, the error in the X axis due to the backlash, AX, can be modeled as follows: A X = - Bxl2sign( dXIdt)

(22)

where sign( ) is the sign function returning the sign of the term inside the bracket, and dX/dt is the time derivative of the position, which is velocity. The minus sign in Eqn (22) is to give the positive backlash when the motion is changed from the forward direction to the reverse direction. Similarly, the backlash errors in the Y and Z axes can be modeled as A Y = - By/2sign(dYIdt)

(23)

A Z = -- Bz/2sign( dZ/dt)

(24)

where By and BZ are the amount of backlash in the Y and Z axes, respectively. 3.5. Squareness error The squareness error is defined as the out-of-squareness between two nominally orthogonal axes, and is mainly due to misalignment, or misassembly, in the orthogonal axes. Let a be the amount of non-squareness of the X axis from the nominal X axis (in the

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Heui Jae Pahk et al.

plane) at a position corresponding to a distance R from the origin point as in Fig. 3. Therefore the Y positional error at X position, Ay, due to the non-squareness of the X axis is as follows:

XY

Ay = -

(25)

aX/R

Similarly, when/31 is the amount of non-squareness error of the Z axis from the nominal Z axis at the R location (in the X Z plane), as in Fig. 3, the positional error at the Z position, AX, is derived as ~;

= -

(26)

/31Z/R

Similarly, when/32 is the amount of non-squareness error of the Z axis from the nominal Z axis at the R location in the Y Z plane, the Y positional error, Ay, is derived as A y = - /32Z/R

(27)

3.6. E r r o r s d u e to s e r v o g a i n m i s m a t c h When the amplifier gains of the servo drives for the axis motion are not properly matched, there exists a steady state following error. If K s x a n d V x are the servo gain and the velocity of the X axis, the steady state following error, AX, between the actual position and the command position in the X axis can be derived from control theory (e.g. Ref. [7]): AX =-

(28)

Vx/Ksx

Similarly, the steady state following error, AY, AZ, in the Y, Z axes can be obtained as Ay =-

Vy/Ksy

(29)

AZ =-

Vz/Ksz

(30)

where Vy, K s y ; Vz, K s z are the velocity and servo gain in the Y, Z axes, respectively. Thus when F is the circumferential feed velocity, V x a n d V y can be obtained as Vx = - F Vx = F

sin 0 and V y = F cos 0 for counterclockwise rotation

(31)

sin 0 and V y = - F cos 0 for clockwise rotation

(32)

where 0 is the angular position along the circular motion. Applying Eqns (31) and (32) to Eqns (29) and (30) then substituting into Eqn (3) gives Y

"~

R

Z

~

d

L~~aX

~X

Z

~y

X' (a)

(b) Fig. 3. Squarenesserrors in three orthogonalplanes.

(c)

Volumetric error assessment in machine tools AR

=

1589

F s!in O(XIR)/Ksx + F cos O(YIR)IKsy

-

= - F sin 0 cos OIKsx + F cos 0 sin OIKsy =

(

-

FKsy + FKsx)/(KsxKsy)cos 0 sin 0

= eFIK cos 0 sin 0 = eF/K(X/R)(Y/R) = Mxy(X/R)(Y/R) for CCW rotation

(33)

where K = ~(KsxKsy), e = (Ksx - Ksy)/K, with Mxy( = eFIK) the coefficient for the gain mismatch between the X - Y axis. A similar equation can be derived for the clockwise rotation: AR = - May(X/R)(Y/R) for CW rotation

(34)

Eqns (33) and (34) represent the circular error which is influenced by the gain mismatch. Similarly, the circular errors due to the gain mismatch between the Y - Z axis and the Z X axis are AR = Myz(Y/R)(ZIR) for CCW rotation = - Myz(Y/R)(ZIR) for CW rotation

(35)

and AR = Mzx(ZdR)(X/R) for CCW rotation = - Mz.x(ZIR)(X/R) for CW rotation

(36)

where Myz, M z x are the coefficients for gain mismatch between the Y - Z axis and Z - X axis, respectively. 4. VOLUMETRIC'.ERROR MODEL AND THE BALL BAR MEASUREMENTOF MACHINETOOLS 4.1. Volumetric ,error equations

Three-dimensional error behavior, that is volumetric error consideration, is vital for error assessment of machine tools in 3D working space. In this paper, the influence of the volumetric error model on the ball bar measurement has been assessed, and an appropriate error diagnosis has also been performed for practical machine tools. Volumetric error equations can generally be derived from the kinematic chain of each machine element when the parametric error components are calibrated (e.g. Ref. [8]), and are dependent on the kinematic configuration of the machine tool. In this paper, two practical machine tools, column type machining center and horizontal machining center, are considered, as they are readily accessible at the Seoul National University. Fig. 4 shows ~the coordinate axis configuration for the two machine tools. For the column type machining center, the volumetric error equations [8] are z2kX-- 8x(X) - &¢(I0 + 8x(Z) + Z[ - Ey(Y) - ~1 - Ey(X)] + Y[Ez(IO + Ez(X)] + rp[Ez(X) + Ez(IO - Ez(Z)] + AY = -

8y(X) + ~y(Y) + ~y(Z)

+ X[ - EZ(X) + a] + Z(Ex(X) + Ex(Y) - /32]

Zp[

-

Ey(X)

-

Ey(l') + Ey(Z)]

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Heui Jae Pahk

et

al.

(a) Column Type Machining Center

(b) Horizontal Machining Center Fig. 4. Coordinate configuration for the machine tools considered.

+ Xp[ - E z ( X ) - E z ( Y ) + Ez(Z)] + Z p [ E x ( X ) + E x ( Y ) - Ex(Z)] AZ = -

8z(X) -

8z(Y) + 8 z ( Z ) + X E y ( X )

+ Y[ - E x ( X ) - Ex(Y)] + X p [ E y ( X ) + E y ( Y ) - Ey(Z)] + Yp[ - E x ( X ) - E x ( Y ) + Ex(Z)]

For the horizontal machining center, the volumetric error equations [8] are At" = 8x(X) + 8x(Y) + a x ( Z ) + Y[Ez(X)

(37)

Volumetric error assessment in machine tools

- Ez(Z)] -

1591

Z[Ey(X)

+/31] + Y p [ E z ( X ) - E z ( Y ) - Ez(LO] + Z p [ - E y ( X ) + Ey(r3 + Ey(Z)] AY = -

S y ( X ) + By(Y)

(38)

+ By(Z) - X [ E z ( X ) + or] + Z [ E x ( X l -

1321+ X p [ - E z ( X ) + E z ( Y )

+ Ez(Z)] + Z p [ E x ( X ) - E x ( Y ) - E x ( Z ) ] AZ

=

~z(X) + 8 z ( Y ) + ~z(Z) + X E y ( X )

-

+ Y[ - E x ( X ) + E x ( Z ) ] + X p [ E y ( X ) - E y ( Y ) - E y ( Z ) ] + Yp[ - E x ( X ) + E x ( Y ) + E x ( Z ) ]

where X, Y, Z are the nominal coordinates from the origin, and Xp, Yp, Zp are the coordinates of tool offset. 4.2. Integration o f the volumetric error equations and the ball bar m e a s u r e m e n t The influence of machine tool volumetric error behavior on the circular error pattern of the ball bar measurement can be considered when Eqns (37) and (38) are applied to the circular error equation (3). Thus, when the volumetric error behavior of the column type machining center is considered, the circular error of ball bar measurement is A R = ( X A X + YA Y + Z A Z ) / R = AX(X/R) + AY(Y/R) + AZ(Z/R) = d ~ I ( X / R ) 2 + dyyl(Y[R) 2 + dzzt(Z/R) z

-- 3dxyl(X/R)(Y/R) 2 + dxz2(X/R)(Z/R) 2 - dyx2(l(IR)(XIR) 2 + dyz2(YIR)(ZIR) 2 + dZX2(JYR)(X/R) 2 + dzy2(ZIR)(Y/R) 2 Bx/26¥/R)sign(dX/dt) - Byl2(Y/R)sign(dY/dt)

-

(39)

Bzl2(ZIR)sign(dZ/dt) - [31(ZIR)(XIR) + a(XIR)(YIR)

-

[32(Y/R)(Z/R)

-

+ Mxy(X/R)(Y/R)Dir + Myz(Y/R)(Z/R)Dir + Mzx(Z/R)(X/R)Dir

where the first terms of polynomial functions are considered for each parametric error component for the efficiency of calculation, and D i r is a function returning plus (clockwise) or minus (counterclockwise) depending on the rotation of the kinematic ball bar. The roll error components and the effect of tool offsets are not considered here for simplicity of modeling. Similarly, for the horizontal machining center, the circular error of ball bar measurement is AR =

d x x l ( X / R ) 2 + dyyl(Y/R) 2 + dzzl(Z/R) 2 + dxy2(X/R')(Y/R) 2 + dxz2(X]R)(Z/R) 2 - dyx2(Y]R)(X/R) 2 - dyz2(Y/R)(Z/R) 2 - dZx2(Z/R)(X/R) 2 + dZyE(Z/R)(Y/R) 2 - Bx/2(X/R)sign(dX/dt) By/2(Y/R)sign(dY/dt) - Bz/2(ZJR)sign(dZ/dt) -

-

[31(7_IR)(X/R) -

a(X/R)(Y/R)

-

[32(Y/R)(Z/R) + Mxy(X/R)(Y/R)Dir

+ Myz(Y/R)(ZIR)Dir + Mzx(Z/R)(X/R)Dir

(40)

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Heui Jae Pahk et al.

4.3. Least-squares technique for the error diagnosis When the ball bar measurement data, AR, are provided, there are several error components involved, as in Eqns (39) and (40). In order to analyze the error components from the ball bar measurement data, the least-squares technique has been effectively applied in this paper. When ARm is the circular error to be modeled, Eqns (39) and (40) can be expressed as a linear combination of polynomial functions: ARm = ZAiFi

(41 )

where Fi is the ith function of error modeling, and Ai is the coefficient of the ith function. Let E be the sum of squares of the deviation between AR and ARm, then E = Z(AR - ARm) 2

(42)

Applying the variational principle to Eqn (42) to minimize E: 3E/OAi = 2Z(ARm - AR)b(ARm)/3Ai = 0

(43)

From Eqn (41), O(ARm)/~Ai = Fi, thus Eqn (43) becomes ZARmF~ = EARF~

(44)

If we set f = [FI,F2..... Fn] r, a = [Alu42,...,A,] v, then Eqn (44) becomes (ffV)a = (ARf)

(45)

That is Ca = d

(46)

where C is the square matrix (ffr), and d is the column vector (ARf). Eqn (46) is a typical linear equation for an unknown matrix, and therefore can be solved by applying numerical methods such as Gauss elimination, etc. 5. PRACTICAL APPLICATION

The developed error analysis technique has been practically applied to a column type machining center and a horizontal machining center, which are installed on the shop floor in the Seoul National University. A commercially available kinematic ball bar of 150 mm nominal length has been used for the circular error measurement, and Fig. 5 shows a typical set-up for the ball bar measurement for the two cases of machine tools. The ball bar has been precalibrated with respect to a calibration fixture made of Zerodur, prior to performing measurement. After the measurement path has been planned, the appropriate CNC code has been generated for circular contouring, and downloaded to the CNC machine tools. The feed rate was 500 mm/min for both applications. The measured data are then analyzed and error diagnosis performed. 5.1. Column type machining center Fig. 6 shows the raw data plot for the circular error measurement in XY, YX, and ZX planes, respectively, where the dotted line indicates a counterclockwise contour and the solid line indicates a clockwise contour. The raw data were analyzed by the developed system and the error components were evaluated as follows: X positional error = 8.4/xm

Volumetric error assessment in machine tools

(a)

Column Type Machining Center

(a) Horizontal Machining Center Fig. 5. Set-up for the ball bar measurement. Y positional error = Z positional error = Y straightness along Z straightness along X straightness alcmg Z straightness along

13.4/xm 11.0/xm X axis = X axis = Y axis = Y axis =

2.5/zm 2.7/~m 2.6 ~ m - 0.6 ~ m

1593

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Heui Jae Pahk et al. ~IFZ--

-

't

•

':

°

i

" ~ ~t.

YZ PLANE

XY PLANE CCW CW

\

///

SCALE 15 [~tm]

;"

J. ÷

:'~/

Z X PLANE

PLOTTED BY S.N.U METROLOGY LAB.

Fig. 6. Raw data plot for the ball bar measurement (column type machining center).

X straightness along Z axis = 1.5/zm Y straightness along Z axis = 0.5/zm Y angular error along X axis = - 36.8/.~rad Z angular error along X axis = 33.6 ~rad X angular error along Y axis = - 9.1/zrad Z angular error along Y axis = - 34.8/zrad X angular error along Z axis = - 7.8/xrad Y angular error along Z axis = 20.3/zrad squareness error between X Y axis = 20.9/zrad squareness error between YZ axis = 133/zrad squareness error between ZX axis = - 88.9/zrad backlash error in X axis = 3.8/zm backlash error in Y axis = 2,7 tzm backlash error in Z axis = 3.2/zm error due to servo gain mismatch between X Y axis = 8.1/zm error due to servo gain mismatch between YZ axis = - 2.0/zm error due to servo gain mismatch between ZX axis = 3.7/zm. The analyzed error components were removed from the raw data of the circular measurement, and the residual circular errors were calculated then plotted. Fig. 7 shows the residual circular errors after removing the error components, giving remarkably reduced error pattem. -rZ

-rY

-rZ .-°.

CCW CW

i

1

±

XY PLANE

YZ PLANE

ZX PLANE

SCALE 15 [lam]

PLOTTED BY S.N.U METROLOGY LAB.

Fig. 7. Residual error plot after analyzing en'or components (column type machining center).

Volumetric error assessment in machine tools

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5.2. Horizontal machining center Fig. 8 shows the raw data plot for the circular error measurement in X¥, YX, and ZX planes, respectively, where the dotted line indicates a counterclockwise contour and the solid line indicates a clockwise contour. The raw data were analyzed by the developed system and the error components were evaluated as follows: X positional error = 14.1/xm Y positional error = 21.7/~m Z positional error = 19.2/.~m Y straightness along X axis = - 2 . 2 / ~ m Z straightness along X axis = - 0 . 7 p , m X straightness along Y axis = - 5 . 7 p , m Z straightness along Y axis = - 7 . 6 / . L m X straightness along Z axis = 0.3/xm Y straightness along Z axis = - 2.3/xm Y angular error along X axis = - 9.6/~rad Z angular error along X axis = - 29.3/xrad X angular error along Y axis = 101.3 p~rad Z angular error along Y axis = 76/zrad X angular error along Z axis = 30.6/xrad Y angular error along Z axis = 4.5/zrad squareness error between XY axis = 7.8/~rad squareness error between YZ axis = 107.3/~rad squareness error between ZX axis = 147.3/xrad backlash error in X axis = 5.9/.~m backlash error in Y axis = 5.1/~m backlash error in Z axis = 5.8/.~m error due to servo gain mismatch between XY axis = - 6.2/~m error due to servo gain mismatch between YZ axis = 1.5/zm error due to servo gain mismatch between ZX axis = - 0.5 ~m. The analyzed error components were removed from the raw data of the circular measurement, and the residual circular errors were calculated then plotted. Fig. 9 shows the residual circular errors after removing the error components, giving remarkably reduced error pattern. Therefore, the developed error analysis system has been found as an efficient tool for the error diagnosis of machine tools based on the kinematic ball bar measurement method.

-rZ

/

+

"....

"4\ t'-,.

*,-T-

//f'/.t'x

\."...

!

YZ PLANE

XY PLANE CCW CW

".

SCALE 15 [l~ml

i

ZX PLANE

PLOTTED BY $.N.U METROLOGY LAB.

Fig. 8. Raw data plot for the ball bar measurement (horizontal machining center).

Heui Jae Pahk et al.

1596

,v

"wZ

.

-rZ

~..~~. .,."°°°°°°" * °°°° ~.~..[ ÷ T XY PLANE CCW CW

SCALE 15 [~m]

±

±

YZ PLANE

ZX PLANE

PLOTTED BY S.N.U METROLOGY LAB.

i

Fig. 9. Residual error plot after analyzing error components (horizontal machining center), 6. CONCLUSION

I. A computer-aided analysis system has been developed for parametric error components, based on the circular error measurement using the kinematic ball bar. 2. A new approach has been proposed and tested such that the 3D volumetric error model is effectively integrated for the ball bar measurement, thus the 3D volumetric error components have been efficiently assessed with the ball bar measurement data. 3. Polynomial based error modeling has been performed for the parametric error components of machine tools. Then the least-squares technique has been applied for evaluation of the coefficients of the modeling functions. The polynomial based error modeling, combined with the least-squares technique, showed good performance for the error analysis. 4. The developed system has been applied to two practical cases of machine tools, demonstrafing high efficiency for the assessment of the three-dimensional error components in a relatively short time when compared with conventional length and angle measuring equipment.

REFERENCES [1] Bryan, J. A., A simple method for testing measuring machines and machine tools. Precision Engineering, 1982, 4(2), 61-69. [2] Knapp, W., Test of three dimensional uncertainty of machine tools and measuring machines and its relation to machine errors. Annals of the CIRP, 1983, 32(1), 459-464. [3] Kunzmann, H., On testing coordinate measuring machines with kinematic reference standards. Annals of the CIRP, 1983, 32(1), 465-468. [4] Kakino, Y., The measurement of motion errors of NC machine tools and diagnosis of their origins by using telescoping magnetic ball bar method. Annals of the CIRP, 1987, 36(1), 377-380. [5] Burdekin, M. and Park, J., A computer aided system for assessing the contouring accuracy of NC machine tools. Proc. 28th MATADOR Conf., Manchester, 1988. [6] Renishaw Ballbar Diagnostic Manual, Renishaw Co., 1992. [7] Koren, Y., Computer Control of Manufacturing Systems. McGraw-Hill, New York, 1983. [8] Pahk, H., Development of computer software for volumetric error map integrating the machine geometric error and the probe error in pursuit of a numerical virtual CMM. Journal of the KSME, 1996, 20(3), 945-959.