modeling, measurement and error compensation of multi-axis machine tools. part i: theory

International Journal of Machine Tools & Manufacture 40 (2000) 1535–1546

Modeling, measurement and error compensation ofmulti-axis machine tools. Part I: theory

Mahbubur Rahman*, Jouko Heikkala, Kauko LappalainenProduction Technology Laboratory, Department of Mechanical Engineering, University of Oulu, P.O. Box 4200,

FIN-90401 Oulu, Finland

Received 29 January 1999; received in revised form 6 September 1999; accepted 26 October 1999

Abstract

Multi-axis machine tools are important units in modern production systems. The complex structures ofthe machine tools produce an inaccuracy at the tool tip caused by kinematics parameter deviation resultingfrom manufacturing errors, assembly errors or quasi-static errors. To find out the error origins of thesemachine tools it is always necessary to have good mathematical models of the machine tools. These modelscan be used for measuring and analyzing the measured data. The diagnosed errors could be taken underconsideration only by the precise description of the actual kinematics of the machine tools. By countingthe new axes values by numerical iteration technique of the defective axes from the cutter location data(CL-data) or from the ideal NC program it may be possible to position the tool tip to the desired position.This paper describes two ways to modify NC programs (implemented in a postprocessor for CL-data orby extra NC program processor for ideal NC program) including modeling and measurement of realmachine tools. Practice and application will be described in part two. 2000 Elsevier Science Ltd. Allrights reserved.

Keywords:Multi-axis machine tools; Accuracy; Error measurement; Error compensation

1. Introduction

Positioning accuracy of multi-axis machine tool depends on different kinds of error originsdepending on the machine tool’s structure, control system etc. In a typical machine tool, thereare multiple error origins including geometric, static and dynamic loading, thermal deviations,mismatches between servo-loop parameters etc. All these error origins affect positioning accuracyat the tool tip in a complex way.

* Corresponding author. Tel.:+358-8-553-2116; fax:+358-8-553-2026.E-mail address:[email protected] (M. Rahman)

0890-6955/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved.PII: S0890-6955(99)00101-7

1536 M. Rahman et al. / International Journal of Machine Tools & Manufacture 40 (2000) 1535–1546

Nomenclature

DR increase in radius for circular interpolationDY positioning error alongY-axisay angular error aroundX-axis during feed motion inY-axis (pitch)by angular error aroundY-axis during feed motion inY-axis (roll)cy angular error aroundZ-axis during feed motion inY-axis (yaw)Ai transformation matrixCL-data cutter location datacx, cy, cz error vector forX, Y andZ-axisd linear movement along an axisDBB double ball barENPP extra NC program processorkyx, kyy, kyz direction cosine ofY-axisl, m, n direction cosine for any axisNC numerical controlO originPP post processorPw workpiece pointPs tooltip pointR radius for circular interpolationX, Y, Z linear movement alongX, Y andZ axisXreal, Yreal, Zreal real coordinate forX, Y andZ-axis.

Substantial work has been performed in the past on the development of error models for geo-metric, thermal etc. error sources for multi-axis machines. Kim and Kim [4] have developed avolumetric error model based on 4×4 homogenous transformation for generalized geometric error.Eman and Wu [2] have developed error model accounts for error due to inaccuracies in thegeometry and mutual relationships of the machine structural elements as well as error resultingfrom the relative motion between these elements. Kakino et al. [3] have measured positioningerrors of multi-axis machine tools in a volumetric sense by Double Ball Bar (DBB) device. Takeu-chi and Watanabe [6] have shown five-axis control collision free tool path and post processingfor NC-data. In this paper, three-dimensional volumetric error model has been used for compen-sation of errors of the considered machine tools. The separated error parameters obtained bydifferent measurement such as positional, straightness, squareness, angular error etc. have beencombined to find out volumetric errors that could be compensated by techniques described inthis paper.

2. Error origins of multi-axis machine tools

Numerous error origins affect the tool tip position. In this paper we mainly consider the geo-metric error of typical horizontal machine tools (Fig. 1).

1537M. Rahman et al. / International Journal of Machine Tools & Manufacture 40 (2000) 1535–1546

Fig. 1. Horizontal type three-axis machine tool.

Some common error origins in multi-axis machine tools are:

O Link geometric errorO Resolution and accuracy of the linear measuring systemO Elastic deformation of drive componentsO Inertia forces when braking/acceleratingO Friction and stick slip motionO Control systemO Cutting forceO Vibration

Errors in geometric parameters originate from the manufacturing or assembly defects of differentparts of the machine. The error includes link length error, angular error, straightness error, square-ness error, parallelism error, perpendicularity error and zero position error (offsets).

The static loads of a machine tool result from the process forces and the masses of the work-piece and machine components. The static working load and the mass of the workpiece beingmachined produce distortions, which result in positioning errors in the machine tool. Servo controlsystems plays an important role in the accuracy of machining. Stick motion, stick slip, inadequatepitch error, response lag etc. error sources depend on the servo control system. A machine toolusually operates in a thermally non-steady state due to heat generation from various sources. Anychange in the temperature distribution of the machine tool structure causes thermal deformationthat affects the accuracy of the workpiece.


Fig. 2 shows that machining accuracy is mainly dependent on systematic errors and randomerrors. Systematic errors are easy to predict for the tool tip position. We shall neglect thermalerror effects in this paper.

3. Modeling of multi-axis machine tools for volumetric error

For analyzing the measurement result of a multi-axis machine tool we should have a goodmathematical model by which we could separate the error origins as correctly as possible orassemble the individual errors to find out the volumetric error. As has been already mentioned,the structure and control of multi-axis machine tools are complex in nature. Additional errorsources such as thermal, stiffness etc. are contributing their parts thereby making the machinetools accuracy even worse.

3.1. Movement along an arbitrary axis

If the rotational/linear axes are not perpendicular to each other, although they should be, thenthere exists squareness error along an arbitrary axis in the space; the dot product of two orthogonalaxes is not zero [for example, cos(90)=0 but cos(89.99)Þ0].

Though the variation is small, we can not neglect it because small variations in angular errorgives large tool tip deviation from the desired position depending on the length of tool and spindlepivot distance or by the amount movement of an axis [abbe effects].

For linear axis the transformation is given by [5]:

Ai531 0 0 d·l

0 1 0 d·m

0 0 1 d·n

0 0 0 14 (1)

Fig. 2. Diagram of error origins affecting the machining accuracy [1].


whered is the traveled axis value for any axis andl, m, n are direction cosines of the axis.Ai iscalled the homogeneous transformation matrix. The first three columns are for rotational partsand the fourth column is position vector.

3.2. Location of spindle tip for all axes

The machine zero point is a point where the joint displacements are zero. The displacement inthe reference coordinate system (origin O in Fig. 3) is given by

Pw5AOC0·A3·A1 (2)

Ps5AOB0·A2·T0 (3)

WhereAi is the homogeneous transformation matrix in the reference coordinate system and isdescribed by the ideal forward kinematics based on Eq. (1) described in Section 3.1 andT0 is toolcoordinate frame.Pw andPs are workpiece and tooltip points. If we express the above equation interm of joint command we find the tool tip position for Fig. 3

ToolTip53−X

Y

−Z4 (4)

Fig. 3. Structure of horizontal type machining center as shown in Fig. 1.


WhereX, Y andZ are joint commands for an individual axis of ideal machine.X andZ axes feedmotion are obtained by table movement.

3.3. Error model of multi-axis machine tools

Joint axes orientations are varied with the degree of misalignment. If we find the value ofmandn in the reference coordinate system thenl is given by

l5Î1−m2−n2 (5)

In modeling of machine tools for error model, the real point is given by [3,4]:

Yreal531 0 0 Y·kyx

0 1 0 (Y+LY)·kyy

0 0 1 Y·kyz

0 0 0 14·3

1 kyx 0 0

−kyx 1 kyz 0

0 −kyz 1 0

0 0 0 14·3

1 −cy by 0

cy 1 −ay 0

−by ay 1 0

0 0 0 14·3

1 0 0 B0Psx

0 1 0 0

0 0 1 B0Psz

0 0 0 14 (6)

In Eq. (6), first homogenous transformation describes effects of squareness error, seconddescribes rotational effects due to squareness, third describes roll (by), pitch (ay) and yaw (cy)and fourth describes fixed geometrical offset of machine. All transformation is based on smallangle assumption.

Yreal is obtained by multiplying several matrices terms consisting of Roll, Pitch, Yaw, pos-itioning and squareness errors. Normally these errors are obtained by typical machine toolmeasurement such as laser interferometer, double ball bar measurement (DBB), linear comparatoretc. Similarly we can define alsoXreal and Zreal for X-axis andZ-axis. Once we know the realposition and ideal position for a control point we are able to correct the tool tip by redefiningthe tool tip.

4. Measurement of multi-axis machine tools

There are numerous ways available for measuring machine tools with different requirementsand accuracy.

4.1. DBB measurement

From ideal and real machine tool’s coordinate transformation we could develop volumetricerror equations for multi-axis machine tools. When a machine is commanded to make circularinterpolation with a radiusR the increase/decrease in radius is given by [3],


DR5(Cx·X+Cy·Y+Cz·Z)

R(7)

DR is increased/decreased in radius for a circular interpolation with a radiusR. Cx, Cy andCz

are error vectors.X, Y andZ are the coordinates of interpolation points.

4.2. Laser interferometer measurement

By laser interferometer measurement we are able to find Roll, Pitch, Yaw and Positioning errorsof multi-axis machine tools. Once we know the magnitude of error in some discrete points weare able to find other point’s error magnitude by interpolation. In this way we are able to locatethe tool tip for every point in the working space.

4.3. Linear comparator measurement of multi-axes machine tools

The Heidenhain VM 182 linear comparator measures positioning errors of linear axes. It alsomeasures the guideway error perpendicular to the traverse direction of the machine axis. Fromthese traverse directions we could get an idea of the squareness error and rolling angular errorof the axis.

5. Software based correction of NC program

In this section we show examples for error compensation. Once we are able to locate themachine tool’s geometry precisely by measurement methods as described in Section 4, we areable to compensate those errors by counting the new axes values by post processor (PP) or byextra NC program processor (ENPP) which include the information of actual machine geometryand error information. If machine tools users use CAM for NC programming we can implementthe error compensation in postprocessor. Otherwise we have to process the NC programs by usingan extra NC program processor which is a software to be developed.

5.1. Implementation in postprocessor (PP)

To make use of multi-axis control machine tools, it is necessary to generate an NC programfrom CL-data. The preparation of NC-data to drive multi-axis control machine tools inevitablyrequires a post processor converting CL-data to each axis movement of machine tools consideringgeometric errors, feed rate control and spindle rotational control [6]. The post processor requiresthe machine data and then calculates new axes values of all three axes on the basis of desiredtool tip point (Fig. 4).

The inverse kinematics for an axis that has arbitrary direction cosines is difficult, however wecould use redefinition of task points based solution to position the ToolTip to the desired position.


Fig. 4. Correcting NC program by postprocessor (PP) [6].

5.2. Implementation in Extra NC Program Processor (ENPP)

If some machine tool users don’t use postprocessor for their NC programming system, it isstill possible to implement the software correction to correct the NC program. The software correc-tion diagram is shown in Fig. 5.

5.3. Examples with squareness error

The major reasons for squareness error is:

O In multi-axis machine tools where the two guide ways are located on the same structure (inFig. 1 X-axis is located on theZ-axis) not perpendicular to each other.

O The case where the two axes are not perpendicular because the columns tilts forwards or back-wards, or to the right or left when the column base is not set horizontally (Fig. 6).

Fig. 5. Correcting NC program by extra NC program processor (ENPP).


Fig. 6. Squareness errorY-axis tilts an anglea on XY plane.

5.3.1. Linear movement along an axisAn example of recalculation of tool tip point by coordinate transformation based on the square-

ness error: For a movementX, Y and Z axes (each 20 mm, from point 1 to point 2) with asquareness error onXY plane 0.01 degrees CW.

3XYZ45320.004

20.000

20.0004 Actual position3DX

DY

DZ4530.004

0.000

0.0004 Error vector

This error vector is for linear movement.

N 200 G00 X 20.0 Y 20.0 Z 20.0 (Wrong Tool Tip Position)N 200 G00 X 19.996 Y 20.0 Z 20.0 (Correct Tool Tip

position)

The above two program lines are for positioning theX andY axes with rapid movement.

5.3.2. Circular movement on XY planeIf we make a circular interpolation onXY plane we obtain an ellipse (Fig. 7) whereYI(q) =

R·cos(q), XI(q) = R·sin(q), YR(q) = (R+DR)·cos(q) and XR(q) = (R+DR)·sin(q) DR is theincrease/decrease in radius according to Eq. (7).

Fig. 8 shows the principle of error compensation of a circular interpolation motion. The square-ness error has an effect that makes the circle an ellipse on given plane. Now the ellipse is replaced


Fig. 7. Representation of squareness error on XY plane for Fig. 6. Solid line means the ideal circle (XI(q), YI(q))and dotted line the real circle (XR(q), YR(q)) [3].

Fig. 8. Compensation of squareness error onXY plane for circular interpolation.

by four arcsab, bc, cd andda. The direct linearization works because of a small angle assumption.The clockwise rotated ellipse (dashed line) is produced by the machine as a result of squarenesserror and the counter-clockwise rotated ellipse (dotted line) is produced as result of the NC pro-gram correction. The correction by four arcs is programmed as follows:


N200 G91N 201 G02X-150.000Y-150.000I-149.978J-0.022 (arcab)N 202 G02X-150.000Y150.000I0.022J150.022 (arcbc)N 203 G02X150.000Y150.000I149.978J0.022 (arccd)N 204 G02X150.000Y-150.000I-0.022J-150.022 (arcda)

6. Conclusion

This paper has described how to correct the machine tool errors by recalculating the axis coordi-nates in NC program by software with numerical examples in a theoretical way. The idea maybe implemented to improve machine tools accuracy in two ways. We can implement the correctionalgorithm for rapid movement, linear movement and for circular interpolation. The next part(Paper II) of this paper would describe practical results based on these correction approaches withreal machine tools. In conclusion the working steps for correcting NC program for a particularmachine tool are summarized:

O Modeling of machine tool for error diagnosis,O Measurement of the machine tool,O Diagnosis of error origins,O Correction of NC program by postprocessor or by extra processor,O Testing of machine tools with corrected NC program.

Acknowledgements

The authors express their acknowledgement to the following companies and organizations forsupporting this research work.

O Technology Development Center of Finland (http://www.tekes.fi)O Nestix Ltd., Oulu, Finland (http://www.nestix.fi)O Valmet Paper Machines Ltd., Jyva¨skyla, Finland (http://www.valmet.com)O JMC Engine Ltd., Ruukki, FinlandO TL-Tuotanto Ltd., Kemi, Finland

References

[1] P.A. Anderson, Methodology for evaluating the production accuracy of the machine tools. Tampere TechnicalUniversity, 1992.

[2] K.F. Eman, B.T. Wu, A generalized error model for multi-axis machines, Annals of the CIRP 36 (1) (1987)253–256.


[3] Y. Kakino, Y. Ihara, A. Shinohara. Accuracy Inspection of NC Machine Tools by Double Ball Bar Method. HanserPublishers, Munich, Germany 191, 1993.

[4] K. Kim, M.K. Kim, Volumetric accuracy analysis based on generalized geometric error model in multi-axis machinetools, Mech. Mach. Theory 26 (2) (1991) 207–219.

[5] W. Mooring, W. Benjamin. Fundamental of Robot Manipulator Calibration. IBM corporation Austin, Texas. JohnWiley and Sons, Inc., p 329.

[6] Y. Takeuchi, T. Idemura, Generation of five-axis control collision free tool path and post processing for NC-data,Annals of the CIRP 41 (1) (1992) 539–542.

modeling, measurement and error compensation of multi-axis machine tools. part i: theory

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