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Analytical Approach for Design of Alignment Errors of Machine Tools

Nobuhiro Sugimura and Shinya Mitani

College of Engineering, Osaka Prefecture University,1-1 Gakuencho, Sakai, Osaka 599-8351, Japan

E-mail: [email protected]

ABSTRACTThe objectives of the present research are to establish a mathematical model representing alignment

errors of machine tools, and to carry out theoretical analysis of machining errors based on the proposed

model. A mathematical model is proposed based on shape generation process of machine tools to

represent alignment errors of 5-axis machine tools. The model describes the relative motions between thetool and the workpiece by combining 4 by 4 homogeneous transformation matrices. Machining errors of

various types of 5-axis machine tools are investigated through simulation by applying the proposed model,aimed at verifying adaptability of the feed axis configurations of the machine tools from the viewpoint of

the machining errors.

INTRODUCTION

New type manufacturing systems are now required to cope with large-variety small batch

productions with high accuracy and high quality products. Machine tools are recognized as key

components of the manufacturing systems. Much progress has been made in the machine tool

technologies aimed at improving the performances of the machine tools from various viewpoints, such as

accuracy, reliability, productivity, and flexibility. One of the most important revolutions in the later part of

this century is introduction of NC (Numerical Control) machine tools, which are able to carry out various

complicated machining processes without human interaction. Various types of multi-axis NC machine

tools are now being applied to machining process of complicated surfaces of mechanical products. It is

now required to clarify the relationships between the kinematic errors of the machine tools and the

geometric errors of the generated surfaces, in order to generate the complicated surfaces with the high

accuracy and quality.

The objective of the research is to develop a systematic design method of multi-axis machine toolsbased on a mathematical model representing the shape generation processes of the machine tools1),2). The

mathematical model has been applied to various research fields of the analysis and the design of themachine tools. For examples, a systematic method was proposed for the static accuracy test of the

machining center based on a model representing the kinematic errors of the machining centers3)

. Shape

generation motions of the 5-axis machining centers were described by a mathematical model taking intoconsideration of the alignment errors, and the motion errors of machine tools were analyzed by applying

the model4),5)

. A method was proposed to estimate the alignment errors of the machine tools based on

machined part shape measurement

6)

. The model of the shape generation process was also applied to NCdata generation and post-processing for the 5-axis machining 7),8).

In the previous researches, systematic methods have been proposed to design feasible structures of

the machine tools based on the geometric shapes of the products and the machining errors by applying the

model of the shape generation process9)-11)

. This paper deals with theoretical analysis of the machining

errors of the 5-axis machine tools based on the model.

DESIGN OF MACHINE TOOL ACCURACY

Figure 1 summarizes the factors that affect the machining errors of the product surfaces and their

relationships from the viewpoint of the shape generation process of the machine tools. The geometric

errors of the product surfaces are determined mainly from the relative motion errors between the tools and

the workpieces, the geometric errors of the tools, and the errors in the material removal processes. Therequirements on the geometric errors of the product surfaces should be distributed among the factorsabove mentioned in the design process of the machine tools, and finally the accuracy of all the

components of machine tools are designed.


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Three types of 5-axis machine tools shown in Fig. 3 are considered in the research, and the axes of

the feed motions and the main motion are schematically illustrated in Fig. 4. The shape generation

motions of these machine tools are given in the followings;

(a) Tool rotation type

r

r

W

T

A d A x A d A y A d A z A d

A A d A A d A c A d

= 3 1 1 3 2 2 3 3 3 3 46

13

54 3

66

23

7

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) (2)

Z

X

Y

Workplace

C-axis R.S.

A-axis R.S.

X-axis.T.S.

Y axis T.S.

Cutting Tool

Spindle

Z-axis T.S.

Z

X

Y

Workplace

C-axis R.S.

A-axis R.S.

X-axis.T.S.

Y axis T.S.

Cutting Tool

Spindle

Z-axis T.S.

Z

X

Y

Workplace

C-axis R.S.

A-axis R.S.

X-axis.T.S.

Y axis T.S.

Cutting Tool

Spindle

Z-axis T.S.

(a) Tool rotation type (b) Tool & workpiece rotation type (c ) Workpiece rotation type

Fig. 4 Feed and main motion axes of 5-axis machine tools

A

C

Z

YX

Z

A

C

YX

C

A

XY

Z

(a) Tool rotation type (b) Tool & workpiece rotation type (c ) Workpiece rotation type

Fig. 3 Various types of 5-axis machine tools


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(b) Workpiece and tool rotation type

r

r

W

T

A d A A d A x A d A y A d

A z A d A A d A c A d

= 3 1 6 1 3 2 1 3 3 2 3 43 3

54 3

66

23

7

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

(3)

(c) Workpiece rotation type

r

r

W

T

A d A A d A A d A x A d

A y A d A z A d A c A d

= 3 1 6 1 3 2 4 3 31 3

4

2 35

3 36

62

3

7

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

(4)

where,

Aj(*): transformation matrices representing the feed motions and the relative positions among the axes,

Acj(*): transformation matrices representing main motion,

x, y, z: parameters representing linear feed motions along the X, Y, and Z-axis, and

, , : parameters representing rotary main motion and rotary feed motions around the X, Y and Z-axis.

Modeling of Alignment Errors

The alignment errors considered here are the errors of the relative positions and orientations

between pairs of succeeding axes, for examples, C-axis and A-axis, A-axis and X-axis, and X-axis andY-axis of the workpiece rotation type machine tool shown in Fig. 3 (c). The position and orientation

errors in the three-dimensional coordinate space are generally composed of three components of

translation errors and three components of rotational errors. If these errors are very small, the position and

orientation errors between pairs of the axes are given by the following equation.

=

1

1

1

0 0 0 1

x

y

z

(5)

where,

, , : rotational errors around X, Y, and Z-axis, and

x, yy, z: translation errors along X, Y, and Z-axis.

If the three dimensional coordinate systems of the machine tools are set suitable to represent thealignment errors, the error between a pair of axes can be represented more simply, as summarized in the

followings;


Z-axis is set to equal to the rotational axis of the main motion, and Y-axis is set to be perpendicular

to both the Z-axis and the rotary feed axis in X directions.(b) Workpiece and tool rotation type

Z-axis is set to equal to the rotational axis of the main motion, and Y-axis is set to be perpendicular

to both the Z-axis and the rotary feed axis in X directions.


Z-axis is set to equal to the rotational axis of the main motion, and X-axis is set to be perpendicularto both the Z-axis and the linear feed axis in Y directions.

In the cases mentioned above, the individual matrices representing the alignment errors can be

simplified, and the shape generating motions including the alignment errors are described by thefollowing equations;


r

r

W XY YZ ZC

CA AC T

A d A x E A d A y E A d A z E A d

A E A d A E A d A c A d

= 3 1 1 3 2 2 3 3 3 3 46

13

54 3

66

23

7

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) (6)


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(b) Workpiece and tool rotation type

r

r

W CX XY YZ

ZA AC T

A d A E A d A x E A d A y E A d

A z E A d A E A d A c A d

= 3 1 6 1 3 2 1 3 3 2 3 43 3

54 3

66

23

7

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

(7)


r

r

W CA AX XY

YZ ZC T

A d A E A d A E A d A x E A d

A y E A d A z E A d A c A d

= 3 1 6 1 3 2 4 3 31 3

4

2 35

3 36

62

3

7

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

(8)

where,Eij: alignment errors between axis-i and axis-j.

ANALYSIS OF MACHINING ERRORS

Machining errors of the 5-axis machine tools are investigated through the simulation by applyingthe proposed model given in Eqs. (6), (7) and (8), aimed at verifying adaptability of the feed axis

configurations of machine tools from the viewpoint of the machining errors. The analysis conditions

applied here are shown in the followings;

(1) Machined surface is a half sphere shown in Fig. 5 (a). The center of the sphere is set not to equal tothe center of the worktable.

(2) Machining process is an end-milling, and the direction of the tool is controlled to equal to the normal

vector of the machined surface, as shown in Fig. 5 (b).

(3) The errors of the rotary main motion and the geometric errors of the tool are ignored.

In the first step of the analysis, the tool path data representing the feed motions of the individual

feed axes are calculated based on the geometry of the machined surface, by applying the ideal shapegeneration motions given in Eqs. (2), (3) and (4). Following this, the tool path including the alignment

errors is generated by applying the Eqs. (6), (7) and (8), and the positioning errors of the tool against the

workpieces are calculated at a set of points on the machined surface in the second step. The positioningerrors of the tool are measured from the machined surface to the bottom face of the end-milling tool along

the normal directions of the surface.

Only one alignment error is taken into consideration in each analysis in order to clarify the effect of

each error on the geometric errors of the machined surface. That is, one of the error components in Table

2 is set to be the unit value in each analysis in the second step. The unit values of the errors are set to be

10 m for the translation errors and 10 rad for the rotational errors.Table 2 summarizes the analysis results describing the effects of the individual error components on

the geometric errors of the machined surface. In the table, the mean values are the averages of the errorsat all the points on the surface, and the standard deviation values are calculated from the mean values andthe errors at all the points. It is shown in the table that the effects of some alignment errors on the

positioning errors are relatively larger than the other alignment errors. It is also concluded that some ofthe alignment errors do not affect the positioning errors in the normal direction of the machined surface.

-80

-60

-40

-20

0

20

4060

-40

-20

020

4060

80100

120

-10

0

10

20

30

40

50

60

70

80

-10

0

10

20

30

40

50

60

70

80

Machi ned Surf ace

Endmi l l i ng Tool

(a) Machined Surface (b) Machining Process

Fig. 5 Machined surface and machining process


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As shown here, the effects of the individual components of the alignment errors can be easily evaluated

by applying the method proposed here.

CONCLUSIONS(1) A mathematical model was proposed to represent the alignment errors of the feed axes and the main

motion axis of three types of 5-axis machine tools. The proposed model represents the shapegeneration motions between the tools and the workpieces taking into consideration of the alignment

errors.

(2) The effects of the alignment error components on the geometric errors of the machined surface wereanalyzed through the simulation applying the proposed model. The results show that the effects of

some alignment errors on the positioning errors of the tool are relatively larger than the other

alignment errors.

REFERENCES

1) Reshetov, D. N., and Portman, V. T., Accuracy of Machine Tools, APME Press, (1988), p.21.

2) Sugimura, N., Iwata, K., and Oba, F., Formation of Shape Generation Processes of Machine Tools,

Proc. of IFAC 81, Vol. XIV, (1981), p.158.3) Inamura, T., Yasui, T., Misawa, T., and Watanabe, M., An Error-model Reference Method of Static

Accuracy Test for Machining Center, J. of JSPE, Vol. 51, No. 5, (1985), p.1060.4) Sakamoto, S. and Inasaki, I., Analysis of Generating Motion for Five-Axis Machining Centers, Trans.

of NAMRI/SME, Vol. 21, (1993), p.287.

5) Sakamoto, S. and Inasaki, I., Error Analysis of Precision Machine Tools, Proc. of the 3rd Int. Conf. of

Ultraprecision in Manufacturing Engineering, (1994), p.245.6) Kishinami, T., Tanaka, F., and Yamada, M., Estimation Method for Alignment Errors of Machine

Tools Based on Machined Part Shape Measuring, Proc. of ASPE 1993 Annual Meeting, (1993), p.358.

7) Takeuchi, Y. and Idemura, T., 5-Axis Control Machining and Grinding Based on Solid Model, Annals

of the CIRP, Vol. 40, (1991), p.455.

8) Takeuchi, Y., and Watanabe, T., Generation of 5-Axis Control Collision-Free Tool Path and

Postprocessing for NC Data, Annals of the CIRP, Vol. 41, (1992), p.539.

9) Iwata, K., Sugimura, N., and Peng, L., A Mathematical Analysis of Product Surfaces for Machine

Tool Design, Proc. of MI 88, ASME, (1988), p.1.

10) Iwata, K. and Sugimura, N., A Knowledge Based Structural Design of Machine Tools for FMS/FMC,

Proc. of 21st Int. Seminar of Manufacturing Systems, (1989), p.1.1.

11) Moriwaki, T., Sugimura, N., and Miao, Y., A Model Based Design of Kinematic Accuracy of MachineTools, Human Aspects in Computer Integrated Manufacturing, Elsevier Science Publishers, (1992),p.673.

Table 2 Effects of alignment errors on machining errors

(a) Tool Rotation Type (b) Tool & Workpiece Rotation Type (c) Workpiece Rotation Type

Machining Errors( m) Machining Errors( m) Machining Errors( m)

Alignment

Errors

Mean

Value

Standard

Deviation

Alignment

Errors

Mean

Value

Standard

Deviation

Alignment

Errors

Mean

Value

Standard

Deviation

xCA 5.1 2.9 xCX 0.7 0.7 xCA 0.0 0.0 yCA

5.1 2.9 yCX

4.8 2.4 yCA

8.0 2.1

yAS

5.1 2.9 yAS

4.8 2.4 zAX 10.0 0.0

zAS

4.8 2.9 zAS

4.8 2.4 yAX

0.0 0.0XY 0.8 0.4 CX 0.9 0.2 CA 1.0 0.3

XY 1.2 0.8 CX 0.2 0.1 CA 0.2 0.1

YZ 1.5 0.8 XY 0.2 0.2 AX 0.3 0.2

YZ 0.0 0.0 XY 0.0 0.0 AX 0.0 0.0

ZC

4.8 0.8 YZ 6.2 1.2 XY 0.0 0.0

ZC 4.4 2.5 YZ 0.2 0.2 XY 0.0 0.0

CA 3.2 1.7 ZA 3.8 2.2 YZ 1.6 0.4

CA 3.2 1.7 ZA 0.6 0.6 ZS 3.5 1.8

AS

0.0 0.0 AS 0.2 0.2 ZS 0.0 0.0

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