1. Introduction

Automation with cam mechanisms, although theyoffer a low degree of flexibility, are frequentlyused due to their low cost and to the fact thatthey provide the simplest way of achievingalmost any desired follower motion. It has beenwidely or generally used the three-axis CNCmachining center for the manufacturing of diskcams. For the case of manufacturing of combineddisk cams (three or more disk cams) with singlecam shaft, however it is not possible or difficultto manufacture such cams using the traditionalthree-axis CNC machining center. A five-axisCNC machining center is presently one of themost versatile machine tools available for suchdisk cams and they are becoming increasinglycommon. To increase the accuracy capabilities ofsuch a machine, it is crucial to be able to studythe geometric errors of the components and itseffect on the quality of machined products.It should be noted that the paper does not focuson the geometric error of the structural elementsof the milling machine, such as positioning errorsof the machine axes, spindle errors, thermallyinduced geometric errors, etc. We will confineourselves to the influence of the specificgeometric-kinematic factor, namely the particulartool path that guides the cutting tool. In otherwords, we investigate the extent to which themachining tool path evaluated by the CAD/CAMcomputer contributes to the inaccuracy of a

machined disk cam.To improve part quality with minimal machiningtime (i.e. to create the well-compensated NCcode) for the disk cams in a five-axis CNCmachining center, the control of tool pathcalculation with minimal error and itscommunication to the NC unit are imperative.Generally, the tool motion functions can bedescribed utilizing linear, circular, polynomial orspline interpolation algorithms. The application oflinear interpolation algorithms (first-degreefunctions), leads to follower velocity andacceleration discontinuities, which causevibrations, mainly in high-speed mechanisms. Theuse of second-degree functions, as a basis forcircular interpolation algorithms, eliminates thefollower velocity discontinuities and may lead tosatisfactory results regarding the dynamicbehavior of the cam mechanism. Using a circularinterpolation algorithm a certain number of camprofile points, regarding minimum distancesbetween them, is required. Small incrementsbetween successive points may lead toundercutting and the NC code size increasesconsiderably. Furthermore, a successive change ofthe curvature of the circular arcs from convex toconcave is caused and significant deviations ofthe real transfer functions (velocity, acceleration,etc.) from the corresponding calculated onesoccur. Using higher-degree functions, such asinterpolation equations, like polynomials orsplines, continuous second and higher order camtransfer functions can be achieved. The applicable

A Study on Tool Path Error Control for Disk Cams in a Five-Axis

CNC Machining Center

Soon-Man Kwon, Joong-Ho Shin and Geun-Jong Yoo

Department of Mechanical Design and Manufacturing, Changwon National University,

Changwon 641-773, South Korea

Abstract: In this paper, we propose a simple but optimized NC code generating technique for disk

cams by means of tool path error control in a five-axis CNC machining center. Using the geometric

theorem of the triangle made between manufacturing points and error checkpoint, the tool path error has

been studied for disk cams profile generation and an improvement in the profile has been obtained.

Then, based on the present manufacturing approach a computer program is developed on C++

language

to perform and to verify the shape design, the manufacturing simulation, and the optimized generation of

the NC code.

Key Words: Tool Path Error, Disk Cam, five-Axis CNC Machining Center, NC Code

ICCAS2004 August 25-27, The Shangri-La Hotel, Bangkok , THAILAND

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spline degree depends on the interpolationfacilities of the NC machine. To determine thenumber of the cam interpolation regions, the camaccuracy and the permitted transfer functionsdeviations have to be considered.In the present paper, we propose a simple butoptimized NC code generating technique for diskcams by means of tool path error control in afive-axis CNC machining center. Using thegeometric theorem of the triangle made betweenmanufacturing points and error checkpoint, thetool path error has been studied for disk camsprofile generation and an improvement in theprofile has been obtained. Then, based on thepresent manufacturing approach a computerprogram is developed on C

++language to

perform and to verify the shape design, themanufacturing simulation, and the optimizedgeneration of the NC code.

2. The Manufacturing of Disk Cam

Mechanism

There are many kinds of cams with variousfollowers. The general methods for the shapedesign of the cam include graphic layout methodand analytic ones. In a way of the shape designof the cam, recently, Shin et al. [1-3] presentedan approach to the shape design of disk cambased on the relative velocity of a followerversus a cam and the shape of the followerusing the velocity graphics, and to determinationof the curvature radius of disk cams usingacceleration graphics. Following the above-mentioned researches [1-3], the entire shape ofdisk cams can be determined via coordinatetransformation as follows (see Ref. [4] for moredetails)

(1)

where is an angle of cam rotation. Also

and represent contact points on the shape

design-coordinate, given by:

(Case 1) Translating roller follower:

(2)

(Case 2) Translating flat-faced follower:

(3)

(Case 3) Oscillating roller follower:

(4)

(Case 4) Oscillating flat-faced follower:

(5)

where the notations in Eqs. (2)-(5) can be found

in Fig. 1.

(Case 1) (Case 2)

(Case 3) (Case 4)

Fig. 1 Various types of disk cam

A 5-axes machine tool has the ability ofsimultaneously positioning and orienting thecutting tool in some coordinate system defined inthe work space. This makes such a machine toolvery versatile. The configuration of the 5-axesCNC machining center can be assembled inseveral ways. In this study, horizontal 5-axesmachining center is used, which consist of theusual three translation motion axes (Z2, Z3, Z4)and two rotation axes (Z0, Z1), as shown in Fig.2.

fX

fZ

fY

frame

2

1

3

45

0

2Z

2X

1X

1Z

3Z

3X

4Z

54,X X

0Z0X

5Z

Fig. 2 A 5-axes CNC machine centerconfiguration set-up

From the shape design data, the tool location ofrough and finishing manufacturing with generaltools is defined in Fig. 3.The cutter location is calculated from the shapedata of the disk cam, given by:

(6)

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(7)

(8)

where { } represent contact points of Eqs.(2)-(5) in the shape-design-coordinate, =i×TD(i=1,2,3 n) and TD is a tool diameter.

Fig. 3 Cutter location with respect to contactpoint of the disk cam

To extract NC-code data in a 5-axes machiningcenter, the tool path position and orientation willbe defined via coordinate mapping technique. Inperforming a manufacturing, it is necessary tocoincide the machining point with design-coordinate Xd. It is possible to perform arotational transformation between tool-coordinatet(Xt, Yt) and design-coordinate d(Xd, Yd). Also,to obtain the translation motion data, we mapwork-coordinate f into shape-design-coordinate d(see Figs. 4 and 5).

fX

fZ

fY

frame

2

1

3

45

0

2Z

2X

1X

1Z

3Z

54,X X

0Z0X

5Z

Fig. 4 Coordinate systems of workpieceon a 5-axes CNC machining center

After such a consideration, we lead to thefollowing NC-code generation equation

(9)where

(10a)

(10b)

Therefore, the NC-code for the above four types

of the disk cam can be expressed as

(11)

where is the rotation angle of the tool whichis dependent of the cam type, and here is theinterference corrected angle to avoid toolinterference under the consideration of thenegative curvature.

frame

2

1

3

4

50

2Z

1X

1Z

3Z

3X

4Z

54,X X

0Z

5Z

fX

fZ

fY

Fig. 5 Coordinate systems of spindleon a 5-axes CNC machining center

3. Tool Path Error Controlby Biarc Algorithm

In programming the tool path of CNC machinery,fewer arc segments can help to improve theproduction efficiency by reducing the number ofinstructions and tool motions. In particular, thequality of cam depends upon the machine toolerrors. A biarc [5] can be described as twoconnected circular arcs with an identical tangentat the connecting point. At the connecting point,C

1tangent continuity is maintained. Furthermore

C2

curvature continuity can be improved if thedifference between two curvatures at theconnecting point is minimized. The toleranceconstraint can be specified as the largestdeviation distance between the curve and theapproximating biarcs. From the machining pointof view, circular interpolation is the basicfunction in most NC machines. Therefore, biarcsare suitable for tool-path generation and NC codegeneration.A circular arc can be defined by a center, a startpoint, and an end point. Two circular arcs canbe linked if the end-point of the first arc andthe start point of the second arc are connected.If the tangents of the two circular arcs are thesame at the connecting point, then two circulararcs composes a biarc.In the present study, we adopted the biarcalgorithm by proposed by Bolton [5]. In Fig. 6,a biarc composed of two circular arcs isillustrated.

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Fig. 6 A biarc matching G1

Hermite data

As shown in the figure, using the geometrictheorem of the triangle the following equationcan be made

(12)

where is the chord length, and

(i=1,2) denote and ,

respectively. Also the sign ± in the last term ofleft hand side of equation means the unimodal(C-shape) and inflection (S-shape) biarcs,respectively.If we set R1, the radius R2 can be evaluated asfollows

(13)

Using Eq. (13), tool path error can be controlledby standard permitted tool path error. Figure 7shows tool path error defined by vertical distancebetween real tool path and linear tool path.

Pi

Pj Pi+1

Pj+1

Pi+2

Tool Path

Tool PathLinear Tool Path

Tool Path Error Tool Path Error

Fig. 7 Definition of tool path error

A tool path error at an arbitrary point Pj can beevaluated by using the Pythagorean theorem ofthe triangle as follows (see Fig. 8):

ERROR= (14)

where

(15a)

(15b)

(15c)

DiskCamXm

Ym

ToolPi

Pi+1Pj

Tool PathTool Path Error

Fig. 8 Tool path error control in disk cam

4. Numerical Examples

A CAD/CAM program, DiskCam , is developedon C

++language. This program can perform the

shape design, the simulation of the manufacturingprocedure and the kinematics simulation, whichcan make the integral NC-code for the 5-axisCNC machining center.Fig. 9 displays the displacement, velocity andacceleration curves for the disk cam mechanism.

Fig. 9 Displacement, velocity and accelerationcurves for cam mechanism

Fig. 10 depicts the integral simulation forassembled process of disk cam mechanism andfive-axis CNC machine center

Fig. 10 Integral simulation for assembled processin a 5-axes CNC machine center

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5. Conclusions

This paper proposes an approach for the shapedesign and manufacturing of the disk cam viatool path error control in a 5-axes CNC millingmachine. The following conclusions can bedrawn.

(1) The cutter location and cutter orientation aredefined for the manufacturing of the 5-axesmachining center from the shape data basedon the relative velocity method.

(2) The integral NC code for multi-axis CNCmachining center is proposed using thecoordinate transform method from the dataof location and orientation of cutter path ona horizontal 5-axes machining center whichis consist of three linear motion axes andtwo rotation axes.

(3) To improve the quality of the cam, the toolpath error is controlled by biarc algorithmand geometric theorem of the triangle.

(4) A CAD/CAM program, DiskCam , isdeveloped on C

++language. This program

can perform shape design, manufacturing andkinematics simulation, which can makeintegral NC code for multi-axis CNCmachining center.

References

[1] Shin, J.H., Kang, D.W., Kim, J.S. and Kim,D.W., 2000. A study on shape designapproach of disk cams using relative velocityof followers. J. of the Korean Society ofPrecision Engineering, 17 (2), 185-192.

[2] Shin, J.H., Kang, D.W., Kim, J.S. and Kim,D.W., 2000. A study on curvaturedetermination approach of disk cams usingrelative velocity of followers. J. of theKorean Society of Precision Engineering, 17(8), 113-119.

[3] Shin, J.H., Kang, D.W., Yoon, Y.H. andZhang, Y.H., 2001. An approach to shapedesign of disk cams using relative velocityof followers. Proceedigns of the InternationalSymposium on Mechanical Transmissions, 1,347-351.

[4] Kang, D.W., 2000. A study on designautomation technology of complexmechanisms with disk cams and cylindricalcams for multi-axis motion control. Ph.D.Thesis, Changwon National University,Korea.

[5] Bolton, K.M., 1975. Biarc curves, ComputerAided Design, 7(2), 89-92.

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