Ruled Surface Machining in 5 Axis Cnc Machine Tools

7/30/2019 Ruled Surface Machining in 5 Axis Cnc Machine Tools

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Journal of Manufachlrzng ProcessesVol. 2mo. 120000

Ruled Surface Machining onFive-Axis CNC Machine ToolsRong-ShineLin, Dept. of Mechanical Engineering, National Chung Cheng University, Chia-Yi, TaiwanYoram Koren, Dept. of Mechanical Engineering and Applied Mechanics, The University of Michigan,Ann Arbor, Michigan

AbstractTo achieve high precision and high productivity n machin-ing sculptured surfaces, a new architecture for a five-axisCNC interpolator for machining ruled surfaces was devel-oped and demonstrated on a milling machine. The objectiveof the five-axis interpolator is to continuously maintain themilling cutter axis in parallel with the straight lines of the ruledsurface. The cutter position and orientation are calculated ateach sampling period of the interpolator, and correspondingaxial position commands are generated by an inverse kine-matics algorithm. This real-time approach produces precisesurfaces and requires substantially less machining time com-pared to the conventional off-line approach. Two new g-codes are also given in this paper for the new interpolator toproduce part surfaces in CNC milling machines.Keywords: Five-Axis Machining, Five-Axis CNC, RuledSurface, Computer Control

IntroductionParts containing ruled surfaces, such as conic andcylindrical surfaces, can be machined either by ball-end cutters on three-axis milling machines or byflank milling on five-axis milling machines. Usingflank milling enables a tremendous increase in pro-ductivity as well as improvement of the surface fin-ish; however, the conventional methods for five-axismachining utilize the off-line approach by whichthe CAD system divides the curve into a set of linesegments that approximates the curve at the desiredtolerance.'-' These line segments are furtherprocessed by post processors to produce the g-codethat contains the computer numeric control (CNC)commands needed to control the machine. The g-code is fed into the CNC real-time interpolatorwhose major function is to generate coordinatedreference inputs for the servocontrollers that controlthe machine tool axes.Conventional off-line approaches for five-axismachines typically assume a constant tool orienta-tion along each segment. The constant orientationalgorithm causes severe roughness around the end-

points along the surface because the orientationchanges are abrupt at these points. More advancedapproaches assume a linear change in the tool orien-tation between each successive end-point. The linearorientation algorithm produces a better surface butstill interpolates the orientations inaccuratelybetween end-points (because the change of orienta-tion is not necessarily linear), which causes surfaceerrors. An additional drawback of off-line methodsis that the cutter accelerates and decelerates at eachsegment, which increases surface nonuniformity andsubstantially increases cutting time.8.9 Chou andYanglo provided a mathematical formulation for aEuler-angle-type five-axis machine to track a para-metric curve. Their work interpolated the curve cor-rectly, but the required tool orientations wereassumed to be given from CAD files that did not fitreal-time implementations for CNC systems.To overcome these drawbacks, this paper devel-ops an algorithm for a real-time five-axis interpola-tor that continuously controls tool orientation andposition. This interpolator can perform accurate pro-duction of ruled surfaces with flank cutters. Theinterpolator, which is contained in the CNC comput-er, calculates new commands in the same time peri-od used for sampling the control loop feedback (typ-ically 0.5 ms to 5 ms). In each sampling period, theinterpolator performs two steps: ( I ) trajectory plan-ning based on a constant feed rate and (2 ) inversekinematics transformation based on the structure ofthe machine.The developed real-time interpolator for machin-ing ruled surfaces is shown inFigure 1.The input tothe interpolator is the NC g-code, which contains thegeometric information of the part surface as well asthe cutting conditions, such as feed rate, spindlespeed, specific tool type, and so on. The trajectoryplanning portion generates the tool position (X , X Z )and orientation (Ox, O,, 0,) of the cutter in eachsampling period based on the surface geometry and


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.Journalof Munujactiiring ProcessesVol.21No. I2000

Figure IProposed Real-Time Interpolator for Machining Ruled Surfaces

Figure 2Example of the Ruled Surface and Its Flank Millingspecified feed rate. This is a generic, machine-inde-pendent algorithm. Based on the calculated cutter'sposition and orie ntation, the reference values (x,JI, z,a, b) of the five axes are computed by the inversekinematics transformation, an algorithm thatdepends on the structure of each particular machine.The detailed operation of these two algorithms isdescribed in the following two sections. Examplesare given in a later section.

Trajectory PlanningThe trajectory planning algorithm generates thelocations and orientations for the cutting tools basedon part geometry and feed rate, which are obtainedfrom the part program. The special geometric prop-erties of ruled surfa ces are described in this section.Based on these properties, algorithms are presented

to calculate tool positions and tool orientations pre-cisely along the ruled surface with machining at aconstant feed rate.Ruled SurfacesA ruled surface is defined as a surface thatthrough each point of it passes at least one straightline lying entirely on the surfac e." In other words, aruled surface is a surface generated by a family ofstraight lines (for example, a conic surface). Thisspecial geometric property can be used to increasethe productivity of machining ruled surfaces onfive-axis machines. The basic idea of this type ofmachining is moving the cutter's edge to follow thestraight lines of the ruled surface. A ruled surfacecan be expressed by the following equation:

S(u, v)= (1 - ) C,, (u)+ v C, (u) (1 )where Co and C 1 are two boundary curves of theruled surface, which are known as directrices; u andv are parameters, normally 0 5 u, 5 1. An exam-ple of a ruled surface is shown in Figure 2.Position CalculationIn this research, the real-time five-axis interpola-tor is proposed to be implemented on finish cutting,which removes only a small amount of material.Therefore, the tool motion can be assumed as con-stant feed rate machining. To simplify the imple-mentation process, the lower boundary curve of theruled surface is chosen for the tool contact trajecto-ry, C(u), as follows:

where

The interpolator calculates the tool position ateach sampling period (AT) based on this parametriccurve. However, the constant change in the parame-ter Au in the parametric domain does not guaranteethe constant length in the Cartesian domain, andconsequently does not guarantee a constant feed


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rate. To obtain the constant feed rate ( V ) , the con-version between the parameter u and the constantdistance that the tool moves at each sampling peri-od has to be derived. A solution based on Taylor'sexpansion is used to obtain the value of Au that cor-responds to the equal trajectory length of the para-metric curve. Eq. (3) below shows the function of uin terms of the length (V -AT) that the tool movesduring one sampling period. If the machining feedrate V is constant, ( V .AT) is constant as well.

whered x d v , d zx ' = - , y ' = - , a n d z =-du du du

The subscriptsk and k-1 mean the current samplingtime and previous sampling time, respectively. Thecutter contact location at the kth sampling period canbe obtained by substituting uk, from Eq. (3), into Eq.(2), as follows:

The precision of this approach has been ana-lyzed in Shpitalni, Koren, and Lo.12 It has beenshown that the reference positions calculated bythis real-time interpolator algorithm result in verysmall contour errors compared with conventionaloff-line interpolators. The variations in the veloci-ty commands (feed rates in g-codes) by the real-time approach are negligible for most applications.By contrast, the off-line approach causes very largevelocity errors.In addition, in machining by the conventional off-line interpolator, the machine has to follow asequence of short straight lines, which causes accel-eration and deceleration at the beginning and theending of each line. This results in a substantialincrease in machining time. In contrast, the real-timeapproach reads the g-code of the parametric curvedirectly and generates the reference positions for thetool at the same rate needed by the control loops.This eliminates the acceleration and deceleration

effect and consequently results in a substantialreduction in machining time.8To obtain the cutter's location, the cutter contactlocation, Eq. (4), must be offset a distance of the cut-ter radius along the surface normal direction.Therefore, the tool center position at the kth Sam-pling can be calculated as follow^:'^^^^

where r is the cutter radius andS is the ruled surface.Orientation Calculation

The key idea of using five-axis machining forruled surfaces is controlling the milling cutter axisin parallel with the straight lines of the ruled surface.In other words, the cutter not only follows theboundary curve of the ruled surface but also orientsthe cutter axis in the direction of the straight linewhere the cutter is located. Elaborated first is theoff-line approach that varies cutter orientations lin-early between the intermediate points of the curveapproximated by linear segments. With this off-lineapproach, the interpolated orientation n between theintermediate points is as follows:

where nk and nk-lare the orientations of the two end-points of the approximated segment and correspondto the parameters uk and ukl, respectively. Thisapproach interpolates inaccurate orientationsbecause the changes in the tool orientations are notnecessarily linear. The orientation errors subse-quently affect the position accuracy (discussed fur-ther in the section below on inverse kinematicstransformation). As an example, the orientationerrors caused by the off-line interpolation of a ruledsurface in terms of the parametric formS(u, v) = (1 - v) cou) + v C1 (u),0 5 u , v 5 1x ( u )= 2.205u3- 7.425u2+0.22~ 5y (u )= -2.205u3 - 0.81u2+ 8.051z (u )= 20 (continued)


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Journal of Manufacturing ProcessesVol. 21No. 12000

Errors(degree)0.01 5

0.01250.01

0 00750.005

0 0025

0.2 0 4 0 6 0.8 1Parameter (u)

Figure 3Orientation Errors Caused by Inaccurate Off-Line Interpolation

were simulated and shown in Figure 3.The orienta-tion error is defined as the surface normal errorbetween the desired boundary and the interpolatedresult. The orientation error caused by the off-lineapproach is due to thehea r interpolating results ofEq. (6). The lower boundary curve Cl(u) in thisexample was approximated by 22 segments. Themaximum contour error for this off-line approxima-tion was 0.01 rnrn.The simulation result shows thatthe maximum orientation error is 0.016 degrees,which will subsequently cause a position error to43 pm. This is not acceptable for high-precisionmachining systems (see the section on translationmotions for details).In contrast, the real-time interpolator calculatesprecisely the tool's orientation at each sampling peri-od based on surface geometry. For each cutter's posi-tion, u = uk, the cutting tool orientation can be cal-culated by the first derivative of Eq. (I) with respectto v, as shown below:

where [Ox, O,, O,] is the tool orientation at the spe-cific tool position u = uk. As the same exampleshown in Eq. (7), the real-time interpolating resultfor the maximum orientation error caused by thereal-time interpolation is 0.00018 degrees. This cal-

culation was based on a CNC system with a feedrate of 20 mmlsec and a CNC sampling time of0.01 sec. The orientation error brings a positionerror of 0.49 pm, which is much more accurate thanthat of the off-line approach. In the off-lineapproach, one possibility to increase the interpolat-ing accuracy is increasing the number of approxi-mated straight lines; however, this will cause verylarge feed rate variations, which results in a roughsurface and makes machine ~ibrat ion.~The six variables [x, y, z, Ox, O,, O,] of Eqs. (5)and (8) are the solutions of the cutter's position andorientation at each sampling time. The derivation sofar is based on the part surface geometry and a spe-cific machining feed rate; therefore, these solutionsare machine independent. Subsequently, these sixvariables have to be transformed by inverse kine-matics into five reference inputs for the controllersof the five-axis machine.

Inverse Kinematics TransformationThe conversion of the trajectory planning out-puts into control references is performed by inversekinematics in the real-time interpolator. The inversekinematics transformation depends on the structureof the machine. Because the cutting tool is sym-metric, only five degrees of freedom are needed toreach any point with orientations in space. In prin-ciple, there are a considerable number of configu-rations with respect to a five-axis arrangement;however, in practice most five-axis machines havea structure based on a three-axis Cartesian machinewith the addition of two rotary axes. One of thesimplest ways of constructing a five-axis machineis installing rotary and tilting tables on the top ofthe x-y table of the machine and having the z-axismove up and down. In this case, there are only sixkinds of configurations, which are shown in Figure4. In the structures shown in Figures 4e and 4J; therotary axis ( A ) is installed in the same direction asthe Z-axis on the X-Y table and does not add adegree of freedom. With these structures, the tooldoes not have five degrees of freedom and five-face machining in one setup is impossible.Therefore, only the four configurations shown inFigures 4a through 4d are commonly used in five-axis machines. Alternatively, the two rotation axesmay be added on the spindle axis (Z-axis). Again,only four configurations out of the possible six are


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Journal ofManufacturing Proce ssesVol. 2mo. 1

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Figure 4Six Kinds of Co nfigurations for Five-Axis Machines.(a) through (d) are configurations usually appearing in tilt tables; (e) and (f) cannot machine five faces in one setup.


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Journal of Munufacrurzng Proc e~se sVol 2INo. 12000

Figure 5Table Motion for Controlling the Orientation.(a ) i s before cutting; (b) is during cutting

acceptable. In this paper, only the four configura-tions in Figures 4u through 4d are discussed; how-ever, the derivation procedu re can be applied to thecase of adding rotating axes on the Z-axis.The structure shown in Figure 4u is chosen todemonstrate the derivation of the solution for theinverse kinematics. Subsequently, a general solution

for the inverse kinematics of the most com mon fourconfigurations, Figures 4a through 4d, is also pre-sented in this paper.Rotation MotionsThe main idea behind five-axis machining forruled surfaces is keeping the cutting tool axis in par-allel with the straight lines of the ruled surfaces dur-ing machining. In e ach sam pling period, the straightline's position [x, y, z ] and o r i e n t a t i ~ n p ~ ,,, 0;larecalculated while maintaining a constant feed rate,based on Eqs. (4) an d (8). To obtain the maximumcontact between the cutter and the surface duringfive-axis machining, the ruled surface straight lineorientationlo,, O,, O Jh as to be rotated to the direc-tion of [0 ,0 , 11 that is the fixed cutter's direction, asshown in Figure 5a for a 2-D case. For the general3-D case, two simultaneous rotations along the Aand B-axis are required. For the structure shown inFigure 4a , the first rotation is of axis A at angle aabout the X-axis, and the second rotation is of angleb about the Z-axis. The resultant motions at theserotations are shown in Figure 56. The equations ofmotions can be o btained as follows:

where a is the rotating angle and b is the tiltingangle. Equation (9) represents three scalar equa-tions. Angles a an d b are obtained by solving thefirst two equations as follows:

where ( 1 0)

By substituting Eq. (10) into the third scalar equa-tion of Eq. (9), 0 ;+ 0; +O_? 1 is obtained, whichis always true b ecause Or,O,, and 0, are the direc-


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Journal o f Manufacturing ProcessesVal. 2iNo. 12000

t ional cosines of a uni t vector in Cartesiancoordinates.Translation Motions

By the rotations demonstrated in Figures 5a and5b, the correc t tool orientation w as achieved, but notthe correct position because the rotation moved thepart out of the paper p lane (that is, the X direction inFigure 5 ) , if both a and b axes are rotated simuita-neously. To machine the part surface on five-axismachines, both the tool orientation and positionmust be controlled. For easy illustration in the 2-Dplane, an example of a cylindrical surface machin-ing is selected to show the change in the tool posi-tion because of the motion of the rotary table. InFigure 6a, To is a position on the cu tter that contactswith the part surface at the current samp ling time; T,is the fixed cutter center position; and PC s the partcenter position. To is lying in the line at thismoment. P, (Figure 6a) is the required tool contactpoint on the part surface at the next sampling time,which can be calculated by Eq. (4). The requirementfor this particular machining is that the contactpoints between the tool and the part surface must bealways lying in the line . Therefore, to com-plete the machining for the next sampling period, P ,must be rotated to P,' (Figure 6b) and linearly trans-lated back to To by the distance . The distanceT,4 is called the pull-back distance. This translationmak es and and collinear, which satisfiesthe requirement for simultaneous motions. For the3 - 0 case based on the table configuration of Figure4a, the pull-back distance can be calculated by thefollowing equation:

where the detailed coordinate system is shown inFigure 7. P , R, W ,an d OT are the coordinate system sfor the part surface , rotary table, tilt table, and cutter

I IFigure 6lllustration of Pull-Back Distance Due to Rotation Motion.(a) before cutting, (b) rotating table to desired orientation,(c) result after rotations and translations

center, respectively. Th e five variables [X , Z 2, , b],shown in Eqs. (10) and (1 I), are the solutions for theinverse kinematics transform that provides the refer-ence inputs for the servocontrollers to accuratelycontrol the five-axis machine with the configurationof Figure 4a. From Eq. (1 l), it is shown that the tool


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Figure 8Example of th e Conic Surfacewhere (x, y, z) are the initial tool contact positions onthe conic surface; ( i , j, k) define the center of thebottom circle of the cone; h is the height of the conicsurface; r is the radius of the upper circle; d is theangle of the conic surface to be produced; and f isthe surface feed rate. The dimensions in this g-codeare in BLUs, where BL U equals 0.01 mm for trans-lation and 0.001 degree for rotation. According tothe conic surface in this example, fixed coordinatesare assum ed at the center o f the bottom circle; thus,the programmed g-code is as follows:

Example 2A compound ruled surface was machined w ith thereal-time interpolator on a five-axis machine. Acompound ruled surface is a surface constructed bymulti-ruIed surfaces. In this example, each ruledsurface is define d by two Bezier curves," and eachBezier curve is determined by four control points.Figure 9 shows an example of a Bezier curve withfour control points. For a ruled surface, eight con trolpoints are needed.The following new g-code, g52, was assigned todefine the eight control points for machining ruledsurfaces with flank milling:

The first four lines specify the upper ruled surface,and the other four lines define the lower ruled sur-

Figure 9Example of the Bezier Curve and Its Control Points

Figure 10Example of th e Compound Ruled Surfaceface. For machining a compound ruled surface, asequence of g52 codes have to be contained in thepart program. Figure 10 shows the example of thecompound ruled surface. The control points of thesurface are shown below, where dimensions are inBLUs (BLU = 0.01 mm),The first ruled surface:the upper Bezier curvepo~ntl (3800,2200,1000) po1nt2 (2150,-550,1000)

point3 (500,500,1000) po1nt4 (1050,1100,1000)the lower Bezier curvepo~ntl (3800,2200.0) po1nt2 (2 150,-1 100.0)point3 (- 1 58,330,O) point4 (360.1 100,O)

The second ruled surface:the upper Bezier curvepo~n tl (1050,1100,1000) po1nt2 (1 600,1650.1000)point3 (3580.1832.1000) point4 (3800,2200,1000)


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Journal oj Manufacturing ProcessesVol. 2Mo. I2000

the lower Bezier curvep o ~ n t l (360,1100,0) point2: (878,1870,O)polnt3: (28 10,1870,O) point4: (3800,2200,O)

The part program to machine this compound ruledsurface is shown as follows:

ConclusionsRuled surfaces are commonly used in industry forpart surface design, such as in automobile and air-plane components. To benefit from the advantagesof using five-axis machines to produce these parts,developing a precise, real-time interpolator for five-axis CNC machines is required. A real-time five-axis interpolator for machining ruled surfaces withflank cutters has been presented in this paper. Thereal-time approach interpolates accurate tool orien-tations and substantially gives precise tool positions.It consists of two algorithms: (1 ) a generic, machine-independent algorithm for trajectory planning togenerate the tool position and orientation at each

Figure I 1Demonstrated Part Surfaces Produced by Real-TimeApproach on a Five-Axis Machine

sampling period, and (2) a machine-dependent algo-rithm for inverse kinematics transformation thatgenerates corresponding axial position commands.Two demonstrated ruled surfaces have been suc-cesshlly produced from a five-axis milling machineimplemented with the developed real-time interpo-lator, as shown in Figure 11. Using the real-timefive-axis interpolator, the machining time requiredto produce the first demonstrated example (conicsurface) is 6 5 seconds. Only one g-code instructionis needed to describe this conic surface. If this partis produced by conventional three-axis machines,99g-code instructions are required. It took about 7minutes to finish this part surface with the machin-ing feed rate of 400 mrnlmin. To produce the seconddemonstrated example (general ruled surface), themachining time is 60 seconds by using the real-timefive-axis interpolator. It requires 16 g-code instruc-tions for this approach, as shown in the previous sec-tion. If the part is produced by a three-axis machine,hundreds of g-codes are needed.The real-time five-axis interpolator developedin this research assumes that the part surface isgouge-free. A gouging problem occurs when thecutter radius is larger than the radius of a concavesurface curvature or when the cutter collides withthe other part of surface. Several algorithms wereproposed to detect and avoid gougingHowever, these algorithms fit only the off-lineapproach because they require a very large compu-tation load. It is assumed, therefore, that to obtaina gouge-free surface the part surface will be exam-ined off-line before machining it with the real-timeinterpolator.References

1. Y Lee and T Chang, "Machined Surface Error Analysrs for 5-AxisMachlnlng," I t i t 1Jotrrnul ofProduc~lon e5earth (v34. nl, 1996), pp111-135

2. Y Lee, "Admiss~bleTool Orlentatlon Controi of Gouging Avordan~efo r AXIS Complex Surface Machlnlng." Comptrler 4ided Design (v29, n71997). pp507-52 13. Y Wang and X Tang, "FIV~-AXISC M dchlning of Sculp turedSurfaces:' Th e lnr 1 k > t ~ n u /fAdtoncrd M& Gchnology (v l5 , 1999). pp7- 144. H Renker. "Coll~ s~on-F ree ~ve-A XIS llllng of Tw~sted RuledSurfaces: Annalr of the CIRP (v421111991),pp457-4615. C Jensen and D Ander\on. "Accurate Tool Pldcernent dndOnentallon for Finlsh Surface Machlmng.' ASMF W~nterAnnudl Mrg(l 992), pp127- 1456. K Sanbandan dnd K Wang. "F~ve-Ax15Swept Volumes for C~raph~c

NC Slmulatlon dnd Ver~tlcdt~on,' SME 15th Deslgn Automdtlon Conf( D t v19-1. 1989), pp143-1507. W Wang and K Wdng, "Real-Trme Vent~catlonot Multi-AXISN CPrograms w~ th aster Graphr~s,"Ptoc of fP E t Int'l Conf on Robotrcz andAuromat~on I 986).pp 166-171


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8. R. Lin and Y. Koren, "Real-Time Interpolators for Multi-Axis CNCMach ine Tools," CIRP Journal ofM fg. Systems (v25, n2,1996), pp145-149.9. D. Yang and T. Kong, "Parametric Interpolator Versus LinearInterpolator for Precision CNC Machining," Computer Aided Design (v26,n3, 1994), pp225-233.10.J. Chou and D. Yang, "On the Generation of Coordinated Motion ofFive-Axis CNCJCMM Machines," Journal of Engg. for Industry (v114,Feb. 1992), pp15-22.11.Michacle E. Mortenson, Geometric Modeling (New York: WileyPress, 1985).12 . M. S hpitalni, Y. Koren, and C.C . Lo, "Real-Time Curve Interpolator,"

Computer Aided Design (v26, nl I, 1994), pp832-838.13.Y. Koren and R. Lin, "Five-Axis S urface Interpolators," Annals of theCZRP (v44/1/1995), pp379-382.14 . I. Faux and M. Pratt, Computational Geom etry for Design and Mfg.(New York: John Wiley & Sons, 1981).

Journal of Manufacturing ProcessesVol. 2m o. 12000

Authors' BiographiesRong-Shine Lin received his PhD in mechanical engineering andapplied mechanics from the University of Michigan in 1994. He worked forChrysler Corporation as a tool and process engineer in the Dept. ofAdvanced Manufacturing Engineering from 1994 to 1995. He was an assis-

tant professor of industrial engineering at Rutgers University from 1995 to1997. Currently, he is an assistant professor in the Dept. of MechanicalEngineering at National Chung Cheng University (Taiwan). His researchinterests include five-axis CNCs, CAD/C AM/CN C, and rapid prototypingYoram Koren received a BS c and an MSc in electrical engineering anda DSc (1970) in mechanical engineering from the Technion-Israel Instituteof Technology. He is the Paul G . Goebel Professor of MechanicalEngineering and Applied Mechanics at the University of Michigan. He isthe author of many publications on automated manufacturing and is theinventor of work covered by three US patents in robotics.

Ruled Surface Machining in 5 Axis Cnc Machine Tools

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