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1Proceedings of DETC99 :25 th Design Automation ConferenceSeptember 12-15, 1999 in Las Vegas, NevadaDETC/DAC-8579Robustness Evaluation of a Mini a turized Machine ToolNozomu MishimaMechanical Engineering LaboratoryAgency of Industrial Science and TechnologyMinistry of International Trade and Industry1-2 Namiki, Tsukuba, Ibaraki, JAPAN+81-298-58-7227, Fax+81-298-58-7201, email: m4430mel.go.jpKousuke Ishii (Corresponding Author)Department of Mechanical EngineeringStanford UniversityTerman 551, Stanford, California 94305(650) 725-1840, Fax (650) 723-3521, e-mail ABSTRACTThis paper applies the method of robust design to machine tooldesign . The new design focuses on miniaturization that providessignificant for energy and space saving . Our approach combinesa n analytical procedure representing the machining motion s of amachine tool (form-shaping theory) with procedures for robustdesign . The effort identifies the design parameters of a machinetool that significantly influence the machining tolerance andleads to a general design guid e line s for robust mini a turization.Further, this research applies the Taguchi method to the form-shaping function of a prototype miniature lathe. The analysisaddresses five machine tool dimension s as control factors , whiletreating local errors in the machine structure as noise factors.The robustness study seeks to identify the importance of eachfactor in improving performance of the machine tool. The resultshows that the thickness of the feed drive unit a ffects theperformance most significantly. A mong the local errors ,straightness error of the same feed drive unit has a criticalimportance .1. I NTRODUCTIONMachine tools must be capable of machining product s in avariety of different shapes and to the required tolerance . To thisend, m achine tool designers have adopted highly rigid andprecise structural components. However, there has not beensufficient discussion of such basic concepts as a basis forchoosing a particular structure or dimensions when designing amachine tool. Because machine tool design is rather experiencebasis, a fundamental design change is often difficult. Forinstance, the machine tools used to produce watches and otherprecise miniatur e products is ex cessively large compared withthe target components . The machine tool community has notaddressed t he question of whether such size is necessary formachining to the required tolerance . In terms of energyefficiency , a large machine represents considerable waste.Miniaturization of machine tools to size compatible to thetarget product s without compromising machining tolerancelead to enormous savings in energy, space , and resources.Recent years have seen the proposal of a factory comprised ofsuch miniaturized machine tools , a “ micro - factory ”(Kawahara et al, 1997 ) , demonstrated a prototype ultra-miniature machine tool (Kitahara, 1996) that serves as thebasic unit of such a factor y. However, the design of the micro-lathe did not involve a n in-depth evaluation of the required sizeand structure . Machine tool designers will need a generalguideline to appropriately reduce the size of machine tools.This study combines an analytical procedure representing themachining motion s with a procedure for robust design . Theeffort identifies critical design parameters that have significantinfluence on the machining tolerance . The re exist an analyticalprocedure known as form-shaping theory (Reshtov and2Portman, 1988) and stud ies on design evaluation of machinetools using the theory or other numerical representation(Slocum, 1992; Mou, 1997). The robust design tool called theTaguchi method is well - known (Taguchi, 1994; Feng andKusiak, 1997). This paper applies this method to analyze theeffect that the design parameters of aforementioned micro-lathehave on its machin ing performance , and proposed a procedurefor design evaluation. These results lead to the combination ofdesign parameters that optimize the performance of existingmicro-lathe . The paper also relate s these results to guidelinesfor the future systematic miniaturization of machine tools inaccordance with the product size and required machiningtolerance .NOMENCL ATUREk = number of componentsi = component number indexS i = local coordinate system assigned for component iA = homogeneous transformation matrixj i = the kind of transformation between S i and S i+ il i = motion amount for the transformation between S i and S i+ iA(i)( j i )( l i ) = matrix indicates a transformation between S i andS i+ i which has kind j i and motion amount l i0rr = form shaping functiontrr = tool shape vector0err = form shaping function i ncluding errorsA ei = the transformation errors between S i and S i+1 0rr = form shaping error functionls = thickness of the work holderly = shaft - Z feed unit distanceld = shaft - motor distancelz = thickness of the feed unitslt = length of the cutting toolR = radius of the workpieceh = height of the workpiecev = rotational speed of the shafti = angular error about x axis in the transformation betweenS i and S i + ii = angular error about y axisi = angular error about z axisxi = parallel translation error to x axis in the transformationbetween S i and S i+ iyi = parallel translation error to yzi = parallel translation error to zx = deviation from the target to x directiony = deviation from the target to y directionz = deviation from the target to z directionn = number of runf em = average value of the objective functionf e i = value of the objective functionV = varianceSn = Signal-to-noise ratio2. REPRESENT ATION OF FORM-SHAPING MOTIONS2.1 Form-shaping functionThe machine tool structure is considered as a chain ofdirectly linked rigid components extending from the workpiecethrough the cutting tool, fr om component 0 to k , as illustratedin Fig. 1. A n orthogonal coordinate system S i is defined toelement i ( i = 0 to k ). The transformation from S i to S i +1represents a coordinate transformation between component s.Machine tool elements CoordinatesWorkpiece S 0| | | S i-1| |Element i S i| | S i+1| | |Tool S kFig.1 Corresponding elements and local coordinatesWe represent these respective coordinate transformationsby homogeneous transformation matrices ( Paul, 1981) A i . Inthe case of an ordinary machine tool, A i is represented by one ofeither parallel transformation along the x, y or z axes orrotation around the x, y , or z axes , or combination of these .Each of these six coordinate transformations is assigned anumber to distinguish them, with parallel movement along thex-axis being 1, and so on. When the homogeneoustransformation matrices A i are represented by thetransformations j i , (=1 to 6), and the amounts of motion(distance if a parallel transformation, a rotation angle ifrotational motion) are represented by l I we define A(i)( j i )( l i ) asthe expressions of the matrices. V ector 0rr represents t herelative displacement between workpiece and tool, and in thecoordinate system given for the tool, the tool shape vector by trris defined . The relation between 0rr and trr is as given by eq.(1), and 0rr is the definition of the form-shaping function .tkkiiiirlikAljiAljiAljAr rr)()(1()()(1()()()()(0(1111000+=(1)2.2 Form-shaping error functionsIn actual machine tools there are assembly tolerances,thermal deformation, wear, deformation caused by externalforces, and many other sources of error. In order to describeactual form-shaping motions , one must consider these errors.Such errors may appear as transformations within an element,as in deformation of a component, or they may appear as atransformation between elements, as imperfect straightnesserror of guide ways. However, errors may for conveniences3sake be treated as either kind of transformation. In this study,we treat such errors in transformations between elements. Wedefine another homogeneous transformation matrix A i togenerally represent transformation error between elements S iand S i +1 . By inserting the error component matrix A i betweenA(i)(j i )(l i ) and A ( i +1)( j i +1 )( l i +1 ) in eq. (1), the form-shapingfunction including errors, 0err is written as follows.tkkkiiiiirAljkAljiAAljiAAljAr rr+=+111110000)()(1()()(1()()()()(0(eeee (2)The form-shaping error function 0rr , expressing the shiftfrom target values of the form-shaping motion of the machinetool , is defined as the difference between the form-shapingfunction not containing errors that containing the errors .000 rrrrvr =e (3)3. APPLIC ATION TO MINI ATURIZED MACHINE TOOLThe previous section explained the general procedure forderiving the form-shaping function of a machine tool. In orderto derive a strategy for miniaturized design of machine tools,which is the objective of this study, we analyze an existingmicro-lathe, and attempt to evaluate the robustness of itsdesign. By this means we can identify those design parametersand local error sources of the machine tool that have a majorinfluence on the machining performance of machine tools withthe same axial configuration as the micro-lathe.3.1 F orm-shaping function of the micro-latheF i g.2 shows t he construction of the micro-lathe studied inthis paper . In response to requests to reduce the overallmachine dimensions as much as possible, in this machine toolthe spindle unit moves with respect to the fixed tool to performcutting and feed motion s.Fig.2 Schematic view of the micro-latheThe machine is extremely small, with the height, lengthand width dimensions each around 1 inch or so. The machine iscapable to machine small workpiece made from brass or plasticand the machining tolerance is better than an ordinal lathe. Asshown in the figure, w e define five different design parametersfor this machine tool (Table 1) . In addition, although not designparameters, the table define s the height h and the radius R ofthe cylindrical workpiece as variables necessary for analysis.Using these design parameters, Fig. 3 shows the elements of themicro-lathe of Fig. 2, the corresponding local coordinatesystems, and the homogeneous transformation matrices used totransform between each local coordinate system.Table 1 Design parameters of the micro-latheName of the parameters VariableWork holder thickness lsShaft - Z feed unit distance lyShaft - motor distance ldFeed unit thickness lzCutting tool length ltElements Coordinates H TMWorkpiece S 0| | - A (0)(3)(0)S haft S 1| | - A (1)(3)( -ls )Z feed unit S 2| -| A (2)(2)( - ly )A (2)(1)( ld )X feed unit S 3| - A ( 3 )( 2 )( -lz )Base S 4-| | A (4)( 1 )( R+lt-ld )A (4)( 3 )( lz+ly )Tool S 5Fig.3 Corresponding elements, coordinatesand HTMfor the micro-latheAs explained in the preceding section, the form-shapingfunction of the machine tool is defined by linear super -positioning of the transformation matrices between localcoordinates of the machine tool, and then taking the tool shapevector. We express a multi-directional transformation as acombination of homogeneous transformation matrices. Thus,the form-shaping function 0rr of the micro-lathe may beexpressed as follows:trlylzAldltrAlzAlshAldAlyAlsAArrr)(3)(4()(1)(4()(2)(3()(3)(3()(1)(2()(2)(2()(3)(1()0)(3)(0(0+=(4)lzX-Z feed unitShaftls lyldltSlider conectorSliderBaseWork holderPulleyBeltShaft drive unitMotorTool restCutting toolWorkpieceZ XY43.2 Form-shaping function including errorsThere are six elements of the machine tool, including theworkpiece, as shown in Fig. 3. In other words, there exist fivetransformations between elements. We define, f or each of thesetransformations , three parallel-translation errors along the threeorthogonal axes and three rotation errors about the three axes.T hese transformation errors provide express ions of errors in themachine tool arising from thermal deformation, wear, motionerror, tolerances in machined parts, and other causes. Todistinguish these errors from the overall machining error, weshall refer to them as local errors. We let the rotation errorsabout the x, y, z axes in the transformation between element iand i+ 1 be respectively i , i , i, and the parallel translationerrors be xi , yi , zi . Table 2 below lists the defined local errors.The form-shaping function of the micro-lathe including errors ,0err may be expressed as eq. ( 5 ).Table 2 Defined local errorsError parametersH TMRotational error s Transitional error sA e0 a0 , b0 , g0 dx 0 , dy 0 , dz 0A e1 a1 , b, g1 dx 1 , dy 1 , dz 1A e2 a2 , b2 , g2 dx 2 , dy 2 , dz 2A e3 a3 , b3 , g3 dx 3 , dy 3 , dz 3A e4 a4 , b4 , g4 dx 4 , dy 4 , dz 4trAlylzAldltRAAlzAlshAAldAlyAAlsAAArrr+=43210)(3)(4()(1)(4()(2)(3()(3)(3()(1)(2()(2)(2()(3)(1()0)(3)(0(0eeeeee(5)3.3 Form-shaping error functionTo simplify, we introduce the following assumptionsregarding the local errors defined in the preceding subsection.(1) In this study we do not include the workpiece andspindle mounting errors, 0 , 0 , 0 , x0 , y0 , z0 .(2) The errors of the main shaft may both be rotationallysymmetrical , so we set y1 = x1 and 1 = 1 .(3) The rotational error of the spindle about its own axis hasno practical meaning, so that 1 is omitted.(4) The same feed units are used for driving in both the Xand Z directions, so we set 3 = 2 , 3 = 2 , 3 = 2 , x3 = z2 ,z3 = x2 , and y3 = y2 .(5) In mounting the tool on the tool rest, we consider onlythe rotational errors 4 , 4 , 4 , and do not consider theparallel-translation errors x4 , y4 , z4 .Using these assumptions, the number of independent localerrors decreased to 12. Further, in the case of this micro-latheeq uation ( 7) gives the tool-shape vector. Tt ltr 100=r (6)From eqs. ( 3 ), ( 5 ) and ( 6 ), the form-shaping error functionfor the micro-lathe analyzed in this study can be written as ineq. ( 7 ). Each element of the vector indicates error amount to x,y and z direction.+=0)()(2)()(2)()(42221212421222122122210ltlylylzldRRltlshldRlylshlylzrzzxyxzxxbagbadddgaagaddgbaadddr (7)4. APPLIC ATION OF THE TAGUCHI METHODIn eq. ( 7) of the previous section, when all variables areknown the form-shaping error function 0rr can be calculated;but when only the overall machining tolerance is specified, theinverse problem of eq. ( 7) cannot be solved. Hence, thisequation indicates the degree of machining error anticipated,given known local errors, but one cannot readily use it toestimate in advance the machining errors or assembly errors ofcomponent elements at the time of design. On the other hand,in order to obtain machining tolerance that is stable under avariety of machining conditions, a method is needed forobtaining a design which is robust with respect to unknownlocal errors. The Taguchi method is widely used in the fields ofexperimental design and quality engineering, and provides anenvironment for robust design. This study uses the Taguchimethod to evaluate the dimensional effect imposed onmachining errors by the design parameters of the machine tool,when local errors are unknown . Simulations were performedapplying the method to the form-shaping error function.4.1 Control factors and noise factorsThe Taguchi method allows us to calculate combinations ofvalues of control factors to optimize an evaluation function,given noise factors fluctuating within given ranges. In thisstud y, the primary objective is to determine the effect onmachining performance of design parameters , when some localerrors exist in the various components of the machine tool .Therefore , it is appropriate to take the design parameters to becontrol factors and the local errors to be noise factors. Here thedimensions R, h of the workpiece shape are given as fixedvalues and act as constraining conditions. Conversely, bysetting the local errors to be the control factors, one can identifythose parts that require special care in order to improve themachining precision of a miniaturized machine tool.4.2 Objective functionSince, this study focuses on the error amount (distance) ata point, among those points on the workpiece, at a targetdistance R from the shaft center and height h from the workholder surface. We define x , y , z to be the error amounts in5each axis direction when the t ool and t he workpiece aredisplaced relative to each other with this point as the target .Then, we take the absolute length of these errors,( x 2 + y 2 + z 2 ) 1 /2 , to be the evaluation function.4.3 Relation between the factorsPrior to setting the range of variation of design parametersand local errors, fixed relations between local errors and designparameters must be introduced. In miniaturization of a machinetool, if we assume that there is no relation between designparameters and local errors and that local errors are alwaysconstant, then machining precision should be constant andindependent of dimensions. On the other hand, if local errorsare proportional to the design parameters, then machiningprecision should also be nearly proportional to the geometricaldimensions. However, the machining precision of a micro-latheis known to be superior in absolute terms, but inferior inrelative terms to that of an ordinary lathe 2 .This trend is due to the inherent limits on the type andprecision of mechanisms used when reducing the size of amachine tool . Consequently , some parts of the machine sufferloss of precision when reduced in size. For instance, whendimensions are reduced to the order of the micro-lathe,considerable care is required in tool mounting due to the shorttool length. In addition, the small-diameter bearing adopted dueis disadvantageous in terms of precision compared withordinary bearing.There ar