四工位全自动回转工作台设计【定位销式分度工作台】【三维SW模型及CAD图纸】
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ORIGINAL ARTICLEPostprocessor development of a five-axis machine toolwith nutating head and table configurationChen-Hua She&Zhao-Tang HuangReceived: 21 August 2006 /Accepted: 7 June 2007#Springer-Verlag London Limited 2007Abstract The postprocessor is an important interface thattransforms cutter location data into machine control data, andin a five-axis machine tool is highly complex because thesimultaneous linear and rotary motions occur. Since mostworks of the five-axis postprocessor method have dealt onlywith the orthogonal machine tools configuration, this studypresents a postprocessor scheme for two types of five-axismachine tools, each with a nutating head and a table whoserotational axes are in an inclined plane. The benefit of such aconfiguration is that it allows switching from vertical tohorizontal machining by a single machine. The generalanalytical equations of NC data are obtained from the forwardand inverse kinematics and the homogeneous coordinatetransformation matrix. The linearization algorithm for thepostprocessor is developed to ensure the machining accuracy.Thepresentedalgorithmisimplementedusingawindow-basedfive-axis postprocessor with nutating axes, and programmed inBorland C+ Builder and OpenGL. A simulation is performedusing solid cutting software and a trial-cut experiment wasconducted on a five-axis machine tool with a nutating table toelucidate the accuracy of the proposed scheme.Keywords Postprocessor.Five-axismachining.Form-shapingfunction.Nutatingaxis1 IntroductionFive-axis machining is becoming increasingly used bymachine tool users, especially in machining complexfreeform surfaces. The conventional five-axis machine toolhas three orthogonal linear axes and two rotary axes. Therotary axes are typically orthogonal to each other and thecentre line of the rotary axis is parallel to the direction ofthe linear axis. Various machine tool builders, such asMakino, Ingersol and Deckel Maho, incorporate a nutatinghead or a nutating table in the machine tools to improvetheir versatility and flexibility. The word of “nutating”means oscillatory motion about the axis of a rotating body,which is similar to the slow spinning of a coin on a table. Afive-axis machine tool with a nutating unit has a rotationalaxis in an inclined plane 1, and not parallel to the linearaxis, providing the advantage that allows the cutting tool toorient itself toward any angle within a hemisphere 2, 3.Such machines can move continuously between thehorizontal and vertical positions in a single setup on thesame machine. The nutating head provides very useful inmanufacturing aerospace parts because it has no motors onthe head, and is more rigid. The motors for the spindle areon the machine and the motion is transferred to them byhollow shafts and gears 4.Because the linear and rotary axes move simultaneouslyon a five-axis machine, the derivation of the five-axisprogram is more complex than that of the three-axisprogram. A postprocessor must be utilized to translate thecutter location (CL) data from the CAM system into themachine control data. Although the advanced controllerscan accept the CL data to machine the workpiece in real-time without the need of postprocessor 5, they arerelatively expensive and not commonly used in mostindustries. The methods of developing multi-axis postpro-Int J Adv Manuf TechnolDOI 10.1007/s00170-007-1126-5C.-H. She (*)Department of Mechanical and Computer Aided Engineering,National Formosa University,64 Wen-Hua Road, Huwei,Yunlin 632, Taiwan, Republic of Chinae-mail: chshe.twZ.-T. HuangDepartment of Mechanical and Automation Engineering,Da-Yeh University,112 Shan-Jiau Road, Da-Tsuen,Chang-Hua 515, Taiwan, Republic of Chinacessors can be mainly divided into three categories -graphical 6, numerical iterative 7 and coordinatetransformational 810. Since the coordinate transforma-tion method yields the analytical equation of NC data mostefficiently, it has been adopted extensively in recent work.However, almost all of these approaches involve postpro-cessor methods for five-axis machine tools with orthogonalrotary axes. Relatively few studies have addressed non-orthogonal configuration. For example, the authors havedeveloped the postprocessor for the spindle-tilting typefive-axis machine tool with a nutating head 11. Recently,Sorby 12 has presented a closed-form solution for a table-tilting type five-axis machine tool with a nutating table.However, this solution exhibits some limitations. Forexample, the offset vectors such as from the workpieceorigin to the rotary table and from the secondary rotary tothe primary rotary are not defined, and the angle ofinclination of the nonorthogonal axis is fixed at 45 degrees.This study develops a postprocessor for two five-axismachine tools each with a nutating head and table config-uration. Based on the homogeneous coordinate transforma-tion matrix, the general analytical equations of NC data areobtained from the forward/inverse kinematics and themachine tools form-shaping function matrix. The deter-mined equation is in general form because the rotary axes areassumed not to intersect each other; the angle of inclinationof the nonorthogonal axis is variable, and the offset vectorfrom the origin of the workpiece to the rotary table isdefined. Moreover, the linearization algorithm of the post-processor is developed to ensure the machining accuracy.A window-based postprocessor is developed and a graph-ical interface that dynamically displays the surface model andthemotions ofallofthe axesoftheconfiguredmachinetoolispresented to help the user to input relevant parameterscorrectly. Additionally, the generated NC data are verifiedusingthecommercialsolidcuttingsoftwareVERICUT 13and a machining experiment is conducted on a five-axismachine tool with a nutating table to confirm the effective-ness of the proposed postprocessor methodology.2 Configuration and modeling of five-axis machine toolMost five-axis machine tools have two rotary axes as wellas the conventional X, Y and Z axes. Following Sakamotoand Inasaki 14, the configurations of five-axis machinetools can be categorized into three types: spindle-tilting,table-tilting and table/spindle-tilting. Commercial machinetools with the nonorthogonal configuration, as shown inFig. 1, are also of three types. Figure 1 (a) shows thespindle-tilting type with a nutating head, such as theMakino MAG3 2, which is designed with a rotary axis(C-axis) behind a nutating head that rotates about the B-axis. Figure 1 (b) displays the table-tilting type with anutating table, such as the Deckel Maho DMU 70 eVolution15, which has two rotary axes on the table, and one rotaryaxis (C-axis) is parallel to the Z-axis while the non-orthogonal rotary axis is inclined at an angle to the C-axis.Figure 1 (c) presents the table/spindle-tilting type with anutating head, such as the Deckel Maho 200P 15,inwhichone rotary table (C-axis) is on the table and the nutatingrotary head (B-axis) is on the spindle. Since the authors havealready presented the spindle-tilting postprocessor with anutating head 11, this study focuses on developing thepostprocessors with the other two configurations.A five-axis machine tools can be regarded as amechanism with serially connected links with revolute orprismatic joints. Forward kinematic equations must beestablished to describe mathematically the motion of thecutting tool in relation to the workpiece. The fundamentalcoordinate transformation matrices, including the transla-tion matrix Trans and the rotation matrix Rot 16, areintroduced. The translation matrix Trans can be expressedas follows:Transa;b;c 100a010b001c0001266437751where Trans(a, b, c) implies a translation given by thevector a i+b j+c k.The general rotation transformation matrix should beused to describe the rotation of the nutating unit. Thecoordinate system is assumed to rotate through an angle offwaround any arbitrary vector W=Wxi+Wyj+Wzk; therotational transformation matrix can be expressed as:Rot W;w W2xVw CwWxWyVw? WzSwWxWzVw WySw0WxWyVw WzSwW2yVw CwWyWzVw? WxSw0WxWzVw? WySwWyWzVw WxSwW2zVw Cw00001266437752Int J Adv Manuf Technolwhere “C” and “S” are cosine and sine functions,respectively, and Vfw=1Cfw.3 Postprocessor3.1 Table-tilting type with a nutating tableFigure 2 depicts relevant coordinate systems for thisconfiguration. The coordinate system for the workpiece isOwXwYwZwwhile the system OtXtYtZtis attached to thecutting tool. Since the two rotary axes are assumed not tointersect each other, a common normal line is mutuallyperpendicular to both axes. The common normal line inter-sects with the C-axis and B-axis at two points, RC and RB.The offset vector Lxi+Lyj + Lzk is determined from theorigin Owto the pivot point RC, whereas the offset vectorMxi+Myj+Mzk is calculated from the pivot point RC tothe pivot point RB.Since the structural elements of the machine toolcomprise the C rotary table, the B nutating rotary table,the machine bed, the X linear table, the Y linear table, the Zlinear table, the spindle head and the cutting tool. Therelative position and orientation of the cutting tool withrespect to the workpiece can be determined sequentiallystarting from the workpiece and ending at the cutting tooland is referred to as the form-shaping function 17. TheCBXYZaXYZCBcBCYXZbFig. 1 Configuration for five-axis machine tool with nutating head andtable. a spindle-tilting type with a nutating head. b table-tilting typewith a nutating table. c table/spindle-tilting type with a nutating headtOtXtYtZBRBwOwXwYwZC xyzLLL+ijkRCOffset vectorxyzMMM+ijkOffset vectorFig. 2 Coordinate systems of table-tilting type configurationInt J Adv Manuf Technolform-shaping function of this machine tool can bemathematically expressed in matrix form as follows:Trans Lx;Ly;Lz?Rot z;?zTrans Mx;My;Mz?Rot W;?wTrans Px;Py;Pz?0 00 01 00 1266437753wherePx, Pyand Pzdenote the relative translation distances ofthe X, Y and Z linear tables, respectively. The termsfzandfwrepresent the angles of rotation for the C-axis and the B-axis, respectively. The positive rotation is in the direction of anadvancing right-hand screw about the +C and +B axes.Equation (3) specifies the form-shaping function matrix of thismachine tool and the joint parameters Px, Py, Pz,fzandfwshould be determined by the inverse kinematics. The firststep is to calculate the required rotary angles to yield the toolorientation, and the second is to calculate the required positionin relation to the linear axis to determine the position of thecentre of the tool tip using the known rotary angles.Whenthe CLdataincludingthe positionofthecentreof thetool tip Qxi Qyj Qzk and the tool orientation Kxi Kyj Kzk are known, the CL data can be expressed in thematrix form as follows:KQ01?KxKyKz0QxQyQz1266437754Since both Eq. (3) and Eq. (4) represent the samerelationship between the cutting tool and the workpiece, thedesired joint parameters can be determined by equatingthese two matrices. Equating the CL data matrix and theform-shaping function matrix, and taking the correspondingelements of the two matrices yield the following equations:KxKyKz026643775CzWxWz1 ? Cw ? WySw? SzWyWz1 ? Cw WxSw?SzWxWz1 ? Cw ? WySw? CzWyWz1 ? Cw WxSw?W2z1 ? Cw Cw0266437755QxQyQz126643775PxCzW2z1 ? Cw Cw? SzWxWy1 ? Cw ? WzSw?PyCzWxWy1 ? Cw WzSw? SzW2y1 ? Cw CwhinoPzCzWxWz1 ? Cw ? WySw? SzWyWz1 ? Cw WxSw?Lx CzMx SzMyPx?SzW2x1 ? Cw Cw? CzWxWy1 ? Cw ? WzSw?Py?SzWxWy1 ? Cw WzSw? CzW2y1 ? Cw CwhinoPz?SzWxWz1 ? Cw ? WySw? CzWyWz1 ? Cw WxSw?Ly? SzMx CzMyPxWxWz1 ? Cw WySw? PyWyWz1 ? Cw ? WxSw?PzW2z1 ? Cw Cw? Lz Mz126666666666666666666666664377777777777777777777777756Int J Adv Manuf TechnolThe joint anglesfzandfwshould be determined first.Equating the corresponding third element in Eq. (5) yieldsthe following B-axis angle:B w arccosKz? W2z1 ? W2z?0 ? w? 7Notably, there is another possible solution for B-axisangle in the range of ? ? w? 0, which can be obtainedas follows:B w ?arccosKz? W2z1 ? W2z?8If the operating range of the nutating table is in the rangebetween and 0, the solution should be modified asshown in Eq. (8). On the other hand, if the operating rangeof the nutating table meets the two possible solutions, theshortest rotational angle movement of the nutating table isusually chosen in the algorithm.Furthermore, equating the corresponding first andsecond elements in Eq. (5) and solving the simultaneouslinear equations for Sfzand Cfz, yield:Sz?KyWxWz1 ? Cw ? WySw? KxWyWz1 ? Cw WxSw?WxWz1 ? Cw ? WySw?2 WyWz1 ? Cw WxSw?29CzKxWxWz1 ? Cw ? WySw? KyWyWz1 ? Cw WxSw?WxWz1 ? Cw ? WySw?2 WyWz1 ? Cw WxSw?210Since the denominators in Eqs. (9) and (10) are the sameand always positive, the C- axis angle can be determined asfollows:C z arctan2?KyWxWz1 ? Cw ? WySw? KxWyWz1 ? Cw WxSw?;KxWxWz1 ? Cw ? WySw?KyWyWz1 ? Cw WxSw? ? z? 11where arctan2(y,x) is the function that returns angles in therange ?p ? q ? p by examining the signs of both y and x16.In addition, comparing the corresponding elements ofthe matrix on both sides of Eq. (6) yields three simulta-neous equations in three unknowns Px, Pyand Pz. Theprogram coordinate system is assumed to coincide with theworkpiece coordinate system. Accordingly, the expressionsof the desired NC data (denoted as X, Y and Z) can beobtained by considering the two offset vectors Lxi+Lyj+Lzk and Mxi+Myj+Mzk, and are expressed as follows:X Px Lx Mx Qx? CzMx SzMy Lx?CzW2x1 ? Cw Cw? SzWxWy1 ? Cw ? WzSw? Qy? ?SzMx CzMy Ly?SzW2x1 ? Cw Cw? CzWxWy1 ? Cw ? WzSw? Qz? Mz Lz? WxWz1 ? Cw WySw? Lx Mx12Int J Adv Manuf TechnolY Py Ly My Qx? CzMx SzMy Lx?CzWxWy1 ? Cw WzSw? SzW2y1 ? Cw Cwhino Qy? ?SzMx CzMy Ly?SzWxWy1 ? Cw WzSw? CzW2y1 ? Cw Cwhino Qz? Mz Lz? WyWz1 ? Cw ? WxSw? Ly My13Z Pz Lz Mz Qx? CzMx SzMy Lx?CzWxWz1 ? Cw ? WySw? SzWyWz1 ? Cw WxSw? Qy? ?SzMx CzMy Ly?SzWxWz1 ? Cw ? WySw? CzWyWz1 ? Cw WxSw? Qz? Mz Lz? W2z1 ? Cw Cw? Lz Mz143.2 Table/spindle-tilting type with a nutating headThe table/spindle-tilting type configuration has one rotaryaxis on the table and one nutating rotary axis on the spindle.Figure 3 illustrates two pivot points RC and RB on the C andB axes, respectively. The pivot point RC is located arbitrarilyon the C-axis and the pivot point RB is chosen to be theintersection of the nutating rotary B-axis and the axis of thecutting tool. The offset vector Lxi +Lyj +Lzk is calculatedfrom the origin Owto the pivot point RC and the effectivetool length, Lt, represents the distance between the pivotpoint RB and the cutter tip centre. The form-shapingfunction matrix of this configuration can be obtained bythe coordinate transformation matrices:Trans Lx;Ly;Lz?Rot z;?zTrans Px;Py;Pz?Rot W;w00001 ? Lt012664377515Equating Eq. (4) and Eq. (15) leads to the followingequations:KxKyKz026643775CzWxWz1 ? Cw WySw? SzWyWz1 ? Cw ? WxSw?SzWxWz1 ? Cw WySw? CzWyWz1 ? Cw ? WxSw?W2z1 ? Cw Cw02664377516QxQyQz126643775?CzWxWz1 ? Cw WySw? SzWyWz1 ? Cw ? WxSw?LtCzPx SzPy Lx? ?SzWxWz1 ? Cw WySw? CzWyWz1 ? Cw ? WxSw?Lt?SzPx CzPy Ly? W2z1 ? Cw Cw?Lt Pz Lz1266666666666643777777777777517Int J Adv Manuf TechnolThe joint parameters can be evaluated using the sameprocedure similar to the table-tilting configuration. Notably,the reference driving point of NC data in this configuration isassumedtobethepivotpointRB.Thisdefinitionisadoptedtoboth the spindle-tilting and table/spindle-tilting type config-urations, and is consistent with those used in most of thecommercial post-processor software packages. The completeanalytical equations for NC data can be expressed as:B w arccosKz? W2z1 ? W2z?0 ? w? 18C z arctan2 ?KyWxWz1 ? Cw WySw?KxWyWz1 ? Cw ? WxSw?;? KyWxSw? WyWz1 ? Cw?KxWySw WxWz1 ? Cw? ? z? 19X Lx Px LtWxWz1 ? Cw LtWySw SzLy? Qy? CzLx? Qx Lx20Y Ly Py LtWyWz1 ? Cw ? LtWxSw? CzLy? Qy? SzLx? Qx Ly21Z Lz Pz LtW2z1 ? Cw LtCw Qz223.3 Linearization problemTheoretically, the CAD/CAM system generates the CL databased on the assumption that the cutting tool moves linearlybetween two successive points. However, the actual toolmotion trajectory with respect to the workpiece is not linearand becomes curved since the linear and rotary axes movesimultaneously. The curved path deviates from the linearlyinterpolated straight line path between successive pathpoints, and this problem is known as the linearizationproblem. An algorithm must be developed to solve thisproblem.Assume that Pn, Pn+1and Pn+2are three continuousadjacent points in CL data, plotted in Fig. 4. The vector ofPnin matrix form can be expressed as Qn,xQn,yQn,zKn,xKn,yKn,z, where Qn,x, Qn,yand Qn,zare the components ofthe position of the center of the tool tip, and Kn,x, Kn,yandKn,zare the components of the tool orientation. Thecorresponding machine NC code of Pnis Mn=XnYnZnBnCn. As the five axes move simultaneously from thecurrent position Pnto the subsequent position Pn+1, eachaxis is assumed to move linearly between the specifiedpoints 18. Therefore, each point in the actual curved pathcan be expressed as follows:Mm;t Mn t Mn1? Mn23where t is a dummy time coordinate0 ? t ? 1. Thecorresponding CL data Pm,tfor Mm,tcan be determined bythe forward kinematics equations, e.g. Eqs. (5) and (6) forthe table-tilting type and Eqs. (16) and (17) for the table/spindle-tilting type. Moreover, each point in the ideal lineartool path can be determined as follows:Pn;t Pn t Pn1? Pn24wOwXwYwZtOtXtYtZCB xyzLLL+ijkRCRBOffset vectortLFig. 3 Coordinate systems of table/spindle-tilting type configurationInt J Adv Manuf Technol,nnP M,n tn tPM,m tm tMP11,nn+PM1,1,ntnt+PM1,1,mtmt+MP,n+2n+2PMInterpolated tool pathActual curved tool pathIdeal linear tool path,n tdFig. 4 Linearization problem inmulti-axis machining abFig. 5 Initiating dialog for the developed postprocessor. a table-tilting type. b table/spindle-tilting typeInt J Adv Manuf TechnolThe distance between Pm,tand Pn,tdenoted as dn,tformsa chordal deviation. If the maximum deviation (dn,t)maxexceeds the prescribed tolerance, then the additionalinterpolated CL data Pn,tshould be inserted into the originalCL data. Theoretically, the numerical iterative method forcalculating (dn,t)maxmust be adopted. Practically, the middlepoint, t = 0.5, is often selected as the candidate point 10.After the intermediate point Pn,thas been inserted, thecorresponding machine NC code can be generated.4 Discussion1.The main characteristic of the nutating rotary axisconfiguration is the continuous motion between thehorizontal and vertical positions in a single setup onthe same machine. In the current configuration of thecommercial machine tool, the angle of inclination ofthe nutating rotary axis is 45 degrees. This fact can beexplained by the equations derived above. The table-tilting type is used as an example. Equation (5)represents the tool orientation in relation to theworkpiece. The tool orientation relative to the work-piece in the initial position, where the table surface ishorizontal, can be determined by substitutingfz=fw=0 into Eq. (5), and is given by 0 0 1 0T. The nutatingrotary axis is assumed to rotate around X-axis by anangle so that the components of the vector W are Wx=0, Wy=S and Wz=C. Substituting the aboveconditions into Eq. (5), and settingfz=0 and KxKyKz0T=0 1 0 0Tfor the table surface in the verticalposition yields the following equation:0?10026643775SSw?SC 1 ? CwC2 1 ? CwCw02664377525The solutions to Eq. (25) for andfware =/4 andfw=. Therefore, the table surface can be set in the verticalposition when the table is rotated through an angle aboutthe nutating B rotary axis at an angle of inclination of /4.2.The nutating units on the five-axis machine tools canenhance the flexibility of the machining strategy. How-ever, the CL data considered are limited. Equation (7)Fig. 6 Implementation dialog for generating NC data for table-tilting type configurationInt J Adv Manuf Technolshows that the conditionKz?W2z1?W2z? ? 1 should be satisfied.When the nutating axis is set at an angle of 45 degrees,i.e. =/4 and Wz=C45, Kzis in the range 0 Kz 1.Consequently, the CL data generated by the CAD/CAMsystem can not be machined in this configuration if Kzisa negative value.3.The derived analytical equation of NC data is a generalform, which can be reduced to the orthogonal config-uration. The table-tilting type is chosen as an example.If the vector W is in the X-axis direction where Wx=1and Wy=Wz=0, the configuration reduces to the CAtable-tilting type. The analytical equation of the NC data,specifying for example Y-axis values, agrees with thosepresented elsewhere 8 and can be expressed as follows:Y Qx? LxSzCw Qy? Ly?CzCw? Qz? LzSw Ly26Notably, in the cited work, the two rotary axes areassumed continuously to intersect each other, and the offsetvector M 0i 0j 0k is used to derive the aboveequation.4.Based on the solutions forfzandfw, the cutting toolmay traverse the singular point wherefw=0, Qx=Qy=0,Qz=1 andfzis undefined 12, 18. This singularposition occurs whenfw=0 and the C-axis axis isparallel to the cutting tool axis. As displayed in Fig. 4, ifthe current position Pn+1is the singular point, any valueoffzis theoretically acceptable sincefzis undefined.The next point Pn+2should be read further to ensure thatfzvaries linearly between two successive points. Thevalue offzat position Pn+1can be determined byconsidering a linear change between Pnand Pn+2.5.Feedrate control is an important issue in practical multi-axis machining. Most controllers, such as FANUC andCincinnati Milacron, apply the feed rate number (FRN)and G93 code to control the feedrate. FRN isdetermined by the feedrate on the workpiece dividedFig. 7 Implementation dialog for generating NC data for table/spindle-tilting type configurationInt J Adv Manuf Technolby the span length of the resulting path. Whenrotational movements are combined with two or morelinear axes movements, determinations for the pathlength become very complex. In most cases, anadequate approximation of the true path length can bedetermined by using only the linear displacement 19.5 Implementation and verification5.1 Software implementationThe aforementioned postprocessor methodology was imple-mented in the Windows-XP environment using the BorlandFig. 8 Snapshot of VERICUTsimulation for table-tilting typeconfigurationFig. 9 Snapshot of VERICUTsimulation for table/spindle-tilt-ing type configurationInt J Adv Manuf TechnolC+ Builder programming language and the OpenGLgraphics library. A semi-sphere with a radius of 35 mm isused to illustrate machining. The CL data are generated bythe commercial CAD/CAM system, PowerMILL 20. Twotypical configurations of the machine tool, the table-tiltingand the table/spindle-tilting, were tested. Figure 5 (a) showsthe initiating dialogue of the developed postprocessorsoftware for the table-tilting type configuration. The usercan use the mouse button to rotate and zoom in on thesurface model of the machine tool, and the “Animate”button to animate dynamically the machine tool. When theuser enters the relevant parameters, such as the offset vectorfrom the program origin toward the centre of the C-axisrotary table, the system displays a figure to help the user toinput the parameters correctly, as shown in Fig. 6. Finally,the target CL data are opened by clicking the “File” menu,after which the NC data are generated accordingly. Figures 5(b) and 7 show screenshots of the initiating and implemen-tation dialogues,respectively, for the table/spindle-tiltingtype. Notably, the set length from the gauge plane to thetool tip centre and the distance from the gauge plane to thepivot point should be considered in this configuration, sincethe spindle head moves about the rotary axis.5.2 Solid cutting simulationA solid cutting simulation software package, VERICUT isused to confirm the generated NC data. Given the rawmaterial size, the specifications of the cutting tool, NC data,the type of controller, and the kinematics and physicalproperties of an NC machine tool, it can interactivelysimulate the material removal process of NC data. A table-tilting type machine tool with a B-axis nutating rotary tableis built in the simulation environment and the finished partis verified, as shown in Fig. 8. The relevant parametersshown in Fig. 6 should be reflected in the dialogue of theVERICUT hierarchical component tree. For example, theB-axis rotates about the X-axis by 45, so the nutating tableis in an inclined plane and the vector of the B-axis isW 0i ? S450j C450k. The two rotary axes are as-sumed not to intersect each other and the offset vector fromthe C rotary axis to the B rotary axis is ?5i ? 10j ? 15k.Moreover, the raw material should be placed in the desiredposition so that the offset vector from the program origin tothe C rotary axis is 30i 20j 10k. When the machinesetup has been completed, the system performs a realistic3D simulation of the configured machine tool.Figure 9 illustrates the “as-machined” workpiece cut ona table/spindle-tilting type machine tool with a B-axisnutating rotary head and a C-axis rotary table. As before,the relevant parameters should be set according to Fig. 7.The B-axis vector is W 0i S45?j C45?k. The offsetvector from the program origin to the rotary C -axis is?50i ? 60j ? 70k. The distance from the gauge plane to thepivot point of the B- axis is 100 mm and the set length fromthe tool tip to the gauge plane in the tool library is 120 mm.The results shown in Figs. 8 and 9 confirm the effectivenessand feasibility of the proposed postprocessor algorithm.5.3 Experimental verificationThe generated five-axis NC data are further verified byperforming an experimental trial-cut. A table-tilting five-axis machining centre (Deckel Maho DMU 70 eVolution)equipped with a Heidenhain iTNC530 control is used tomachine the semi-spherical workpiece. The experiment isconducted under the following conditions: (1) two ball-endmills with diameters of 10 mm and 4 mm are used forroughing and finishing, respectively; (2) the spindle speedis 5000 rpm and the feedrate is 1000 mm/min; (3) theworkpiece material is made of 7075 aluminum alloy.Notably, the positive direction of the C-axis of this machineFig. 10 Practical machining experiment on the Deckel Maho DMU70 eVolution. a roughing. b finishingInt J Adv Manuf Technoltool is along the negative Z direction. The actual NC valueof the C-axis is the negative value given by Eq. (11).Figure 10 shows the actual machining process, and revealsthat the developed postprocessor can correctly and success-fully generate five-axis NC data.6 ConclusionA postprocessor methodology for the five-axis machinetool with a nutating table and head configuration has beendeveloped. The general analytical equations of NC datawere determined by the homogeneous coordinate transfor-mation matrix, and the forward and inverse kinematics. Thederived NC data equation is useful for such machine tools,whose rotary axes need not intersect each other and forwhich the angle of inclination of the non-orthogonal axis isvariable. Introducing the variable angle of inclination canincrease the usefulness of the derived equations since theNC data can be reduced to the orthogon
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