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Simulation of three-dimensional free-formsurface normal machining by 3SPSRRPUand 2SPSRRPRR parallel machine toolsY Lu* and J-Y XuCollege of Mechanical Engineerig, Yanshan University, Hebei, Peoples Republic of ChinaThe manuscript was received on 4 May 2007 and was accepted after revision for publication on 11 December 2007.DOI: 10.1243/09544054JEM900Abstract: A novel 5-degrees-of-freedom (DOF) 3SPSRRPU parallel machine tool (PMT) anda novel 5-DOF 2SPSRRPRR PMT are proposed for a three-dimensional (3D) free-form surfacenormal machining. A computer aided design (CAD) variation geometry approach is adopted forsolving the extension/rotation of linear/rotational actuators and the pose of the two PMTsduring machining. First, two simulation mechanisms are created by the CAD variation geome-try technique for the 3SPSRRPU PMT and the 2SPSRRPRR PMT, respectively. Second, a 3Dfree-form surface and a guiding plane of tool path are constructed above the moving platformof the simulation mechanism. Third, the tool axis of the simulation PMT is kept normal to the3D free-form surface, and two 5-DOF simulation PMTs are created. Finally, in the light of thetwo prescribed tool paths, the extension/rotation of linear/rotational actuators, and the poseof the PMTs are solved automatically and visualized dynamically.Keywords: computer simulation, parallel machine tool, three-dimensional free-form surface,normal machining1 INTRODUCTIONRecently, parallel kinematic machines (PKMs) havebeen studied extensively 1, 2. PKMs, especiallythe six-degree-of-freedom (6-DOF) hexapods, possessnovel features of closed-loop and symmetricalmechanism, and optimized low moving weight. Thehexapod-based parallel machine tool (PMT) hasbeen often credited with high speed, high rigidity,high dynamic bandwidth, high accuracy, and lowcost. It has been successfully used to machine a three-dimensional (3D) workpiece 24. In order to simplifystructure and control processes, some limited-DOFPMTs have been developed, such as a 3-DOF tripodmachine tool 5, a 3-DOF 3-prizmatic-revolute-sperical (3-PRS) serialparallel machine tool 6, 7, avariax five-axis parallel kinetic machining centre 8,a three-axis PMT 9, 10, a high-speed three-axisPMT 11, 12. In order to improve machining quality,a 3D free-form surface normal machining is requiredand can be completed by some PMTs with more than4-DOFs 2, 3. Fang and Tsai synthesized a class of 5-DOF PKMs by screws theory 13; Li and colleaguessynthesized 3R2T 5-DOF PKMs by Lie group of dis-placements 14, 15; Gao et al. synthesized new kine-matic structures for 25-DOF PKMs 16; Zhang andGosselin 17 and Alizade and Bayram 18 studiedsome 5-DOF PKMs and n-DOF PKMs with a passiveconstraint leg.In the traditional milling and computer numericalcontrol (CNC) processes, tool axis is required per-pendicular to the 3D free-form surface for improvingmachining quality. However, it is not easy to com-pile the NC code for machining a complicated 3Dfree-form surface, such as a model of an automobilewindshield, an impeller blade of a ship, a launch, ora turbine 12, 1922. Lu proposed a computer aideddesign (CAD) variation geometric approach for kine-matic analysis of some PKMs 2325, and success-fully simulated the 3D free-form surface verticalmachining by 6- or 3-DOF PMTs 2629. Upto now, there has been no effort towards the 3Dfree-form surface normal machining by 5-DOF*Corresponding author: College of Mechanical Engineering,Yanshan University, Qinhuangdao, Hebei 066004, PeoplesRepublic of China. email: 485JEM900 C211 IMechE 2008 Proc. IMechE Vol. 222 Part B: J. Engineering ManufacturePMTs with fewer than five legs. The current paperfocuses on a 3D free-form surface normal machiningby a 3SPSRRPU PTM and a 2SPSRRPRR PTM. Inaddition, a novel CAD variation geometry approachis developed for the 3D free-form surface normalmachining by this two PMTs without compilingany NC code.2 THE 3SPSRRPU PKM AND ITSSIMULATION MECHANISM2.1 The 3SPSRRPU PKM and its DOFA3SPSRRPU PKM includes a moving platform m,afixedbaseB,and3SPS(sphericaljointactiveprismaticjoint-sphericaljoint)activelegsri(i1,2,3)withlinearactuators,andonecentralRRPU(activerevolutejointrevolute jointactive prismatic jointuniversal joint)active leg rowith a linear actuator and a rotationalactuator; see Fig. 1(a) where m is an equilateral ternarylink Da1a2a3with three sides lil, three verticesai,andacentrepointo; B is an equilateral ternary linkDA1A2A3with three sides LiL, three vertices Ai,andacentrepointO.Letm be a coordinate systemo-xyz fixed on m at o. Let B be a coordinate systemO-XYZ fixed on B at O. ? and k denote a perpendicularconstraint and a parallel constraint respectively. Eachof riconnects m to B by a spherical joint S at ai,anactive leg riwith a prismatic joint P, and a sphericaljoint S at Ai. The central RRPU active leg roconnectsm to B by a universal joint U,alegrowith anactive prismatic joint P,arevoluteRB2, and an activerevolute joint RB1. U is attached to m at o,andiscomposed of two cross revolute joints Rm1and Rm2.RB1is attached to B at O and is connected with theaxis of the motor. RB2and RB1are crossed and con-nected by a link. Some geometric constraints (RB1and Z being collinear, RB1?RB2, Rm2kRB2, Rm1?Rm2,and Rm1and x being collinear) are satisfied. Sinceeach of the SPS active legs ribears only the axial forcealong ri,the3SPSRRPU PKM obviously has a relativelarge capacity of load-bearing.In the 3SPSRRPU PKM, the number of links arek11foroneplatform,fourcylinders,fourpistonrods,one link for connecting RB1and RB2,andonebase;thenumber of joints is g13forfourprismaticjoints,tworevolute joints, one universal joint, and six sphericaljoints; f11 for the prismatic or revolute joint, f22for the universal joint, f23 for the spherical joint; theredundant DOF is F03forthreeSPS-typeactivelegsrotating about their own axes, and F0has no influenceon the kinematic characteristics. Thus, the DOF F ofthe 3SPSRRPU PKM can be calculated by a revisedKutzbachGrubler equation 1asF 6kC0gC01Xgi1fiC0F0 611C013C01412263C03 52.2 The simulation 3SPSRRPU PKMIn order to construct a simulation 3SPSRRPU PMTand verify its DOF, a simulation 3SPSRRPU PKM(see Fig. 1(b) is created by CAD variation geometrytechniques of some CAD softwares such as Solid-Works, Solid-Edge, Solid-Designer, MechanicalDesktop, Pro/E, and so on. Some basic techniquesof CAD variation geometry techniques for construct-ing simulation mechanisms and verifying the DOF ofFig. 1 The 3SPSRRPU PKM and its simulation mechanism486 Y Lu and J-Y XuProc. IMechE Vol. 222 Part B: J. Engineering Manufacture JEM900 C211 IMechE 2008the PKM are described in Appendix 2 and references23, 28, and 29.The creation procedures of the simulation3SPSRRPU PKM are outlined below.1. Construct thebaseB in a two-dimensional sketch.The sub-procedures are:(a) construct an equilateral triangle DA1A2A3bythe polygon command;(b) coincide its centre point Owith origin of defaultcoordinate, set its one side horizontally, andgive its one side a fixed dimension in length;(c) transform DA1A2A3into a plane by the planarcommand.2. Construct the platform m in a 3D sketch. Thesub-procedures are:(a) create three lines li(i1, 2, 3), and connectthem to form a closed triangle Da1a2a3;(b) give each of lithe same initial dimension;(c) create a line y, and connect its two ends to a2and l2; (d) create a line c, and connect its twoends to a1and y at point o;(e) set y?c and c?l2.3. Construct three SPS-type active legs. The sub-procedures are:(a) construct three lines ri(i1, 2, 3), and con-nect their two ends to m at aiand to B at Aiby the point-to-point coincident constraint;(b) give each of rithe initial driving dimension inlength for linear actuators.4. Construct a RRPU-type active leg. The sub-pro-cedures are:(a) construct a line ro, and connect its two ends tom at o and to B at O;(b) construct an auxiliary line E for two crossedrevolute joints RB1and RB2, and connect itsone end to B at O;(c) construct an auxiliary line e for the universaljoint U, and connect its one end to m at o;(d) set E?Z, e?l2, ro?e, ro?E, and ekE;(e) give the angle C18 between line E and line L2adriving dimension for the rotational actuatorof RB1, and give line roa driving dimension inlength for the linear actuator.Thus, a simulation 3SPSRRPU PKM is constructed.The pose of m in B can be solved by the proce-dures outlined below.1. Construct a line Zo, connect its two ends to m at oand to B at point Ao,andsetZo?B. Construct a linez1,connectitsoneendtoB at O and set z1?m.2. Take the default coordinate O-XYZ as a fixedcoordinate on B, give each of the distances fromAoto Y, X, and o the driven dimensions respec-tively. Give the angles (a, b, g) between z1andX, Y, Z the driven dimensions respectively.3. When varying the driving dimensions of (ror1r2r3C18), the pose parameters (Xo, Yo, Zo, a, b, g)ofm in B are solved automatically and visualizeddynamically. Therefore, it is verified that the3SPSRRPU PKM has five DOFS.3 3SPSRRPU PMT AND ITSSIMULATION MECHANISM3.1 The guiding plane P0of tool pathand 3D free-form surface sWhen a tool T, such as a milling cutter or a ground-ing wheel, is installed perpendicularly onto the plat-form m of the 3SPSRRPU PKM, a 3SPSRRPUPMT is formed. When a 3D free-form surface s anda guiding plane P0of the tool path are attached onB above m of the simulation mechanism, and T iskept perpendicular to s at any point, a simulation3SPSRRPU PMT is created; see Fig. 2. Before creat-ing simulation PMT, the guiding plane P0and 3Dfree-form surface s must be created by the 3Dmodelling technique. How to fix s and P0on B ofPMT and arrange them above m of PMT is a key issueto be solved. A guiding plane P0of the tool path and a3D free-form surface s are created as follows.1. Modify B of the simulation mechanism, and con-struct a datum plane P0by the reference planecommand. Set P0kB, and give the distance fromP0to B a fixed dimension h3000mm. Constructa rectangle on P0, and transform it into a guidingplane of the tool path by the plane-forming com-mand; see Fig. 2.2. Modify B of the simulation mechanism, constructseveral datum planes (Pj, j1, 2,.k), and setthem parallel to each other and perpendicularto B by the reference plane command.3. Based on the prescribed curve data or curveequation, construct a spline ujon the jth planePj(j1, 2,.k) by the sketching spline commandor data table, and arrange each spline curve withrespect to P0above m of the simulation mechan-ism; see Fig. 2(a).4. Construct a smooth and continuous 3D free-formsurface s from all uj(j1, 2,.k)bysomespecialmodelling techniques, such as loft, swept, extrude,rotation commands, etc. Here, s is constructed bya loft modelling technique and attached on Babove m and under P0; see Figs 2 and 3(b) (later).3.2 The simulation 3SPSRRPU PMTGenerally, there are two kinds of tool path formachining s. One is a linear reciprocation toolpath; the other is a rectangle or circle spiral toolSimulation of three-dimensional free-form surface normal machining 487JEM900 C211 IMechE 2008 Proc. IMechE Vol. 222 Part B: J. Engineering Manufacturepath. Based on the simulation 3SPSRRPU PKMin Fig. 1(b), a simulation 3SPSRRPU PMT with alinear reciprocation tool path for machining s iscreated; see Fig. 2(a).The creation procedures are outlined below.1. Transform all the driving dimensions of exten-sion/rotation (r1, r2, r3, ro, C18) of the linear/rotational actuators in the simulation 3SPSRRPU PKM into the driven dimensions by thedimension command, and give each of (r1, r2,r3, ro, C18) a dimension name.2. Coincide the tip of T with s at point p by the coin-cident constraint command.3. Construct a guiding line g, connect its two endsto s at point p and to P0at point d respectively,Fig. 2 Two configurations of the simulation 3SPSRRPU PMT to machine s (a) along a linear reciproca-tion tool path, and (b) along a rectangle spiral tool pathFig. 3 The 2SPSRRPRR PKM and its simulation PMT488 Y Lu and J-Y XuProc. IMechE Vol. 222 Part B: J. Engineering Manufacture JEM900 C211 IMechE 2008by the coincident constraint command, andset g?P0.4. Construct two short lines e1and e2, connect theirone ends to point p, set e1and e2tangent to s at p,and set e1?e2, T?e1, T?e2by the geometric con-straint command. Thus, a geometric constraintT?s at p is always satisfied.5. Give each of the distances from d to the left sideand the lower side of P0a driving dimension d1and d2respectively. When varying the drivingdimensions of d1and d2, the driven dimensionsof the active legs ro, ri, rotational angle C18 arevaried automatically.3.3 Machining s by PMT along alinear reciprocation tool pathWhen a linear reciprocation tool path w on P0is usedto machine s, see Fig. 2(a), its constitution proceduresare described as follows.1. Determine the machining range (dymin, dxmin,dymax, dxmax) and the feed rate (dx, dy) for eachincrement. Set dymindxmin300, dymax1500,dxmax1400mm, increment ddxdy10mm.2. Retain d2dymin, and gradually increase d1by dxeach time from dxminto dxmaxby the dimensionautomatic fill command.3. Retain d1dxmax, and gradually increase d2by dyeach time from dyminto dyminn1dy, n12.4. Retain d2dyminn1dy, and gradually decrease d1by C0dxeach time from dxmaxto dxmin.5. Repeat steps 2 to 4 above, until d2dymaxandd1dxmax.3.4 Machining s by PMT alonga rectangular spiral tool pathWhen a rectangular spiral tool path w on P0is used inthe simulation 3SPSRRPU PMT, see Fig. 2(b), themachining procedures are described as follows.1. Construct a set of vertical and horizontal lines onP0, connect them to form a rectangular spiralcurve on P0, and transform it into a rectangularspiral spline w without any split points.2. Coincide the free end point d of guiding line gwith w.3. Construct two driving lines d1and d2,andconnecttheir one end to point p and the other end to thetwo vertices (v1, v2) of plane P0respectively.4. Give d1and d2a driving dimension or a drivendimension alternately, and gradually vary d2ord1by using the automatic fill function to moved outwards along w. The sub-steps are:(a) give d2and d1a driving dimension and a dri-ven dimension respectively, and graduallyvary the driving dimension of d2by using theautomatic fill function to move d outwardsalong w from the middle of one line to themiddle of the next line;(b) gived2and d1a driven dimensionand a drivingdimension respectively, and gradually vary thedriving dimension of d1by using the automaticfill function to move d outwards along w fromthe middle of one line to the middle of thenext line; see Fig. 2(b) and Fig. 3(b).5. Repeat step 4 until the required machining of s isfinished.4 A 2SPSRRPRR PKM AND ITSSIMULATION PMTA 2SPSRRPRR PKM is similar to the 3SPSRRPUPKM except that the central RRPU active leg roisdeleted, and the SPS active leg r2is replaced by anew RRPRR (active revolute jointrevolute jointac-tive prismatic jointrevolute jointactive revolutejoint) active leg r2with two rotational actuators andone linear actuator; see Fig. 3(a). The RRPRR activeleg r2connects m to B by an active revolute jointRm1, a revolute joint Rm2, a leg r2with an activeprismatic joint P, a revolute joint RB2, and an activerevolute joint RB1. Where, Rm1is attached to m at a2and connected with the motor 1. RB1is attached toB at A2and connected with the motor 2. Rm1andRm2are crossed and connected by a link. RB1andRB2are crossed and connected by a link. In addition,some geometric constraints are satisfied: RB1?B,RB1?RB2, Rm2?r2, Rm2kRB2, Rm1?Rm2, Rm1and y beingcollinear. Since there are only three active linearactive legs, the 5-DOF 2SPSRRPRR PKM isobviously simple in structure.In the 2SPSRRPRR PKM, the number of links arek10 for one platform, three cylinders, three pis-ton-rods, two links, and one base; the number ofjoints is g11 for three prismatic joints, four revolutejoints, and four spherical joints; the redundant DOFsis F02 for two SPS active legs rotating about theirown axes, and F0has no influence on the kinematiccharacteristics of the PKM. Therefore, the DOFs F ofthe 2SPSRRPRR PKM is calculated asF 6kC0gC01Xgi1fiC0F0 610C011C01312243C02 5A simulation 2SPSRRPRR PKM is similar to thesimulation 3SPSRRPU PKM, except that some addi-tional creation procedures are conducted as follows.1. Delete the central RRPU active leg roof the latter.2. Transform a SPS active leg r2of the latter into aRRPRR active leg r2. The sub-procedures are:(a) create two auxiliary lines E and Eafor revolutejoints RB1and RB2, connect their one end to Bat A2and O respectively;Simulation of three-dimensional free-form surface normal machining 489JEM900 C211 IMechE 2008 Proc. IMechE Vol. 222 Part B: J. Engineering Manufacture(b) create two auxiliary lines e and eafor revolutejoints Rm1and Rm2, and connect their oneend to m at a2and a3respectively;(c) set E?Z, e?y, EkEa, ekea, r2?e, and r2?E;(d) give an angle C181between line Eaand line Y adriving dimension for motor 2, give a line r2a driving dimension in length for a linearactuator, and give an angle C182between line l3and line eaa driving dimension for motor 1.Thus, a simulation 2SPSRRPRR PKM is con-structed.Similarly, the pose of m with respect to B can besolved as follows: when varying the driving dimen-sions of (C181, C182, r1, r2, r3), the pose parameters (Xo,Yo, Zo, a, b, g)ofm in B are solved automatically.Therefore, it is verified that the 2SPSRRPRR PKMhas 5 DOFs.Based on the simulation 2SPSRRPRR PKM, anovel simulation 2SPSRRPRR PMT is created for a3D free-form surface normal machining along therectangle spiral tool path; see Fig. 3(b). Similarly, asimulation 2SPSRRPRR PMT is created for s normalmachining along a linear reciprocation tool path.5 SIMULATION RESULTS OF 3D FREE-FORMSURFACE NORMAL MACHININGSince a 3D free-form surface s is smooth and con-tinuous, and may be any prescribed