Computers & Industrial Engineering 59 (2010) 764–769

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Computers & Industrial Engineering

journal homepage: www.elsevier .com/ locate/caie

Simulation of normal machining of 3D free-form surface by an orthogonal3-leg parallel machine tool with 5-DOF q

Yi Lu a,⇑, Shuyan Li b, Chongjie Du a, Jianping Yu c, Jiayin Xu a

a College of Mechanical Engineering, Yanshan University, Qinhuangdao, Hebei 066004, PR Chinab Capital Engineering & Research Incorporation Limited (CERI), Beijing, PR Chinac College of Foreign Studies, Yanshan University, Qinhuangdao, Hebei, PR China

a r t i c l e i n f o a b s t r a c t

Article history:Received 29 February 2008Received in revised form 20 March 2010Accepted 4 August 2010Available online 12 August 2010

Keywords:SimulationParallel machine tool3D Free-form surface

0360-8352/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.cie.2010.08.001

q This manuscript was processed by Area Editor Gu⇑ Corresponding author.

E-mail address: luyi@ysu.edu.cn (Y. Lu).

A novel orthogonal 3-leg parallel machine tool (PMT) with 5 DOFs is designed for normal machining a 3Dfree-form surface s. A CAD variation geometry approach is adopted for pre-solving the extension/rotationof the linear/rotational actuators and the pose of this PMT. First, a simulation mechanism of this PMT iscreated by using the CAD variation geometry technique. Second, a s and a guiding plane of the tool pathare constructed, the tool axis of the PMT is kept perpendicular to s, and then a simulation PMT is created.Third, in the light of the two kinds of prescribed tool paths, the extension/rotation of the linear/rotationalactuators and the pose of the PMT are pre-solved automatically and visualized dynamically.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

In the conventional process of machining a complex workpieceand a 3D free-form surface s, the NC milling machine tools and theCNC technology are widely used (Chang, Man Kim, & Park Sang,2009; Kim & Sarma, 2002; Koparkar & Mudur, 1986; Makhanovet al., 2002). In the CNC process, the mathematical relation be-tween the tool path and the rotation of each driving motor mustbe determined. A numerical control program must be compiledbased on the profile of the workpiece and s. Currently, some com-puter aided design (CAD) softwares can be used for compile codesof conventional NC milling machine tools when tool is kept per-pendicular to base or datum plane, such as SolidCAM, Mastercam,MSC Adams and IDEAS (Date, Krishnaswami, Satish, & Motipalli,2009; Farin, 1990; Masood, Bagam, & Chantanabubpha, 2002).However, since many complex profiles of workpiece and s cannot be prescribed in a mathematic function, it is uneasy to compilethe numerical control program and the code for machining them,such as the model of an automobile windshield, the impellerblades of ships and turbines. (Date et al., 2009; Farin, 1990; Hartley& Judd, 1980; Masood et al., 2002). Moreover, the tool axis is re-quired perpendicular to s in order to improve the machining qual-ity and the force situation of the tool. Thus, the compilation of thenumerical control program becomes more difficult, even some CNCprograms and codes can not be compiled by these CAD. Recently,

ll rights reserved.

rsel A. Suer

some PMTs have been developed and applied to the machiningof complex workpiece and s (Huang, Fang, & Kong, 1998; Kim &Choi, 2000). PMTs, especially the hexapods with 6-DOF (degreeof freedom), possess novel features of closed-loop mechanismand optimized low moving weight. A hexapod based PMT isclaimed of high speed, high rigidity, high dynamic bandwidth, highaccuracy and low cost, and has been successfully used to machine3D complex workpiece and s (Myriam, Arnaud, & Jean-Yves, 2004;Patel Amit & Ehmann Kornel, 2000; Zhang & Heisel, 2003). In orderto simplify the structure and the control processes, some PMTswith less than 6-DOF are developed, such as the tripod PMT with3-DOF (Chen & Hsu, 2004), the 3-PRS serial–parallel machine tool(Fan, Wang, Zhao, & Chang, 2003; Wang & Fan, 2004), the variax5 axis parallel kinetic machining center (Geldart et al., 2003), the3-axis PMT (Cai, Wang, Hu, Kang, & Kim, 2001; Company & Pierrot,2002), and the high-speed 3-axis PMT (Dong, Yuan, Stori, & Ferre-ira, 2004). In order to improve the machining quality and the forcesituation of the tool, the tool is required to be perpendicular to s,and this process of normal machining s can be realized by somePMTs with 5-DOF or more (Farin, 1990; Koparkar & Mudur,1986). Hence, the PMTs with 5-DOF have attracted more attentionsthan the tripod PMTs with 3-DOF and the hexapod based PMT with6-DOF. In the aspect of synthesis of the parallel mechanisms (PMs),Fang and Tsai synthesized a class of 5-DOF PMs by screw theory(Fang & Tsai, 2002); Lu and Hu analyzed stiffness and elastic defor-mation of some 3–5-DOF PKMs with SPR legs (Lu & Hu, 2008). Gaoet al. synthesized new kinematic structures for 2–5-DOF PMs (Gao,2002); Zhang et al. studied some 5-DOF PMs and n-DOF PMs with apassive constraint leg (Alizade & Bayram, 2004; Zhang & Gosselin,

Nomenclature

PMT parallel machine toolPM parallel mechanismB, m the base and the moving platforms 3D free-form surfaceP0 the guiding plane of tool path curveP, R prismatic joint and revolute jointU, S universal joint and sphere jointF degree of freedom (DOF)T cutter toolp tip of toolg guiding line of toold1, d2 two driving dimensionsri the active legs i = 1, 2, 3 or its length

L, li sides of B and mw the prescribed tool path curveuj prescribed spline on Pj j = 1, 2, . . . kPj datum plane for sketching spline uj

O, o the central points of B and ma, b, c the 3 orientation components of Tci binary linki = 1, 2, . . ., 6Xo Yo, Zo the 3 translation components of mF0 passive degree of freedomh, h1, h2 active angles of rotational actuators?; k perpendicular and parallel symbols

Y. Lu et al. / Computers & Industrial Engineering 59 (2010) 764–769 765

2001). These 5-DOF PMs have become the key mechanisms for thedesign of the 5-DOF PMTs. The CAD variation geometry techniqueis a common and basic technique of many CAD softwares. It can beused to take the place of some complex and expensive CAD soft-wares for constructing the parallel machine tool, analyzing kine-matics and machining of s, such as the 3D solid modeling, 3Dassembly model, 3D motion analysis model. Therefore, the CADvariation geometry approach is economical and it is easy to learn.By this approach, Lu et al. designed some PMs, analyzed their kine-matics (Lu, 2004, 2006), and simulated the machining of s by somePMTs with the tool perpendicular to the base (Lu, 2002, 2005; Lu &Leinonen, 2002; Lu & Xu, 2007).

This paper focuses on the design of a novel orthogonal 3-leg and5-DOF PMT with 2 SPS legs and a UPU composite leg. It also studiesa simulation approach of the normal machining of s by this PMTbecause this PMT has the following merits: (1) the tiny self-motionof the platform can be eliminated by the UPU composite leg; (2)the interferences between the legs are avoided due to the fewerlegs; (3) the workspace is enlarged and 4) its structure is simple.In order to overcome the programming and coding difficulties,the CAD variation geometric approach is used for the normalmachining of s by this PMT without compiling any programs.

Fig. 1. The orthogonal 3-leg PMT with 5 DOFs.

2. Design of the 3-leg and 5-DOF PMT and its DOF

An orthogonal 3-leg PMT with 5 DOFs is designed, as shown inFig. 1.

This PMT is composed of a moving platform m with a tool T, afixed base B, 2 SPS (spherical joint-active prismatic joint-sphericaljoint) active legs with the linear actuators, and a UPU (active uni-versal joint-active prismatic joint-active universal joint) compositeleg r2 with 2 rotational actuators and 1 linear actuator. Here, B is aquarter of cylinder, which provides 3 connected vertices Ai (i = 1, 2,3). m is a cub, which provides 3 connected vertices ai. Let {m} be acoordinate frame o-xyz fixed onto m, {B} be a coordinate frame O-XYZ fixed onto B, ? and || be perpendicular and parallel constraints,respectively. Each of the SPS active legs ri (i = 1, 3) connects m withB by a spherical joint S on m at ai, a prismatic joint P, and a spher-ical joint S on B at Ai. The UPU composite active leg r2 connects m toB by an universal joint Um attached to m at a2, an active leg r2 witha prismatic joint P, and an universal joint UB attached to B at A2. UB

is composed of 2 crossed revolute joints RB1 and RB2. Um is com-posed of 2 crossed revolute joints Rm1 and Rm2. Some geometricconstrains (RB1 ? the tope plane of B, RB1 ? RB2;Rm2 ? RB2;Rm1 ?Rm2, and Rm1ka1a3) are satisfied. In addition, a motor 1 is attachedonto B at its tope plane and its axis is connected and coincided withRB1. A motor 2 is attached onto m at its tope plane, and its axis is

connected and coincided with Rm1. In the initial configuration,three active legs are orthogonal each other. Thus, a larger horizon-tal cutting force can be balanced by the active legs r1 and r3. Sincethere are only three linear active legs, this PMT is simple in struc-ture obviously. In this PMT, the number of links are q0 = 8 for oneplatform, three cylinders, three piston-rods, and one base; thenumber of joints is q = 9 for 3 P, 2 U, and 4 S; the passive DOFs isF0 = 2 for two SPS-type active legs rotating about their won axes,and F0 has no influence on the kinematic characteristics. Therefore,the DOF of this PMT is calculated as below,

F ¼ 6ðq0 � q� 1Þ þXq

i¼1

fi � F0

¼ 6� ð8� 9� 1Þ þ ð3� 1þ 2� 2þ 4� 3Þ � 2 ¼ 5:

Since the UPU-type composite active leg has 5 DOFs (corre-sponding to 2 DOFs of UB and 2 DOFs of Um and one DOF of P),the DOF of the UPU active leg is the same as that of the platformof the PMT. Thus, the motion of the platform is completely con-strained by the UPU active leg. Therefore, the tiny self-motion ofthe PMT can be eliminated. This tiny self-motion is caused by thestructural coupling constraint which is highly sensitive to manu-facturing error (Zhang & Gosselin, 2001).

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3. Basic techniques and creation procedures of the simulationPMT

A simulation PMT of the orthogonal 3-leg and 5-DOF PMT is cre-ated by CAD variation geometry, see Fig. 2. Some basic techniquesof the CAD variation geometry approach for creating the simula-tion PMT (Lu, 2005; Lu & Leinonen, 2002; Lu & Xu, 2007) are ex-plained as follows:

1. The dimensions in the simulation mechanism are classified intothe driving dimension, the driven dimension and the fixeddimension. The driving dimensions are given to the active leglimbs for driving m to move. The driven dimensions are givento the pose of m in the respect to B for solving kinematic param-eters of mechanism. The fixed dimensions are given to the sidesof m and B for modifying the size and configuration of the sim-ulation mechanism.

2. Construct a line, when giving it a fixed dimension, a drivingdimension, and a driven dimension, the line is equivalent a bin-ary link, an active leg with active prismatic joint P, and a passiveconstraint leg with passive prismatic joint P, respectively.

3. Construct a line and a link, connect the one end of line to anyvertex point of link (such as B, m, and another line) by thepoint-point coincident command. Thus, the connecting pointis equivalent to a spherical joint S.

4. The constraints of the simulation mechanism in Soildworks areclassified into the complete constraint with black color, theover-constraint with red color, and the lack-constraint withblue color. When a simulation mechanism is in the completeconstraint, it can be moved normally, and its DOFs can be veri-fied. When a simulation mechanism is in the over-constraint, itcan not be moved. When a simulation mechanism is in the lackconstraint, it can not be moved normally.

The creation procedures of the simulation PMT are explained asfollows.

1. Construct the base B in 2D sketch. The sub-procedures are: (a)Construct a quarter of cylinder with sides L = 140 cm, height

Fig. 2. The simulation PMT of orthogonal 3-leg PMT with 5 DOFs.

h = 140 cm. (b) Transform a quarter of cylinder into a shellframe. (c) Take the default coordinate frame O-XYZ as a fixedcoordinate frame on B. (d) Coincide Z with the central line ofcylinder, and coincide X, Y with two sides of the shell frame.

2. Construct the platform m in 3D sketch. The sub-procedures are:(a) Create 3 lines li (i = 1, 2, 3), and connect their one end to apoint o, set l1 ? l3; l1 ? l2, and l2 ? l3. (b) Give each of li the sameinitial dimension l = 25 cm. (c) Construct a line b, connect itstwo ends to l1 at a1 and to l3 at a3. (d) Create a line c, connectits two ends to o and b. (e) Set c ? b. (f) Construct a line ca, con-nect its one end to l2 at a2, and set ca||c.

3. Construct two SPS-type active legs. The sub-procedures are: (a)Construct two lines ri (i = 1, 3), and connect their two ends to mat ai and to B at Ai by point to point coincident constraint. (b)Give each of ri the initial driving dimension in length.

4. Construct a UPU-type active leg. The sub-procedures are: (a)Construct a line r2, and connect its two ends to m at a2 and toB at A2. (b) Construct an auxiliary line E for the universal jointUB, and connect its one end to B at A2. (c) Construct an auxiliaryline e for the universal joint Um, and connect its one end to m ata2. (d) and set E ? Z; e ? b; r2 ? e; r2 ? E, and ekE. (e) Give theangle h1 between line E and side L a driving dimension, givethe angle h2 between line e and line ca a driving dimension,and give line r2 a driving dimension in length.

5. Construct a tool T. The sub-procedures are: (a) Construct a lineT, and connect its one end to m at o. (b) set T||l2. (c) Give T a driv-ing dimension T = 30 cm.

Thus, a 3-leg 5-DOF simulation PMT is constructed.The pose of m in {B} can be solved by following processes:

Step 1. Step 1 Construct a line Zo, connect its 2 ends to m at o andto B at point Ao, and set Zo ? B. Construct a line z1, connectits one end to B at O and set z1||l2.

Step 2. Step 2 Give each of distances from Ao to Y, X, and o the dri-ven dimensions. Give the angles (a, b, c) between z1 and X,Y, Z the driven dimensions, respectively.

Step 3. Step 3 When varying the driving dimensions of (r1, r2, r3,h1, h2), the pose parameters (Xo, Yo, Zo, a, b, c) of m in {B}are solved automatically and visualized dynamically.Therefore, it is verified that the 3-leg 5-DOF PMT has 5DOFs.

4. The simulation PMT and its key parts

The simulation PMT includes the following key parts: (1) a sim-ulation PM; (2) a tool T (such as a milling cutter or a grindingwheel); (3) a 3D free-form surface s; (4) a guiding plane Po withprescribed tool path. When T, s, and Po are installed onto this sim-ulation PM, a whole simulation PMT is created. In this simulationPMT, T is fixed onto m of PM and kept perpendicular to s at anypose; s is located under m at a machining position, P0 is attachedonto B and kept perpendicular to Z and located under s by Zo = �H,see Fig. 1b and Fig. 2.

4.1. The guiding plane P0 and 3D free-form surface s of the simulationPMT

Before creating the simulation PMT, P0 and s must be con-structed by the 3D modeling technique in Solidworks (Li,2009). The construction processes of P0 and s are described asfollows:

1. Modify B of the simulation mechanism, construct a datum planeP0 by the reference plane command, set P0\Z, and give the dis-tance from P0 to O a fixed dimension H = 200 cm. Construct a

Fig. 3. A simulation PMT of the orthogonal 3-leg and 5-DOF PMT with a rectanglespiral tool path wr.

Y. Lu et al. / Computers & Industrial Engineering 59 (2010) 764–769 767

rectangle on P0, transform it into a guiding plane of tool path bythe plane forming command.

2. Modify B of the simulation PM, construct several datum planes(Qj, j = 1, 2, . . . k), and set them parallel to each other and per-pendicular to P0 by the reference plane command.

3. Based on the prescribed curve data or curve equation, constructa spline uj on the jth plane Qj (j = 1, 2, . . ., k) by using sketchingspline command or data table, and arrange each spline curvewith respect to P0 above m of the simulation mechanism, seeFig. 2a.

4. Construct a smooth and continuous s from all uj (j = 1, 2, . . ., k)by some special modeling techniques, such as loft, swept,extrude, rotation commands, etc. Here, s is constructed by a loftmodeling technique and attached on B above P0 and under m,see Fig. 1b and Fig. 2.

4.2. The tool T of the simulation PMT

Generally, there are two kinds of the tool paths for normalmachining of s. One is a linear reciprocation tool path; the otheris a rectangle or circle spiral tool path. Base on Fig. 1, the simula-tion PMT with a linear reciprocation tool path is created for normalmachining of s, see Fig. 2. The creation procedures are explained asfollows:

1. Transform all driving dimensions of extension/rotation (r1, r2,r3, h1, h2) of linear/rotational actuators in the simulation PMTinto the driven dimensions by using the dimension command,and give each of (r1, r2, r3, h1, h2) a dimension name.

2. Coincide the tip of T with s at point p by using coincident con-straint command.

3. Construct a guiding line g, connect its two ends to s at point pand to P0 at point d, respectively, by the coincident constraintcommand, and set g ? P0.

4. Construct two short lines e1 and e2, connect their one ends topoint p, set e1 and e2 tangent to s at p, and sete1 ? e2; T ? e1; T ? e2 by the geometric constraint command.Thus, a geometric constraint T ? s at p is satisfied.

5. Give each of the distances from d to the left side and the lowerside of P0 the driving dimensions d1 and d2, respectively. Whenvarying the driving dimensions of d1 and d2, the driven dimen-sions of active legs ri, rotational angles h1, h2 are variedautomatically.

4.3. Two tool paths for normal machining of s

Generally, the two kinds of the tool paths can be used for nor-mal machining of s. One is a linear reciprocation tool path wl, otheris a rectangle spiral tool path wr, see Fig. 3.

When a prescribed wl on P0 is used to machine s (see Fig. 2), itsconstruction procedures are described as follows:

1. Determine the machining range (dymin, dxmin, dymax, dxmax) andthe feed speed (dx, dy) each time. Set dymin = dxmin = 30,dymax = 150, dxmax = 140 cm, increscent dx = dy = 5 cm each time.

2. Retain d2 = dymin, and gradually increase d1 by dx each time fromdxmin to dxmax by using the dimension automatic fill command.

3. Retain d1 = dxmax, gradually increase d2 by dy each time fromdymin to dymin + n1dy, n1 = 2.

4. Retain d2 = dymin + n1dy, gradually decrease d1 by �dx each timefrom dxmax to dxmin.

5. Repeat the steps 2–4 above, till d2 = dymax and d1 = dxmax.

When wr on P0 is used to machine s, a simulation PMT with wr iscreated, see Fig. 3.

The machining procedures are described as follows:

(1) Construct a set of vertical and horizontal lines on P0, connectthem to form a rectangle spiral curve on P0, and transform itinto a rectangle spiral spline wr without any split points.

(2) Coincide the free end point d of guiding line g with wr.(3) Construct two driving lines d1 and d2, and connect their one

ends to point p and the other ends to the two vertices (v1, v2)of plane P0, respectively.

(4) Give d1 and d2 a driving dimension or a driven dimensionalternately, and gradually vary d2 or d1 by using the auto-matic fill function to move d outwards along wr. The sub-steps are: (a) Give d1 and d2 a driving dimension(d1 = 106.67 cm) and a driven dimension (d2 = 92.68 cm),respectively, gradually vary the driving dimension of d2 byusing the automatic fill function to move d outwards alongw from the middle of one line to the middle of the next line.(b) Give d2 and d1 a driven dimension and a driving dimen-sion, respectively, and gradually vary the driving dimensionof d1 by the automatic fill function to move d outwards alongwr from the middle of one line to the middle of the next line,see Fig. 3b.

(5) Repeat the step 4 until the required machining of s isfinished.

5. The simulation results

When given prescribed s, wr and wl, the poses and length of ac-tive legs of simulation PMT are solved, see Fig. 4.

When given prescribed s and wr, (a, b, k, h1, h2) of this simulationPMT are solved, see Fig. 4a.

When given a prescribed s and wr, (r1, r2, r3, Xo, Yo, Zo) of the sim-ulation PMT are solved, see Fig. 4b.

Fig. 4. The simulation results of the simulation PMT by the 2 tool paths.

768 Y. Lu et al. / Computers & Industrial Engineering 59 (2010) 764–769

When given prescribed s and wl, (a, b, k, h1, h2) of the simulationPMT are solved, see Fig. 4b.

When given prescribed s and wl, (r1, r2, r3, Xo, Yo, Zo) of the sim-ulation PMT are solved, see Fig. 4d.

6. Some special techniques of normal machining of s

6.1. Machining larger area s

Generally, the workspace of m is limited by the extension of lin-ear actuator (Lu, 2005; Lu & Xu, 2007). When the area of s is largerthan the workspace of m, the extra part of s must be moved into theworkspace of m. For this reason, the position of s in X and Y direc-tions should be varied gradually by the dimension command untilthe extra part of s is moved into the workspace of m. Meanwhile,retain the original necessary geometric constraints, such as T\s,g\P0, P0||B, the surface-point coincident between s and tip pointp of T, the curve-point coincident between w and the end point dof g, the fixed distance from P0 to B, the fixed dimension of tool Tin length, and the all geometric constraints of this simulation PMT.

6.2. Combining some simple machining processes

In general, the whole processes of the normal machining of sshould be divided into several sub-processes, such as the toolentering into machining position, the tool retreating from machin-ing position, rough machining, rough finishing, and finishingmachining. For this purpose, the feeding depth of the tool mustbe varied. Therefore, only vary the dimension of tool T in length,the issue of feeding normal to s can be solved. In order to completeevery sub-machining processes, several processes are required tocomplete as follows:

1. When starting machining, the tool T of the simulation PMTmust be moved from a preparing position to a machining posi-tion. Therefore, give T an increment dz at each step from zmin tozmax by using the dimension command for its feeding operation.Next solve all driven dimensions of active legs and pose of m foreach step.

2. When completing machining, T must be retreated from s to asuitable position. Therefore, give T an increment dz at each stepfrom zmax to zmin by using the dimension command for itsretreating operation, and solve all driven dimensions of activelegs and pose of m for each step.

3. During rough machining, rough finishing, and finishing machin-ing, T must be moved into s at the different depth. For this pur-pose, the dimension of T should be reduced at the differentincrement dT in length for different machining processes.

4. Combine all solved data in all sub-processes together, produce awhole machining data.

6.3. Inspecting and avoiding interference

During machining s by the simulation PMT, the interferenceamong the active legs and the tool path can be checked in the envi-ronment of the Solidwork software (Li, 2009).

From the simulation curves, the workspace of the simulationPMT can be inspected and calculated for determining the positionof s in respect to m.

When the interferences occur, the tool path or the simulationPMT should be varied or modified.

If a tool path is not suitable for machining s, its size, coveredarea, position, and path density can be modified by varying dimen-sions of the start point and the end point of tool path on P0 in re-spect to B. Thus, a tool path can be used repeatedly.

6.4. Reducing machining error

Since the tip of the tool T is always remained coincident with s,there is no machining error at the starting point and the end point

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at each feed. However, there may be machining error on the feedline from the starting point to the end point at each feed. Whenreducing feed increscent d at each time, the machining error willbe reduced. Since s is smooth and continuous, the machining erroron the feed line at each small feed must be very small.

7. Conclusions

A novel orthogonal 3-leg and 5-DOF parallel machine tool(PMT) is designed. It is characterized by 2 SPS-type active legsand one UPU-type serial active leg. It has relative large capacityof load bearing and is simple in structure. It can be used for thenormal machining of any smooth and continuous 3D free-formsurface s or normal carving of any complicated letters on s. Themachining quality of s and the force situation of tool of the PMTcan be improved greatly.

By using a CAD variation geometry approach, a simulation PMTof this PMT can be created; any prescribed s can be constructedfrom several precision splines; the extensions of the active legsand the pose of the tool can be solved and visualized automaticallybased on s. In the whole processe of simulation machining of s, nonumerical control programs are required. Therefore, this approachis straightforward and simple.

In addition, a complicated process of machining s by the simu-lation PMT can be divided into several simple processes to produceits corresponding simulation data. These simulation data in eachsimple process are orderly input into the active legs, so that thecomplicated machining process can be simulated easily.

The simulation results of this simulation PMT can be used tocheck the workspace of the PMT and to inspect any interference.By comparing the actual workspace and the simulation workspaceof the PMT, the desired size and location of the workpiece in thesimulation PMTs can be determined.

8. Future studies

The future studies will focus on the analyses of the singularity,kinematics and dynamics of this PMT by CAD variation geometryapproach as follows:

1. Pre-solve the velocity/acceleration of the actuators and the plat-form under the given tool feeding speed, and to reduce themwhen they become quite large or are larger than the limitation.

2. Pre-determine the singularities of this PMT at some positionsand avoid them before the normal machining of s.

3. Pre-determine the workloads applied onto the platform and theactive forces of active legs and modify them when they becomequite large.

Acknowledgements

The authors would like to acknowledge the financial support ofthe Natural Sciences Foundation Council of China (NSFC)50575198. The authors would like to acknowledge the financialsupport of Doctoral Fund from National Education Ministry No.20060216006.

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