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Using CAD functionalities for the kinematics analysisof spatial parallel manipulators with 3-, 4-, 5-, 6-linearlydriven limbsYi Lu*Robotics Research Center, Department of Mechanical Engineering, Yanshan University, Qinhuangdao,Hebei 066004, PR ChinaAbstractA novel computer aided geometric approach is put forward for designing the computer simulationmechanisms of spatial parallel manipulators with 3-, 4-, 5-, 6-driving limbs. Several new spatial parallelmanipulators with 3-, 4-, 5-, 6-driving limbs are synthesized. Some common computer aided geometryconstraints and dimension driving techniques and definitions for designing the simulation mechanisms arepresented. Based on some new and original spatial parallel manipulators with 3-, 4-, 5-, 6-driving limbs, the12 types of simulation mechanisms are created, respectively, by applying these techniques. When the drivingdimensions of driving limbs are modified by using the dimension driving technique, the configurations ofthe simulation mechanisms are varied correspondingly, and the kinematic parameters of the movingplatform are solved. The results of computer simulation prove that the computer aided geometric approachis not only fairly quick and straightforward, but is also advantageous from viewpoint of accuracy andrepeatability.? 2003 Elsevier Ltd. All rights reserved.Keywords: Computer aided geometry; Spatial parallel manipulator; Simulation mechanism1. IntroductionSome spatial parallel manipulators with 3- or 6-driving limbs 1,2 have been utilized for manypractical applications, in which the good kinematic and dynamic performance are adopted for therobot manipulator, the parallel machine tool, and the legs of walking machine, high load car-rying capacity is used for the flight simulator, the automobile or tank simulator, the earthquake*Tel.: +86-35-8057070; fax: +86-335-8061449.E-mail address: luyip0736 (Y. Lu).0094-114X/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0094-114X(03)00103-4Mechanism and Machine Theory 39 (2004) 4160/locate/mechmtsimulator, and so on 46. Tsai 3 proved that in general, the number of driving limbs of thespatial parallel manipulator equals to the number of its DOF. In conducting the synthesis, ki-nematic analysis, and optimum design of the spatial parallel manipulators, some analytic ap-proaches (such as the influence coefficient matrix approach 7, the screw approach 8, and thespatial vector analytic approach 9,10) have been applied for studying the kinematic and dynamicperformances of the spatial parallel manipulators with 3- or 6-driving limbs 717. Up to now, thespatial parallel manipulators with 4- or 5-driving limbs have not been used for many practicalapplications, since the studies on them have not been perfect and mature.The analytic approaches are suitable for computer programming and have the advantages ofaccuracy and repeatability. However, the calculation in the analytic processes of the spatialparallel mechanism is quite complicated, such as conducting the dimension synthesis, analyzingthe kinematic and dynamics, and determining the singularity configuration. In terms of CADmechanism analyses, based on the analytic approaches, some suitable programs are studied andcompiled 18,19. Since these analytic approaches are complex, the programming processes arecomplicated and not straightforward, especially in the case of multiple-solutions and the spatialparallel mechanisms. Therefore, the application of analytic approaches is limited. Currently, thecomputer aided geometric technology is an effective tool for feature designing, concept designing,3D modeling, and synthesis and analysis of planar mechanism 2025. However, how to use thistool to solve kinematic and the SC analyses problems for spatial parallel manipulators with 3-, 4-,5-, 6-driving limbs is still a key question, which has not been solved.In order to solve the above problems, some simulation mechanisms of the spatial parallelmanipulators with 3-, 4-, 5-, 6-driving limbs are designed, respectively, by using a CAD softwarewith the functions of geometric constraint, equation constraint, and dimension driving functions.A novel computer aided geometric approach without modeling and assembling of 3D solid isexplored for dynamically solving kinematic parameters.2. The common technique for creating the simulation mechanismsBefore creating the simulation mechanisms of spatial parallel manipulators, some commontechniques and definitions are described below.Step 1. The dimensions in the simulation mechanisms are classified into the driving dimension,the driven dimension, and the fixed dimension. The driving dimensions are given to the drivinglimbs for driving the moving platform. The driven dimensions are given to the position andorientation of the moving platform in the respect to the base in order to solve kinematic pa-rameters of mechanism. The fixed dimensions are given to the sideline of the moving platform, thesideline of the base, and the line for connecting any two joints in order to modify the size andconfiguration of the simulation mechanism.Step 2. Some basic equivalent links in simulation mechanism are constituted as follows:(a) Constitute a line l, and give it an initial fixed dimension. Thus, a line with fixed dimensionsequivalent to a binary link.(b) In 2D sketch environment of advanced CAD software, constitute an equilateral triangle, anequilateral quadrangle, and an equilateral hexagon, respectively, by using the polygon command.Transform them into an equilateral triangle plane, an equilateral quadrangle plane, and an equi-42Y. Lu / Mechanism and Machine Theory 39 (2004) 4160lateral hexagon plane, respectively, by using the planar area command. Give one sideline ofpolygon plane an initial fixed dimension by using dimension command. Thus, these polygon planescan be used as either the base or the moving platform, but they cannot be used as both, because thebase and the moving platform cannot be constituted in 2D sketch environment at the same time.Therefore, if the base is constituted in 2D sketch environment, the moving platform must beconstituted in 3D sketch environment by adopting steps 2c, 2d and 2e below, and vice versa.(c) Constitute three lines lii 1;2;3, and connect them to form a closed triangle Da1a2a3byadopting the pointpoint coincident command. Next, give each sideline liof Da1a2a3an initialdriving dimension. Constitute a line c1connect its two ends to point a1and sideline l3by adoptingthe pointpoint coincident command and the pointline coincident command, respectively.Constitute a line c2and connect its two ends to point a2and line c1at point a0. Set c1perpen-dicular to l3and set c2perpendicular to l1In this way, an equivalent planar ternary link in 3Dsketch environment is constituted, and its center point a0is determined, as shown in Fig. 1a.(d) Constitute four lines lii 1;2;3;4, and connect them to form a closed quadrangle(a1a2a3a4) by using pointpoint coincident command. Set l1perpendicular to both l2and l4andset l1parallel to l3. Thus, the four points (a1;a2;a3;a4) are always retained onto the same plane.Constitute a line c1and connect its two ends to a1and a3, respectively. Constitute a line c2andconnect its two ends to a2and a4, respectively. Set c1perpendicular to c2. Give one sideline of thequadrangle an initial dimension by using dimension command. In this way, an equivalent planarquadrangle link (a1a2a3a4) in 3D sketch environment is constituted, and the central point a0of thelink can be determined from the crossover point of c1and c2as shown in Fig. 1b.(e) Constitute six lines lii 1;2;.;6, and connect them to form a closed hexagon(a1a2a3a4a5a6) by adopting the pointpoint coincident command. Constitute a line c1and connectits two ends to a3and a6, respectively. Constitute a line c2connect its two ends to a1and line c1ata0, respectively, by using the pointpoint and pointline constraint command. Set c1parallel toboth l2and l5and set c2parallel to both l3and l6. Give same dimension to line c2and each of thesideline li. Thus, the six points aii 1;2;.;6 are always retained on the plane Da1a6a0. In thisway, an equilateral hexagon plane (a1a2a3a4a5a6) in 3D sketch environment is constituted, and itscentral point a0can be determined from the crossover point of c1and c2as shown in Fig. 1c.Step 3. Some basic equivalent joints in simulation mechanism are constituted as follows:(a) Constitute a line l, and give it a driving dimension in length. Thus, l is equivalent to aprismatic joint p or a driving limb with a prismatic joint P.a1 a0 a2 a3 l1 l3 l2 c1 c2 a2 a1 a6 a3 a4 a5 c2 a0 c1 l1 l6 l3 l5 l4 l2 a4 a3 a2 a1 a0 l3 l2 l1 c2 c1 l4 (a) (b) (c)Fig. 1. (a) Equilateral ternary link; (b) equilateral quadrangle link and (c) equilateral hexagon link. The equivalentplanar equilateral ternary, quadrangle, and hexagon links.Y. Lu / Mechanism and Machine Theory 39 (2004) 416043(b) Follow the step 2 above, constitute a link and a line l, connect one end of l to any point p onthe link (such as the base, the moving platform, or another line) by using the pointpoint coin-cident command. Thus, the connecting point p is equivalent to a spherical joint S.(c) Follow the step 2 above, constitute a link and a line l, connect one end of l to any vertex pon the link (such as the base, or the moving platform) or the end of another line by using thepointpoint coincident command, and set l perpendicular to another line on this link. Thus, theconnecting point p is equivalent to a revolving joint R.(d) Constitute two lines l1and l2connect one end of l1onto l2at point p by using the pointlinecoincident command, and set l1perpendicular to l2. Thus, the connecting point p is equivalent toa cylinder joint C.(e) Constitute two lines l1and l2connect one end of l1onto l2at point p by using the pointlinecoincident command, as shown in Fig. 2a. Thus, the connecting point p is equivalent to a com-posite joint PS.Step 4. Follow the step 2 above, constitute a base B and a moving platform m, respectively.Follow the steps 3a and 3b above, constitute a line r1give r1a driving dimension in length, andconnect its two ends to B at point A1and m at point a1respectively, by using the pointpointcoincident command. Thus, the two equivalent spherical joints S at point A1and point a1areconstituted, respectively, and one equivalent SPS driving limb r1for connecting m and B is alsoconstituted.Step 5. Follow the step 4 above, set the driving limb r1perpendicular to one sideline of the baseB. Thus, an equivalent revolving joint R for connecting r1and B is constituted. In this way, oneequivalent SPS driving limb r1is transformed into an equivalent SPR driving limb r1.Step 6. Follow the step 4 above, constitute an auxiliary line B1and connect its one end to thebase B at point A1. Set B1perpendicular to both driving limb r1and a sideline of B (L3for thesituation of the 3-driving limbs, and C1for the 4 driving limbs) by using perpendicular constraintcommand. Thus, one equivalent universal joint U at point A1on B is constituted. Similarly,constitute an auxiliary line b1connect its one end to m at point a1and set b1perpendicular to bothr1and a sideline of m (l3for the 3-driving limbs, and c1for the 4-driving limbs). Thus, oneequivalent universal joint U at point a1is constituted. Next, set B1parallel or perpendicular to b1.In this way, one equivalent SPS driving limb r1is transformed into one equivalent UPU drivinglimb r1as shown in Fig. 2b.b1 a1 r1 A1 B1 l3 L3 B m p l1 l2 P S l1 l2 P U U m B c1C1 a1 C1 r1 b1 A1 c1 B1 B m A1 a1 r1 (a)(b) Fig. 2. (a) The equivalent a composite joint PS and (b) the equivalent UPU-driving limb r1. The equivalent planarequilateral ternary, quadrangle, and hexagon links.44Y. Lu / Mechanism and Machine Theory 39 (2004) 4160Step 7. Follow steps 4 and 6, constitute an equivalent spherical joint S at point A1on B and anequivalent universal joint U at point a1on m. In this way, one equivalent SPS driving limb r1istransformed into an equivalent SPU driving limb r1. In fact, the SPU driving limb r1is equivalentto the SPS driving limb r1in simulation mechanism, because one local redundant DOF, which isproduced from the driving limb rotating about itself axis, does not influence the motion of themoving platform. Therefore, the UPS driving limb r1in simulation mechanism can be replaced bythe equivalent SPS driving limb r1.The DOF of the spatial parallel mechanism can be calculated by Kutzbach Grubler equation24 belowF kk ? j ? 1 Xni1fi? F01where k is the number of links, j is number of joints, k is the degrees of the space within which themechanism operates for spatial motions k 6; fiis the degree of freedom of the ith joint, F0is thelocal redundant DOF, which do not influence the motion of mechanism.3. The spatial parallel manipulator with 3-driving limbs3.1. The 3-RPS parallel manipulator and its simulation mechanismAn existing spatial 3-RPS parallel manipulator has three DOF 2,3. It includes a movingplatform m, a base B, and three extendable driving limbs riwith their hydraulic cylinders andpiston-rods, as shown in Fig. 3a. Where, m is a regular triangle Da1a2a3with a0as its center, and B(a) (b) R, A1S, a1P4BS, a2a3, SR, A2A3, Rm13625PP130(130)(130)85103103(95.28)(37.87)(64.41)(79.18)(96.22)(192.52)A2a3dya1a2A1A3mr3r2r1dzdxa0d0l3L3l2l3L2L1nn1xyNn2BzFig. 3. (a) The spatial 3-SPR parallel manipulator and (b) its simulation mechanism. The spatial 3-SPR parallelmanipulator and its simulation mechanism.Y. Lu / Mechanism and Machine Theory 39 (2004) 416045is a regular triangle DA1A2A3with A0as its center. Three identical limbs connect m to B by aspherical joint S at point ai, a driving limb with a prismatic joint P, and a revolute joint R at pointAi(i 1;2;3), respectively.In the 3D sketch environment, a simulation mechanism of the 3-SPR spatial manipulatoris created, as shown in Fig. 3b. The creation processes are explained as follows.1. Follow the step 2 in the common technique, constitute the moving platform Da1a2a3witha sideline (80 cm) in length, and the base DA1A2A3with a sideline (130 cm) in length.2. Follow step 5 in common technique, constitute an equivalent SPR driving limb r1.3. Repeat step 2 above, but (r1and L3) are replaced by (r2and L1) and (r3and L2), respectively.Thus, the other two equivalent SPR driving limbs (r2r3) can be constituted. In this way,a 3-SPR parallel simulation mechanism is created.3.2. The position-orientation of the moving platformThe position-orientation of the moving platform in the respect to the base of the 3-SPR sim-ulation mechanism can be determined by following processes.1. Constitute a line n, connect its one end to the center point a0on the moving platform m, and setline n perpendicular to m by adopting the perpendicular constraint command. In this way, anormal line n of m is constituted.2. Constitute a line N, set it perpendicular to both sideline L1and sideline L2of the base B, andconnect its one end to point a0on m by adopting the pointpoint coincident command.3. Constitute a line dy, connect its two ends to line N at point d0and sideline L1, respectively, byadopting the pointline coincident command. Next, set dyperpendicular to both N and L1.4. Give the distance from point a0to point d0an initial driven dimension dz(95.28 cm), give thedistance from point A1to line dyan initial driven dimension dx(64.41 cm), and give line dyaninitial driven dimension (37.87 cm) in length, respectively, by using dimension command. Thus,the three translation components (dxdy;dz) of m are constituted, as shown in Fig. 3b.5. Constitute line n1and line n2and connect their one ends to point A1on B, set n1parallel to n,and set n2perpendicular to both L1and N. Give the angle between n1and L1an initial drivendimension bx(79.18?), give the angle between n1and n2an initial driven dimension by(96.22?),give the angle between n and N an initial driven dimension bz(192.52?), respectively. Thus, thethree rotation components (bx;by;bz) of m are constituted, as shown in Fig. 3b.6. When varying or modifying each driving dimension of the driving limb ri, the driven dimen-sions of the three translation components (dxdy;dz) and the three rotation components(bx;by;bz) of m can be varied correspondingly. In this way, the position-orientation of the mov-ing platform in respect to the base of the 3-SPR simulation mechanism can be solved dynam-ically.3.3. The spatial 3-UPU parallel manipulator with 3-driving limbsAn existing spatial 3-UPU parallel manipulator has three DOF 3,4. It includes a movingplatform m, a base B, and three extendable driving limbs riwith their hydraulic cylinders and46Y. Lu / Mechanism and Machine Theory 39 (2004) 4160piston-rods as shown in Fig. 4a. Where, m is a regular triangle Da1a2a3with a0as its center, and Bis a regular triangle A1A2A3with A0as its center. Three identical UPU limbs connect m to B by auniversal joint U at point ai, a driving limb riwith a prismatic joint P, and a universal joint U atpoint Aifor i 1, 2, and 3, respectively.In the 3D sketch environment, a simulation mechanism of the spatial 3-UPU manipulator iscreated, as shown in Fig. 4b. The creation processes are explained as follows.1. Follow the step 2 in the common technique, constitute a moving platform Da1a2a3with sideline(80 cm) in length, and a base DA1A2A3with sideline (130 cm) in length, respectively.2. Follow the step 6 in the common technique, constitute an equivalent UPU driving limb r1. Sim-ilarly, replace (r1and L3) by (r2and L1) and (r3and L2), respectively. Thus, the other two equiv-alent UPU driving limbs (r2and r3) are constituted. In this way, a 3-UPU parallel simulationmechanism is created.3. The position-orientation of the moving platform of the 3-UPU simulation mechanism can besolved by adopting similar processes in the Section 3.2.3.4. The spatial 3-URPR parallel manipulator with 3-driving limbsA new type of the spatial 3-URPR parallel manipulator is designed, as shown in Fig. 5a. Itincludes a moving platform m, a base B, three binary links, and three extendable driving limbswith their hydraulic cylinders and piston-rods. Where, m is a regular triangle Da1a2a3with a0asits center, and B is a regular triangle A1A2A3with A0as its center. Three identical RPRU drivinglimbs connect m to B by a universal joint U at point ai, a binary link gi, a revolute joint R atpoint ei, a driving rod riwith a prismatic joint P, and a revolute joint R at point Aii 1;2;3,respectively.(a) (b) 1010101010(130)130(130)100103103(99.84)(31.67)(75.15)10A2a3dya1a2A1A3mr3r2r1dzdxBb2b1b3B1B3B2a0A0l3L3l2l3L2L1mBa3, UU, a2U,a1A3, UU, A1U, A2PPPFig. 4. (a) The spatial 3-UPU parallel manipulator and (b) its simulation mechanism. The spatial 3-UPU parallelmanipulator and its simulation mechanism.Y. Lu / Mechanism and Machine Theory 39 (2004) 416047By inspecting the whole 3-URPR parallel mechanism, we know that k 11 for one movingplatform, three binary links, three cylinders, and three piston-rods, and one base; j 12 for threeuniversal joints U, six revolute joints R, three prismatic joints P; f1 2 for the universal joint,f2 1 for the prismatic joint, f3 1 for the revolute joint; F0 0. Therefore, the DOF of thewhole mechanism isF kk ? j ? 1 Xni1fi? F0 6 ? 11 ? 12 ? 1 3 ? 2 6 ? 1 3 ? 1 ? 0 32In the 3D sketch environment, a simulation mechanism of the spatial 3-URPR parallel manip-ulator is created, as shown in Fig. 5b. The creation processes are explained as follows.1. Follow the step 2 in the common technique, constitute a moving platform Da1a2a3withsideline (80 cm) in length, and a base DA1A2A3with sideline (120 cm) in length, respec-tively.2. Follow the step 3c in the common technique, set line r1perpendicular to line g1and set line r1perpendicular to a sideline L3of B. Thus, the two revolute joints R at point e1and point A1areconstituted, respectively.3. Follow the step 6 in the common technique, set an auxiliary line b1perpendicular to both l1andg1and set b1parallel to r1. Thus, one equivalent universal joint U at point a1on m is consti-tuted. In this way, an equivalent URPR driving limb is constituted.4. Similarly, follow the steps 2 and 3 above, constitute the other two equivalent URPR drivinglimbs. In this way, a spatial 3-URPR simulation mechanism is created.5. The position-orientation of the moving platform of the 3-URPR simulation mechanism can bedetermined by adopting the similar processes in Section 3.2.(a) (b) A3R, A1PR, e1U, a1A2e2Bma2a3e3PPg3g1g2706540404080658080(8.44)(4.87)(73.63)101010120A1A3a2a3A2b2b1b3c1c2a0A0e1e2e3dxdydznr1r2r3a1d0g3g2g1l3l1l2L3L1L2Fig. 5. (a) The spatial 3-URPR parallel manipulator and (b) its simulation mechanism. The spatial 3-URPR parallelmanipulator and its simulation mechanism.48Y. Lu / Mechanism and Machine Theory 39 (2004) 4160From the 3-URPR simulation mechanism, we obtain two interesting conclusions:(1) If the three lines gii 1;2;3 are simultaneously reduced to 0 in length, then the spatial 3-URPR simulation mechanism is transformed into the spatial 3-SPR simulation mechanism.(2) If the three lines giare exchanged with the 3-driving rods rii 1;2;3 with a prismatic P,respectively, then the spatial 3-URPR simulation mechanism is transformed into a spatial3-UPRR parallel manipulator and its DOF is not change. In this case, if the three linesgii 1;2;3 are simultaneously reduced to 0 in length, then the spatial 3-UPRR simulationmechanism is transformed into a spatial 3-UPU simulation mechanism.3.5. The spatial 3-SRP parallel manipulator with 3-driving limbsA new type of spatial 3-SRP parallel manipulator is designed, as shown in Fig. 6a. It includes amoving platform m, a base B, three binary links, and three extendable driving limbs with theirhydraulic cylinders and piston-rods, as shown in Fig. 5a. Where, m is a regular triangle Da1a2a3with a0as its center, and B is a regular triangle DB1B2B3with B0as its center. Three identical SRPdriving limbs connect m to B by a spherical joint S at point ai, a binary link gi, a revolute joint R atpoint Aiand a driving limb riwith a prismatic joint P, for i 1;2, and 3, respectively. The drivinglimb riwith the prismatic joint P are retained coincident with the axis of revolute joint Riand thesideline Liof B i 1;2;3, respectively.By inspecting the whole mechanism of Fig. 6a, we know that k 8 for one moving platform,three cylinders, and three piston-rods, and one base; j 9 for three sphere joints S, three revolutejoints R, and three prismatic joints P; f1 3 for the sphere joint, f2 1 for the prismatic joint,(a) (b) S2, a1S2, a2S3, a3R2, A2R1, A1R3, A3g1P1, r1P3, r3P2, r2B1B3B2mB808080403050707070(69.17)(10)(5.78)(89.94)(90.85)(179.15+)S, a1a2a3g1g2g3B1A1B2B3AA3d0mBa0dzdydxnn1xyzr2r3r1L1L2L3Fig. 6. (a) The spatial 3-SPR parallel manipulator and (b) its simulation mechanism. The spatial 3-SPR parallelmanipulator and its simulation mechanism.Y. Lu / Mechanism and Machine Theory 39 (2004) 416049f3 1 for the revolute joint; F0 0. Therefore, based on Eq. (1), the DOF of the whole mech-anism isF kk ? j ? 1 Xni1fi? F0 6 ? 8 ? 9 ? 1 3 ? 3 3 ? 1 3 ? 1 ? 0 33In the 3D sketch environment, a simulation mechanism of the spatial 3-SPR manipulator iscreated, as shown in Fig. 6b. The creation processes are explained as follows.1. Follow the step 2 in the common technique, constitute the moving platform Da1a2a3with side-line (80 cm) in length, and the base DB1B2B3with sideline (130 cm) in length, respectively.2. Follow the step 4 in common technique, constitute a line g1connect its two ends to point a1onm and sideline L1at A1on B, and set g1perpendicular to L1. Give the distance from point B1topoint A1an initial driving dimension (40 cm). Thus, an equivalent SRP driving limb g1r1is con-stituted.3. Repeat the step 2 above, but (r1, g1and L1) are replaced by (r2, g2and L2) and (r3, g3and L3),respectively. Thus, other two equivalent SPR driving limbs (g2r2and g3r3) are constituted. Inthis way, a 3-SRP parallel simulation mechanism is created.4. The position-orientation of the moving platform of the 3-SRP simulation mechanism can bedetermined by adopting the similar processes in Section 3.2.4. Spatial parallel manipulator with 4-driving limbsA new type of spatial parallel mechanism with 4-driving limbs is designed, as shown in Fig. 7a.It includes a quaternary moving platform m, a quaternary base B, two extendable UPU limbs, andtwo extendable SPU limbs. Where, m is an equilateral quadrangle (a1a2a3a4) with a0as its center,and B is an equilateral quadrangle plane (A1A2A3A4) with A0as its center. Two identical SPUdriving limbs connect m to B by a spherical joint S at point ai, a driving limb riwith a prismaticjoint P, and a universal joint U at point Aifor i 2;4, respectively. Two identical UPU driving(a) (b) (c) PU, A1U, a1mBc2C2C1S, a2PPa3,US, a4A3,UPA4,UU, A2SPUmBUUUPPSUPUL1L3UUPUUSPPBmUUPSFig. 7. (a) 2-SPUUPU mechanism I; (b) 2-SPUUPU mechanism II and (c) 2-SPUUPU mechanism III. The spatialparallel mechanism with 4-driving limbs.50Y. Lu / Mechanism and Machine Theory 39 (2004) 4160limbs connect m to B by a universal joint U at point ai, a driving limb riwith a prismatic joint P,and a universal joint U at point Aifor i 1;3, respectively. We define it as a 2-SPUUPUmechanism I.When 2-UPU limbs and 2-SPU limbs are arranged in different positions, the other two types of2-SPUUPU mechanism II and 2-SPUUPU mechanism III can be synthesized, respectively, asshown in Fig. 7b and c.By inspecting the whole mechanisms of Fig. 7, we know that k 10 for one moving platform,four cylinders, and four piston-rods, and one base; j 12 for six universal joints U, four prismaticjoints P, and two sphere joints S; f1 2 for the universal joint, f2 1 for the prismatic joint,f3 3 for the sphere joint; F0 0. Therefore, based on Eq. (1), the DOF of the whole mechanismisF kk ? j ? 1 Xni10fi? F0 6 ? 10 ? 12 ? 1 6 ? 2 4 ? 1 2 ? 3 ? 0 44A simulation mechanism I of the spatial 2-SPUUPU parallel mechanism is created in the 3Dsketch environment, as shown in Fig. 8a, and its creation processes are described below.1. Follow the step 2 in the common technique, constitute a moving platform (a1a2a3a4) with side-line (60 cm) in length, and a base (A1A2A3A4) with sideline (100 cm) in length, respectively.2. Follow the steps 6 and 7 in the common technique, constitute two equivalent UPU drivinglimbs (r1and r3) and two equivalent SPU driving limbs (r2and r4), respectively. In this way,a simulation mechanism of the spatial 2-SPUUPU parallel mechanism I can be created.3. The position-orientation of the moving platform of the 2-SPUUPU simulation mechanism Ican be determined by adopting similar processes in Section 3.2.Similarly, a 2-SPUUPU simulation mechanisms II is also created, as shown in Fig. 8b.(a) (b)60607270727010101020(85.03)(46.45)(50)10101060A1A3A4A2C1dxdza2a1b1B1B2B3B4b3anaA0ac2c1L4C2dyr1r2r4r3Bm60601010101020108083801083(76.20)(50)(56.11)mBdydxdzFig. 8. (a) The simulation mechanism I and (b) the simulation mechanism II. The simulation mechanisms of the2-SPUUPU parallel mechanism with 4-driving limbs.Y. Lu / Mechanism and Machine Theory 39 (2004) 4160515. Spatial parallel manipulator with 5-driving limbs5.1. The 4-SPU and 1-UPU parallel manipulator and its simulation mechanismA new type of the spatial 4-SPU and 1-UPU parallel manipulator with 5-driving limbs is de-signed, as shown in Fig. 9a. It includes a pentagonal moving platform m, a pentagonal base B, oneextendable UPU driving limb with a hydraulic cylinder and a piston-rod, and four extendableSPU driving limbs with their hydraulic cylinders and piston-rods. Where, m is an equilateralquadrangle (a1a2a3a4) with a0as its center, and B is an equilateral quadrangle plane (A1A2A3A4)with A0as its center. Four identical SPU driving limbs connect m to B by a spherical joint S atpoint ai, a driving limb riwith a prismatic joint P, and a universal joint U at point Ai(i 1;2;3;4), respectively. 1-UPU driving limb connects m to B by a universal joint U at point a0a driving limb r5with a prismatic joint P, and a universal joint U at point A0.By inspecting this whole mechanism, we know that k 12 for one moving platform, fivecylinders, and five piston-rods, and one base; j 15 for six universal joints U, five prismaticjoints P, and four sphere joints S; f1 2 for the universal joint, f2 1 for the prismatic joint,f3 3 for the sphere joint; F0 0. Therefore, based on Eq. (1), the DOF of the whole mech-anism isF kk ? j ? 1 Xni1fi? F0 6 ? 12 ? 15 ? 1 6 ? 2 5 ? 1 4 ? 3 ? 0 55A simulation mechanism of the spatial 4-SPU and 1-UPU parallel manipulator is created inthe 3D sketch environment, as shown in Fig. 9b. Its creation processes are described as fol-lows.1=i(a) (b)PU, A1S, a1mBc2C2C1S, a2PPa3, SS, a4A3,UPA4, UU, A2U, A0nPa06060106999890301313941013(91.26)(53.40)(72.29)131 3(92.15)(99.61)(170.15,)100A1A3A4A2a1a3a4a2B2B1B3B4BA5a0d0dydzdxc1c2C2C1xyznn1r1r2r3r4r5mFig. 9. (a)The spatial 4-SPU and 1-UPU parallel manipulator and (b) its simulation mechanism. The 4-SPU and1-UPU parallel manipulator with 5-driving limbs and its simulation mechanism.52Y. Lu / Mechanism and Machine Theory 39 (2004) 41601. Follow the step 2 in the common technique, constitute a moving platform (a1a2a3a4) with side-line (60 cm) in length, and a base (A1A2A3A4) with sideline (100 cm) in length, respectively.2. Follow the steps 6 and 7 in the common technique, constitute four equivalent SPU drivinglimbs (r1, r2, r3and r4) and one equivalent UPU driving limb r5, respectively. In this way, a sim-ulation mechanism of the spatial 4-SPU and 1-UPU parallel manipulator is created.3. The position-orientation of the moving platform of the spatial 4-SPU and 1-UPU parallelmanipulator can be determined by adopting similar processes in Section 3.2.5.2. The 4-PSU and 1-UPU parallel manipulator and its simulation mechanismA new type of spatial 4-PSU and 1-UPU parallel manipulator with 5-driving limbs is designed,as shown in Fig. 10a. It includes a moving platform m, a base B, 1-UPU driving limb, and 4-PSUdriving limbs. Where, m is an equilateral quadrangle (a1a2a2a4) with a0as its center. B is cubicframe, in which an equilateral quadrangle plane (A1A2A2A4) is connected onto another equilateralquadrangle plane (B1B2B2B4) by the four vertical columns AiBii 1;2;3;4, respectively. Thefour prismatic joints Piare retained coincident with the four vertical columns AiBii 1;2;3;4,respectively.By inspecting the whole mechanism, we know that k 12 for one moving platform, five cyl-inders, and five piston-rods, and one base; j 15 for six universal joints U, five prismatic joints P,and four sphere joints S; f1 2 for the universal joint, f2 1 for the prismatic joint, f3 3 for thesphere joint; F0 0. Therefore, based on Eq. (1), the DOF of the whole mechanism isF kk ? j ? 1 Xni1fi? F0 6 ? 12 ? 15 ? 1 6 ? 2 5 ? 1 4 ? 3 ? 0 56A simulation mechanism of the spatial 4-PSU and 1-UPU parallel manipulator is created in the3D sketch environment as shown in Fig. 10b, and its creation processes are described below.(a)(b) A3A2A1P3S, e3A4a4a3U, a1r2e1S, e1P2P1P4e4S, A0BC1a2r3r1r4mr5a0B4B3B1B2ggggP560606060404070707070120(57.61)(50.12)(119.76)10050e2e3e1A3A2A1a1a4e4a3a2A4A0d0C1a0mBdzdxdyr1r2gr3gr4ggr5Fig. 10. (a) The spatial 4-PSU and 1-UPU parallel manipulator and (b) its simulation mechanism. The 4-PSU and1-UPU parallel manipulator with five DOF and its simulation mechanism.Y. Lu / Mechanism and Machine Theory 39 (2004) 4160531. Follow the step 2 in the common technique, constitute a moving platform (a1a2a3a4) with side-line (50 cm) in length, and a base plane (A1A2A3A4) with sideline (100 cm) in length, respectively.2. Constitute four lines Aiei, connect their one ends to point Aii 1;2;3;4 of the base plane(A1A2A3A4), respectively, by using the pointpoint coincident command, and set Aieiperpendic-ular to the base plane.3. Follow the steps 3e, 6, and 7 in the common technique, constitute four equivalent PSU drivinglimbs (r1, r2, r3and r4) and an equivalent UPU driving limb r5, respectively. In this way, a sim-ulation mechanism of the 4-PSU and 1-UPU spatial parallel manipulator is created.4. The position-orientation of the moving platform of the 4-PSU and 1-UPU spatial parallelmanipulator can be determined by adopting similar processes in Section 3.2.6. Spatial parallel manipulator with 6-driving limbs6.1. The 3/6-SPS parallel manipulator and its simulation mechanismAn existing Stewart 3/6-SPS spatial parallel manipulator has six DOF 1,13,14. It includes amoving platform m, a base B, and six extendable driving limbs riwith their hydraulic cylinders andpiston-rods, as shown in Fig. 11a. The moving platform is a regular triangle Da1a2a3with a0as itscenter. The base is a regular hexagon (A1A2A3A4A5A6) with A0as its center. Six identical SPSdriving limbs riconnect m to B by a spherical joint S at point akk 1;2;3, a driving limb riwitha prismatic joint P, and a spherical joint S at point Aii 1;2;.;6, respectively.A simulation mechanism of the spatial 3/6-SPS parallel manipulator is created as shown inFig. 11b. The creation processes are explained as follows.(a) (b)S, a3S, a1S, a2A4A3A2S, A1A5A6Bmr1r2r3r4r5PPPPP5050808080909090(78.91)(38.64)(0.14)(98.08)(89.81)8.081a3a1a2A4A2A1A5A6a0r1r4r3r6r5r2c2c1n1ndzdxdyxzyxd0A0mBr6PA3Fig. 11. (a) The 3/6-SPS parallel manipulator and (b) its simulation mechanism. The 3/6-SPS parallel manipulator andits simulation mechanism.54Y. Lu / Mechanism and Machine Theory 39 (2004) 41601. Follow the step 2 in the common technique, constitute an equilateral hexagon plane(A1A2A3A4A5A6) with sideline (50 cm) in length, and take it as the base. Constitute an equilateraltriangle Da1a2a3with sideline (50 cm) in length, and take it as the moving platform.2. Follow the step 5 in the common technique, constitute 6-SPS driving limbs rit 1;2;.;6and give each driving limb an initial driving dimension (90 cm) in length. In this way, a simu-lation mechanism of the 3/6-SPS parallel manipulator is created.3. The position-orientation of the moving platform of the 3/6-SPS simulation mechanism can bedetermined by adopting similar processes in Section 3.2.By inspecting this whole mechanism, we know that k 14 for one moving platform, six cyl-inders, and six piston-rods, and one base; j 18 for six prismatic joints P, six sphere joints S on B,and six sphere joints S on M which are transformed into three composite sphere joints S; f1 1for the prismatic joint, f2 3 for the sphere joint; F0 6 for 6-driving limb rotating aboutthemselves axis. Therefore, based on Eq. (1), the DOF of the whole mechanism isF kk ? j ? 1 Xni1fi? F0 6 ? 14 ? 18 ? 1 6 ? 1 12 ? 3 ? 6 676.2. The 6-SPS spatial parallel manipulator and its simulation mechanismAn existing StewartGough 6-SPS spatial parallel manipulator has six DOF 1,13,14. It in-cludes a moving platform m, a base B, and six extendable driving limbs riwith the hydrauliccylinder and the piston-rod, as shown in Fig. 12a. Where, m is a regular hexagon (a1a2a3a4a5a6)with a0as its center, and B is the regular hexagon (A1A2A3A4A5A6) with A0as its center. Sixidentical SPS driving limbs riconnect m to B by a spherical joint S at point ai, a driving limb riwith a prismatic joint P, and a spherical joint S at point Aii 1;2;.;6, respectively.(a)(b)A2S, A1A5A6A3a2A4a4a5S, a1a0a6r6r1r2A0r3r4cbr6Pr5a3PPPPPmB82838250505050505082828310030(74.96)(57.86)(15.94)(93.28)(197.06#)106.72#25A2A1A5A6A3a2A4a4a3a5a1a0a6dznxzyn1ydydxxcd0A0mBFig. 12. (a) The 6-SPS parallel manipulator and (b) its simulation mechanism. The StewartGough 6-SPS parallelmanipulator and its simulation mechanism.Y. Lu / Mechanism and Machine Theory 39 (2004) 416055By inspecting this whole mechanism, we know that k 14 for one moving platform, six cyl-inders, and six piston-rods, and one base; j 18 for six prismatic joints P, six sphere joints S on B,and six sphere joints S on M; f1 1 for the prismatic joint, f2 3 for the sphere joint; F0 6.Therefore, the DOF of the whole mechanism is the same as that of Stewart 3/6-SPS spatial parallelmanipulator.A simulation mechanism of the 6-SPS spatial parallel manipulator with 6-driving limbs iscreated, as shown in Fig. 12b. The creation processes are explained as follows.1. Follow the step 2 in the common technique, constitute an equilateral hexagon (A1A2A3A4A5A6)with sideline (50 cm) in length, and take it as the base. Constitute an equilateral hexagon plane(a1a2a3a4a5a6) with sideline (25 cm) in length, and take it as the moving platform.2. Follow the step 5 in the common technique, constitute 6-SPS driving limbs rii 1;2;.;6and give each driving limb an initial driving dimension (82 cm) in length. In this way, a simu-lation mechanism of the 6-SPS parallel manipulator with 6-driving limbs is created.3. The position-orientation of the moving platform of the 6-SPS simulation mechanism can be de-termined by adopting similar processes in Section 3.2.6.3. The 6-SSP spatial parallel manipulator and its simulation mechanismA new type of the 6-SSP spatial parallel manipulator is designed, as shown in Fig. 13a. Itincludes a moving platform m, a base B, six binary links, and six extendable driving limbs riwiththe hydraulic cylinder and the piston-rod. Where, m is a regular hexagon (a1a2a3a4a5a6) with a0asits center, and B is an equilateral quadrangle link (B1B3B4B6) with three translation paths. Sixidentical SSP driving limbs connect m to B by a spherical joint S at point ai, binary link gi, a(a)(b)S, A1PPPS, A6S, A3A2A5S, A4PPPS, a3S, a2S, a1S, a6S, a5S, a4mBL1L2L3150302819202830100100100106100100110(22.63)(88.58)(96.44)50(86.26)(114.09k)(24.419)A6A2A5A1A3A1a5a6a4a3a2a1g4r2g1g5g6g3dydxdza0L1L2L3xyzmBB6B1B5B2B3B4nn1Fig. 13. (a) The 6-SSP parallel manipulator and (b) its simulation mechanism. The 6-SSP parallel manipulator and itssimulation mechanism.56Y. Lu / Mechanism and Machine Theory 39 (2004) 4160spherical joint S at point Ai, and a driving limb riwith a prismatic joint P at point Ai,i 1;2;.;6, respectively.By inspecting the whole mechanism of Fig. 13a, we know that k 14 for the six cylinders, sixpiston-rods, one base, and one moving platform; j 18 for the 12 spherical joints S, and sixprismatic joints P; f1 3 for spherical joint, f2 1 for prismatic joints; F0 6 for the six binarylinks rotating about their axis. Therefore, the DOF of the 6-SSP parallel manipulator is deter-mined as follows.F kk ? j ? 1 Xni1fi? F0 6 ? 14 ? 18 ? 1 12 ? 3 6 ? 1 ? 6 68A simulation mechanism of the spatial 6-SSP parallel manipulator with 6-driving limbs is createdas shown in Fig. 13b. The creation processes are explained as follows.1. Follow the step 2b in the common technique, constitute an equilateral hexagon plane(a1a2a3a4a5a6) with sideline (50 cm) in length, and take it as the moving platform.2. Follow the step 2d in the common technique, constitute an equilateral quadrangle link(B1B3B4B6) with sideline (100 cm) in length, and take it as the base. Constitute a line L2andconnect its two ends to the middle point B2of sideline B1B3and the middle point B5of sidelineB4B6, respectively, by using the pointline coincident command.3. Follow the steps 3e and 7 in the common technique, constitute six lines gi, (i 1;2.;6), giveeach of them an initial fixed dimension (100 cm) in length. Connect their two ends to the mov-ing platform at point aiand the base at point Ai, respectively. Thus, 6-SSP driving limbs areconstituted. In this way, a simulation mechanism of the spatial 6-SSP parallel manipulator with6-driving limbs is created.4. The position-orientation of the moving platform of the 6-SSP simulation mechanism can be de-termined by adopting similar processes in Section 3.2.6.4. The spatial 6-SPRP parallel manipulator and its simulation mechanismFrom the spatial 3-SRP parallel manipulator of Fig. 6a, replace the three binary links giby theextendable driving limbs riwith the hydraulic cylinder and the piston-rod (i 1;2;3), respectively.In this way, a new type of the spatial 6-SPRP parallel manipulator is designed. It includes amoving platform m, a base B, and six extendable driving limbs with their hydraulic cylinders andpiston-rods, as shown in Fig. 14a. Moreover, this mechanism is partially parallel and partiallyserial, which is often referred to as hybrid structure.By inspecting the whole mechanism of Fig. 14, we know that k 11 for the six cylinders,three piston-rods, one base, and one moving platform; j 12 for the three spherical joints S,and six prismatic joints P, three revolute joints R; f1 3 for spherical joint, f2 1 for pris-matic joints, f3 1 for revolute joints; F0 0. Therefore, the DOF of the 6-SSP parallelmanipulator isF kk ? j ? 1 Xni1fi? F0 6 ? 11 ? 12 ? 1 3 ? 3 6 ? 1 3 ? 0 69Y. Lu / Mechanism and Machine Theory 39 (2004) 416057A simulation mechanism of the spatial 6-SPRP parallel manipulator with 6-driving limbs, asshown in Fig. 14b is similar to that of spatial 3-SRP parallel manipulator, as shown in Fig. 6b,except that the fixed dimension of the three binary links gii 1;2;3 are transformed intodriving dimension.7. Solving displacement of the moving platformIn the light of the simulation mechanism of the spatial 3-UPU parallel manipulator as shown inFig. 4b, when set li 50 cm, Li 120 cm (i 1;2;3), its original configuration is changed into anew one as shown in Fig. 14a. Therefore, once a simulation mechanism is created, it can be usedrepeatedly for synthesizing same type of spatial parallel mechanism with different sizes andconfigurations. The three position components of the moving platform of the 3-UPU simulationmechanism can be solved, and the solving processes are explained below.In the environment of the 3-UPU simulation mechanism, a data sheet of Microsoft Excel isembedded into the sketching interface of the CAD software. Next, click each driving dimension ofdriving limbs, thus their dimension names (r1, r2and r3) are automatically listed in the first row ofthe data sheet. After that, fill the three driving dimensions (r1, r2and r3) from 80 to 120 cm withdifferent increments (1, 1.5 and 2) and the number of configurations into their correspondingcolumn, respectively, by using automatically filling function. Thus, the 20 different configurationsof the simulation mechanism are constituted. Each configuration corresponds to each row drivingdimensions of (r1, r2and r3). When any one of all configurations is active, the 3-UPU simulationmechanism will be varied and can be directly visualized on screen. Thus the position componentsof the moving platform of spatial 3-UPU parallel manipulator can be solved automatically. The(a) (b)808080403050707070(69.17)(10)(5.78)(89.94)(90.85)(179.15+)S, a1a2a3g1g2g3B1A1B2B3AA3d0mBa0dzdydxnn1xyzr2r3r1L1L2L3S2, a1S2, a2S3, a3R2, A2R1, A1R3, A3P3,g1P1, r1P3, r3P2, r2B1B3B2mBP6,g3P2, g2Fig. 14. (a) The 6-SPRP parallel manipulator and (b) its simulation mechanism. The 6-SPRP parallel manipulator andits simulation mechanism.58Y. Lu / Mechanism and Machine Theory 39 (2004) 4160simulation results of position components of the spatial 3-UPU parallel manipulator are retainedthe same as that of analytics? approach, as shown in Fig. 15.8. ConclusionsBy applying the computer aided geometry constraint and dimension driving techniques in 3Dsketch environment of the advanced CAD software, the simulation mechanisms of the some newand original spatial parallel manipulators with 3-, 4-, 5-, 6-driving limbs can be designed. Whenmodifying the driving dimensions of driving limbs by using the dimension driving technique, theconfigurations of the simulation mechanisms are varied correspondingly. In this way, the kine-matic parameters of the moving platform of each simulation mechanism can be solved.When all the driving dimensions and the fixed dimensions of simulation mechanism aremodified, all the geometric constraints and dimension constraints in each simulation mechanismare always retained. Therefore, once a simulation mechanism is created, it can be used repeatedlyfor synthesizing same type of spatial parallel mechanism with different sizes and configurations.When modifying the driving dimension in simulation mechanism, the driven dimension will bevaried correspondingly. In this way, based on the DOF of parallel spatial mechanism, the numberof driving dimensions of the simulation mechanism can be determined.By applying the driving and driven dimension technique and recording function, the position-orientation of the moving platform in respect to the base can be solved. The approximate
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