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Robotica (2003) volume 21, pp. 645653. 2003 Cambridge University PressDOI: 10.1017/S0263574703005198Printed in the United KingdomA three translational DoFs parallel cube-manipulatorXin-Jun Liu*, Jay il Jeong, and Jongwon KimRobust Design Engineering Lab. (Room 210, Building 301), School of Mechanical and Aerospace Engineering,Seoul National University, (South) KOREA 151742(Received in Final Form: April 5, 2003)SUMMARYThis paper concerns the presentation and analysis of a typeof three translational degrees of freedom (DoFs) parallelcube-manipulator. The parallel manipulators are the topol-ogy architectures of the DELTA robot and Tsaismanipulator, respectively, which have three translationalDoFs. In the design, the three actuators are arrangedaccording to the Cartesian coordinate system, which meansthat the actuating directions are normal to each other, andthe joints connecting to the moving platform are located onthree sides of a cube, for such reason we call this type ofmanipulator the parallel cube-manipulator. The kinematicsproblems, singularity, workspace, compliance characteristicof the manipulator are investigated in the paper. Theanalysis results show that the manipulators have theadvantages of no singularities in the workspace, relativelymore simple forward kinematics, and existence of acompliance center. The parallel cube-manipulator can beapplied to the elds of micro-motion manipulators, remotecenter compliance (RCC) devices, assembly, and so on.KEYWORDS: Parallel manipulators; Singular congurations;Compliance; Kinematics; Workspace.1. INTRODUCTIONAfter a motion simulator with parallel kinematics chainswas introduced in 1965,1parallel manipulators receivedmore and more attention because of high stiffness, highspeed, high accuracy, compact and high carrying capability.And more and more parallel manipulators with speciednumber and type of degrees of freedom (DoFs) have beenproposed. Behi2described a 6-DoF conguration with threelegs where each leg consists of a PRPS chain. Hudgens andTesar3investigated a device with six inextensible legs whereeach leg is driven by a four-bar mechanism mounted on thebase. Romiti and Sorli4introduced a 6-DoF parallelmanipulator named TuPaMan with three legs, each of whichis composed of a double parallelogram, one sliding joint, theconnecting bar, and a ball joint. Austad5mentioned a hybridarchitecture with 5 DoFs based on two parallel mechanisms.Pierrot and Company6presented a new family of parallelmanipulators with 4 DoFs, which are 3 translations and 1rotation. Kim and his colleague7proposed a 6-DoF parallelmechanism named Eclipse-II, which has the advantage ofenabling continuous 360-degree spinning of the platform* Corresponding author: xinjunlbased on redundant actuators. And Liu et al.8proposed aspatial parallel manipulator with two translational DoFs andone rotational DoF, which has high mobility. They havebeen applied widely to the elds of motion simulator, force/torque sensor, packaging, micro-motion manipulators,medical devices, compliance devices and machine tools. Inthese designs, parallel manipulators with three translationaldegrees of freedom have been playing important roles in theindustrial applications,911especially, the DELTA robot,9which is covered by a family of 36 patents.12Generally, most of these manipulators suffer from bothsingular congurations within the workspace and multiplesolutions of the direct geometric model. The objective of thepaper is to design a three translational DoFs parallelmanipulator which avoids these drawbacks and which canbe well adapted to the applications of micromotionmanipulators, remote center compliance (RCC) devices, andprecision assembly machines. In the design, the threeactuators are arranged according to the Cartesian coordinatesystem, which means that the actuating directions arenormal to each other, and the joints connecting to themoving platform are located on three planes being perpen-dicular to each other too; hence, we call this type ofmanipulator a parallel cube-manipulator. The kinematicsproblems, singularity, workspace and compliance character-istics of the manipulator are investigated in the paper. Thekinematics analysis of the manipulator shows that thesolution for the inverse and forward kinematics is unique forthe reason of interference. The manipulators have theadvantages of high compactness and stiffness, no singular-ities in the workspace, relatively more simple forwardkinematics, and existence of a compliance center. Otheradvantages of this type of parallel manipulator will beinvestigated in the future work.2. PROPOSAL OF THE PARALLEL CUBE-MANIPULATORIn the eld of parallel manipulators, an interesting problemis to nd a method to design a mechanical architecture fora parallel manipulator being given its number and type ofdegree of freedom. After Gough established the basicprinciples of a mechanism with a closed-loop kinematicsstructure in 1947, many other parallel manipulators withspecied number and type of degrees of freedom have alsobeen proposed. Fig. 1 is the general architecture of 6-DoFparallel manipulators. Theoretically speaking, the arrange-ment of the six legs of the manipulator could be at will,中国科技论文在线646Fig. 1. A general 6-DoF parallel manipulator.which will lead to some potential 6-DoF parallel manip-ulators, such as a manipulator as shown in Fig. 2, where thelegs are arranged as 3-2-1 style. This architecture has theapplication advantage in micro system.13The arrangementdisposal of six legs, as shown in Fig. 3, will make themanipulator move freely along a specied direction, whichis very useful for the application in industry.14Letting every two legs of the six legs of the manipulator,as shown in Fig. 1 be parallel to each other will lead to a6-DoF parallel manipulator similar to that with revolutelactuators presented by Pierrot shown in Fig. 4.15Thenumber of DoFs of the manipulator will be different ifinputs of the two parallel legs are identical. This is actuallya topology mechanism of the DELTA with linear actuators.Similarly, the manipulator shown in Fig. 4 can also beevolved into the DELTA robot. The fact that the outputs oftwo nearby revolute actuators in Fig. 4 are always same toeach other will result in the different output of the movingplatform. Actually, this is the design of the well-knownDELTA robot, as shown in Fig. 5.Apart from the DELTA robot, there are also many otherparallel manipulators with three DoFs. For example, theFig. 2. The manipulator with 3-2-1 arrangement.Fig. 3. The Hexaglide manipulator.A three translational DoFs parallel cube-manipulatorFig. 4. The Pierrots manipulator.spatial 3-DoF 3-RPS parallel manipulator presented byHunt in the early of 80s.16The output of the manipulator isthe complex motion, which are one translation and tworotations with continuously changed axes.17For suchreason, the manipulator has less good applications exceptfor the micro-motion device.18The similar manipulatorincludes CaPaMan in Italy,19which has complex DoFs aswell. When we consider three translational parallel manip-ulators, Tsais parallel manipulator11is worth mentioning.Although Tsais manipulator has translations identical withthat of DELTA, this manipulator is not the version ofDELTA. Tsais manipulator is the rst design to solve theproblem of UU chain. There is also another 3 translationalDoFs parallel manipulator, Star, which is designed by Hervbased on group theory.10Such a kind of parallel manipulatorhas wide applications in industrial world, e.g. pick-and-place application, parallel kinematics machines, andmedical devices.12In the family of spatial 3-DoF parallelmanipulators, the manipulators with three rotational DoFsreceived much more attention.2023 They have been appliedto the elds of camera-orienting and haptic devices.24,25Another type of 3-DoF parallel manipulator is that themoving platform is connected to the base through four legs.For such a mechanism, it usually consists of 3 identicalactuated legs with 6 DoFs and one passive leg with 3 DoFsconnecting the platform and the base, i.e. the degree offreedom of the mechanism is dependent on the passive legsdegree of freedom. One can improve the rigidity of this typeof mechanism through optimization of the link rigidities toreach a maximal global stiffness and precision.26This typeof mechanism has been applied to the design of a hybridmachine tool.27There are few 3-DoF fully parallel manip-ulators that can combine the spatial rotational andtranslational DoFs and be further with denite motion(being opposite to the complex motion) except for that ofHALFwith three non-identical legs.8All these spatialFig. 5. The DELTA robot.中国科技论文在线A three translational DoFs parallel cube-manipulatorFig. 6. The three translational DoFs parallel cube-manipulators.parallel mechanisms have some disadvantages in common,i.e. singularity existing in the workspace, more complexforward kinematics problem, and no compliance center,which are very important for some applications. This paperattempts to present a three translational DoFs parallelmanipulator which avoids these drawbacks.The supposed design conceives for the above-mentionedmanipulators can not only make us to understand themanipulators easily, but provide us new conceive to designa manipulator as well. For example, the 6-DoF parallelmanipulators described in references 28,29 can be taken asthe topology architectures of the manipulator shown inFig. 1 by rearranging the six legs. This inspires us to designnew parallel manipulators by rearranging the three legs ofDELTA9and Tsais parallel manipulator,11which are shownin Fig. 6. In Fig. 6(a), the moving platform is connected tothe base by three legs with PRPSchains, where PSstands forthe spatial four-bar parallelogram with four spherical joints,P prismatic joint and R revolute joint. In the design ofFig. 6(b), the three legs are with PRPRR chains, where PRdenotes the planar four-bar parallelogram with four revolutejoints. In those two designs, the three actuators are arrangedaccording to the Cartesian coordinate system, which meansthat the actuating directions are normal to each other. Andthe joints connecting to the moving platform are located inthree sides of a cube, for such reason we call this type ofmanipulator the parallel cube-manipulator. The manip-ulators also have three translational DoFs as in the cases ofDELTA robot and Tsais manipulator. But, they have someimportant characteristics being different from that ofDELTA robot and Tsais manipulator, which makes the newdesigns novel.3. KINEMATICS PROBLEMSA kinematics model of such parallel cube-manipulators canbe developed as shown in Fig. 7. The center of the link ineach of the three legs connected to the base by prismatic647joints is denoted asBi(i = 1, 2, 3). And the center of theinterval between the two spherical joints connecting themoving platform for each chain is denoted as Pi(i = 1, 2, 3),which is in one of three sides of the cubic moving platform,respectively. Biand Piare the centers of the revolute jointsin each leg for the design of Fig. 6(b). A xed globalreference system: oxyzis located at the intersectionpoint of three axes of actuated prismatic joints with the z-axis and they-axis directed along the rst and thirdactuation directions, respectively, as shown in Fig. 7.Another reference system: ox y zis located at thecenter of the cubic moving platform. The z -axis and y -axisdirected alongP1oandP3o , respectively, as shown inFig. 7. Related geometric parameters areo Pi= randBiPi= L, where i = 1, 2, 3.Fig. 7. A kinematics model of the parallel cube-manipulators.中国科技论文在线6483.1. Inverse kinematics problemThe objective of the inverse kinematics is to nd the inputsof the manipulator with the given position of reference pointo . The position vector cof pointoin framecan bewritten asA three translational DoFs parallel cube-manipulatoramongx, yandzis with one degree. This allows us toexpress the two of three variables x, y and z with the thirdone very easily, e.g. x and y can be described as the functionof z, that is2c= xyzT(1)x =1+ rz(1 + r)+2+ r(12)As shown in Fig. 7, The coordinate of the point Piin theframecan be described by the vectorPi(i = 1, 2, 3),which can be expressed as0r02 + r + ry =1z3 + r 2(2+ r)(1+ r)22(3+ r)2+ r+32(13)p1=0r,p2=00,p3=0r(2)Substituting Eqs. (12) and (13) to Eq. (6) and rearranging itwill lead toaz2+ bz + c = 0(14)Then vectors pi(i = 1, 2, 3) in frame Oxyz can be writtenaspi=pi+ c,(3)As shown in Fig. 7, vector bi(i = 1, 2, 3) is dened as theposition vector of point Biin frame, and0x20Then there isin whichz =b b24ac2a(15)b1=0z1,b2=00,b3=y30(4)a = 1 +1+ r2+1+ r2where z1, x2and y3are the inputs of the manipulator, whichwill be denoted as1,2and3respectively. Then theinverse kinematics of the parallel cube-manipulator can besolved by writing following constraint equation2+ rb =(1+ r)313+ r+1that ispibi= L,i = 1, 2, 3(5)x2+ y2+ (z1r)2= L2(6)2(2+ r)22(3+ r)24(x2r)2+ y2+ z2= L2x2+ (y3r)2+ z2= L2(7)(8)c =(2+ r)4+(3+ r)4+(1+ r)41(2+ r)2+1(3+ r)2L2from which, if the position of the moving platform isspecied, the inputs of the manipulator can be obtained asx and y will be reached substituting Eq. (15) to Eqs. (12) and(13), respectively, if inputs1,2and3are given. And fromEqs. (12), (13) and (15) one can see that there are only two21= L22= L23= Lx2y2+ zrz2y2+ xrx2z2+ yr(9)(10)(11)forward kinematics solutions for the manipulator. To obtainthe foward conguration as shown in Fig. 7, the sign “ ” inthe Eq. (15) should be “ + ”. The second forward kinematicssolution, in which the sign “ ” in the Eq. (15) is “”, willlead to the interference between the moving platform andfrom which one can see that there are eight inversekinematics solutions for the manipulator. To get the inverseconguration as shown in Fig. 7, the sign “ ” in each of theEqs. (9)(11) should be “”. Actually, any other assemblymodes except for the one as shown in Fig. 7 willundoubtedly result in the interference between the movingplatform and the legs. Therefore, the cube-manipulator canonly be assembled as the conguration shown in Fig. 7,which means that there is only one inverse kinematicssolution for the manipulator.3.2. Forward kinematics problemFrom Eqs. (6)(8), we can see that, in each equation, themaximum degree of the polynomial is two, the coeffcient ofwhich is 1, and, what is more, there is only one variablelegs. Comparing the forward kinematics problem with theresult of a DELTA robot with linear actuators described inreferences 30,31, it is not difcult to nd that the problemof the cube-manipulator studied in this paper is relativelymore simple.4. VELOCITY EQUATIONS AND JACOBIANMATRIXEquations (6)(8) can be differentiated with respect to timeto obtain the velocity equations, which leads to(x2r)x + yy + zz = (x2r)2xx + (y3r)y + zz = (y3r)3xx + yy + (z1r)z = (z1r) 1中国科技论文在线A three translational DoFs parallel cube-manipulatorRearranging above three equations leads to an equation ofthe formA = B (16)where is the vector of output velocities dened as649only conguration- but architecture-dependent aswell.32This corresponds to congurations in which thechain can undergo nite motions when its actuators arelocked.From Eqs. (19) and (20), one can see that the rst kind of = xyzTand is the vector of input velocities dened as = 231T(17)(18)singularity will occur if any one of the three points Biislocated in the plane deterrnined by the side of the cubicmoving platform where Piis located. In this condition, theparallelogram will also reach its singular conguration. Butthe manipulator cannot reach these congurations owing toMatrices Aand Bare, respectively, the 33 inverse andforward Jacobian matrices of the manipulator and can beexpressed asmechanical limitations, which means that there is no suchsingularity within the workspace of a parallel cube-manipulator. It is more important that the second kind ofA = diag(a11, a22, a23)(19)singularity will not occur in the case of the parallel cube-manipulator and onlyL= 0 will lead to the third kind ofB =a11xxya22yzza33(20)singularity. All in all, the parallel cube-manipulator has nosingularities practically.6. WORKSPACE ANALYSISwith a11= x2r, a22= y3r and a33= z1r. Andif the matrix A is nonsingular, Eq. (16) can be rewritten asWorkspace is an important index to evaluate the workingperformance of a manipulator. How to determine the = Jwhere J is the Jacobian matrix of the manipulator,1y/a11z/a11J = A1B =x/a221z/a22(21)(22)workspace is one of the most important issues in the designof a manipulator, especially for a parallel manipulator whichis well known to have relatively small useful workspace.Therefore, the geometric determination of the workspacefor a parallel manipulator attracts more and more research-ers attention.3436The workspace of translational parallel5. SINGULARITIESx/a33y/a331manipulators has been studied by many researchers.30,3739Ingeneral, it can always be determined geometrically for suchmanipulators, which is also same to the parallel cube-manipulators considered in this paper.Singularities are particular congurations where the manip-ulator becomes uncontrollable. As it is well known, thereare three kinds of singularities in multi-loop kinematicsmechanisms,32,33which are determined by the analysis ofinverse and forward Jacobian matrices A and B:(i) The rst kind of singularity occurs when AbecomesFrom Eqs. (6)(8), one can see that if the inputs of thecube-manipulator are given, workspaces of the three legsare three spheres, which are centered at points (0, 0,1+ r),(2+ r, 0, 0)and (0,3+ r, 0), respec