Singularities Analysis of Paradoxical Mechanism and Parallel Manipulators

Jouma/ of Donghua Uhiversity (Eng. Ed.) Vol.23, No.6 (2006) 5SingularitiesAnalysis of Paradoxical Mechanism and ParallelManipulatorsHAO Kuang-rong (郝矿荣)College of Information Science and Technology, Donghua University, Shanghai 201620Parallel manipulatorassociated with a set of functionsconstructed based on this principle. Liu et al. [12] analyzeddefined by its closure constraints. In this paper, using Lieconfiguration space singularities by constructing a Morsealgebra method, we provide a study on the singularities offunction and defined the parameterization singularities.parallel manipulators, their relations with the second orderThey also gave an intrinsic classification of theof the closure functions, and the tangent space of theparameterization singularities and defined their topologicalconfiguration space of the manipulator. The transverseorders.condition criterion is applied to analyze the behavior of theAn interesting example of manipulators offered bysingularities. This gives a downright explication why the 6RSeoul National University (SNU), is a 3-DOF translationalparadoxical mechanisms work in their singularmanipulator with the joints of its three sub-chains arrangedconfigurations, and allows to gain insight on configurationin the order of universal- -prismatic -universal (UPU).space singulrities and to choose the adequate designTsaifDO initiated design of the general 3-UPU manipulatorsparameters for the parallel manipulator.and his work was later generalized by Gregorio. andKeywords: Lie algebra, screws, singularities, parallelParenti-aselif[5g. Gregorio et al. [10.1] analyzedmani pulator.singularities of Tsai's manipulator and divided them intorotational and translational singularities. Han et al.C18]identified a configuration space singularity of the SNUIntroductionmanipulator and observed that the manipulator at its homeposition exhibited finite motions even with all activeIn 1990, Gosselin and AngelesH were perhaps the firstprismatic joints locked. Zlatanov et al.In studied thisto define and study singularities of closed loop kinematicsingularity by using screw theory and lassifiedit as achains. Based on some derived Jacobian relations, theyconstraint singularity. The same singularity was alsintroduced several notions of singularities which formed aidentified by Joshi and Tsai[o] , Simaan and Shohamf2] ,basis of later research. For example, Sugimoto et al.c]and Wolf et al. [2] by using an augmented Jacobian matrixapplied the results of Hunt臼to the identification of specialwhich took the constraints into account, By interpretingconfigurations of single loop 1-DOF mechanisms.he rows of this matrix as lines， linear geometryIn 1978， Herved] proposed that all articularmethod:23), could be used to efficiently find all possibledisplacements could be represented by Lie group, andsingularities. Liu et al. 03] introduced coordinate invariantKargerSs.0 firstly described the robot manipulator usingdefinitions of configuration and parameterizationgeometry language. Then, Chevller7,U introduced thesingularities with actuator and end-effector singularities.Lie algebra and dual mumber method in the use of robotEven though a good explanation of the singularities of thekinematics. Based on the above works, Lerbet et al.[回SNU manipulator has been offered in the above studies, aestablished the robot kinematics in second-order andconcise mathematical presentation for describing theanalyzed the singularities of closed-loop chain by usinggeometric behavior of these singularities is not available.transverse condition criterion. Park and Kim[10] usedThe purpose of this paper is stated as follows, in viewdifferential geometric tools to study the singularities 。of the fact that a parallel manipulator is most naturallyparallel mechanisms and provided a finer classification ofdefined by the constraint function, and the differentialsingularities. In their later woirks.1, they proposed theforms associated with the manipulator's closure constraints,use of redundant actuation as a means of eliminatingwe develop a generalized method for detecting singularitiesactuator singularities and improving the manipulatorof parallel mechanisms by using Lie algebra method, asperformances. A six axis parallel machine platform wassuchintrineie nrnnertie nf the mechanisms can be中国煤化工Received Aug.20, 2006MYHCNMH._Supported in part by the Key Projecet of the National Nature Science Foundauuno unna TIU. wwJvcui, uc National Nature ScienceFoundation of China (No. 60474037), and the National 863 Plan of China (No.2002AA755026)Correspondence should be addressed to HAO Kuang-rong, E mail: krhao@dhu. edu. cn6Joumal of Donghua University (Eng. Ed.) Vol. 23, No.6 (2006)discovered. We introduce the general form of mechanismsXe(q)= A.(q1 )2.(2...rqo)G .(2)in second order, which permits to analyze the closedThe fllwing set is defined, rank (Jn' (q)) = rankmechanism in their tangent space without using the analytic(f(q))= rank(X;(q), .. Xn(q)), which is equivalentsolution. We prove in a special case, when the number ofto Jacobian matrix J = (Xi (q), . X. (q)). Thejoints n =6, and the rank of their Jacobian matrices isr=singularity analysis of the closed mechanisms is transformedn- 1, the transverse condition of the closed mechanism isto the study of rank (JI). The DOF is equal ton-r in thealways satisfied. That is the reason for that when n= 6 andregular cases, ie., r is locally constant.r= 5, the closed chain has one DOF even in their singularThe mechanism in the position q is characterized by nconfiguration. We develop the second order form ofvectors X,(q). Let F,=(X(q), . Xn(q)), the sub-closure condition to the parallel manipulator which ivector space of dimension r of D in which X.(q) is a basiscomposed of more than one closed chain, and give a newofF, with a∈1={1, . r}, the other vectors aremethod to analyze dfferent types of singularities of parallelexpressed by X(q)=习CqX.(q) withi∈Iz={1, .manipulators.The paper is organized as follows: in the next section,n-r}, the cofficients ci are functions of q. LetSq bewe give some notations of screw theory. In the thirdthe sub-algebra of dimension s generated by Fq andG, asub-vector space, supplementary of Fq in Sq, Up(q) issection, we firstly consider the paradoxical mechanism[24.252by using the differential geometric forms, then we provechosen asa basis with P∈Is={1, " s-r}.Proposition1 x= (X(q), .. X,(q)) is anthat only the first order analysis is sufficient to determinethe behavior of the singular configuration. Next, we defineapplication of R* in L(R* , Sq) of which differential formis as follows.the set of closure functions of the well-known SNU parallelmanipulator and its Jacobian matrix. According to the rankVx, y∈R° : Hxy =(X'(q). x)(y)of the Jacobian matrix, we distinguish the singular= 2 E*'y*[X, X.]∈D (3)configuration. After the result of the second-order of theclosure function, we find that the system of the SNUparallel manipulator is bifurcated, that joins the IIMSingularity Analysis of Parallel Manipulator(increased instantaneous mobility) singularity defined byZlatanov et al.[20, Finally, we give a brief conclusion of1 Paradoxical closed mechanismsthe paper.The paradoxical mechanisms mean that they can workonly in their singular configurations, for example, BennettNotations of Screw Theoryand Bricard mechanism (see Fig.1).Proposition2 While r= rank (J(q))=n-1 and s=We denote the three-dimensional affined Euclideann, for n=3, 4, 6, the transverse condition is alwaysspace and the associated vector space by A and E,satisfied, then 2"(f)= x~'(E') is a sub-manifold of R ,respectively: The set of all skew-symmetric vector fields isa vector space over the real number field which will bethe mechanism has a DOF at q.denoted by D. The vector subspace of constant vector fieldsProof (1) If f isa sub-inmersion, (X;(q), .over E will be denoted by C (the vector field X is constantXn-1(q))is formed a Lie sub-algebra, then the result isover A whenever wx=0). If wx≠0 (i.e. X∈C), theevident.compliment vector subspace Z of C in C is defined by D=C(2) Ifr= rank (J(q))=n-1, (X:(q), .. X.-④z. The set of the point p, which has X(p)Xwx=0, is(q)) is linearly independent, a∈l={1, . r} and P∈a straight line Ax directed as wx in A(the axis of X) (seeIs={1}. According to Proposition 2, there exists at leastone Lie bracket [X1 |X2]4Fq, a basis of G, is defined asRefs. [7] and [9] for more details).Definition The equation for a closed chain mechanismU,= {[X; Xz]}. Moreover, there exists C;≠0 and theis the relation f(q)=e withtransverse condition is satisfied.Considering n = 6，the kinematic behavior can bef(q) = exp(q1 )...... exp(qnsn),q=(q1, . qn)∈R°(1)characterized by the set ofF,=(X(q), "" Xo(q)),We study the set, namely configuration space M =f^"(e) withwhe中国煤化工cobian Matrix J =(Xie mobile for a closedM={q=(q1, . qn)∈R*If(q)=e}RYH.CNMHGoisthattherankofThe explicit definition of the derivative f' (q), wethe Jacobian matrix is smaller than 6, that is :to say, theregive firstly:exists a“transversal" [20], an element of Lie algebra r∈DJoumal of Donghua University (Eng. Ed.) Vol.23, No. 6 (2006) 7such that[r|Xx]=0, k=1, .6. The case rank (J(q))q' with=5 for all q∈M is studied in this sub-section, theTqM= Ker (f)={x∈R|η= -x，xz=-好，mechanism possess a 1-DOF, on the other words, thexg二一始,如二始，xs=xo}acceptable kinematic configuration set M is a sub-manifoldof one dimension.We should explain, in Proposition2, the case n=5 isnot considered, because it is proved that it does not existany Lie sub algebra of dimension 5 for the generated sub-algebra from Fq饲.Proposition3 In order that M∈R' is a sub manifold)of one dimension for all q∈M, the necessary andf TMsufficient condition is that one of the three following termsis satisfied.Fig. 1 Bricard mechanism(1) rank (J(q))=5, for all q∈M;(2) M= 2'();23-UPU manipulatorIn this sub-section, we will study the configuration(3) There exists a "transversal", such that [r|X]=space singularities of a well studied spatial mechanism:0, for all q∈M.the 3-UPU manipulator as shown in Fig. 2. TheProol After Ref. [9], it is sufficient to prove thatmanipulator consists of three serial chains with their jointsconditions (1)-(3) are equivalent.arranged in the order of UPU, where only the three(1)#(2), sinceM= 2(f)U-.U习"(f) andprismatic joints are actuated. The mechanism has three2' = σ with i≠1.DOFs.3-UPU manipulators with their simple kinematicstructures have attracted many researchers. They found(1)=>(3), while rank (J(q))=5, we havethat for the axis arrangement as the SNU manipulator,det (J)=det(X1, Xz, X，X, Xs, Xo)=0.the mechanism will exhibit a“strange" singularity at theThere exits r=(r，I2, rs, r, Is, ro)T∈Rhome position 14, 15, 18, where the end-effector is freediferent from zero, such thatto rotate even when all the prismatic joints are locked.They pointed out that“strangeness" of the singularity liesnXx + r2Xx2+ IsXs+ r.Xu+ IsXs+ rsXx=in the fact that it cannot be detected from the Jacobian0, fork=1, ..6matrix defined from the generalized velocity of the end-It can be represented byeffector. In this sub-section, we will show this singularity断. X+(p)+wxp●f(p)=0= >[r|Xk]=0, for pEAby using Lie language and detecting the transversecondition and its behavior in second order of the set ofand wr=(I, Tz, r)T∈E, then (1)=>(3). .closure functions.Letq1 =[qml, " qms]', m=1, 2, 3, be the joint(1)←=(3), while rank (J(q))- <5, there exists at leasttwo“transversals". In the case rank (J(q)) = 6, det .angles of the three serial chains. The parallel mechanism isdescribed by three families of screws X= {Xm1，Xm2, ...(J(q))≠0, we have r=(0, 0, 0,0, 0, 0)T∈R.Considering Bricard mechanism, max(rank (1)) =Xms}, m=1,2,3.6, it works in its singular configuration. We have theplatformJacobian matrix in the reference configuration consistedof 6 screws:0lAx=x=| ,x=,x=|,|,x=| 。,X=|中国煤化工rank (J")= (好，Xi，X好，Xi, xg，X) = 5, the .CNMHGulatorFig. z 1 ne SNU manupuator wIn mne { first, second} and“transversal" r=(1, -1, 1, 0, 1, 1), M∈R° isathe {fourth, fifth} axles of the three serial chainssub-manifold of one dimension in the neighborhood oflying in two parallel planes8Joumal of Donghua University (Eng. Ed.) Vo1.23, No. 6 (2006)Defining f: RI5→D, a set of closure functions is:Imposing H.r∈Fq, i.e.. Has=0≈H: = H2 =0, withf(q)={f;(q)=eH2ue= 0.5xz\Xz + 2xz xz4 - 2xmXs - 0. 5xx4 xs一\f2(q)=e .0. 433 +0.433x* -0.433 + 0.4赌3I exp(qn X**.x.. Xis)ep '(qzs Xs )... '(qnXn)=eH2 =0.5xsnx + 000xgnx + 2xsnXx + 0.002x -I exp( 911 X*.. q1sX1s)ep^ '(qxs Xs )... '(q3nXg)=e2xxs +0.0002xgXxu + 0.000 3xgXs - 0.5xyXxs +0.433x品-0. 433x温+ 0.433x强- 0.433x茹The configuration space of such a mechanism is thefollowing set:They are two quadratic equations, it means that thesolution will be bifurcated into two branches. The systemM={q∈R'°|f(q)=e}of mechanism is degenerated.This is a 'constraintsingularity defined by Zlatanov et al.[2]， It isM= M:UM2= {q∈R° |f(q)=e}U{q∈R'° |fz(q)=e}demonstrated that the transverse condition is not satisfied,When the same reference frame is taken as Liu etand the condition of Hx∈Fq cannot be used. It is provedal.C四), the Jacobian matrix is obtained J(0)= [Xn，...that the mechanism cannot possess any DOF in thisXis, Xn, .. Xs](sce Ref. [13] for details). Amongconfiguration.While for Tsai's manipulator (see Fig. 3) , the Jocobianthe screws, X;s represents prismatic displacement. rankmatrix (see Ref. [13] for details) has full rank:(J(0))= 5<6 is the maximal rank, so the mechanism is ina singular position. In order to analyze the relationshiprank (J(0))= rank ([XI, .. Xis, Xz, . Xs])=6among the velocity of the screws, a basis X. (0) =In this case, F, = D, and the transverse condition is[XIn, ..,,, a∈I={1, ... 5}, of F, based on thesealways satisfied. The manipulator is not singular at itsscrews is introduced, the other screws can be expressedhome position,with this base, i.e. [Xz， *. Xz]= X.(O)[C，".C;]T. The joint instantaneous velocity can be analyzedplatformthrough the kernel of f'(0).Ker(f(0))={x∈R'S |x。the fllowing conditions aresatisfied}= {x∈R'5 |Ker(f'(0)∩Ker(fr'(0))}A2三More precisely,来AKer(f"(0))={x∈R°|x =Krxz∈R, x =Kgxs∈R}Fixed basewhere K2, Ks are 5X5 matrices.Fig.3 Tsai's manipulaterP0.50000.8660-0.00020-0.814 0.4100-0.003 -0.0700 0.0404ConclusionK2=| 223.4899 -387.1135 -0.98-301.0883 173.8288 |0.0520 0.0900 0.0003 0.5700This paper presented a geometric framework for00.0002 0.86600.5000analyzing the singularities of a closed chain and the parallelP0.5000-0.8600.0002 0manipulator. Using the differential geometric forms| 0.81400.4100-0.0030 -0.0700 1.6685associated with the closure functions, we derived someKg=| -223.4899 -387.1135 -0.9800 -301.0883 -273.1681conditions in second order for singularities. Analysis on| 0.05200.09000.0003 0. 5700-0.8025transverse condition criterion was also investigated. We-0.002 -0.8660 0.5000gave an intrinsic method to distinguish singularitiecs, anMoreover, F, is not a Lie sub algebra, the generatedshowed that the results did not depend on the selection ofinput-output parameters. First, we applied the proposedLie algebra D is of dimension 6. A basis of D can be built as{X.(0), U(0)}=[Xn, ... Xs, [Xn, Xr]], up(q)=approach to the paradoxical closed chain which is a non-{[Xn, Xr2]} is a basis of G, which is the supplementary ofdegenerate singularity, and have Hux∈F, Then, weF, in D. The second order Hu of this mechanism is thenapp中国煤化工the analysis of thecalculated from Proposition 1. Hxx =[Hx●H] can bebsYH_CNMHGSNU manipulator andhol,larity and an irregularexpressed in {X.(0), Up(0)}. We are interested in thesingularity, and we have Huz 史F. The system iscofficient of [Xn，X2] which is quadratic form.bifurcated into two branches and is degenerated, as suchJoumal of Donghua Uhiversity (Eng. Ed.) Vvol.23, No.6 (2006) 9this configuration should be avoided in the work space.[14] L. W. Tsai, Kinematics of a Three-DOF Platform withThree Extensible Limbs, Recent Advances in RobotTheoretically Hu∈F. is only a necessary condition in orderthe manipulator to having a DOF. But as far as we know, weKinematicst Analysis and Control, J. Lenaric and M. L.Husty Eds. , Norwell, MA, Kluwer, 1996; 49 - 58d not find that an example stisying Hs∈F。dos not15] R. Di Gregorio, V. Pareni-aseli, A Translationalpossess a DOF, it is not theoretically demonstrated yet.3-DOF Pralle Manipulator, Recent Advances in RobotKinematics, MA, Kluwer, 1998; 401 - 410References[16] R. Di Gregorio,Statics and Singularity Loci of the 3-UPUWrist, Proc. IEEE/ASME Int. Conf. Advanced Inelligent[1] C. Ooselin, J. Angeles, IEEE Trans. on Robot.Mechatronics, 2001: 470-475Auomat, 1990, 6 (3); 281 - 290[2] K. Sugimoto, J. Duffy, K. H. Hunt, J. Mech. Mach.[17] R. Di Gregorio, V. Prenti-atelli, ASME J. 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