International audienceThe Accurate calculation of the workspace and joint space for 3 RPS parallel robotic manipulator is a highly addressed research work across the world. Researchers have proposed a variety of methods to calculate these parameters. In the present context a cylindrical algebraic decomposition based method is proposed to model the workspace and joint space. It is a well know feature that this robot admits two operation modes. We are able to find out the set in the joint space with a constant number of solutions for the direct kinematic problem and the locus of the cusp points for the both operation mode. The characteristic surfaces are also computed to define the uniqueness domains in the workspace. A simple 3-RPS parallel with similar base and mobile platform is used to illustrate this method

Chablat, Damien

Jha, Ranjan

Moroz, Guillaume

Rouillier, Fabrice

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HAL Id: hal-01006614https://hal.archives-ouvertes.fr/hal-01006614Submitted on 16 Jun 2014HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Workspace and joint space analysis of the 3-RPS parallelrobotDamien Chablat, Ranjan Jha, Fabrice Rouillier, Guillaume MorozTo cite this version:Damien Chablat, Ranjan Jha, Fabrice Rouillier, Guillaume Moroz. Workspace and joint space analysisof the 3-RPS parallel robot. ASME 2013 International Design Engineering Technical Conferences &Computers and Information in Engineering Conference, Aug 2014, Buffalo, United States. pp.1-10.￿hal-01006614￿Proceedings of the ASME 2014 International Design Engineering Technical Conferences &Computers and Information in Engineering ConferenceIDETC/CIE 2014August 17-20, 2014, Buffalo, USADETC2014/MR-34593WORKSPACE AND JOINT SPACE ANALYSISOF THE 3-RPS PARALLEL ROBOTD. Chablat, R. JhaInstitut de Recherche en Communicationset Cybernétique de Nantes(UMR CNRS 6597) FranceEmail addresses:Damien.Chablat@irccyn.ec-nantes.frRanjan.Jha@irccyn.ec-nantes.frF. RouillierINRIA Paris-RocquencourtInstitut de Mathématiques de Jussieu(UMR CNRS 7586), FranceEmail addresses:Fabrice.Rouillier@inria.frG. MorozINRIA Nancy-Grand Est,FranceEmail addresses:Guillaume.Moroz@inria.frABSTRACTThe Accurate calculation of the workspace and joint spacefor 3 RPS parallel robotic manipulator is a highly addressed re-search work across the world. Researchers have proposed a va-riety of methods to calculate these parameters. In the presentcontext a cylindrical algebraic decomposition based method isproposed to model the workspace and joint space. It is a wellknow feature that this robot admits two operation modes. We areable to find out the set in the joint space with a constant num-ber of solutions for the direct kinematic problem and the locus ofthe cusp points for the both operation mode. The characteristicsurfaces are also computed to define the uniqueness domains inthe workspace. A simple 3-RPS parallel with similar base andmobile platform is used to illustrate this method.INTRODUCTIONThe workspace of parallel robots mainly depends upon theactuated joint variables, the range of motion of the joints andthe mechanical interferences between the bodies of mechanism.There are different techniques based on geometric, discretiza-tion, numerical and algebraic methods which are used to calcu-late the workspace of parallel robot. The main advantage of thegeometric approach is that, it establish the nature of the bound-ary of the workspace [1]. Also it allows the computation of thesurface and volume of the workspace while being very efficientin terms of storage space, but if the rotational motion is included,it becomes more complex. The interval analysis based methodcan be used to compute the workspace but the computation timedepends on the complexity of the robot and the accuracy re-quested. The ALIAS library is a good implementation for theparallel robots [2]. Discretization methods are usually less com-plex and take into account all kinematic constraints, but requiremore space and computation time for higher resolutions. Themajority of numerical methods which is used to determine theworkspace of parallel manipulators includes the discretization ofthe pose parameters for the determination of workspace bound-aries [3]. Algebraic methods are used in [4–6] to study planaror spatial parallel robots. Two main teps are necessary to per-form the workspace and jointspace analysis. First, the discrim-inant variety is computed to characterize the boundaries of theworkspace and jointspace as well as the singularities. Second,the Cylindrical Algebraic Decomposition (CAD) is used to de-fine the connected regions where there exists a constant numberof real solutions to the inverse and direct kinematic problem andno parallel or serial singularities [5–7].For the design or the trajectory planning, the workspace ofthe parallel manipulator is divided into singularity-free regions[8]. The singularities divide the workspace into aspects and thecharacteristic surfaces induce a partition of each aspect into a setof regions (the basic regions) [9]. For the parallel robots withseveral inverse and direct kinematic solutions, the aspects are de-1 Copyright c© 2014 by ASMEfined as the maximal singularity-free sets in the workspace or thecross-product of the joint space by the workspace. An assemblymode is associated with a solution for the Direct Kinematic Prob-lem (DKP) and a working mode for the Inverse Kinematic Prob-lem (IKP). Practically, a change of assembly mode may occurduring the execution of a trajectory between two configurationsin the workspace which are not necessarily associated with thesame input for a given working mode. The uniqueness domainscan be defined as the maximal regions of the workspace whereall the displacements of the end-effector can be accomplishedwithout changing of assembly mode and working mode [5]. Themain goal of this paper is to generalize the notions of the unique-ness domains for parallel robot with several operation modes anda single working mode.The outline of the paper is as follows. Firstly, the kinematicequations of the 3-RPS parallel robot under study are introduced.The singular configurations are written for both operation modes.The cylindrical algebraic decomposition algorithm is presentedto explain the study of the workspace and the joint space. Then,the characteristic surface is defined for the parallel robots withseveral operations. Finally, the basic regions and the basic com-ponents are computed to define the uniqueness domains for thedirect kinematic problem.KINEMATICS OF 3 RPSThe robot under study is the 3-RPS parallel robot with threedegrees of freedom and has been studied by many researchers[10, 11]. It is the assembly of two equilateral triangles (the base,moving platform) by three identical RPS legs where R is a rev-olute passive joint, P an actuated prismatic joint and S a passivespherical joint. Thus, the revolute joint is connected to the fixedbase and the spherical joint to the mobile platform.A1A2A3B1B2B3POghxyzuvwρ1ρ2ρ3FIGURE 1. 3-RPS parallel robotConsidering the 3-RPS parallel manipulator, as shown in fig-ure 1, the fixed base consists of an equilateral triangle with ver-tices A1, A2 and A3, and circumradius g. The moving platformis another equilateral triangle with vertices B1, B2 and B3 andcircumradius h, and circumcenter P. The two design parametersg and h are positive numbers. Connecting each of the vertices’pairs Ai, Bi (i = 1,2,3) by a limb, a rotational joint lies at Ai anda spherical joint lies at Bi. ρi denotes the length of each limb andthe adjustment is done through an actuated prismatic joint. Thus,there are five parameters, namely g, h, ρ1, ρ2 and ρ3. The g and hparameters determine the design of the manipulator whereas thejoint parameters ρ1, ρ2 and ρ3 determine the motion of the robot.To simplify the equations, we will study a robot with g = h = 1,which permits to have simple constraint equations.Kinematic equationsSpatial rotations in three dimensions can be parametrizedusing Euler angles [12], unit quaternions [13] or dual quater-nions [10]. The quaternion representation is used for modelingthe orientation as quaternions do not suffer from singularities asEuler angles do. Moreover, to transform the trigonometric equa-tions to algebraic equations, we may either introduce the singu-larity of the transformation t = tan(α/2) or replace the angle αby two parameters cosα and sinα with cos2α +sin2α −1. In addi-tion, it is easier to represent workspace sections with the quater-nions than the dual quaternions.A quaternion q is defined byq = q1 + q2i+ q3j+ q4k (1)The quaternion rotation matrix for the parallel robot is thenQ =2q12 + 2q22 − 1 − 2q1q4 + 2q2q3 2q1q3 + 2q2q42q1q4 + 2q2q3 2q12 + 2q32 − 1 − 2q1q2 + 2q3q4−2q1q3 + 2q2q4 2q1q2 + 2q3q4 2q12 + 2q42 − 1(2)with q21 +q22 +q23 +q24 = 1. The transformation from the movingframe to the fixed frame can be described by a position vectorp =OP and a 3×3 rotation matrix R. Let u, v and w be the threeunit vectors defined along the axes of the moving frame, then therotation matrix can be expressed in terms of the coordinates of u,v and w as:R =ux vx wxuy vy wyuz vz wz (3)2 Copyright c© 2014 by ASMEThe vertices of the base triangle and mobile platform triangle areA1 =g00 A2 =−g/2g√3/20 A3 =−g/2−g√3/20 (4)b1 =h00 b2 =−h/2h√3/20 b3 =−h/2−h√3/20 (5)The coordinates of bi with respect to fixed frame reference areobtained by Bi = P+Rbi for i = 1,2,3. Also the coordinates ofthe centre of the mobile platform in the fixed reference is P =[x y z]T . The distance constraints yields:||Ai −Bi||= ρ2i with i = 1,2,3 (6)As Ai are revolute joints, the motion of the Bi are constrained inplanes. This leads to the three constraint equations:uyh+ y = 0 (7)y− uyh/2+√3vyh/2+√3x−√3uxh/2+ 3vxh/2 = 0 (8)y− uyh/2−√3vyh/2−√3x+√3uxh/2+ 3vxh/2 = 0 (9)Solving with respect to x and y we get:y = −huy (10)x = h(√3ux −√3vy − 3uy+ 3vx)√3/6 (11)In Equations 6, 8, 9, we substitute x,y using relations 10 and 11,and u,v,w by quaternion expressions using 2. Then (8) and (9)become q1q4 = 0. Thus, we have either q1 = 0 or q4 = 0. Thisproperty is associated with the notion of operation mode. Toobtain the algebraic equations, we replace√3 by the variable S3and add the equation S23 − 3 = 0 and the constraint s3 > 0.Operation ModesNotion of operation mode (OM) was introduced in [14] toexplain the behavior of the DYMO robot. An operation mode isassociated with a specific type of motion. For the DYMO robot,there are five operation modes: translational, rotational, planar(two types) and mixed motions. In the workspace W , for eachoperation mode, WOM j is defined such that• WOM j ⊂W• ∀X ∈WOM j , OM is constantFor a parallel robot with several operating modes, the posecan be defined by fixing the same number of parameters as thedegree of freedom of the mobile platform. Given an operationmode OM j, if we have a single inverse kinematic solution, wecan define an application that maps X onto q:g j(X) = q (12)Then, the images in WOM j of a posture q in the joint space Q aredefined by:g−1j (q) = (X/(X,q) ∈ OM j) (13)where g−1j stands to be the direct kinematic problem for the op-eration mode j.SingularitiesDifferentiating the constraints equations with respect to timeleads to the velocity model:At+Bq̇ = 0 (14)where A and B are the parallel and serial Jacobian matrices, re-spectively, t is the velocity of P and q̇ joint velocities. The paral-lel singularities occur whenever det(A) = 0. Let OM1 be the op-eration mode where q1 = 0 and OM2, where q4 = 0, then SOM1and SOM2 represent the singularity locus and are characterizedby:SOM1 : q4(8q2q23q64 + 2q2q84 − 64zq63q4 − 96zq43q34 − 36zq23q54−6zq74 − 24z2q2q23q24 − 6z2q2q44 − 32q2q23q44 − 10q2q64 + 2z3q34+96zq43q4 + 72zq23q34 + 23zq54+ 16z2q2q23 + 10z2q2q24 + 8q2q44−z3q4 − 36zq23q4 − 21zq34 − 4z2q2 + 4zq4) = 0SOM2 : q21(6q71q3 + 8q51q33 − 2zq61 + 36zq41q23 + 96zq21q43+64zq63 − 18z2q31q3 − 24z2q1q33 − 18q51q3 − 16q31q33 + 2z3q21+3zq41 − 72zq21q23 − 96zq43 + 18z2q1q3 + 12q31q3 − z3+3zq21 + 36zq23− 4z) = 0The serial singularities occur whenever det(B) = 0 i.e ρ1ρ2ρ3 =0. The common coordinates for the both operation modes are z,q2 and q3. Figure 2(a) represents the singularity curve for OM13 Copyright c© 2014 by ASMEq2q3q4q2q3Real singularitySpurious singularity(a) (b)FIGURE 2. Singularity curves for z = 3, q1 = 0 (a) and its 2D projec-tion under (q2,q3) (b)q1q2q3q2q3Real singularitySpurious singularity(a) (b)FIGURE 3. Singularity curves for z = 3, q4 = 0 (a) and its 2D projec-tion under (q2,q3) (b)for a fixed value of z (i.e z = 3) and Figure 2(b) shows its projec-tion in the two dimensional coordinate space (q2 − q3).Due to the redundancy of the quaternion representation thereexists two triplets based on these three coordinates to representthe same pose in the same operation mode. To overcome thisproblem, we set q1 > 0 and q4 > 0. We can then depict a sliceof this surface by fixing one parameter as shown in Figures 2(a)and 3(a). Figures 2(b) and 3(b) shows the singularity curves inthe projection space z, q2 and q3, where the red curve is a real sin-gular locus, which represents the positive values of q4 or q1 andthe green curve is a spurious singularity curve, which representsthe negative values of q4 or q1.Cylindrical algebraic decompositionThe workspace (resp. Joint space) analysis classifies thenumber of solutions of the parametric system associated with theInverse (resp. Direct) Kinematic Problem (IKP). This methodwas introduced for parallel robots in [5]. We will recall here themain steps of the computation and the new step for a robot withseveral operation modes. For such operations, both workspace aswell as joint space is decomposed into cells C1,...,Ck, such that:• Ci is an open connected subset of the workspace;• for all pose values in Ci, the direct (resp. inverse) kinematicsproblem has a constant number of solutions;• Ci is maximal in the sense that if Ci is contained in a set E ,then E does not satisfy the first or the second condition.• The Ci are disjoint and their complementary is a set of nullmeasure.The three main steps involved in the analysis are:• Computation of a subset of the joint space (resp. workspace)where the number of solutions changes: the DiscriminantVariety [5].• Description of the complementary of the discriminant va-riety in connected cells: the Generic Cylindrical AlgebraicDecomposition.• Connecting the cells belonging to the same connected com-ponent in the counterpart of the discriminant variety: inter-val comparisons.From a general point of view, the discriminant variety is de-fined for any system of polynomial equations and inequalities.The union of the Discriminant variety and of the cells thus de-fine a partition of the considered space. Let p1,...,p1 and q1,...,qlbe polynomials with rational coefficients depending on the un-knowns X1,...,Xn, and on the parameters U1,...,Ud . Let us con-sider the constructible set:C ={v ∈Cn+d , p1(v) = 0, ..., pm(v) = 0,q1(v) 6= 0, ...,ql(v) 6= 0}(15)If we assume that C is a finite number of points for almost allthe parameter values, a discriminant variety VD of C is a varietyin the parameter space Cd such that, over each connected openset U satisfying U ∩VD = /0, C defines an analytic covering.In particular, the number of points of C over any point of U isconstant.Let us now consider the following semi-algebraic set:S ={v ∈Cn+d , p1(v) = 0, ...,pm(v) = 0,q1(v)≥ 0, ...,ql(v)≥ 0}(16)If we assume that S has a finite number of solutions overat least one real point that does not belong to VD, then VD ∩Rdcan be viewed as a real discriminant variety of S ∩ R\+⌈,with the same property: over each open set U ⊂ R⌈ such thatU ∩VD ∩Rd = /0, C defines an analytic covering. In particu-lar the number of real points of S over any point of U is con-stant. Discriminant varieties can be computed using basic andwell known tools from computer algebra such as Groebner bases(see [15], Chapter 3) and a full package computing such objects4 Copyright c© 2014 by ASMEin a general framework is available in Maple software throughthe RootFinding[Parametric] package.The cylindrical algebraic decomposition (CAD) imple-mented in the SIROPA library has been used to compute theaspects into a set of cells where algebraic equations define itsboundaries and a sample point in each one [5] for the 2PRR–RPR parallel robot. For example, the CAD can provide a formaldecomposition of the joint space in cells where the polynomialsdet(A) and det(B) have a constant sign and the number for theDKP is constant [16].WORKSPACE AND JOINT SPACE ANALYSISJoint Space AnalysisThe joint space analysis allows the characterization of the re-gions where the number of real solutions for the direct kinematicmodel is constant. Using CAD, we can do this study on sectionsof the joint space. The calculation for the full joint space is pos-sible, but the number of cells obtained is too large for the displaycapabilities of Maple. Without taking into account the notion ofoperation mode, Figure 4 depicts the regions with 4, 8, 12 or 16solutions for the DKP. The maximum number for the DKP ofeach operation mode is 8.4 solutions8 solutions12 solutions16 solutionsρ2ρ3FIGURE 4. Slice of the joint space for ρ1 = 3 and the number ofsolution for DKPFigures 5 and 6 show several slices of the joint space forOM1 and OM2, where the DKP admits four and eight real solu-tions in the blue and red region respectively.Cuspidal configurations are associated with second-orderdegeneracies that appear for triply coalesced configurations.These configurations play an important role in the path planningbecause they are directly linked to the non-singular assemblymode changing trajectories [6, 17–19]. A state of the art for thecomputation of the cusp points is given in [20]. Here we makeuse of the Jacobian criterion on the singular locus to extract thepoints of multiplicity greater than or equal to three. Figures 5and 6 show that the number of cusp points is changing accordingto the value of ρ1. Figure 7 depicts the locus of the cusp pointsin the joint space. These curves are obtained as the intersectionof 11 surfaces for OM1 and 8 surfaces for OM2. However, forthe 3-RPS parallel robot is very difficult to remove the multiplic-ity greater than two by using the saturation method as introducedin [21].Workspace AnalysisThe workspace of the robot is a cylinder in the projectionspace (z, q2, q3) if there are no joint limits on the actuated joints.The workspace analysis can be done by dividing it into aset of aspects. The notion of aspect, previously defined in [22]for serial robots and in [8] for parallel robot with one operationmode can be extended for a parallel robot with several operationmodes such that:• WA ji ⊂WOM j• WA ji is connected• ∀X ∈WA ji , det(A) 6= 0 and det(B) 6= 0In other words, an aspect WAji is the largest connected regionwithout any singularity of the OM j.The analysis of the workspace is done in the projection space(z, q2, q3). We found out four aspects as shown in Figure 8.4 solutions8 solutionsρ2ρ3 4 solutions8 solutionsρ2ρ3(a) (b)4 solutions8 solutionsρ2ρ3 4 solutions8 solutionsρ2ρ3(c) (d)FIGURE 5. Slice of the joint space for OM1 for ρ1 = 1 (a), ρ1 = 2(b), ρ1 = 3 (c) and ρ1 = 4 (d) and the number of solution for DKP5 Copyright c© 2014 by ASME

Workspace and joint space analysis of the 3-RPS parallel robot

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Workspace and joint space analysis of the 3-RPS parallel robot

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