The aim of this paper is the kinematic study of a 3-RPR planar parallel
manipulator where the three fixed revolute joints are actuated. The direct and
inverse kinematic problem as well as the singular configuration is
characterized. On parallel singular configurations, the motion produce by the
mobile platform can be compared to the Reuleaux straight-line mechanism

Bonev, Ilian

Chablat, Damien

Wenger, Philippe

English

arXiv

International audienceThe aim of this paper is the kinematic study of a 3-RPR planar parallel manipulator where the three fixed revolute joints are actuated. The direct and inverse kinematic problem as well as the singular configuration is characterized. On parallel singular configurations, the motion produce by the mobile platform can be compared to the Reuleaux straight-line mechanism

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Kinematic analysis of the 3-RPR parallel manipulatorDamien Chablat, Philippe Wenger, Ilian BonevTo cite this version:Damien Chablat, Philippe Wenger, Ilian Bonev. Kinematic analysis of the 3-RPR parallelmanipulator. 10th International Symposium on Advances in Robot Kinematics, Jun 2006,Ljubljana, Slovenia. Kluwer Academic Publishers, pp.221-228, 2006. <hal-00168606>HAL Id: hal-00168606https://hal.archives-ouvertes.fr/hal-00168606Submitted on 29 Aug 2007HAL 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, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.hal-00168606, version 1 - 29 Aug 2007KINEMATIC ANALYSIS OFTHE 3-RPR PARALLEL MANIPULATORD. Chablat, Ph. WengerInstitut de Recherche en Communications et Cyberne´tique de Nantes, 1, rue de la Noe¨,44321 Nantes, FranceDamien.Chablat@irccyn.ec-nantes.fr, Philippe.Wenger@irccyn.ec-nantes.frI. BonevDe´partement de ge´nie de la production automatise´eE´cole de Technologie Supe´rieure1100 rue Notre-Dame Ouest, Montre´al (Que´bec) Canada H3C 1K3Ilian.Bonev@etsmtl.caAbstract The aim of this paper is the kinematic study of a 3-RPR planar parallelmanipulator where the three fixed revolute joints are actuated. The di-rect and inverse kinematic problem as well as the singular configurationis characterized. On parallel singular configurations, the motion produceby the mobile platform can be compared to the Reuleaux straight-linemechanism.Keywords: Kinematics, Planar parallel manipulators, Singularity1. Introduction2. PreliminariesA planar three-dof manipulator with three parallel RRR chains, theobject of this paper, is shown in Fig. 1. This manipulator has beenfrequently studied, in particular in Merlet, 2000; Gosselin, 1992; Bonev,2005. The actuated joint variables are the rotation of the three revolutejoints located on the base, the Cartesian variables being the positionvector p of the operation point P and the orientation φ of the platform.The trajectories of the points Ai define an equilateral triangle whosegeometric center is the point O, while the points B1, B2 and B3, whosegeometric center is the point P , lie at the corners of an equilateral tri-angle. We thus have ||a2 − a1|| = ||b2 −b1|| = 1, in units of length thatneed not be specified in the paper. .A1B1( )x,y B2A2A3B3θ3θ1 θ2φρ3ρ1ρ2Figure 1. A three-DOF parallel manipulator3. KinematicsThe velocity p˙ of point P can be obtained in three different forms,depending on which leg is traversed, namely,p˙ = ρ˙i(bi − ai)||bi − ai|| + θ˙iE(bi − ai) + φ˙E(p− bi) (1)with matrix E defined asE =[0 − 11 0]We would like to eliminate the three idle joint rates ρ˙1, ρ˙2 and ρ˙3 fromEqs.(1), which we do upon dot-multiplying the former by (Evi)T , thusobtaining(Evi)T P˙ = θ˙iρi + φ˙(Evi)TE(p− bi) (2)withvi =(bi − ai)||bi − ai|| =[cos(θi)sin(θi)]and ρi = (Evi)TE(bi − ai)Equations (2) can now be cast in vector form, namely,At = Bθ˙ with t =[p˙φ˙]and θ˙ = θ˙1θ˙2θ˙3 (3)with θ˙ thus being the vector of actuated joint rates.Moreover, A and B are, respectively, the direct-kinematics and theinverse-kinematics matrices of the manipulator, defined asA = (Ev1)T − (Ev1)TE(p− b1)(Ev2)T − (Ev1)TE(p− b2)(Ev3)T − (Ev3)TE(p− b3) B = ρ1 0 00 ρ2 00 0 ρ3 (4)When A and B are nonsingular, we obtain the relationst = Jθ˙, with J = A−1B and θ˙ = Ktwith K denoting the inverse of J.3.1 Parallel SingularitiesParallel singularities occur when the determinant of matrix A van-ishes (Chablat, 1998 and Gosselin, 1990). At these configurations, it ispossible to move locally the operation point P with the actuators locked,the structure thus resulting cannot resist arbitrary forces, and control islost. To avoid any performance deterioration, it is necessary to have aCartesian workspace free of parallel singularities.For the planar manipulator studied, the direct-kinematic matrix canbe written as function of θiA = − sin θ1 cos θ1 0− sin θ2 cos θ2 cos(θ2)− sin θ3 cos θ3 cos(θ3)/3 + sin(θ3)√3/2 (5)and its determinant is the follows,det(A) = −2 sin(θ3 − θ1 − θ2)− sin(θ3 − θ1 + θ2)− sin(θ3 + θ1− θ2)+√3 cos(θ3 + θ1− θ2)−√3 cos(θ3 − θ1 + θ2) (6)The parallel singularities occur whenever the three axes normal to theprismatic joint intersect. In such configurations, the manipulator cannotresist a wrench applies at the intersecting point P , as depicted in Fig. 2A special case exists when θ1 = θ2 = θ3 because the intersection pointis in infinity. Thus, the mobile platform can translate along the axes ofthe prismatic joints, as depicted in Fig. 3.3.2 Serial SingularitiesSerial singularities occur when det(B) = 0. In the presence of thesessingularities, there is a direction along which no Cartesian velocity canbe produced. Serial singularities define the boundary of the CartesianA1B1B2A2A3 B3v1v3v2PFigure 2. Parallel singularity when the three axes normal to the prismatic jointintersectA1B1 B2A2A3B3Figure 3. Parallel singularity when θ1 = θ2 = θ3workspace. For the topology under study, the serial singularities occurwhenever at least one ρi = 0 (Figure 4). When ρ1 = ρ2 = ρ3 = 0 notany motion can be produce by the actuated joints.3.3 Direct kinematicsThe position of the base joints are defined in the base reference frame,a1 =[00]a2 =[10]a3 =[1/2√3/2](7)A1B1B2A2A3B3v1Figure 4. Serial singularity when ρ1 = 0In the same way, the position of the mobile platform joints are definedin the mobile reference frame,b′1=[00]b′2=[10]r′3=[1/2√3/2](8)The rotation matrix R describes the orientation of the mobile framewith respect to the base frame.R =[cos(φ) − sin(φ)sin(φ) cos(φ)]The position of p can be obtain in three different ways:{x = cos(θ1)ρ1y = sin(θ1)ρ1(9a)or{x = − cos(φ) + cos(θ2)ρ2 + 1y = − sin(φ) + sin(θ2)ρ2 (9b)or{x = − cos(φ+ pi/3) + cos(θ3)ρ3 + 1/2y = − sin(φ+ pi/3) + sin(θ3)ρ3 +√3/2(9c)To remove ρi from the previous equations, we multiply sin(θi) (respec-tively cos(θi)) the equations in x (respectively in y) and we subtract thefirst one to the second one, to obtain three equations,sin(θ1)x− cos(θ1)y = 0 (10a)sin(θ2)x− cos(θ2)y + sin(θ2 − φ)− sin(θ2) = 0 (10b)sin(θ3)x− cos(θ3)y − cos(θ3 − φ+ pi/6)− sin(θ3)/2 + cos(θ3)√3/2 = 0 (10c)We obtain x and y as function of φ by using Eqs. 10a-bx =sin(θ2 − φ+ θ1)− sin(θ1 − θ2 + φ)− sin(θ1 + θ2) + sin(θ1 − θ2)2(sin(θ1 − θ2))y =cos(θ1 − θ2 + φ)− cos(θ2 − φ+ θ1)− cos(θ1 − θ2) + cos(θ1 + θ2)2(sin(θ1 − θ2))Thus, we substitute x and y in Eq. 10c to obtainm cos(φ) + n sin(φ)−m = 0 (12a)withm = sin(θ3) cos(θ2) sin(θ1) +√3 cos(θ3) sin(θ2) cos(θ1)− 2 cos(θ3) sin(θ2) sin(θ1)−√3 cos(θ3) cos(θ2) sin(θ1)+ sin(θ3) sin(θ2) cos(θ1) (12b)n = cos(θ3) sin(θ2) cos(θ1) +√3 sin(θ3) sin(θ2) cos(θ1)− 2 sin(θ3) cos(θ2) cos(θ1)−√3 sin(θ3) cos(θ2) sin(θ1)+ cos(θ3) cos(θ2) sin(θ1) (12c)Equation 12a admits two rootsφ = 0 and φ = tan−1(2nmn2 +m2,−−m2 + n2n2 +m2)(13)A trivial solution is φ = 0 which exists for any values of the actuatedjoint values where x = y = 0. This means that when configuration of themobile platform associated to the trivial solution to the direct kinematicproblem is also a serial singularity because ρ1 = ρ2 = ρ3 = 0.Such a behavior is equivalent to that of the agile eye....3.4 Inverse kinematicsFrom Eqs. 10, we can easily solve the inverse kinematic problem andfind two real solutions in ]− pi pi] for each leg,θ1 = tan−1(yx)+ kpiθ2 = tan−1(y + sin(φ)x+ cos(φ)− 1)+ kpiθ3 = tan−1(y + sin(φ+ pi/3)−√3/2x+ cos(φ+ pi/3)− 1/2)+ kpifor k = 0, 1 and from Eqs. 9, we can easily find ρi,ρ1 =√x2 + y2ρ2 =√(x+ cos(φ)− 1)2 + (y + sin(φ))2ρ3 =√(x+ cos(φ+ pi/3)− 1/2)2 + (y + sin(φ+ pi/3)−√3/2)23.5 Full cycle motion: Cardanic curveA planar Cardanic curve is obtained by the displacement of one pointof one body whose position is constrained by making two of its points lieon two coplanar lines. This curve is obtained when we set for example θ1and θ2 and we observe the location of B3. A geometrical method to solvethe direct kinematic problem is find the intersection points between thiscurve and the line passing through the third prismatic joint whose ori-entation is given by v3. For the manipulator under study, the Cardaniccurve always intersects A3 and, apart from the singular configurations,I.The loop (A1, B1, B2, A2) can be written,ρ1 cos(θ1) + cos(φ) − 1− ρ2 cos(θ2) = 0 (16a)ρ1 sin(θ1) + sin(φ)− ρ2 sin(θ2) = 0 (16b)We solve the former system to have ρ1 and ρ2 as a function of φ and θiρ1 =− cos(φ) sin(θ2) + sin(θ2) + cos(θ2) sin(φ)cos(θ1) sin(θ2)− cos(θ2) sin(θ1) (17a)ρ2 =− sin(θ1) cos(φ) + sin(θ1) + sin(φ) cos(θ1)cos(θ1) sin(θ2)− cos(θ2) sin(θ1) (17b)The position of B3 in the base frame can be written as follows,xB3 = ρ1 cos(θ1) + cos(φ+ pi/3) (18a)yB3 = ρ1 sin(θ1)− sin(φ+ pi/3) (18b)Thus, by substituting the values of ρ1 and ρ2 obtained in Eqs. 17a-b inEqs. 18, we obtain the equations of the Caradanic curve,xB3 =(− cos(θ1) cos(θ2)cos(θ2) sin(θ1)− cos(θ1) sin(θ2) −√32)sin(φ)+(cos(θ1) sin(θ2)cos(θ2) sin(θ1)− cos(θ1) sin(θ2) +12)cos(φ)−(cos(θ1) sin(θ2)cos(θ2) sin(θ1)− cos(θ1) sin(θ2))(19a)yB3 =( − cos(θ2) sin(θ1)cos(θ2) sin(θ1)− cos(θ1) sin(θ2) +12)sin(φ)+(sin(θ1) sin(θ2)cos(θ2) sin(θ1)− cos(θ1) sin(θ2) +√32)cos(φ)− sin(θ1) sin(θ2)cos(θ2) sin(θ1)− cos(θ1) sin(θ2) (19b)Cardanic curveIA1B1B2A2B3A3v3Figure 5. Cardanic curve of B3 whenB1 andB2 sliding along v1 and v2, respectivelyThe Cardanic curve degenerates for specific actuated joint values (Hunt, 1982). In this case, the manipulator under study is equivalentto a Reuleaux straight-line mechanism (Nolle, 1974). This mechanismis composed by two prismatic joint and a mobile platform assembledwith the prismatic joint via two revolute joints as depicted in Fig. 6.The angle between the axes passing through the prismatic joints is pi/3.The mobile platform is an unit equilateral triangle. The displacementmade by P is a straight line whose length is two. The magnitude of thisdisplacement is the same for A1 and A2.Whenever θ2 − θ1 = pi/3 or θ1 − θ2 = pi/3 and θ3 − θ1 = pi/3 orθ1 − θ3 = pi/3 with θ1 6= θ2 6= θ3, there exists a infinity of solutionsto the direct kinematic problem. This feature exists whenever we havethe same condition to have a parallel singularity and the three vector viintersect in one point. The displacement of Ai are around this point Pand is magnitude is 4√3/3.4. ConclusionsA kinematic analysis of a planar 3-RPR parallel manipulator was pre-sented in this paper. The parallel and serial singularities have been char-pi/3PA1A2Figure 6. Reuleaux straight-line mechanismA1B1B2A2A3B3PFigure 7. Degenerated Cardanic curve when θ2 − θ1 = pi/3acterized as well as the direct and inverse kinematics. This mechanismfeatures two direct kinematic solutions whose one is a trivial singularconfiguration. For some actuated joint values associated to a parallelsingularity, the motion made by the mobile platform is equivalent to aReuleaux straight-line mechanism where the amplitude of the motion iswell known.ReferencesGosselin, C. and Angeles, J., Singularity Analysis of Closed-Loop Kinematic Chains,IEEE, Transaction on Robotics and Automation, Vol. 6, pp. 281-290, June 1990.Chablat, D. and Wenger, Ph., Working Modes and Aspects in Fully-Parallel Manip-ulator, Proceedings IEEE International Conference on Robotics and Automation,1998, pp. 1964-1969.Merlet, J-P., “Parallel robots,” Kluwer Academic Publ., Dordrecht, The Netherland,2000.Gosselin, C., Sefrioui, J., Richard, M. J., Solutions polynomiales au proble`me de lacine´matique des manipulateurs paralle`les plans a` trois degre´ de liberte´, Mechanismand Machine Theory, Vol. 27, pp. 107-119, 1992.Bonev, I. and Gosselin C.M., “Singularity Loci of Spherical Parallel Mechanisms”,Proceedings IEEE International Conference on Robotics and Automation, 2005.Tischler C.R., Hunt K.H., Samuel A.E., “A Spatial Extension of Cardanic Movement:its Geometry and some Derived Mechanisms”, Mechanism and MachineTheory,No. 33, pp. 1249–1276, 1998.Nolle H., “Linkage coupler curve synthesis: A historical reviewII. Developments after1875”, Mechanism and Machine Theory, Vol. 9, Issues 3-4, pp. 325–348, 1974Hunt K.H. and Herman P.M., “Multiple Cusps, Duality and Cardanic Configurationsin Planar Motion”, Mechanism and Machine Theory, Vol. 17, No. 6, pp. 361–368,1982

Kinematic analysis of the 3-RPR parallel manipulator

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Kinematic analysis of the 3-RPR parallel manipulator

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