A Classification of 3R Orthogonal Manipulators by the Topology of their Workspace

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1 A Classification of R Orthogonal Manipulators by the Topology of their Workspace Maher aili, Philippe Wenger an Damien Chablat Institut e Recherche en Communications et Cybernétique e Nantes, UMR C.N.R.S. 6597, rue e la Noë, P 90, Nantes Ceex 0 France Maher.aili@irccyn.ec-nantes.fr 9 Abstract A classification of a family of -revolute (R) positining manipulators is establishe. This classification is base on the topology of their workspace. The workspace is characterize in a half-cross section by the singular curves. The workspace topology is efine by the number of cusps an noes that appear on these singular curves. The esign parameters space is shown to be ivie into nine omains of istinct workspace topologies, in which all manipulators have similar global kinematic properties. Each separating surface is given as an explicit expression in the DH-parameters. Keywors Classification, Workspace, Singularity, Cusp, noe, orthogonal manipulator. I. INTRODUCTION A positioning manipulator may be use as such for positioning tasks in the Cartesian space or as the regional structure of a 6R manipulator with spherical wrist. Most inustrial regional structures have the same kinematic architecture, namely, a vertical revolute joint followe by two parallel joints, like the Puma. Such manipulators are always noncuspial (i.e. must meet a singularity to change their posture) an they have four inverse kinematic solutions (IKS) for all points in their workspace (assuming unlimite joints). This paper focuses on alternative manipulator esigns, namely, positioning R manipulators with orthogonal joint axes (orthogonal manipulators). Orthogonal manipulators may have ifferent global kinematic properties accoring to their link lengths an joint offsets. They may be cuspial, that is, they can change their posture without meeting a singularity [, ]. In 998, A-Robotics launche the IR 600C, a 6R manipulator to be use in the car inustry an esigne to minimize the swept volume. The only ifference with the Puma was the permutation of the first two link axes, resulting in a manipulator with all its joint axes orthogonal, an cuspial. Commercialization of the IR 600C was finally stoppe one year later. Exact reason is beyon the knowlege of the authors but the cuspial behavior is likeable to have isappointe the en users. Cuspial robots were unknown before 988 [], when a list of conitions for a manipulator to be noncuspial was provie [, 5]. This list inclues simplifying geometric conitions like parallel an intersecting joint axes [] but also nonintuitive conitions [5]. A general necessary an sufficient conition for a -DOF manipulator to be cuspial was This work was supporte in part by C.N.R.S. MathStic program "Cuspial robots an triple roots". establishe in [6], namely, the existence of at least one point in the workspace where the inverse kinematics amits three equal solutions. The wor cuspial manipulator was efine in accorance to this conition because a point with three equal IKS forms a cusp in a cross section of the workspace [, 7]. The categorization of all generic R manipulators was establishe in [8] base on the homotopy class of the singular curves in the joint space. [9] propose a proceure to take into account the cuspiality property in the esign process of new manipulators. More recently, [0] applie efficient algebraic tools to the classification of R orthogonal manipulators with no offset on their last joint. Five surfaces were foun to ivie the parameters space into 05 cells where the manipulators have the same number of cusps in their workspace. The equations of these five surfaces were erive as polynomials in the DH-parameters using Groebner ases. A kinematic interpretation of this theoretical work was conucte in [] : the authors analyze general kinematic properties of one representative manipulator in each cell. Only five ifferent cases were foun to exist. However, the classification in [] i not provie the equations of the separating surfaces in the parameters space for the five cells associate with the five cases foun. On the other han, [] i not take into account the occurrence of noes, which play an important role for analyzing the number of IKS in the workspace. The purpose of this work is to classify a family of R positining manipulators accoring to the topology of their workspace, which is efine by the number of cusps an noes that appear on the singular curves. The esign parameters space is shown to be ivie into nine omains of istinct workspace topologies, in which all manipulators have similar global kinematic properties. This stuy is of interest for the esign of new manipulators. The rest of this article is organize as follows. Next section presents the manipulators uner stuy an recalls some preliminary results. The classification is establishe in section III. Section IV synthesizes the results an section V conclues this paper. II. PRELIMINARIES A. Manipulators uner stuy The manipulators stuie in this paper are orthogonal with their last joint offset equal to zero. The remaining lengths parameters are referre to as,,, an r while the angle parameters α an α are set to 90 an 90, respectively. The

2 three joint variables are referre to as θ, θ an θ, respectively. They will be assume unlimite in this stuy. Figure shows the kinematic architecture of the manipulators uner stuy in the zero configuration. The position of the entip (or wrist center) is efine by the three Cartesian coorinates x, y an z of the operation point P with respect to a reference frame (O, x, y, z) attache to the manipulator base as shown in Fig.. z. O θ θ y r x θ P If, the operation point can meet the secon joint axis whenever θ =±arccos(- / ) an two horizontal lines appear, which may intersect S an S epening on,, an r []. The number of aspect epens on these intersections. Note that if <, no aitional curve appears in the workspace cross-section but only two points where the operation point meets the secon joint axis an the manipulator has an infinite number of IKS. Fig. shows the singularity curves when =, =, = an r =. The singular line efine by θ =+arccos(- / ) maps onto one singular point in the workspace crosssection, which is locate at the self-intersection of the internal singular bounary. The remaining singular line θ = arccos(- / ) maps onto an isolate singular point in the workspace. The workspace topology of this manipulator features two cusps an three noes, two regions with two IKS an two regions with four IKS. In the following section, the complete classification is establishe. Figure. Orthogonal manipulators uner stuy.. Singularities an aspects The eterminant of the Jacobian matrix of the orthogonal manipulators uner stuy is et(j) = ( + c )(s + c (s c r )) () where c i =cos(θ i ) an s i =sin(θ i ). A singularity occurs when et(j)=0. Since the singularities are inepenent of θ, the contour plot of et(j)=0 can be isplaye in π θ < π, π θ < π where they form a set of curves. If >, the first factor of et(j) cannot vanish an the singularities form two istinct curves S an S in the joint space []. S an S ivie the joint space into two singularityfree open sets A an A calle aspects []. The singularities can be also isplaye in the Cartesian space [, ]. Thanks to their symmetry about the first joint axis, a -imensional representation in a half cross-section of the workspace is sufficient. The singularities form two isjoint sets of curves in the workspace. These two sets efine the internal bounary WS an the external bounary WS, respectively, with WS =f(s ) an WS =f(s ). Fig. (left) shows the singularity curves when =, =, =.5 an r =. For this manipulator, the internal bounary WS has four cusp points. It ivies the workspace into one region with two IKS (the outer region) an one region with four IKS (the inner region). Figure. Singularity curves in joint space (left) an workspace (right, number of IKS in each region is inicate). Figure. Singularity curves when <. The two horizontal singular lines maps onto isolate singular points in the workspace. III. WORKSPACES CLASSIFICATION A. Classification criteria The classification is conucte on the basis of the topology of the singular curves in the workspace, which we characterize by (i) the number of cusps an (ii) the number of noes or intersecting points. A cusp (resp. a noe) is associate with one point with three equal IKS (resp. with two pairs of equal IKS). These singular points are interesting features for characterizing the workspace shape an the accessibility in the workspace.. Number of cusps For now on an without loss of generality, is set to. Thus, we nee hanle only three parameters, an r. Efficient computational algebraic tools were use in [0] to provie the equations of five separating surfaces, which were shown to ivie the parameter space into 05 cells. ut [] showe that only 5 cells shoul exist, which means that one or more surfaces among the five ones foun in [0] are not relevant. However, [] i not try to fin which surfaces are really separating. To erive the equations of the true separating surfaces, we nee to investigate the transitions between the five cases. First, let us recall the five ifferent cases foun in []. The first case is a binary manipulator (i.e. it has only two IKS) with no cusp an a hole (Fig. ). The remaining four cases are quaternary manipulators (i.e. with four IKS). The secon case is a manipulator with four cusps on the internal bounary. Fig. 5 shows a manipulator of this case with a hole an two noes. Note that the manipulator shown in Fig. is another instance of

3 case, although it has no noe an no hole (see section C). Transition between case an case is a manipulator having a pair of points with four equal IKS, where two noes an one cusp coincie [5]. + where A is given by (). The fourth case is a manipulator with four cusps. Unlike case, the cusps are not locate on the same bounary (Fig. 7). A () Figure. Manipulator of case. Figure 7. Manipulator of case. Figure 5. Manipulator of case. Deriving the conition for the inverse kinematic polynomial to have four equal roots yiels the equation of the separating surface [5] where ( + r ) + r = + r A () A= ( + ) + r an = ( ) + r. () The thir case is a manipulator with only two cusps on the internal bounary, which looks like a fish with one tail (Fig. 6). As shown in next section, an intermeiate state exists between the manipulator shown in Fig. 5 an the one epicte in Fig. 6. This intermeiate state is a variant of case with two noes an no hole (the upper an lower segments of the internal bounary cross, forming a -tail fish, see Fig. ). Transition between case an case is characterize by a manipulator for which the singular line given by θ = arccos(- / ) is tangent to the singularity curve S [5]. Expressing this conition yiels the equation of the separating surface an > where is given by (). As shown in next section, an intermeiate state exists between the manipulator shown in Fig. 6 an the one epicte in Fig. 7. This intermeiate state is a variant of case, which features two aitional noes that result from the intersection of the two workspace bounaries (like in Fig. ). Last case is a manipulator with no cusp. Unlike case, the internal bounary oes not boun a hole but a region with IKS. The two isolate singular points insie the inner region are associate with the two singularity lines. (5) Figure 8. Manipulator of case 5. Figure 6. Manipulator of case. As shown in [5], transition between case an case is characterize by a manipulator for which the singular line given by θ = arccos(- / ) is tangent to the singularity curve S. Expressing this conition yiels the equation of the separating surface Transition between case an case 5 is characterize by a manipulator for which the singular line given by θ =+arccos(- / ) is tangent to the singularity curve S [5]. Expressing this conition yiels the equation of the separating surface an < We have provie the equations of four surfaces that ivie the parameters space into five omains where the number of cusps is constant. Fig. 9 shows the plots of these surfaces in a section (, ) of the parameter space for r =. Domains,,, an 5 are associate with manipulators of case,,, an 5, respectively. C, C, C an C are the right han sie of (6)

4 (), (), (5) an (6), respectively. Figure 9. Plots of the four separating surfaces in a section (, ) of the parameter space for r =. It is interesting to see the corresponence between the equations foun with pure algebraic reasoning in [0] an those provie in this paper. The five equations foun in [0] are r + = (7) + r = (8) r r r r r r r 0 r r = (9) r = 0 (0) r = 0 () Equation (9) is a secon-egree polynomial in. Solving this quaratics for shows that (9) can be rewritten as ( + r ) + r = + r or A ( + r ) + r = + r + A where A an are efine in (). The first branch is the separating surface =C between omains an. Equation (0) is a secon-egree polynomial in. y solving this quaratics for an assuming strictly positive values for an r, (0) can be rewritten as ( an > ) or ( an < ) where is efine in (). These two branches are the separating surfaces =C an =C, respectively. In the same way, () can be rewritten as, + which is the separating surface =C. Thus, (7) an (8) foun in [0] o not efine separating surfaces, an only one branch of (9) efines a separating A surface. C. Number of noes In this section, we investigate each omain accoring to the number of noes in the workspace. ) Domain Since all manipulators in this omain are binary, they cannot have any noe in their workspace. Thus, all manipulators in omain have the same workspace topology, namely, 0 noe, 0 cusp an a hole insie their workspace. This workspace topology is referre to as WT. ) Domain Figures 5 an show two istinct workspace topologies of manipulators in omain, which feature noes an 0 noe an which we call WT an WT, respectively. Transition between these two workspace topologies is one such that the two lateral segments of the internal bounary meet tangentially (Fig. 0). Figure 0. Transition between WT an WT. Equation of this transition can be erive geometrically an the following equation is foun [5] = ( A ) () where A an are efine in (). As note in section, a thir topology exists in this omain, where the internal bounary exhibits a -tail fish. This workspace topology, which we call WT, features two noes like in Fig. 5, but these noes o not play the same role. They coincie with two isolate singular points, which are associate with the two singularity lines efine by θ =±arccos(- / ) (the operation point lies on the secon joint axis an the inverse kinematics amits infinitely many solutions). Also, the noes o not boun a hole like in Fig. 5 but a region with four IKS (Fig. 0). Figure. Workspace topology WT. Transition between WT an WT is a workspace topology

5 such that the upper an lower segments of the internal bounary meet tangentially (Fig. ). No subcase exist in this omain [5]. Such topologies are referre to as. IV. RESULTS SYNTHESIS Figure. Transition between WT an WT. As shown in [5], this transition is the occurrence of the aitional singularity + c = 0, that is = () ) Domains an 5 The internal bounary has either cusp (omain ) or 0 cusp (omain 5). This bounary is either fully insie the external bounary (like in Figs 6 an 8), or it can cross the external bounary, yieling two noes as in Fig. an. Thus, omain (resp. omain 5) contains two istinct workspace topologies, which we call ( noe) an (resp. an ). A. Parameter space partition Taking into account the noes in the classification results in a new partition of the parameter space, as shown in Fig. 5, where E, E an E are the right han sie of (), () an (), respectively. Figure 5 epicts a section (, ) of the parameter space for r =. 7 Figure 5. Parameter space partition accoring to the number of cusps an noes (in a section r =). Plots of the separating surfaces in sections for ifferent values of r are shown in Fig. 6. Figure. Workspace topology. Transition between an an transition between an are such that the internal bounary meets the external bounary tangentially (Fig. ). WT WT WT WT WT WT r =0. r =0.7 WT WT WT WT 8 WT WT WT Figure. Transition between an (left) an between an right). WT WT r =.5 r = This transition can be erive geometrically an the following equation is foun [5] = ( A+ ) () where A an are efine in (). ) Domains Manipulators in omain have four cusps an four noes. Figure 6. Separating surfaces for ifferent values of r. The areas associate with WT, WT, an ecrease when r increases. The area associate with WT is very tiny, especially for small values of r. This means that few manipulators have a topology of the WT type.

6 . Classification tree A multi-level classification of the R orthogonal manipulators uner stuy can be establishe by the classification tree shown in Fig. 7. For more legibility, only the generic cases are reporte on this tree (i.e. manipulators on the separating surfaces of the parameter space are not reporte). The root of the tree is the set of all manipulators uner stuy an each leave is the set of manipulators with a completely specifie workspace topology. The first level of the classification tree shows that a R orthogonal manipulator has either aspects (if > ), or it is quaternary an has no hole in its workspace (if < ). The secon level shows that (i) a R orthogonal manipulator with aspects is either quaternary with cusps (if >C ), or binary with no cusp, no noe an a hole (if <C ) an (ii) a R orthogonal quaternary manipulator may have cusps an 6 aspects (if >C or <C ), or cusps an 5 aspects (if C < <C an <C ), or 0 cusp an aspects (if >C ). < C 0 cusp 0 noe (,, 0) binary WT Fig. C R orthogonal manipulator with r =0 < E 0 noe (0,, ) aspects < > 0 hole quaternary > C ( > C ) or ( < C ) (C < < C ) an ( < C ) cusps quaternary < E cusps 6 aspects > C cusps 0 cusp 5 aspects aspects > E > E Fig. 8 noes Fig. (0,, ) < C < E > E noes (,, ) 0 noe (0,, ) > C noes (0,, ) noe (0,, ) noes (0,, ) noes (0,, ) WT WT WT Fig. 5 Fig. Fig. Fig. 6 Fig. Fig. 7 E E C E C C r C A + C C ( + r ) + r A = + E = ( A ) E = E = ( A + ) A = ( + ) + r an = ( ) + r (,, 0) means hole region with IKS 0 region with IKS Figure 7. Classification tree. V. CONCLUSIONS A family of R manipulators was classifie accoring to the topology of the workspace, which was efine as the number of cusps an noes. The esign parameters space was shown to be ivie into nine omains of istinct workspace topologies. Each separating surface was given as an explicit expression in the DH-parameters. Further work will investigate each omain accoring to various interesting esign criteria. REFERENCES [] C.V. Parenti an C. Innocenti, "Position Analysis of Robot Manipulators: Regions an Sub-regions," in Proc. Int. Conf. on Avances in Robot Kinematics, pp 50-58, 988. [] J. W. urick, "Kinematic analysis an esign of reunant manipulators," PhD Dissertation, Stanfor, 988. [] P. orrel an A. Liegeois, "A stuy of manipulator inverse kinematic solutions with application to trajectory planning an workspace etermination," in Proc. IEEE Int. Conf. Rob. an Aut., pp 80-85, 986. [] J. W. urick, "A classification of R regional manipulator singularities an geometries," Mechanisms an Machine Theory, Vol 0(), pp 7-89, 995. [5] P. Wenger, "Design of cuspial an noncuspial manipulators," in Proc. IEEE Int. Conf. on Rob. an Aut., pp 7-77., 997 [6] J. El Omri an P. Wenger, "How to recognize simply a non-singular posture changing -DOF manipulator," Proc. 7th Int. Conf. on Avance Robotics, p. 5-, 995. [7] V.I. Arnol, Singularity Theory, Cambrige University Press, Cambrige, 98. [8] P. Wenger, "Classification of R positioning manipulators," ASME Journal of Mechanical Design, Vol. 0(), pp 7-, 998. [9] P. Wenger, "Some guielines for the kinematic esign of new Manipulators," Mechanisms an Machine Theory, Vol 5(), pp 7-9, 999. [0] S. Corvez an F. Rouiller,"Using computer algebra tools to classify serial manipulators,"in Proc. Fourth International Workshop on Automate Deuction in Geometry, Linz, 00. [] M. aili, P. Wenger an D. Chablat, "Classification of one family of R positioning manipulators, "in Proc. th Int. Conf. on Av. Rob., 00. [] J. El Omri, 996, Kinematic analysis of robotic manipulators, PhD Thesis, University of Nantes (in french). [] D. Kohli an M. S. Hsu, "The Jacobian analysis of workspaces of mechanical manipulators," Mechanisms an Machine Theory, Vol. (), p , 987. [] M. Ceccarelli, "A formulation for the workspace bounary of general n- revolute manipulators," Mechanisms an Machine Theory, Vol, pp 67-66, 996. [5] M. aili, "Classification of R Orthogonal positioning manipulators, " technical report, University of Nantes, September 00.