Kinematic Analysis of a Family of 3R Manipulators

Transcription

1 Kinematic Analysis of a Family of R Manipulators Maher Baili, Philippe Wenger an Damien Chablat Institut e Recherche en Communications et Cybernétique e Nantes, UMR C.N.R.S , rue e la Noë, BP 92101, 4421 Nantes Ceex 0 France Abstract The workspace topologies of a family of -revolute (R) positioning manipulators are enumerate. The workspace is characterize in a half-cross section by the singular curves. The workspace topology is efine by the number of cusps that appear on these singular curves. The esign parameters space is shown to be ivie into five omains where all manipulators have the same number of cusps. Each separating surface is given as an explicit expression in the DHparameters. As an application of this work, we provie a necessary an sufficient conition for a R orthogonal manipulator to be cuspial, i.e. to change posture without meeting a singularity. This conition is set as an explicit expression in the DH parameters. Keywors Workspace, Singularity, R manipulator, Cuspial 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 [1, 2]. Cuspial robots were unknown before 1988 [], when a list of conitions for a manipulator to be noncuspial was provie [4, 5]. This list inclues simplifying geometric conitions like parallel an intersecting joint axes [4] but also nonintuitive conitions [5]. A general necessary an sufficient conition for a - DOF manipulator to be cuspial was 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 [4, 7]. The categorization of all generic quaternary 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, [10] applie efficient algebraic tools to the classification of R orthogonal manipulators with no offset on their last p1

2 joint. Five surfaces were foun to ivie the parameters space into 105 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 Bases. A kinematic interpretation of this theoretical work was conucte in [11] : 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 [11] i not provie the equations of the separating surfaces in the parameters space for the five cells associate with the five cases foun. The purpose of this work is to classify a family of R positioning manipulators accoring to the topology of their workspace, which is efine by the number of cusps appear on the singular curves. The esign parameters space is shown to be ivie into five omains where all manipulators have the same number of cusps. As an application of this work, a necessary an sufficient conition for a R orthogonal manipulator to be cuspial is provie as an explicit expression in the DH parameters. 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 states the necessary an sufficient conition an section V conclues this paper. II. MANIPULATOR UNDER STUDY The manipulators stuie in this paper are orthogonal with their last joint offset equal to zero. The remaining lengths parameters are referre to as 2,, 4, an r 2 while the angle parameters α 2 an α are set to 90 an 90, respectively. The three joint variables are referre to as θ 1, θ 2 an θ, respectively. They will be assume unlimite in this stuy. Figure 1 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 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. 1. z y 2 O θ 2 r 2 θ 1 θ. P x 4 Figure 1 : Orthogonal manipulators uner stuy. p2

3 The singularities of general R manipulators can be etermine by calculating the eterminant of the Jacobian matrix. For the orthogonal manipulators uner stuy, the eterminant of the Jacobian matrix takes the following form [15]: et(j) = ( + c 4 )(s 2 + c 2 (s c r 2 )) (1) where c i =cos(θ ) ι an s i =sin(θ ). ι A singularity occurs when et(j)=0. Since the singularities are inepenent of θ 1, the contour plot of et(j)=0 can be isplaye as curves in π θ < π, π θ < π. The singularities can also be isplaye in the Cartesian space by 2 plotting the points where the inverse kinematics has ouble roots [1]. Thanks to their symmetry about the first joint axis, it is sufficient to raw a half cross-section of the workspace by plotting 2 2 the points ( ρ = x + y, z). If > 4, the first factor of et(j) cannot vanish an the singularities form two istinct curves S 1 an S 2 in the joint space [15]. When the manipulator is in such a singularity, there is line that passes through the operation point an that cuts all joint axes [4]. The singularities form two isjoint sets of curves in the workspace. These two sets efine the internal bounary WS 1 an the external bounary WS 2, respectively, with WS 1 =f(s 1 ) an WS 2 =f(s 2 ). Figure 2(a) shows the singularity curves when =2, 4 =1.5 an r 2 =1. For this manipulator, the internal bounary WS 1 has four cusp points, where three IKS coincie. It ivies the workspace into one region with two IKS (the outer region) an one region with four IKS (the inner region). 4 IKS 2 IKS (a) =2, 4 =1.5, r 2 =1 (b) =, 4 =4, r 2 = Figure 2 : Singularity curves when > 4 (a) an when < 4 (b) If 4, the operation point can meet the secon joint axis whenever θ = ±arccos(- / 4 ) an two horizontal lines appear in the joint space. No aitional curve appears in the workspace cross-section but only two points. This is because, since the operation point meets the secon p

4 joint axis when θ = ±arccos(- / 4 ), the location of the operation point oes not change when θ 2 is rotate. Figure 2 (b) shows the singularity curves of a manipulator such that =, 4 =4, r 2 =. III. WORKSPACE TOPOLOGIES The workspace is efine by the topology of the singular curves, which we characterize by the number of cusps. A cusp is associate with one point with three equal IKS. These singular points are interesting features for characterizing the workspace shape an the accessibility in the workspace. For now on an without loss of generality, 2 is set to 1. Thus, we nee hanle only three parameters, 4 an r 2. Efficient computational algebraic tools were use in [10] to provie the equations of five separating surfaces, which were shown to ivie the parameter space into 105 cells. But [11] showe that only 5 cells shoul exist, which means that one or more surfaces among the five ones foun in [10] are not relevant. However, [11] 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 [11]. 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. Figure 4 shows a manipulator of this case with a hole. Transition between case 1 an case 2 is a manipulator with two points with four equal IKS, where one noe an two cusps coincie [15]. Figure : Manipulator of case 1 Figure 4 : Manipulator of case 2 Deriving the conition for the inverse kinematic polynomial to have four equal roots yiels the equation of the separating surface [15] ( + r2 ) + r 2 4 = + r2 2 AB (2) where p4

5 A= ( + 1) + r an B = ( 1) + r () Note that there exist two other instances of case 2: the manipulator shown in Fig. 2a with no hole, an a manipulator where the upper an lower segments of the internal bounary cross, forming a 2-tail fish [15]. The thir case is a manipulator with only two cusps on the internal bounary, which looks like a fish with one tail (Fig. 2b). As shown in [15], transition between case 2 an case is characterize by a manipulator for which the singular line given by θ = arccos(- / 4 ) is tangent to the singularity curve S 1. Expressing this conition yiels the equation of the separating surface, where A is given by () : 4 1+ = A (4) The fourth case is a manipulator with four cusps. Unlike case 2, the cusps are not locate on the same bounary (Fig. 5). Transition between case an case 4 is characterize by a manipulator for which the singular line given by θ = arccos(- / 4 ) is tangent to the singularity curve S 2 [15]. Expressing this conition yiels the equation of the separating surface, where B is given by (): = B an > (5) Last case is a manipulator with no cusp. Unlike case 1, the internal bounary oes not boun a hole but a region with 4 IKS. The two isolate singular points insie the inner region are associate with the two singularity lines. Transition between case 4 an case 5 is characterize by a manipulator for which the singular line given by θ = +arccos(- / 4 ) is tangent to the singularity curve S 1 [15]. Expressing this conition yiels the equation of the separating surface: = B an < (6) where B is given by (). p5

6 Figure 5: Manipulator of case 4 Figure 6 : Manipulator of case 5 We have provie the equations of four surfaces that ivie the parameters space into five omains where the number of cusps is constant. Figure 7 shows the plots of these surfaces in a section (, 4 ) of the parameter space for r 2 =1. Domains 1, 2,, 4 an 5 are associate with manipulators of case 1, 2,, 4 an 5, respectively. C 1, C 2, C an C 4 are the right han sie of (2), (4), (5) an (6), respectively. Fig. 7 : Plots of the four separating surfaces in a section (, 4 ) of the parameter space for r 2 =1. IV. NECESSARY AND SUFFICIENT CONDITION FOR A MANIPULATOR TO BE CUSPIDAL The above classification provies a means to erive an explicit DH parameter conition for an orthogonal manipulator to be cuspial, i.e., to change posture without meeting a singularity. In effect, as shown in Fig. 7, any cuspial manipulator belongs to omains 2, or 4. Thus, the DHparameters must satisfy 4 > C 1 an ( 4 < C 4 or > 1). Thus, a necessary an sufficient conition for an orthogonal manipulator to be cuspial is (by iviing the parameters by 2, one gets the general formula for manipulators such that 2 1) p6

7 r ( + r2 ) ( + r2 ) 2 4 > ( + 2) + r2 ( 2) + r2 an or < 2 an 4 < ( 2) + r2 ) 2 (7) This conition is explicit an can be checke very easily. V. CONCLUSION A family of R manipulators was classifie accoring to the topology of the workspace, which was efine as the number of cusps. The esign parameters space was shown to be ivie into five omains where all manipulators have the same number of cusps. Each separating surface was given as an explicit expression in the DH-parameters. An interesting application result of this work is the establishment of a necessary an sufficient conition for a manipulator to be cuspial, i.e., to change posture without meeting a singularity. This conition was set as an explicit expression in the DH parameters. References [1] C.V. Parenti an C. Innocenti, "Position Analysis of Robot Manipulators: Regions an Subregions," in Proc. Int. Conf. on Avances in Robot Kinematics, pp , [2] J. W. Burick, "Kinematic analysis an esign of reunant manipulators," PhD Dissertation, Stanfor, [] P. Borrel 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 , [4] J. W. Burick, "A classification of R regional manipulator singularities an geometries," Mechanisms an Machine Theory, Vol 0(1), pp 71-89, [5] P. Wenger, "Design of cuspial an noncuspial manipulators," in Proc. IEEE Int. Conf. on Rob. an Aut., pp , 1997 [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, pp , [7] V.I. Arnol, Singularity Theory, Cambrige University Press, Cambrige, [8] P. Wenger, "Classification of R positioning manipulators," ASME Journal of Mechanical Design, Vol. 120(2), pp 27-2, [9] P. Wenger, "Some guielines for the kinematic esign of new Manipulators," Mechanisms an Machine Theory, Vol 5(), pp , [10] S. Corvez an F. Rouiller, "Using computer algebra tools to classify serial manipulators,"in Proc. Fourth International Workshop on Automate Deuction in Geometry, Linz, [11] M. Baili, P. Wenger an D. Chablat, "Classification of one family of R positioning manipulators, "in Proc. 11th Int. Conf. on Av. Rob., 200. [12] J. El Omri, 1996, Kinematic analysis of robotic manipulators, PhD Thesis, University of Nantes (in french). [1] D. Kohli an M. S. Hsu, "The Jacobian analysis of workspaces of mechanical manipulators," Mechanisms an Machine Theory, Vol. 22(), p , p7

8 [14] M. Ceccarelli, "A formulation for the workspace bounary of general n-revolute manipulators," Mechanisms an Machine Theory, Vol 1, pp , [15] M. Baili, "Classification of R Orthogonal positioning manipulators, " technical report, University of Nantes, September 200. p8

9 SUMMARY Kinematic Analysis of a Family of R Manipulators Maher Baili, Philippe Wenger an Damien Chablat Institut e Recherche en Communications et Cybernétique e Nantes, u.m.r. C.N.R.S , rue e la Noë, BP 92101, 4421 Nantes Ceex 0 France Abstract The workspace topologies of a family of -revolute (R) positioning manipulators are enumerate. The workspace is characterize in a half-cross section by the singular curves. The workspace topology is efine by the number of cusps that appear on these singular curves. The esign parameters space is shown to be ivie into five omains where all manipulators have the same number of cusps. Each separating surface is given as an explicit expression in the DHparameters. As an application of this work, we provie a necessary an sufficient conition for a R orthogonal manipulator to be cuspial, i.e. to change posture without meeting a singularity. This conition is set as an explicit expression in the DH parameters. Keywors Workspace, Singularity, R manipulator, Cuspial manipulator. p9