Workspace Topologies of Industrial 3R Manipulators

A mathematical analysis is used to characterize workspace topologies of industrial 3R manipulators. A level-set reconstruction of the workspace is formulated to identify characteristic points with fairly simple algebraic expressions. Thus, industrial 3R manipulators are classified as functions of workspace kinematic properties. Examples are illustrated to show practical usefulness of the proposed workspace characterization.


Introduction
Workspace analysis of serial manipulators is of great interest since the workspace geometry can be considered a fundamental issue for manipulator design, robot placement in a working environment, and trajectory planning. Nowadays the majority of manipulators for industrial applications are of serial type. They often have geometric design simplifications, such as intersecting joint axes, orthogonal or parallel joint axes. Moreover, most of the industrial manipulators are wrist-partitioned, that is they consist of a concatenation of a 3R (Revolute) arm, i.e., regional structure, and a spherical wrist that is attached to the terminal link of the arm. The workspace analysis of such manipulators can be performed by considering both the positioning and orienting tasks, and the singularities determination. Early studies have been developed for 3R manipulators for position workspace only (Freudenstein, F. & Primrose, E.J.F., 1984), ( Roth, B., 1975). An algebraic formulation for determining the workspace of 3R manipulators has been presented by using the geometry of ring generation as described in a transversal plane in (Freudenstein, F. & Primrose, E.J.F., 1984), or in a cross-section plane in (Ceccarelli, M., 1989). Only the last approach has been generalized for nR manipulators in (Ceccarelli, M., 1996). The determination of the workspace boundary in Cartesian Space has been proposed also by using identification of singularities in workspace boundary, like for example in (Smith, D.R. & Lipkin, H., 1993). Other papers are related to the singularity of the Jacobian matrix that is usually expressed in the Joint Space.
Regions that are free of singularities in the Joint Space have been named C-sheets, (Burdick, J.W., 1995). In Csheets it is possible to determine how to change posture without passing through singularities (Parenti-Castelli, V. & Innocenti, C., 1988). Manipulators that can change posture without meeting a singularity have been named cuspidal manipulators in (Wenger, P.,2000). The analysis and characterization of geometric singularities of the cross-section boundary curve has been proposed in (Ottaviano, E., Ceccarelli, M. & Lanni, C., 1999), (Ottaviano, E., Husty, M. & Ceccarelli, M., 2004). Several authors have grouped manipulators into classes, as reported in (Burdick, J.W., 1995), (Wenger, P.,2000), (Zein, M., Wenger, P. & Chablat D., 2005), by considering special architectures, such as cuspidal or orthogonal manipulators, which have simplification in the architecture. In this paper we present a classification of industrial 3R manipulators as based on kinematic properties of the workspace, but not only on parameters simplifications. A level-set reconstruction is used to analyze workspace topology by using algebraic expressions, as outlined in (Ottaviano, E., Husty, M. & Ceccarelli, M., 2006a). The two-parameter set of curves is used to characterize the workspace cross-section and gives an interesting insight of the internal structure of the workspace boundary as obtained as an envelope of generating circles. Characteristic points are determined for a fairly simple classification of industrial 3R manipulators with a direct kinematic interpretation, as pointed out in (Ottaviano, E., Husty, M. & Ceccarelli, M., 2006b). In this paper we have focused specific attention to workspace analysis for industrial manipulators.

A Formulation for Workspace Determination
Figures have to be made in high quality, which is suitable A general 3R manipulator is sketched in Fig.1, in which the kinematic parameters are denoted by the Hartenberg and Denavit (H-D) notation. Without loss of generality the base frame is assumed to be coincident with X1Y1Z1 frame when θ1= 0, a0=0 and d1=0. The end-effector point H can be usually chosen as either the center of the endeffector, or the tip of a finger. Point H is placed on the X3 axis at a distance a3 from O3, as shown in Fig.1. The general 3R manipulator is described by the H-D parameters a1, a2, d2, d3, α1 and α2, and θi, for (i = 1,…,3), as shown in Fig.1. The workspace of a general 3R manipulator can be expressed in the form of radial and axial reaches, r and z respectively, with respect to the base frame. r is the radial distance of point H from the Z1-axis and z is the axial reach; both can be expressed as function of H-D parameters. Reaches r and z can be evaluated as function of reach coordinates in the form Equation (3) represents a 2-parameter family of curves, which gives the cross-section workspace in a cross-section plane (Freudenstein, F. & Primrose, E.J.F., 1984), (Ceccarelli, M., 1989), as function of the H-D parameters through H1 x , H1 y and H1 z coefficients.

A Level-set Analysis for 3R Manipulators
In the following the above-mentioned two-parameter set is interpreted as a level-set. The level-set of a differentiable function ℜ → ℜ n : f corresponding to a real value c is the set of points, (Sethian, J.A., 1996) The potentiality of the level-set method is now applied to the workspace analysis of 3R manipulators, as outlined in (Ottaviano, E., Husty, M. & Ceccarelli, M., 2006 b and c). In particular, the level-set reconstruction for a serial manipulator can be obtained by using the 2 parameterfamily of curves that are expressed in Eq.(3). The level-sets belonging to constant values of θ3 are curves in the RZ-plane. Therefore, this one parameter set of curves can be viewed as the contour map of a surface S, which conveniently can be used to analyze the workspace of the manipulator. The surface S can be defined though the functions By performing the half-tangent substitution v = tan (θ2/2) in Eq. (5) where S1 represents four double planes parallel to XY plane, in which the height depends on the H-D parameters; S2 is the graph of the level-set function in which the parameter lines belong to θ2 = const or θ3 = const. Geometrically S is generated by taking a crosssection of the workspace that is parameterized by θ2 and θ3 and explode the overlapping level-set curves in the direction of Z-axis, as illustrated in the example of Fig It is to point out that in a workspace cross-section two different one-parameter sets of level-curves can be traced as function of θ3 = const and θ2 = const, respectively. On Fig.2b) the corresponding surface S is displayed. Geometrically the level-set curves of Fig.2a) are the orthogonal projections of the intersection curves with planes Z = const and the surface S onto the XY-plane. The level-set curves for θ3 = const in Fig.2a) are therefore the contour curves for the surface S. Additionally we have displayed in Fig.2b) a gross line parallel to the Z-axis. This line shows clearly four intersection points with the surface S. Therefore, the corresponding point in the levelset plane in Fig.2a) corresponds to a four fold solution of the IK. In Figs.2a) and 2b) the level-set curves are shown as function of different values of angles θ2 and θ3. In the displayed cross-section of the workspace the two different one-parameter sets of level-curves are displayed in order to show those lines that permit to built the levelset surface from the parametrization of the workspace cross-section.. The meridian ones, which are shown as horizontal in the 3D plot, belongs to θ3 = const, and the other radial-like ones belongs to θ2 = const. In order to determine the algebraic degree of S2 one has to homogenize and intersect with the plane at infinity. The resulting intersection is completely independent of the H-D parameters. It consists of an eight fold line Z = 0 and two complex double lines. Thus, the surface is of algebraic degree 12.
Manipulators having singularities on the surface S can be considered as an algebraically closed set. Singularities of the surface S can be found by considering the implicit equation of S, together with its partial derivatives with respect to X, Y, and Z, respectively, (Gibson, C.G., 1998). The graph S of the level-set function reveals a very different nature of these highly interesting singular points that are related with the ring void boundary. Some of them arise just from the projection of S into the level-set plane and some of them come from singularities of the surface S. A geometrical interpretation for the singularities of the graph of the level-set function is that there is a value of θ3 for which the operation point H lies on Z2 axis. Therefore, there is a free motion about Z2 axis, which is completely independent by θ2 angle. In this paper we have focused our analysis on this kind of singularity for the industrial 3R manipulators, which have been grouped in orthogonal and ortho-parallel manipulators.

A Formulation for Orthogonal Manipulators
Orthogonal manipulators are characterized by having three revolute joint axes, which are orthogonal to each other. Therefore, kinematic parameters can be identified as a1, a2, a3, d2, d3, twist angles α1 and α2 are set equal to -90 and 90 deg. Joint variables are identified as θ1, θ2 and θ3, respectively, and they will be assumed unlimited in this work. A kinematic scheme is displayed in Fig. 3. The surface S of Eq.(6) can be studied by looking at factors S1 and S2 of S. For orthogonal manipulators the surface S1 can be expressed in the form where the coefficients ki depend on a2, a3 and d3 only, in the form In general the equation for S1 can have real solutions. The necessary and sufficient condition for having real solutions is: d3 = 0 and a3 >a2. Other singularities can be found by analyzing surface S2. Zeros of the set of equations S2 = 0; S2X = 0; S2Y = 0; and S2Z = 0, yield the geometric singularities of the surface S2. Singularities of S2 surface can be can expressed by the product of two polynomials in the form The zeros of the set of equations: S2 = 0; S2X = 0; S2Y = 0; and S2Z = 0, yield the geometric singularities of S2. The herein proposed classification allows obtaining design information related to workspace properties. Geometrically, when α2 is equal to 90 deg. then a member of the θ3 parameter set of curves belonging to different values of a3 can intersect the Z2 axis only when d3 is equal to zero. In particular, each possible intersection of a θ3 curve represents a singularity of the level-set graph. Only three cases can arise: no intersection, two distinct intersections and two coincident intersections. The three cases represent three classes of industrial-type manipulators.

A Formulation for Ortho-Parallel Manipulators
Ortho-parallel manipulators are characterized by having the first two revolute joint axes orthogonal to each other, and the last revolute joint axis is parallel to the second one. Therefore, kinematic parameters can be identified as a1, a2, a3, d2, d3, and twist angles α1 and α2 are set equal to -90 and 0 deg. Joint variables are identified as θ1, θ2 and θ3, respectively, and they will be assumed unlimited in this work. A kinematic scheme is displayed in Fig. 4. The factors S1 and S2 of S are analyzed separately. S1 can be expressed in the form in which ki coefficients depend on a2 and a3 only, in the form A necessary and sufficient condition for having real solutions can be determined through only when there are changes in the signs of coefficients ki. In particular, the number of real roots is equal to the number of changes of sign in the ki coefficients. Therefore, S1 has no real solutions. Other singularities can be found by analyzing surface S2. Zeros of the set of equations S2 = 0; S2X = 0; S2Y = 0; and S2Z = 0, yield the geometric singularities of the surface S2. Singularities of S2 surface are given in the form Geometrically, when α2 is equal to 0 deg then the θ3 parameter set of curves can intersect the Z2 axis only when a2 is equal to a3. For this case, a θ3 curve is in a plane, which is orthogonal to Z2 axis. Only two cases can arise: no intersection and two coincident intersections. These two cases yield two different classes of industrialtype manipulators.

A Classification for Industrial 3R Manipulatros
The following groups contain all possible topologies of orthogonal and ortho-parallel manipulators, which can be characterized by the presence of singularities on the surface S. Furthermore, if the surface has real singularities then they also correspond to singularities in the crosssection of the boundary curve. A classification of industrial 3R manipulators can be proposed as shown in Table 1 by referring to the general scheme in Fig.5, in agreement with the following discussion. In particular, the shape of S surface function is characterized by the relative location of the characteristic points A and B that represent singularities for S function.

Class 1: A General Manipulator
A manipulator that belongs to the Class 1 has no (real) singularities on the surface S since both the characteristics points A and B are at infinite and in opposite directions. It may have either a changing posture behavior or it can present a void within the workspace. A characteristic shape with corresponding cross-section figures is reported in the examples of Figs. 6 and 7 for orthogonal manipulators and ortho-parallel manipulators, respectively. The corresponding cross-section boundary curve can have only cusps and/or double points. Such general manipulators are characterized to have no singularities on the graph of the level-set. In addition, it can be observed that in general cuspidality behavior is not strictly related to special designs.

Class 2 Industrial Manipulators
A manipulator that belongs to the Class 2 has only one singularity on the surface S that is determined by the coincidence of the characteristic points A and B. Class 2 manipulators can be characterized by the presence of 4solution regions for the IK as identified in the S area that is delimited by characteristic points A and B. The crosssection boundary curve for Class 2 manipulators contains one acnode as a singular point, .
A Class 2 orthogonal manipulator is characterized by having a2 = a3; and d3 = 0. If a2 a3, the operation point H can meet the second joint axis whenever cos θ3 = ± (a2/a3), which was found also in (Zein, M., Wenger, P. & Chablat D., 2005). Characteristic shape with corresponding crosssections for Class 2 orthogonal manipulators is reported in the example of Fig. 8. For class 2 orthogonal manipulators, S1 expression vanishes and singularities can be found by checking S2 polynomial expression. Class 2 orthogonal manipulators are characterized to have two coincident singular configurations that depend on θ3 parameter only. A Class 2 ortho-parallel manipulator is characterized by having a2 = a3 . Characteristic shapes of workspace crosssection and surface S are shown in the example of Fig. 9.

Class 3 Industrial Manipulators
This class of manipulators is characterized by having two distinct singularities on the surface S and in general 4solution regions for the IK. Class 3 orthogonal manipulators have a3 > a2 and d3=0. The meaning of a singularity is that for a 3 value exists a line passing through the operation point H and intersecting one of the manipulator axes. If d3 is not equal to zero then the generating curve has no real solutions; if a3 is less than a2 then no singularities are on the S surface. A characteristic shape and corresponding cross-section of manipulator workspace is reported in the example of Fig. 10 Figure 11 refers to the manipulator arm of the Miller Hybrid Arc Welding Robot System, (Miller, 2001), that shows an orthogonal architecture. The results of the workspace analysis through the proposed level-set reconstruction are shown in Fig.11b) from which the manipulator can be recognized as a Class 3 type manipulator.

Numerical examples
In Figs. 12 and 13 the results of the proposed analysis are shown for the KUKA KR30 robot, (KUKA 2006), by considering two structure configurations that are related to the cases of freezing a joint in the 4R manipulator chain with the aim to simulate operations with unused or damaged mobility of joints. An ortho-parallel configuration of the manipulator of industrial robot KUKA KR30 is obtained by considering the first three joints of the manipulator chain only when the last joint is frozen. The workspace characteristics are shown in Fig.12, indicating a Class 2 manipulator. An orthogonal architecture of the manipulator of industrial robot KUKA KR30 is obtained by considering by freezing the third joint of the robot arm and by considering the last joint as the extreme joint of the 3R manipulator chain. The workspace characteristics are shown in Fig.13, identifying it as a Class 1 manipulator. Such an investigation of sub-structures of manipulator arms can be of interest for practical operation of the robot when for any reason not all the joints are used or when partial motions refer to those sub.-structures.

Conclusions
Workspace topologies of 3R industrial manipulators are characterized by using a level-set reconstruction. The proposed algebraic expressions have been useful to classify industrial 3R manipulators and to identify design conditions for avoiding singularities in the workspace.