933  Abstract —   A   classification   of   a   family   of   3-revolute   (3R) positining manipulators is established. This classification is based on   the   topology   of   their   workspace.   The   workspace   is characterized in a half-cross section by the singular curves. The workspace topology is defined by the number of cusps and nodes that appear on these singular curves. The design parameters space   is   shown   to   be   divided   into   nine   domains   of   distinct workspace topologies, in which all manipulators have similar global kinematic properties. Each separating surface is given as an explicit expression in the DH-parameters. Keywords —Classification,   Workspace,   Singularity,   Cusp, node, orthogonal manipulator.  I.   I NTRODUCTION A   positioning   manipulator   may   be   used   as   such   for positioning tasks in the Cartesian space or as the regional structure   of   a   6R   manipulator   with   spherical   wrist.   Most industrial   regional   structures   have   the   same   kinematic architecture, namely, a vertical revolute joint followed by two parallel joints, like the Puma. Such manipulators are always  noncuspidal   (i.e.   must   meet   a   singularity   to   change   their posture) and they have four inverse kinematic solutions (IKS) for all points in their workspace (assuming unlimited joints). This paper focuses on alternative manipulator designs, namely, positioning   3R   manipulators   with   orthogonal   joint   axes (orthogonal manipulators). Orthogonal manipulators may have different global kinematic properties according to their link lengths and joint offsets. They may be   cuspidal , that is, they can change their posture without meeting a singularity [1, 2]. In 1998,   ABB-Robotics   launched   the   IRB   6400C,   a   6R manipulator to be used in the car industry and designed to minimize the swept volume. The only difference with the Puma was the permutation of the first two link axes, resulting in a manipulator with all its joint axes orthogonal, and   cuspidal . Commercialization of the IRB 6400C was finally stopped one year later. Exact reason is beyond the knowledge of the authors but the cuspidal behavior is likeable to have disappointed the end users. Cuspidal robots were unknown before 1988 [3], when a list of conditions for a manipulator to be noncuspidal was provided [4, 5]. This list includes simplifying geometric conditions like parallel and intersecting joint axes [4] but also nonintuitive conditions [5]. A general necessary and sufficient condition   for   a   3-DOF   manipulator   to   be   cuspidal   was  This work was supported in part by C.N.R.S. MathStic program "Cuspidal robots and triple roots".  established in [6], namely, the existence of at least one point in the workspace where the inverse kinematics admits three equal solutions. The word “cuspidal manipulator” was defined in accordance to this condition because a point with three equal IKS forms a cusp in a cross section of the workspace [4, 7]. The   categorization   of   all   generic   3R   manipulators   was established in [8] based on the homotopy class of the singular curves in the joint space. [9] proposed a procedure to take into account the cuspidality property in the design process of new manipulators. More recently, [10] applied efficient algebraic tools to the classification of 3R orthogonal manipulators with no offset on their last joint. Five surfaces were found to divide 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 derived as polynomials in the   DH-parameters   using   Groebner   Bases.   A   kinematic interpretation of this theoretical work was conducted in [11] : the   authors   analyzed   general   kinematic   properties   of   one representative manipulator in each cell. Only five different cases were found to exist. However, the classification in [11] did not provide the equations of the separating surfaces in the parameters space for the five cells associated with the five cases found. On the other hand, [11] did not take into account the occurrence of nodes, 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 3R positining manipulators according to the topology of their workspace, which is defined by the number of cusps and nodes that appear on the singular curves. The design parameters space is shown to be divided into nine domains of distinct workspace topologies,   in   which   all   manipulators   have   similar   global kinematic properties. This study is of interest for the design of new manipulators. The rest of this article is organized as follows. Next section presents   the   manipulators   under   study   and   recalls   some preliminary results. The classification is established in section III. Section IV synthesizes the results and section V concludes this paper. II.   PRELIMINARIES  A.   Manipulators under study  The manipulators studied in this paper are orthogonal with their last joint offset equal to zero. The remaining lengths parameters are referred to as   d 2 ,   d 3 ,   d 4 , and   r 2   while the angle parameters  α 2   and  α 3   are set to –90° and 90°, respectively. The A Classification of 3R Orthogonal Manipulators by the Topology of their Workspace Maher Baili, Philippe Wenger and Damien Chablat Institut de Recherche en Communications et Cybernétique de Nantes, UMR C.N.R.S. 6597 1, rue de la Noë, BP 92101, 44321 Nantes Cedex 03 France Maher.Baili@irccyn.ec-nantes.fr
three   joint   variables   are   referred   to   as  θ 1 ,  θ 2   and  θ 3 , respectively. They will be assumed unlimited in this study. Figure 1 shows the kinematic architecture of the manipulators under study in the zero configuration. The position of the end- tip   (or   wrist   center)   is   defined   by   the   three   Cartesian coordinates   x ,   y   and   z   of the operation point   P   with respect to a reference frame (O,   x ,   y ,   z ) attached to the manipulator base as shown in Fig. 1. z   y O x P  θ 1  θ 2  θ 3  d 2  r   2  d 4  d 3 . Figure 1.   Orthogonal manipulators under study.  B.   Singularities and aspects  The determinant of the Jacobian matrix of the orthogonal manipulators under study is det( J ) = ( d 3   + c 3 d 4   )(s   3 d 2   + c 2   (s   3 d 3   – c   3 r 2   ))   (1) where   c i =cos(  θ i ) and   s i =sin(  θ i ). A singularity occurs when det( J )=0. Since the singularities are independent of  θ 1 , the contour   plot   of   det( J )=0   can   be   displayed   in 2   3 ,  π   θ   π   π   θ   π −   ≤   <   −   ≤   <   where they form a set of curves. If  d 3 > d 4 ,   the   first   factor   of   det( J )   cannot   vanish   and   the singularities form two distinct curves   S 1   and   S 2   in the joint space [12].   S 1   and   S 2   divide the joint space into two singularity- free open sets A 1   and A 2   called   aspects   [1]. The singularities can be also displayed in the Cartesian space [13, 14]. Thanks to their   symmetry about the   first joint axis, a 2-dimensional representation   in   a half   cross-section   of   the workspace   is sufficient. The singularities form two disjoint sets of curves in the workspace. These two sets define the internal boundary  WS 1   and   the   external   boundary   WS 2 ,   respectively,   with  WS 1 =f( S 1 ) and   WS 2 =f( S 2 ). Fig. 2 (left) shows the singularity curves   when   d 2 =1,   d 3 =2,   d 4 =1.5   and   r 2 =1.   For   this manipulator, the internal boundary   WS 1   has four cusp points. It divides the workspace into one region with two IKS (the outer region) and one region with four IKS (the inner region).  4   2 Figure 2.   Singularity curves in joint space (left) and workspace (right, number of IKS in each region is indicated). If   d 3 ≤ d 4 , the operation point can meet the second joint axis whenever  θ 3 = ± arccos(- d 3 / d 4 ) and two horizontal lines appear, which may intersect   S 1   and   S 2   depending on   d 2 ,   d 3 ,   d 4   and   r 2 [12]. The number of aspect depends on these intersections. Note that if   d 3 < d 4 , no additional curve appears in the workspace cross-section but only two points where the operation point meets the second joint axis and the manipulator has an infinite number of IKS. Fig. 3 shows the singularity curves when   d 2 =1,  d 3 =3,   d 4 =4 and   r 2 =2. The singular line defined by  θ 3 =+arccos(-  d 3 / d 4 ) maps onto one singular point in the workspace cross- section, which is located at the self-intersection of the internal singular boundary. The remaining singular line  θ 3 =–arccos(-  d 3 / d 4 ) maps onto an isolated singular point in the workspace. The workspace topology of this manipulator features two cusps and three nodes, two regions with two IKS and two regions with   four   IKS.   In   the   following   section,   the   complete classification is established.  Figure 3.   Singularity curves when   d 3   < d 4   . The two horizontal singular lines maps onto isolated singular points in the workspace. III.   WORKSPACES CLASSIFICATION  A.   Classification criteria  The classification is conducted on the basis of the topology of the singular curves in the workspace, which we characterize by ( i ) the number of cusps and ( ii ) the number of nodes or intersecting points. A cusp (resp. a node) is associated 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 and the accessibility in the workspace.  B.   Number of cusps  For now on and without loss of generality,   d 2   is set to 1. Thus, we need handle only three parameters   d 3 ,   d 4   and   r 2 . Efficient computational algebraic tools were used in [10] to provide the equations of five separating surfaces, which were shown to divide the parameter space into 105 cells. But [11] showed that only 5 cells should exist, which means that one or more surfaces among the five ones found in [10] are not relevant. However, [11] did not try to find which surfaces are really separating. To derive the equations of the true separating surfaces, we need to investigate the transitions between the five cases. First, let us recall the five different cases found in [11]. The first case is a binary manipulator (i.e. it has only two IKS) with no cusp and a hole (Fig. 4). The remaining four cases are quaternary manipulators (i.e. with four IKS). The second case is a manipulator with four cusps on the internal boundary. Fig. 5 shows a manipulator of this case with a hole and two nodes. Note that the manipulator shown in Fig. 2 is another instance of
case 2, although it has no node and no hole (see section   C ). Transition between case 1 and case 2 is a manipulator having a pair of points with four equal IKS, where two nodes and one cusp coincide [15].  Figure 4.   Manipulator of case 1.  Figure 5.   Manipulator of case 2. Deriving the condition for the inverse kinematic polynomial to have four equal roots yields the equation of the separating surface [15] 2   2   2   2 2 2   2   3   2   3   2 4   3   2 (   ) 1 2  d   r   d   r d   d   r   AB  ⎞ ⎛   +   −   + =   +   −   ⎟ ⎜ ⎝   ⎠  (2) where 2   2 2   2 3   2   3   2 (   1)   and   (   1) A   d   r   B   d   r =   +   +   =   −   +   .   (3) The third case is a manipulator with only two cusps on the internal boundary, which looks like a fish with one tail (Fig. 6). As shown in next section, an intermediate state exists between the manipulator shown in Fig. 5 and the one depicted in Fig. 6. This intermediate state is a variant of case 2 with two nodes and no hole (the upper and lower segments of the internal boundary cross, forming a ‘2-tail fish’, see Fig. 11).  Figure 6.   Manipulator of case 3. As shown in [15], transition between case 2 and case 3 is characterized by a manipulator for which the singular line given by  θ 3 =–arccos(- d 3 / d 4 ) is tangent to the singularity curve  S 1 .   Expressing   this   condition   yields   the   equation   of   the separating surface 3 4 3 1  d d   A d  =   ⋅ +   (4) where   A   is given by (3). The fourth case is a manipulator with four cusps. Unlike case 2, the cusps are not located on the same boundary (Fig. 7).  Figure 7.   Manipulator of case 4. Transition between case 3 and case 4 is characterized by a manipulator for which the singular line given by  θ 3 =–arccos(-  d 3 / d 4 ) is tangent to the singularity curve   S 2   [15]. Expressing this condition yields the equation of the separating surface 3 4   3 3 and   1 1  d d   B   d d  =   ⋅   > −   (5) where   B   is   given   by   (3).   As   shown   in   next   section,   an intermediate state exists between the manipulator shown in Fig. 6 and the one depicted in Fig. 7. This intermediate state is a variant of case 3, which features two additional nodes that result from the intersection of the two workspace boundaries (like in Fig. 3). Last case is a manipulator with no cusp. Unlike case 1, the internal boundary does not bound a hole but a region with 4 IKS. The two isolated singular points inside the inner region are associated with the two singularity lines.  Figure 8.   Manipulator of case 5. Transition between case 4 and case 5 is characterized by a manipulator for which the singular line given by  θ 3 =+arccos(-  d 3 / d 4 ) is tangent to the singularity curve   S 1   [15]. Expressing this condition yields the equation of the separating surface 3 4   3 3 and   1 1  d d   B   d d  =   ⋅   < −   (6) We have provided the equations of four surfaces that divide the parameters space into five domains where the number of cusps is constant. Fig. 9 shows the plots of these surfaces in a section ( d 3 ,   d 4 ) of the parameter space for   r 2 =1. Domains 1, 2, 3, 4 and 5 are associated with manipulators of case 1, 2, 3, 4 and 5, respectively.   C 1 ,   C 2 ,   C 3   and   C 4   are the right hand side of
(2), (4), (5) and (6), respectively.  Figure 9.   Plots of the four separating surfaces in a section ( d 3 ,   d 4   ) of the parameter space for   r 2   =1. It is interesting to see the correspondence between the equations found with pure algebraic reasoning in [10] and those provided in this paper. The five equations found in [10] are 2 3   4   2   4   0 d   d r   d −   +   +   =   (7) 2   2   2 3   4   2   0 d   d   r −   +   =   (8) 2   6   4   4   2   4   2   2   4   4   2   4   2   2 4   3   4   3   4   3   2   4   3   4   3   4   3   2 2   2   2   2   4   2   2   4   2   4   4   4   2   6 4   3   4   3   2   3   2   4   2   4   2   4   4   2 2   2   2   4 4   2   4   2 3   2   2   2 3   2 2   0  d   d   d   d   d   d   r   d   d   d   d   d   d   r d   d   d   d   r   d   r   d   r   d   r   d   d   r d   r   d   r  −   +   −   +   − +   +   −   −   −   −   + +   +   =  (9) 2   2   2   3   4   2   2   2   2 3   2   3   3   3   4   3   4   3   4 2   2   0 d   r   d   d   d   d   d d   d   d +   −   +   −   +   −   =   (10) 2   2   2   3   4   2   2   2   2 3   2   3   3   3   4   3   4   3   4 2   2   0 d   r   d   d   d   d   d d   d   d +   +   +   −   −   −   =   (11) Equation (9) is a second-degree polynomial in   d 4   2 . Solving this quadratics for   d 4   shows that (9) can be rewritten as 2   2   2   2 2 2   2   3   2   3   2 4   3   2 (   ) 1 2  d   r   d   r d   d   r   AB  ⎞ ⎛   +   −   + =   +   −   ⎟ ⎜ ⎝   ⎠  or 2   2   2   2 2 2   2   3   2   3   2 4   3   2 (   ) 1 2  d   r   d   r d   d   r   AB  ⎞ ⎛   +   −   + =   +   +   ⎟ ⎜ ⎝   ⎠  where   A   and   B   are defined in (3). The first branch is the separating surface   d 4   = C 1   between domains 1 and 2. Equation (10) is a second-degree polynomial   in   d 4 . By solving this quadratics for   d 4   and assuming strictly positive values for   d 4   and   r 2 , (10) can be rewritten as 3 4   3 3 (   and   1) 1  d d   B   d d  =   ⋅   > −   or   3 4   3 3 (   and   1) 1  d d   B   d d  =   ⋅   < −  where   B   is   defined   in   (3).   These   two   branches   are   the separating surfaces   d 4 = C 3   and   d 4 = C 4 , respectively. In the same way, (11) can be rewritten as, 3 4 3 1  d d   A d  =   ⋅ +  which is the separating surface   d 4   = C 2 . Thus, (7) and (8) found in [10] do not define separating surfaces, and only one branch of (9) defines a separating surface.  C.   Number of nodes  In this section, we investigate each domain according to the number of nodes in the workspace.  1)   Domain 1  Since all manipulators in this domain are binary, they cannot   have   any   node   in   their   workspace.   Thus,   all manipulators in domain 1 have the same workspace topology, namely, 0 node, 0 cusp and a hole inside their workspace. This workspace topology is referred to as   WT 1 .  2)   Domain 2  Figures 5 and 2 show two distinct workspace topologies of manipulators in domain 2, which feature 2 nodes and 0 node and which we call   WT 2   and   WT 3 , respectively. Transition between these two workspace topologies is one such that the two lateral segments of the internal boundary meet tangentially (Fig. 10).  Figure 10.   Transition between   WT 2   and   WT 3   . Equation of this transition can be derived geometrically and the following equation is found [15] 4 1 (   ) 2  d   A   B =   −   (12) where   A   and   B   are defined in (3). As noted in section   B , a third topology exists in this domain, where the internal boundary exhibits a ‘2-tail fish’. This workspace topology, which we call   WT 4 , features two nodes like in Fig. 5, but these nodes do not play the same role. They coincide with two isolated singular points, which are associated   with   the   two   singularity   lines   defined   by  θ 3 = ± arccos(- d 3 / d 4 ) (the operation point lies on the second joint axis   and   the   inverse   kinematics   admits   infinitely   many solutions). Also, the nodes do not bound a hole like in Fig. 5 but a region with four IKS (Fig. 10).  Figure 11.   Workspace topology   WT 4   . Transition between   WT 3   and   WT 4   is a workspace topology
such   that   the   upper   and   lower   segments   of   the   internal boundary meet tangentially (Fig. 12).  Figure 12.   Transition between   WT 3   and   WT 4   . As shown in [15], this transition is the occurrence of the additional singularity   d 3   + c 3 d 4   = 0, that is  d 4 = d 3   (13)  3)   Domains 3 and 5  The internal boundary has either 2 cusp (domain 3) or 0 cusp (domain 5). This boundary is either fully inside the external boundary (like in Figs 6 and 8), or it can cross the external boundary, yielding two nodes as in Fig. 3 and 13. Thus,   domain   3   (resp.   domain   5)   contains   two   distinct workspace topologies, which we call   WT 5   (1 node) and   WT 6 (resp.   WT 8   and   WT 9 ).  Figure 13.   Workspace topology   WT 9   . Transition between   WT 5   and   WT 6   and transition between  WT 8   and   WT 9   are such that the internal boundary meets the external boundary tangentially (Fig. 14).  Figure 14.   Transition between   WT 5   and   WT 6   (left) and between   WT 8   and   WT 9 right). This   transition   can   be   derived   geometrically   and   the following equation is found [15] 4 1 (   ) 2  d   A   B =   +   (14) where   A   and   B   are defined in (3).  4)   Domains 4  Manipulators in domain 4 have four cusps and four nodes. No subcase exist in this domain [15]. Such topologies are referred to as   WT 7 . IV.   RESULTS SYNTHESIS  A.   Parameter space partition  Taking into account the nodes in the classification results in a new partition of the parameter space, as shown in Fig. 15, where   E 1 ,   E 2   and   E 3   are the right hand side of (12), (13) and (14), respectively. Figure 15 depicts a section ( d 3 ,   d 4 ) of the parameter space for   r 2 =1.  7  Figure 15.   Parameter space partition according to the number of cusps and nodes (in a section   r 2   =1). Plots of the separating surfaces in sections for different values of   r 2   are shown in Fig. 16.  d 3  d 4   WT 3  WT 9  WT 1  WT 8   WT 2  WT 5  WT 6  WT 7  d 3  d 4   WT 3  WT 9  WT 1  WT 8   WT 2  WT 5  WT 6  WT 7  r 2 =0.3   r 2 =0.7  d 3  d 4  WT 4  WT 3  WT 9  WT 1  WT 8  WT 2  WT 5  WT 6  WT 7  d 3  d 4  WT 4  WT 3  WT 9  WT 1  WT 8  WT 2  WT 5  WT 6  WT 7  r 2 =1.5   r 2 =2 Figure 16.   Separating surfaces for different values of   r 2   . The areas associated with   WT 1 ,   WT 2 ,   WT 7   and   WT 9   decrease when   r 2   increases. The area associated with   WT 4   is very tiny, especially   for   small   values   of   r 2 .   This   means   that   few manipulators have a topology of the   WT 4   type.
B.   Classification tree  A   multi-level   classification   of   the   3R   orthogonal manipulators   under   study   can   be   established   by   the classification tree shown in Fig. 17. For more legibility, only the generic cases are reported on this tree (i.e. manipulators on the separating surfaces of the parameter space are not reported). The root of the tree is the set of all manipulators under study and each leave is the set of manipulators with a completely specified   workspace   topology.   The   first   level   of   the classification tree shows that a 3R orthogonal manipulator has either 2 aspects (if   d 3 > d 4 ), or it is quaternary and has no hole in its workspace (if   d 3 < d 4 ). The second level shows that ( i ) a 3R orthogonal manipulator with 2 aspects is either quaternary with 4 cusps (if   d 4 > C 1 ), or binary with no cusp, no node and a hole (if   d 4 < C 1 ) and ( ii ) a 3R orthogonal quaternary manipulator may have 4 cusps and 6 aspects (if   d 4 > C 3   or   d 4 < C 2 ), or 2 cusps and 5 aspects (if   C 2 < d 4 < C 3   and   d 4 < C 4 ), or 0 cusp and 4 aspects (if  d 4 > C 4 ). 3R orthogonal manipulator with r 3 =0  d 4   <   d 3  d 4   >   d 3 • 2 aspects •   0 cusp • 0 node • (1, 1, 0) • binary • 4 cusps • quaternary • 2 nodes • (1, 1, 2) • 0 node • (0, 1, 1)  d 4   <   C 1  d 4   <   E 1  d 4   >   E 1  d 4   >   C 1 • 2 cusps • 5 aspects • 0 cusp • 4 aspects • 4 cusps • 6 aspects • 2 nodes • (0, 1, 3) • 3 nodes • (0, 2, 2) • 1 node • (0, 1, 2) ( d 4   >   C 3   ) or ( d 4   <   C 2   ) (C 2   < d   4   < C   3 ) and (d   4   < C   4 )  d 4   >   C 3  d 4   <   C 2  d 4   <   E 3  d 4   >   E 3  d 4   >   C 4  d 4   <   E 3  d 4   >   E 3 • 2 nodes • (0, 2, 1) • 0 node • (0, 1, 1) • 0 hole • quaternary  WT 1  WT 2  WT 3  WT 4  WT 5  WT 7  WT 9  WT 6  WT 8  • 4 nodes • (0, 2, 3)  2   2   2   2   2 2   2   3   2   3   2 1   3   2 (   ) 1 2  d   r   d   r C   d   r   AB  ⎛   ⎞ +   −   + =   +   − ⎜   ⎟ ⎝   ⎠  2   2   2   2 3   2   3   2 (   1)   and   (   1) A   d   r   B   d   r =   +   +   =   −   +  1 1 (   ) 2  E   A   B =   −  2   3 E   d =  3 2 3 1  d C   A d  =   ⋅ +  3 1 (   ) 2  E   A   B =   +  3 3 3   1  d C   B d  =   ⋅ −  3 4 3 1  d C   B d  =   ⋅ −  (1, 1, 0) means • 1 hole • 1 region with 2 IKS • 0 region with 4 IKS  d 4  C 1  E 1  E 2  C 2  E 3  C 3 Fig. 4 Fig. 5 Fig. 2 Fig. 7 Fig. 3 Fig. 6 Fig. 11 Fig. 8   Fig. 13 3R orthogonal manipulator with r 3 =0  d 4   <   d 3  d 4   >   d 3 • 2 aspects •   0 cusp • 0 node • (1, 1, 0) • binary • 4 cusps • quaternary • 2 nodes • (1, 1, 2) • 0 node • (0, 1, 1)  d 4   <   C 1  d 4   <   E 1  d 4   >   E 1  d 4   >   C 1 • 2 cusps • 5 aspects • 0 cusp • 4 aspects • 4 cusps • 6 aspects • 2 nodes • (0, 1, 3) • 3 nodes • (0, 2, 2) • 1 node • (0, 1, 2) ( d 4   >   C 3   ) or ( d 4   <   C 2   ) (C 2   < d   4   < C   3 ) and (d   4   < C   4 )  d 4   >   C 3  d 4   <   C 2  d 4   <   E 3  d 4   >   E 3  d 4   >   C 4  d 4   <   E 3  d 4   >   E 3 • 2 nodes • (0, 2, 1) • 0 node • (0, 1, 1) • 0 hole • quaternary  WT 1  WT 2  WT 3  WT 4  WT 5  WT 7  WT 9  WT 6  WT 8  • 4 nodes • (0, 2, 3)  2   2   2   2   2 2   2   3   2   3   2 1   3   2 (   ) 1 2  d   r   d   r C   d   r   AB  ⎛   ⎞ +   −   + =   +   − ⎜   ⎟ ⎝   ⎠  2   2   2   2 3   2   3   2 (   1)   and   (   1) A   d   r   B   d   r =   +   +   =   −   +  1 1 (   ) 2  E   A   B =   −  2   3 E   d =  3 2 3 1  d C   A d  =   ⋅ +  3 1 (   ) 2  E   A   B =   +  3 3 3   1  d C   B d  =   ⋅ −  3 4 3 1  d C   B d  =   ⋅ −  (1, 1, 0) means • 1 hole • 1 region with 2 IKS • 0 region with 4 IKS 2   2   2   2   2 2   2   3   2   3   2 1   3   2 (   ) 1 2  d   r   d   r C   d   r   AB  ⎛   ⎞ +   −   + =   +   − ⎜   ⎟ ⎝   ⎠  2   2   2   2 3   2   3   2 (   1)   and   (   1) A   d   r   B   d   r =   +   +   =   −   +  1 1 (   ) 2  E   A   B =   −  2   3 E   d =  3 2 3 1  d C   A d  =   ⋅ +  3 1 (   ) 2  E   A   B =   +  3 3 3   1  d C   B d  =   ⋅ −  3 4 3 1  d C   B d  =   ⋅ −  (1, 1, 0) means • 1 hole • 1 region with 2 IKS • 0 region with 4 IKS  d 4  C 1  E 1  E 2  C 2  E 3  C 3  d 4  C 1  E 1  E 2  C 2  E 3  C 3 Fig. 4 Fig. 5 Fig. 2 Fig. 7 Fig. 3 Fig. 6 Fig. 11 Fig. 8   Fig. 13  Figure 17.   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