1  L'Institut de Recherche en Communications et Cybernétique de Nantes ( IRCCyN ), École Central de Nantes, 1 rue de la Noë, 44321 Nantes, Cedex 3, France. Emails: {Abdel-Kader.Zaiter, Philippe.Wenger, Damien.Chablat}@irccyn.ec-nantes.fr}  Résumé —   Dans une section de l'espace articulaire des ma- nipulateurs parallèles, les points cusps et noeuds ont un rôle essentiel dans le changement de mode d'assemblage. Ce travail présente une étude détaillée de ces points dans   l’ espace articu- laire et de travail. On démontre qu'un point cusp (resp. un noeud) définit un point de tangence (resp. un point d'intersec- tion), dans l'espace de travail, entre les courbes singulières et les courbes associées aux surfaces caractéristiques. L'étude est illustrée sur le manipulateur planaire 3- RPR   mais son champ d'application est général.  Abstract —   Cusps and nodes on plane sections of the singu- larity locus in the joint space of parallel manipulators play an important role in nonsingular assembly-mode changing mo- tions. This paper analyses in detail such points, both in the joint space and in the workspace. It is shown that a cusp (resp. a node) defines a point of tangency (resp. a crossing point) in the workspace between the singular curves and the curves associated   with   the   so-called   characteristics   surfaces.   The study is conducted on a planar 3- RPR   manipulator for illustr- ative purposes. Key-words:   parallel manipulator/ assembling mode/ characteristic surfaces/ singularity/ cusp.  1.   INTRODUCTION  Most parallel manipulators have singularities that limit the motion of the moving platform. The most dangerous ones are the singularities associated with the direct kinematics, where two direct kinematic solutions (DKSs) or assembly modes (AM) coalesce. Indeed, approaching such a singular- ity results in large actuator torques or forces, and in a loss of stiffness. Planar parallel manipulators have received a lot of attention [1-4, 6, 7, 9, 10, 15-16] because of their relative simplicity with respect to their spatial counterparts. Moreo- ver, studying the former may help understand the latter. Planar manipulators with three extensible leg rods, referred to as 3- RPR   manipulators, have often been studied. Such manipulators may have up to six assembly modes (AM) [1]. The direct kinematics can be written in a polynomial of degree six. Moreover, the singularities coincide with the set of   configurations   where   two   direct   kinematic   solutions coincide. It was first pointed out that to move from one assembly mode to another, the manipulator should cross a singularity [2]. However, [3] showed, using numerical ex- periments, that this statement is not true in general. More precisely, this statement is only true under some special geometric conditions, such as similar base and mobile plat- forms [4]. Relying on geometric arguments, [5] conjectured that the workspace of 3- RPR   parallel manipulator is divided into two singularity-free regions called aspect and that there should be 3 solutions in each aspect. Recently, [6] provided a mathematical proof of the decomposition of the work- space into two aspects using geometric properties of the singularity surfaces. Also, non-singular AM changing mo- tions were described in a 3-D representation. McAree and Daniel [4] pointed out that a 3 -RPR   planar parallel manipu- lator can execute a non-singular change of assembly-mode if a point with triple direct kinematic solutions exists in the joint space. The authors established a condition for three direct kinematic solutions to coincide and showed that the encirclement of a cusp point is a sufficient condition for a non-singular AM change. This condition was exploited by [7], where authors provided an algorithm to detect the cusp points of a 3- RPR   parallel manipulator in a section of the joint space, which corresponds to an input joint variable set to a constant value. Wenger and Chablat [8] investigated the question of whether a change of assembly-mode must occur or not when moving between two prescribed poses in the workspace. They defined the uniqueness domains in the workspace as the maximal regions associated with a unique assembly-mode and proposed a calculation scheme for 3 - RPR   planar   parallel   manipulators   using   octrees.   They showed that up to three uniqueness domains exist in each singularity-free region. When the starting and goal poses are in the same singularity-free region but in two distinct uniqueness domains, a non-singular change of assembly- mode is necessary. However they did not investigate the kind of motion that arises when executing a non-singular change of assembly-mode. [9] analyzed the variation of the topology of singularity curves and the distribution of the cusp points from one section of the joint space to another. By calculating the image in the workspace of a trajectory Abdel Kader Zaiter , Philippe Wenger   and Damien Chablat  A study of the sing u larity locus in the joint   space of planar parallel  manipul a tors: special focus on cusps and nodes
2 that encircles a cusp point, [10] described all corresponding trajectories. Also, the authors introduced the notion of re- duced configuration space for an aspect and used this no- tion   to   show   which   cusp   points   must   be   encircled   to achieve a non singular AM change in the associated aspect. [11] showed the singularity surface in the 3D workspace for the 3-RPR parallel manipulator, with the six solutions cor- responding to the same point in the joint space. Moreover, they studied   the   coincidence   of DKSs on the   singular curves in the reduced configuration space. Another type of AM change was reported by [12], which corresponds to the encirclement of an   α -curve in the joint space. Moreover, the authors showed the possibility to produce an AM change when approaching a singularity for micro mechanisms that present relatively large joint clearances. Later, [13] related the encirclement of a   α -curve to the encirclement of a loop characterized by a node, which corresponds to the simulta- neous coalescence of two couples of DKSs. [14] provided a mathematical condition of the existence of cusps and nodes. [15] provided a tool to calculate the practical workspace and, therefore, the reduced configuration space, keeping one input variable constant. This work was extended in [16] with more explanations and more examples. In [17], the authors showed, using numerical experiments, that not any cusp point may be encircled to perform a nonsingular AM changing motion. However, their work did not make it possible to identify definite rules. In this work, a detailed analysis is provided to explain the role of the singular curves in the joint space and in the workspace as pertained to the loss of solutions. More atten- tion is drawn to the study of cusps and nodes. It is shown that a cusp (resp. a node) defines a point of tangency (resp. a crossing point) in the workspace between the singular curves and the curves associated with the so-called charac- teristics surfaces. This study is illustrated with a   3-RPR  planar parallel manipulator. It is a first step towards a ri- gorous and complete analysis of nonsingular changing mo- tions in parallel manipulators.  2.   ILLUSTRATIVE MANIPULATOR  A   3-RPR   planar parallel manipulator is used in this paper to illustrate the analysis. As showed in figure 1, this manipula- tor has triangular base and platform denoted 3 2 1   A A A   and 3 2 1   B B B  , respectively.  Figure 1. The 3-RPR planar parallel manipulator.  The base and the platform are related by three similar legs. Each leg consists of three joints; the first and third ones are passive revolute joints while the second one is an actuated prismatic joint. i     , (i=1, 2 and 3) represents the passive variable that corresponds to the angles between the leg axis and the x-axis. The input variables are defined by i     (i=1, 2 and 3); the lengths of legs. The output variables are usual- ly defined by the Cartesian coordinates (   ,   ) x   y   of 1 B   and   ; the orientation of the platform in the plane with respect to the x-axis. In this paper, the position coordinates will be more conveniently defined by ( 1     , 1     ) i.e., the cylindrical coordinates of 1 B   . With these parameters, indeed, it is possible   to   simplify   the   problem:   we   consider   2- dimensional slices of the joint space and of the workspace by fixing the joint parameter 1     . The projection of a ( 1     ,   )-slice onto the ( 2     , 3     ) plane is consistent with the pro- jection of the configuration space onto the joint space [4]. The geometric parameters of the illustrative manipulator are the same as those used in [3, 10, and 8], namely: 04 . 17 1    d  , 54 . 16 2    d   , 84 . 20 3    d   , ) 0 , 0 ( 1    A   , ) 0 , 91 . 15 ( 2    A  and ) 10 , 0 ( 3    A   .  3.   S INGULARITIES  3.1   Singularities and aspects  The   3-RPR   parallel manipulator is in a singular configura- tion whenever the axes of its three legs are concurrent (the platform orientation gets out of control) or parallel (the translation along a direction orthogonal to the two parallel legs is out of control). 1   1  A  x y 2  A 3  A 2   3   3   1  d  1   2   2  d 3  d 2  B 1  B 3  B
3 The singular configurations of the   3-RPR   planar parallel manipulator can be calculated without any difficulty. They define surfaces both in the workspace and in the joint space. The singular surfaces divide the workspace into two singularity-free domains called aspects [6, 5], referred to as 1 WA  and 2 WA   . When mapped into the joint space, these two aspects coincide and thus define two coincident sets. Since we are considering slices by fixing 1     , the singulari- ty locus can be depicted as curves in the workspace (resp. the joint space) in the ( 1     ,   )-plane (resp. in the ( 2     , 3     )- plane). At a point on a singularity curve in the workspace, the manipulator is necessarily in a singular configuration. At a point on a singularity curve that bounds the joint space, the manipulator is in a singular configuration. On an internal singularity curve, the point admits several non- singular DKSs in addition to the singular solution. Near a  singular point, there are two “mirrored” DKSs in the wor k- space on each part of the singular curve [3]. When a singu- lar point of a singular curve is met in the joint space, the two mirrored DKSs coincide and one DKS is then lost in each aspect (see [3, 4] for example). Plots of the singularity curves for 1     =17 are shown in fig- ure 2. A   3-RPR   planar parallel manipulator has up to 6 as- sembly modes [1]. For a given point in the joint space that is not on a singular curve, there are 2, 4 or 6 DKSs in the workspace. The number of DKSs in each region of the joint space is indicated in figure 5.  3.2   Cusps and nodes  Cusps and nodes appear on the singular curves as a conse- quence of the projection of the manipulator configuration space folds onto its joint space [4]. As described by Whit- ney [18 ], a cusp arises when a fold is “folded”.   Figure 3 shows the local model of a cusp where the singular locus is shown in red lines. Three DKSs coincide at a cusp point. A node appears on the singular curves in the joint space at a crossing point. A node arises from the projection of two distinct folds. There are two pairs of coincident DKSs at a node.  3.3   Characteristic surfaces, basic regions - basic components  The notion of   characteristic surfaces   was introduced in [5]. Its definition is recalled thereafter. Let i WA    define the boundary of aspect i WA   .   The charac- teristic surfaces of i WA   , denoted by ) (   i c   WA S   , are defined as follows: i i i c   WA WA g g WA S           )) ( ( ) (   1  (1) where: g  maps all points of a given set of the workspace into the joint space through the manipulator inverse kinematics. 1   g  maps all points of a given set of the joint space in the workspace through the manipulator direct kinematics. Note that since we are considering slices by fixing 1     , the characteristic surfaces are defined by curves in this paper. ) (   i c   WA S  divides i WA   into different   basic regions ik WAb   , ) (   i c i ik K k   WA S WA WAb        (2) where K   is the set indexing the basic regions. The   basic components ik QAb   are defined as the images of ik WAb  in the joint space: ) (   ik ik   WAb g QAb     (3) Since the   3-RPR   planar parallel manipulator has 2 aspects and up to 3 DKSs in each aspect, as many as 3 distinct points in each aspect map onto one unique point in the joint space. Thus, as many as 3 of the basic regions in the work- space map onto 3 coincident basic components in the joint- space. In figure 2a, the basic components are the domains bounded by parts of the singularity curves. Some of the basic regions are shown in figure 6. The following notation will be used throughout the paper. - k D   : direct kinematic solution number k. Let us label 1, 2, 3 the solutions in aspect 1 WA   and 4, 5, 6 those in 2 WA   . - k C   : curve segment on the singularity curves in the joint space where solution k D   is lost when crossing k C   . Note that since one solution is lost in each of the two aspects, there will be always a mirrored solution n D   in the second aspect that is lost simultaneously. Before disappearing, n D  and k D   coincide on k C   . - m k CP      : the cusp point that enables a nonsingular assem- bly-mode change between k D   and m D   . - m k N      : node defined at the crossing point between k C   and m C  . - ) ( i Sc   m k   : the nonsingular curve image of k C   in aspect i WA  at the boundary of the basic region im WAb   . - k S   : singular curve image of k C   in the workspace.
4 Figure 2a shows the curve segments k C   for our   3-RPR  planar parallel manipulator when 1     =17.  4.   S OLUTION LOSS ON THE SINGULARITY CURVES  Let take a first point initial P   in a domain associated with 6 DKSs, 3 in each aspect. There are 3 coincident basic com- ponents associated with each aspect in this domain. Let take a second point final P   in a neighboring domain with 4 DKSs. Point initial P   (resp. final P   ) is defined by 1     =17, 2    =15   and 3     =15   (resp. 1     =17, 2     =13.25   and 3    =20.39). Figure 2b shows why solutions 1 D   and 5 D   are lost when crossing 1 C   = 5 C   when going from initial P   to final P  . We can determine the solution loss on all the other curve segments in a similar manner.  Figure 2. Loss of solutions when crossing singular curves in the joint space  5.   C ORRESPONDENCE OF CUSPS AND NODES IN THE WORKSPACE  In this section, it is shown that one image of a cusp point, which corresponds to the coalescence of three DKSs in the workspace, defines a tangency point between the curves associated with the characteristic surfaces and the singular curves in the workspace. Also, the other images will cor- respond to cusp points formed by the characteristic surfac- es. For a node, two images will correspond to a crossing point between the curves associated with the characteristic surfaces and the singular curves in the workspace, where these two images correspond to the coalescence of two couples of DKSs. The other images of a node will corres- pond to cross points between curves associated to characte- ristic surfaces. For more simplicity, we consider a manipu- lator with two aspects, like the illustrative manipulator shown in figure 1.  5.1   Case of a cusp point  As shown in figure 4, a cusp point m k CP      appears where curve segments k C   and m C   meet. Both are parts of the boundary of coincident basic components. Since m k CP     allows a non singular AM change between solutions k D  and m D   , these two solutions lie necessarily in the same aspect [4], say 1 WA   . In the workspace, k C   defines one singular curve   segment k S   and one nonsingular curve segment ) 1 (  m k Sc   in aspect 1 WA   . ) 1 (  m k Sc   lies on the cha- racteristic surface ) (   1 WA S   c   and defines a part of the boun- dary of the basic region m WAb   1   . k S   lies on the singular curve, on which k D   is lost, and defines a part of the singu- lar boundary of the basic region k WAb   1   . Similarly, m C  gives one non-singular curve segment ) 1 (  k m Sc   defining a part of the boundary of the basic region k WAb   1   in aspect 1 WA  and one singular curve segment m S   defining a part of the singular boundary of the basic region m WAb   1   . As (a) 1  C  = 5  C  P - initial  P - fin al 1  C  = 4  C 3  C  = 4  C 3  C  = 6  C 2  C  = 4  C 2  C  = 6  C 1  C  = 6  C initial  D    3 3  C  = 5  C final  D    3 final  D    6 final  D    2 final  D    4 initial  D    5 initial  D    6 initial  D    1 initial  D    4 initial  D    2 k  C  a n d m  C  (b)  Coalescence of sol u tions 1  D  and 5  D   on   the   si n-  gular curve
5 shown in figure 4, another solution n D   , that belongs to a second aspect 2 WA   , disappears at curves k C   = n C   and m C  = n C   . Also, the singular curve segments k S   = n S   and m S  = n S   , defined by k C   and m C   , respectively, define two parts of the singular boundary of n WAb   2   , where n WAb   2  lies in the second aspect 2 WA   . Since m k CP      belongs to both k C   and m C   , its images in the workspace belong to ) 1 (  m k Sc  , ) 1 (  k m Sc   , k S   and m S   . At m k CP      , solutions k D   , m D  and n D   coalesce in the workspace, therefore this im- age of m k CP      is the point where the three basic regions k WAb   1  , m WAb   1   and n WAb   2   meet in the workspace. Since m k CP     has one image k D   in k WAb   1   having ) 1 (  k m Sc   and k S  as parts of its boundary, where m k CP      has also an im- age on ) 1 (  k m Sc   and k S   , this image corresponds to the intersection point between ) 1 (  k m Sc   and k S   . By the same way, we can conclude that solution m D   of m k CP      corres- ponds to the intersection point between ) 1 (  m k Sc   and m S   at the boundary of m WAb   1   , and n D   corresponds to the inter- section point between k S   and m S   at the boundary of n WAb   2  . Consequently, the image of m k CP      that corres- ponds to the coalescence of k D   , m D   and n D   is the inter- section point between ) 1 (  m k Sc   , ) 1 (  k m Sc   , k S   and m S   (see figure 4). From Whitney [18] and Corvez [19], the local model of a cusp always defines a point of tangency between the   singular   curve f C   associated   with   f   and   the   set f f   C C f f   \ )) ( (  1   , as illustrated in figure 3. In our exam- ple, the sets f C   and f f   C C f f   \ )) ( (  1    correspond to the curves m k   S S      and ) 1 ( ) 1 (   k m m k   Sc Sc      ,   respectively. Thus ) 1 ( ) 1 (   k m m k   Sc Sc      is tangent to m k   S S      at the im- age of m k CP      in the workspace.  Figure 3. Tangency at a cusp point (from [19]) .  Figure 4. The correspondence of the three coincident solutions   of   CP k-m .  5.2   Example on the cusp point  The 3-RPR planar parallel manipulator described in section 2 is chosen as illustrative example. We will explain the correspondence, in the workspace, of cusp point 6 4    CP  shown in figure 5. This cusp point is formed by the parts of curves 4 C   and 6 C   colored in black and light blue, respec- tively. Near 6 4    CP   , each of these curves separates two do- mains in the joint space. The first domain corresponds to the   coalescence   of   3   basic   components 1 1  Q A b  = 1 2  Q A b   = 1 3  Q A b   for   aspect   1   plus   3 24 QAb  = 25 QAb   = 26 QAb   for aspect 2. In the second do- main, 2 basic components coalesce (the number of DKSs is shown in figure 5). The 6 associated basic regions, which correspond to the first region, 11 WAb   , 12 WAb   and 13 WAb  in the first aspect and 24 WAb   , 25 WAb   and 26 WAb   in the second aspect, are shown in figure 6 and colored in dark green and orange, respectively. Any point on 4 C   and 6 C  yields 5 different solutions, one of which is a double solu- tion that lies on the DKP singular curves. Therefore, the image of each one of the two curves yields 5 curve images in the workspace, where one curve image is a part of the singular curves and the four remaining images are parts of the characteristic surfaces. In figure 6, the curve images of curves 4 C   and 6 C   are represented by colors black and light blue, respectively. Each curve gives the expected number and types of curve images. -Image at the boundary of m WAb   1   : Intersection point between ) 1 (  m k Sc   and m  S   . -Image at the boundary of k WAb   1   : Intersection point between ) 1 (  k m Sc   and k  S   .   The coalescence of these three images is the inter- section   point   between ) 1 (  m k Sc  , ) 1 (  k m Sc   , k  S   and m  S  . -Image at the boundary of n WAb   2   : Intersection point between k  S   and m  S   . The   three coincident images   of m k  CP     . m k  CP    k  C  = n  C m  C  = n  C
6  Figure 5. Cusp point   CP 4-6   and node   N 1-2   in the joint space. Number of DKSs in each region is indicated (orange disks)  Curve 4 C   yields the curve image ) 2 (  6 4 Sc   at the boundary of 26 WAb   and 4 S   = 3 S   at the common singular boundary of 24 WAb   and 13 WAb   . Also, curve 6 C   yields the curve image ) 2 (  4 6 Sc   at the boundary of 24 WAb   and 6 S   = 3 S   at the common singular boundary of 26 WAb   and 13 WAb   . Fig- ure 6 shows that these four curve images have one intersec- tion point that represents the image of 6 4    CP   correspond- ing to the coalescence of three DKSs in the workspace. As expected, this image is a tangent point between the DKP singular curves and the characteristic surfaces. It is known that, in the reduced configuration space, the different   DKSs   corresponding   to   same   input   variables represent the intersection points of a vertical line with the configurations space [10, 11]. For a cusp point, the vertical line is tangent on the singular curve folded in a specific way, where three DKSs coalesce. The projection of this fold on the joint space yields the cusp point, also the others images of the cusp points will be projections of this fold on the configuration space, where these images correspond to a cusp point formed by the characteristic surfaces. Figure 6 shows that the other images of 6 4    CP   correspond to 3 in- tersection points between the other images of curves 4 C  and 6 C   , marked by (*), (**) and (***). As we have 6 cusp points in the section of the joint space represented in figure 5 (the red dots), figure 6 shows 6 tangent points (large red dots) between the DKP singular curves and the curves as- sociated with the characteristic surfaces. Also, the studied cusp 6 4    CP   has 4 distinct DKSs and the other 5 cusps have only 2 distinct DKSs. There are formed by singular curves that separate two regions in the joint space, where one re- gion has 4 DKSs and the other has 2 DKSs. Figure 6 show 3+5=8 cusp points formed by the characteristic surfaces in the workspace (small red dots).  Figure 6. The images of   CP 4-6   . Red points show the images of the all cusp points. The largest points are those associated with a tangency point.  5.3   Case of a node  In the joint space, a point where two singular curves cross is called a node. Such a point corresponds to the simultane- ous coalescence of two couples of DKSs. For the illustra- tive manipulator, the image in the workspace of a node yields two couples of coincident DKSs lying on the singu- lar curves, plus, possibly, two nonsingular DKSs. Assume that curve segments h C   = m C   and k C   = n C   inter- sect in the joint space at node k h N      . Since a node comes from the projection of two distinct folds of the configura- tion space, generically the distribution of DKSs in the four domains when turning around k h N      is   n ,   n -2,   n -4 and   n -2, where   n   is 4 or 6 for the manipulator under study (Figure 7). Let h WAb   1   , k WAb   1   , m WAb   2   and n WAb   2   be the basic regions associated with the four coincident basic compo- nents in h QAb   1   , k QAb   1   , m QAb   2   and n QAb   2   in the do- main with   n      4 solutions. Note that, h WAb   1   and k WAb   1  (resp. m WAb   2   and n WAb   2   ) belong to aspect 1 WA   (resp. to aspect 2 WA   ). ) 2 (  6 4  Sc 4  S 13  WAb  The correspondence  of 6 4     CP  Singular  curves  Characteristics  surfaces 25  WAb 11  WAb 12  WAb 24  WAb 26  WAb 6  S ) 2 (  4 6  Sc 6 4     CP 2 1   N  2 k  C  a n d m  C  6 k  C  a n d m  C  4 k  C  a n d m  C  4 k  C  a n d m  C  2 k  C  a n d m  C  4 k  C  a n d m  C  2 k  C  a n d m  C 4  C 4  C 6  C  (**)  (***)  (*) 1  C 2  C
7  Figure 7. The correspondence of the two couples of coincident DKSs of   N h-k   in the workspace.  Among the images of curve h C   = m C   in the workspace, we have h S   = m S   that defines the singular boundary between basic   regions h WAb   1   and m WAb   2   .   Also, h C   yields ) 1 (  k h Sc  and ) 2 (  n h Sc   at the boundaries of k WAb   1   and n WAb   2  , respectively. Similarly, curve k C   = n C   has one image k S   = n S   at the singular boundary between k WAb   1  and n WAb   2   , and ) 1 (  h k Sc   and ) 2 (  m k Sc   at the boundaries of h WAb   1   and m WAb   2   , respectively. Since node k h N     belongs to curves h C   and k C   , k h N      has images on all the images of these two curves, especially on h S   , ) 1 (  k h Sc   , ) 2 (  n h Sc  , k S   , ) 1 (  h k Sc   and ) 2 (  m k Sc   . On the other hand, as k h N      has one image in h WAb   1   where h S   and ) 1 (  h k Sc  form parts of its boundary, this image is the intersection point between h S   and ) 1 (  h k Sc   . By the same way, we can demonstrate that the image of k h N      at the boundary of m WAb   2  is   the   intersection   point   between m S   and ) 2 (  m k Sc  . Since these two images of k h N      coalesce at h S  = m S   , this coalescence represents the intersection point between h S   = m S   , ) 1 (  h k Sc   and ) 2 (  m k Sc   (see figure 7). As ) 1 (  h k Sc  and ) 2 (  m k Sc   lie in two different aspects and represent a part of the curve associated with the characteris- tic surface, also h S   lies on the singular curves in the work- space, so the first correspondence of a node, representing the coalescence of the first couple of DKSs ( h D   , m D   ), is a crossing point between the characteristic surfaces and the singular   curves.   Similarly,   the   second   image   of k h N     representing the coalescence of k D   and n D   at k C   = n C   is the   intersection   point   between k S   = n S   , ) 1 (  k h Sc   and ) 2 (  n h Sc  (see figure 7). Therefore, the second correspon- dence of a node is also a crossing point between the curves associated with the characteristic surfaces and the singular curves.  5.4   Example on the node  Node 2 1  N   in figure 5 is formed by curves 1 C   and 2 C   . Each one of 1 C   and 2 C   yields 5 curve images in the work- space, which are colored in green and red, respectively, in figure 8. The curve images colored in blue and light brown are   the   images   of   the   parts   of   curves 1 C   and 2 C   , represented by the same colors in figure 5. 1 C   = 5 C   yields 1 S  = 5 S   at the singular boundary between basic regions 11 WAb  and 25 WAb   . Also, 1 C   yields ) 1 (  2 1 Sc   and ) 2 (  6 1 Sc  at the boundaries of 12 WAb   and 26 WAb   , respectively. Si- milarly, curve 2 C   = 6 C   has one image 2 S   = 6 S   at the singu- lar boundary between 12 WAb   and 26 WAb   , and ) 1 (  1 2 Sc   and ) 2 (  5 2 Sc  at the boundaries of 11 WAb   and 25 WAb   , respec- tively.  Figure 8. Images of   N 1-2   in the workspace.  - I mage at the boundary of n  WAb   2   : Inte r section  point between k n   S S      and ) 2 (  n h  Sc   .  - Image at the boundary of k  WAb   1   : Inte r section  point between k  S   and ) 1 (  k h  Sc   .  - Image at the boundary of m  WAb   2   : Inte r section  point between h m   S S      and ) 2 (  m k  Sc   . k  C  = n  C  The   coalescence   of  these   two   images   is  the   intersection   point  of h  S   , ) 1 (  h k  Sc   and ) 2 (  m k  Sc  .  The   coalescence   of  these   two images   is the  intersection   point   of k  S  , ) 1 (  k h  Sc   and ) 2 (  n h  Sc  .  - Image at the boundary of h  WAb   1   : Inte r section  point between h  S   and ) 1 (  h k  Sc   .  DKP k h  N     n - 4 sol u tions n - 2 sol u tions  n sol u tions   n - 2 sol u tions h  C  = m  C  (**)  0 ) 1 (  1 2  Sc 13  WAb  The two cross points corresponding to 2 1   N 25  WAb 11  WAb 12  WAb 6 24  WAb 26  WAb 1  S 2  S ) 2 (  5 2  Sc ) 2 (  6 1  Sc ) 1 (  2 1  Sc  (*)  0
8 Figure 8 shows the two cross points corresponding to 2 1  N   ; the first one is the intersection point between 1 S   , ) 1 (  1 2 Sc  and ) 2 (  5 2 Sc   , where the second one is the intersection point between 2 S   , ) 1 (  2 1 Sc   and ) 2 (  6 1 Sc   . Note that 2 1  N   has two other image points, which define crossing points between the curves associated with charac- teristic surfaces. These images are represented by (*) and (**) in figure 8. Due to space limitation, the demonstration is not presented in this paper. We have 6 double points in the joint space (figure 5), three of them have 4 distinct images and the three remaining ones have 2 distinct images. Figure 8 shows 12 intersection points (the large blue dots) between the singular curves and the curves associated with the characteristic surfaces, and 6 intersection points (the small blue dots) between the curves associated with the characteristic surfaces.  6.   C ONCLUSION  A detailed analysis of singularities in the joint space and in the workspace was presented in this paper. It was shown that the image of a cusp point corresponding to the coales- cence of three DKSs, and the image of a double point cor- responding to the coalescence of two couples of the DKSs are, respectively, a tangent point and a crossing point be- tween a singular curve and a curve associated with the cha- racteristic surfaces. The demonstration was based on a comprehensive description of the images of the singular curves reflecting the coalescence of the DKSs on it. This preliminary work will help study in detail non-singular as- sembly-mode changing motions in parallel manipulators.  R EFERENCES  [1]   Hunt K. H.,   “ Structural Kinematics of In Parallel-Actuated Robot- Arms, ”   Journal of Mechanisms, Transmissions and Automation in Design, Vol. 105, pp. 705-712, December 1983. [2]   Hunt K. H, Primrose E. J. F.,   “ Assembly Configurations of some In- Parallel-Actuated Manipulators, ”   Mechanism and Machine Theory, Vol. 28, N° 1, pp. 31-42, 1993. [3]   Innocenti C., Parenti-Castelli V.,   “ Singularity-free evolution from one configuration to another in serial and fully-parallel manipula- tors ” , ASME Robot, Spatial Mechanisms and Mechanical Systems, vol. 45, pp. 553-560, 1992. [4]   P.R. Mcaree, R.W. Daniel,   “ An explanation of never-special assem- bly changing motions for 3 – 3 parallel manipulators ” , The Interna- tional Journal of Robotics Research 18 (6) (1999) 556 – 574. [5]   Wenger Ph., Chablat D.,   “ Definition Sets for the Direct Kinematics of Parallel Manipulators ” , th  8   International Conference Advanced Robotics, pp. 859-864, 1997. [6]   L.   Husty M.,   “ Non-singular assembly mode change in   3-RPR- parallel manipulators ” . In Proceedings of the 5 th   International Work- shop on computational Kinematics, pp. 43-50, Kecskeméthy, A., Müller, A. (eds.), 16 April 2009. [7]   M. Zein, P. Wenger, D. Chablat,   “ An algorithm for computing cusp points in the joint space of 3-RPR parallel manipulators ,”   in: Pro- ceedings of EuCoMeS, First European Conference on Mechanism Science, Obergurgl Austria, February 2006. [8]   Wenger Ph., Chablat D.,   “ Workspace and Assembly modes in Fully- Parallel Manipulators: A Descriptive Study ,”   Advances in Robot Ki- nematics: Analysis and Control, Kluwer Academic Publishers, pp. 117-126, 1998. [9]   M. Zein, P. Wenger, D. Chablat,   “ Singular curves and cusp points in the joint space of 3-RPR parallel manipulators ,”   in: Proceedings of IEEE International Conference on Robotics and Automation, Orlan- do, May 2006. [10]   M . Zein, P. Wenger, D. Chablat, “ Non-singular assembly-mode changing motions for 3-RPR parallel manipulators ,”   Mechanism and Machine Theory 43(4), 480 – 490, Jun 2007. [11]   E. Macho, O. Altuzarra, C. Pinto, and A. Hernandez, “Singular ity free change of assembly mode in parallel manipulators: Application  to the 3RPR planar platform,” presented at th e 12th IFToMM World Congr., Besancon, France, Jun. 18 – 21, 2007, Paper 801. [12]   Bamberger, H., Wolf, A., and Shoham, M.,   “ Assembly mode chang- ing in parallel Mechanisms ,”   IEEE Transactions on Robotics, 24(4), pp. 765-772, 2008. [13]   E. Macho, E. Altuzarra, O., Pinto, C., and Hernandez, A,   “ Transi- tions between multiple solutions of the Direct Kinematic Problem ,”  11th International Symposium Advances in Robot Kinematics, June 22-26, 2008, France. [14]   A. Hernandez, O. Altuzarra, V. Petuya, and E. Macho.,   “ Defining conditions for non singular transitions of assembly mode, ”   IEEE Transactions on robotics, 25(6):1438-1447, 2008. [15]   M. Urizar, V. Petuya, O. Altuzarra and A. Hernandez,   “ Computing the Configuration Space for Motion Planning between Assembly Modes, ”   Computational Kinematics Proceedings of the 5th Interna- tional Workshop on Computational Kinematics, Thursday 6 October 2009. [16]   M. Urizar, V. Petuya, O. Altuzarra, E. Macho and A. Hernandez,  “ Analysis of the Direct Kinematic Problem in 3-DOF Parallel Mani- pulators, ”   SYROM 2009 Proceedings of the 10th IFToMM Interna- tional Symposium on Science of Mechanisms and Machines, held in Brasov, Romania, October 12-15, 2009. [17]   M. Urizar, V. Petuya, O. Altuzarra and A. Hernandez,   “ Researching into nonsingular transitions in the joint space, ”   Advances in Robot Kinematics: Motion in Man and Machine, Piran Portoz, Slovenia, Springer 45-52. Monday, June 28, 2010. [18]   H. Whitney.   “ On singularities of mappings of euclidean spaces. I. Mapping of the plane into the plane, ”   Ann. Of Math. (2), 62: 374- 410, 1955. [19]   S. Corvez,   “ Études de systèmes polynomiaux: contributions à la classification d'une famille de manipulateurs et au calcul des inter- sections de courbess A-splines ,”   PhD Thesis, Rennes, École docto- rale MATISSE U.F.R. de Mathématique, Avril 2005.