Uniqueness Domains in the Workspace of Parallel Manipulators  Philippe WENGER, Damien CHABLAT Laboratoire d’Automatique de Nantes URA CNRS 823 (École Centrale de Nantes / Université de Nantes) 1, rue de la Noë, BP 92101, 44321 Nantes Cedex 3 France email : Philippe.Wenger@lan.ec-nantes.fr  ABSTRACT  This   work   investigates   new   kinematic   features   of parallel manipulators.   It is well known   that parallel manipulators admit generally several direct kinematic solutions for a given set of input joint values. The aim of this paper is to characterize the uniqueness domains in the workspace of parallel manipulators, as well as their image in the joint space. The study focuses on the most usual case of parallel manipulators with only one inverse kinematic solution. The notion of   aspect   introduced for serial manipulators in   [Borrel 86]   is redefined for such parallel manipulators. Then, it is shown that it is possible to link several solutions to the forward kinematic problem without meeting a singularity, thus meaning that the aspects are not uniqueness domains. An additional set of surfaces, namely the   characteristic surfaces , are characterized which divide the workspace into   basic regions   and yield new uniqueness domains. This study is illustrated all along the paper with a 3-RPR planar parallel manipulator. An octree model of spaces is used to compute the joint space, the workspace and all other newly defined sets.  Key-words   : Parallel Manipulator, Workspace, Singularity, Aspect, Uniqueness domains, Octree.  1.   INTRODUCTION  A well known feature of parallel manipulators is the existence of multiple solutions to the direct kinematic problem.   That   is,   the   mobile   platform   can   admit several positions and orientations (or configurations) in the workspace for one given set of input joint values  [Merlet   90] .   The   dual   problem   arises   in   serial manipulators,   where   several   input   joint   values correspond to one given configuration of the end- effector. To cope with the existence of multiple inverse kinematic solutions in serial manipulators, the notion of   aspects   was introduced   [Borrel 86] . The aspects were defined as the maximal singularity-free domains in   the   joint   space.   For   usual   industrial   serial manipulators,   the   aspects   were   found   to   be   the maximal sets in the joint space where there is only one inverse   kinematic   solution.   Many   other   serial manipulators,   referred   to   as   cuspidal   manipulators, were shown to be able to change solution without passing through a singularity, thus meaning that there is more than one inverse kinematic solution in one aspect.   New   uniqueness   domains   have   been characterized for cuspidal manipulators   [Wenger 92] ,  [El Omri 96] . It is also of interest to be able to characterize   the   uniqueness   domains   for   parallel manipulators, in order to separate and to identify, in the workspace, the different solutions to the direct kinematic problem. To the authors knowledge, the only   work   concerned   with   this   issue   is   that   of  [Chételat 96] , which proposes a generalization of the implicit   function   theorem.   Unfortunately,   the hypothesis of convexity required by this new theorem is   still   too   restrictive.   This   paper   is   organized   as follows. Section 2 describes the planar 3-RPR parallel manipulator which will be used all along this paper to illustrate the new theoretical results. Section 3 restates the notion of   aspect   for parallel manipulators. New surfaces, the   characteristic surfaces , are defined in section 4, which, together with the singular surfaces, further divide the aspects into smaller regions, called  basic regions . Finally, the uniqueness domains are defined in section 5. The workspace, the aspects, the characteristic and singular surfaces, and the uniqueness domains are calculated for the planar 3-RPR parallel manipulator using octrees. The images in the joint space of the uniqueness domains are also calculated. It is shown that the joint space is composed of several subspaces with different numbers of direct kinematic solutions.  2.   PRELIMINARIES 2.1   P   ARALLEL MANIPULATOR STUDIED  This   work   deals   with   those   parallel   manipulators which have only one inverse kinematic solution.   In addition, the passive joints will be always assumed unlimited in this study. For more legibility, a planar manipulator will be used as illustrative example all along this paper. This is a planar 3-DOF manipulator, with 3 parallel RPR chains (Figure 1). The input joint variables are the three prismatic actuated joints. The output variables are the positions and orientation of the platform   in   the   plane.   This   manipulator   has   been frequently   studied,   in   particular   by   [Merlet   90] ,  [Gosselin 91]   and   [Innocenti 92] . The   kinematic   equations   of   this   manipulator   are  [Gosselin 91]   :  ρ 1 2   2   2  =   + x   y   (1)  (   ) (   )   (   ) (   ) ρ   φ   φ 2 2 2   2 2 2 2  =   +   −   +   + x   l   c   y   l cos   sin   (2)
(   ) (   )   (   ) (   ) ρ   φ   θ   φ   θ 3 2 3   3 2 3   3 2  =   +   +   −   +   +   +   − x   l   c   y   l   d cos   sin  (3)  Figure 1 : 3-RPR planar manipulator  The dimensions of the platform are the same as in  [Merlet 90]   and in   [Innocenti 92]   :  •   A   1   = (0.0, 0.0)   B   1 B 2 = 17.04  •   A   2   = (15.91, 0.0)   B   2 B 3 = 16.54  •   A   3   = (0.0, 10.0)   B   3 B 1 = 20.84 The limits of the prismatic actuated joints are those chosen in   [Innocenti 92]   : 10 0   32 0 .   . ≤   ≤ ρ i  The passive revolute joints are assumed unlimited.  2.2   O CTREE MODEL  The octree is a hierarchical data structure based on a recursive subdivision of space   [Meagher 81] . It is particularly   useful   for   representing   complex   3-D shapes, and is suitable for Boolean operations like union, difference and intersection. Since the octree structure has an implicit adjacency graph, arcwise- connectivity analyses can be naturally achieved. The octree model of a space S leads to a representation of S with cubes of various sizes. Basically, the smallest cubes lie near the boundary of the shape and their size determines the accuracy of the octree representation. Octrees have been used in several robotic applications  [Faverjon 84] ,   [Garcia 89] ,   [El Omri 93] . In this work,   the   octree   models   are   calculated   using discretization and enrichment techniques as described in   [Chablat 96] . The octree models of the reachable joint space (in the space  ρ 1 ,  ρ 2 ,  ρ 3 ) and of the workspace (in the space   x ,   y  et  φ ) of the 3-RPR parallel manipulator are shown in figures 2 and 3 (the workspace and the reachable joint space are defined in section 3.1). The reachable joint space is not a complete parallelepiped, since not any joint vector can lead to an assembly configuration of the manipulator.  ρ 1   ρ 2  ρ 3  X   Y  φ  Figure 2 : Octree model of the joint space Figure 3 : Octree model of the workspace  2.3   S   INGULARITIES  The vector of input variables and the vector of output variables for a n-DOF parallel manipulator are related through a system of non linear algebraic equations which can be written as :  (   ) F q X ,   =   0   (4)  where   0   means here the n-dimensional zero vector. Differentiating (4) with respect to time leads to the velocity model :  A X   B q &   & +   =   0   (5)  where   A   et   B   are   n  ×   n   Jacobian matrices. These matrices are functions of   q   and   X   :  A   F X   B   F q  =   = ∂ ∂ ∂ ∂   (6)  These matrices are useful for the determination of the singular configurations   [Sefrioui 92] .  2.3.1   Type-1 singularities  These singularities occur when   det(B) = 0 . For   the   planar   manipulator,   this   condition   can   be satisfied only when  ρ 1   = 0   or  ρ 2   = 0   or  ρ 3   = 0 . In practise, the type-1 singularities are attained when one of the actuated prismatic joints reaches its limit [ Gosselin 90] . The corresponding configurations are located at the boundary of the workspace. For parallel manipulators which may have more than one inverse kinematic solution, type-1 singularities are configurations   where   two   solutions   to   the   inverse kinematic   problem   meet.   By   hypothesis,   type-1 singularities will be always associated with joint limits in this paper.  2.3.2   Type-2 singularities  They occur when   det(A) = 0 . Unlike the preceding ones, such singular configurations occur inside the workspace.   They   correspond   to   configurations   for which two branches of the direct kinematic problem meet.   They   are   particularly   undesirable   since   the manipulator   cannot   be   steadily   controlled   in   such configuration where the manipulator stiffness vanishes in some direction. For the planar manipulator, such configurations are reached whenever the axes of the three prismatic joints
intersect (possibly at infinity). In such configurations, the manipulator cannot resist a wrench applied at the intersecting point (Figure 4). The resulting singular surface is built and modelled using octrees (Figure 5). The equation of   Det(A)   can be put in an explicit form   y = s(x,  φ ) , only two variables need to be swept in the octree enrichment process  [Chablat 96] .  X   Y  φ  Figure 4 : Type-2 singular configurations Figure 5 : Octree model of the set of type-2 singularities in the workspace  3.   NOTION   OF   ASPECT   FOR PARALLEL MANIPULATORS  The notion of aspect was introduced by   [Borrel 86]   to cope with the existence of multiple inverse kinematic solutions   in   serial   manipulators.   An   equivalent definition was used in   [Khalil 96]   for a special case of parallel manipulators, but no formal, more general definition   has   been   set.   The   aspects   are   redefined formally in this section.  3.1   D EFINITION  Let   OS   m   and   JS   n   denote the operational space and the joint   space,   respectively.   OS   m   is   the   space   of configurations of the moving platform and   JS   n   is the space of the actuated joint vectors. Let   g   be the map relating the actuated joint vectors to the moving platform configurations :  g   OS   JS X   q   g X m   n : (   )  → →   =   (7)  It   is   assumed   m = n ,   that   is,   only   non-redundant manipulators will be studied in this paper. Finding the solutions   X   to equation   q= g(X)   means solving the direct   kinematic   problem.   For   our   planar   parallel manipulator,   it   has   been   shown   that   the   direct kinematic   model   can   admit   six   real   solutions  [Gosselin 91] . Let   W   be the reachable workspace, that is, the set of all positions and orientations reachable by the moving platform   [Kumar 92] ,   [Pennock 93] . Let   Q   be the reachable joint space that is, the set of all joint vectors reachable by actuated joints.  {   } Q   q   JS   i   n q   q   q   Q   JS n   i   i   i   n  =   ∈   ∀   ≤   ≤   ≤   ⊂ ,   ,   , min   max   (8)  (   ) Q   g W   W   OS   m  =   ⊂ ,   (9)  Definition 1:  The aspects   WA i   are defined as the maximal sets such that :  •   WA i   ⊂   W   ;  •   WA i   is connected ;  •  ∀   X  ∈   WA i ,   (   ) Det A   ≠   0   . In   other   words,   the   aspects   are   the   maximal singularity-free connected regions   in the workspace . The   aspects   are   computed   as   the   connected components   of   the   set   obtained   by   removing   the singularity surfaces from the workspace, which can be done easily with the octree model :  ∪   =   − WA   W   S i   (10)  Application :  For the planar manipulator studied, we get two aspects ( WA 1   and   WA 2 ), where   Det(A) > 0   and   Det(A) < 0 , respectively. The singular surface of Figure 5 is the common boundary of the two aspects (Figure 6).  X   Y  φ  Figure 6 : Octree model of the aspects  3.2   N ON -   SINGULAR   CONFIGURATION CHANGING TRAJECTORIES  In   [Innocenti   92] ,   a   non-singular   configuration changing   trajectory   was   found   for   the   planar manipulator. However, it appears that this trajectory passes close to a singular configuration. We have been able   to   confirm   that   non-singular   configuration changing trajectories do exist for this robot. For the following input joint values :  ρ   ρ   ρ 1   2   3 14 98   15 38   12 0 =   =   = .   .   .   (11)  The   solutions   to   direct   kinematic   problem   are (Table 1) :  x   y   φ   (rad)  1   -8.715   12.183   -0.987 2   -5.495   -13.935   -0.047 3   -14.894   1.596   0.244 4   -13.417   -6.660   0.585 5   14.920   -1.337   1.001 6   14.673   -3.013   2.133  Table 1 : Six direct kinematic solutions for the planar manipulator  It can be verified that solutions -2-, -3-, -6- are in the same aspect ( WA 1 ) (Figure 7).
S olution 3 Solution 6 Solution 2 X   Y  φ  Figure 7 : Three direct kinematic solutions in one single aspect  This example clearly shows that the aspects are not the uniqueness domains. Additional surfaces have to be defined for separating the solutions.  4.   CHARACTERISTIC SURFACES 4.1   D EFINITION  The characteristic surfaces were initially introduced in  [Wenger 92]   for serial manipulators. This definition is restated here for the case of parallel manipulators.  Definition 2 :  Let   WA   an aspect in workspace   W . The   characteristic surfaces   of the aspect   WA , denoted   S   c   (WA) , are defined as the preimage in   WA   of the boundary   WA   that delimits   WA   (Figure 8) :  (   )   (   ) (   ) S   WA   g   g   WA   WA c   =   ∩ − 1   (12)  where :  •   g   is defined as in (15)  •   g   -1   is a notation. Let   B  ⊂   Q   :  (   ) {   } g   B   X   W   g X   B −   =   ∈   ∈ 1   (   )   /  The boundaries   WA   of   WA   are composed of :  •   the type-2 singularities ;  •   the type-1 singularities (limits of the actuated joints).  Figure 8 : Definition of the characteristic surfaces  4.2   C ASE   OF   THE   PLANAR   3 - RPR MANIPULATOR  The   characteristic   surfaces   are   computed   using definition (12). The singular surfaces are scanned and their preimages in the joint space are calculated using  g . The resulting inverse singularities are mapped back into the workspace using the direct kinematic model. We get one characteristic surface for each aspect, denoted   S   c1   and   S   c2 , respectively. Figure 9 depicts the singularity   surface   S   along   with   the   characteristic surface   S   c1 .  Singularity surfaces S Characteristic surfaces S c1 X   Y  φ  Figure 9 : Octree model of the singularities and of the characteristic surfaces   S c1  5.   UNIQUENESS DOMAINS 5.1   B ASIC COMPONENTS AND BASIC REGIONS Definition 3 :  Let   WA   be   an   aspect.   The   basic   regions   of   WA , denoted   {   } WAb   i   I i   ,   ∈   , are defined as the connected components of the set   (   ) WA   S   WA C & −   (   & −   means the difference between sets). The   basic regions   induce a partition on   WA   :  (   )   (   ) WA   WAb   S   WA i   I   i   C = ∪   ∪ ∈   (13)  Definition 4 :  Let   QA bi   = g(WAb i ) ,   QA bi   is a domain in the reachable joint space   Q   called   basic components . Let   WA   an aspect   and   QA   its   image   under   g .   The   following relation holds :  (   )   (   ) (   ) QA   QAb   g S   WA i   I   i   C  = ∪   ∪ ∈   (14)  Proposition 1 :  The   basic components   of a given aspect are either coincident, or disjoint sets of   Q .  Theorem 1 :  The restriction of   g   to any basic region is a bijection. In other words, there is only one direct solution in each basic region.  Proof :  We define the function   F   of   WA   QA ×   in   QA   (where  QA = g(WAb) ) such that :  (   )   (   )   (   ) ∀   ∈   ×   =   − X q   WA   QA F X q   g X   q ,   ,   ,   (15)  F   is   a   continuous   and   differentiable   function   on  WA   QA ×   . Let   QAb   i   be any basic component and let  WAb   i   be such that   (   ) QAb   g WAb i   i  =   . Let   (X   0, q   0)   be an arbitrary point in   WAb   QAb i   i ×   such that   (   ) F X   q 0   0   0 ,   =   . Since   (   ) X   q   WAb   QAb i   i 0   0 ,   ∈   ×   ,   (   ) Det   F X   X   q  ∂ ∂   0   0   0 ,  ⎛ ⎝ ⎜   ⎞ ⎠ ⎟ ≠   . Then, the implicit function theorem tells us that there exists   a   neighbourhood   U   of   X 0   in   WAb   i   and   a neighbourhood   V   of   q   0   in   QAb   i   such that, for any   q   in
V , equation   F(X, q)   has one unique solution   X = f 1(q)  in   U . Let   f 2   be another solution to equation   (   ) (   ) F   f   q   q 2   0 ,   =  and such that   (   )   (   ) f   q   f   q 1   0   2   0  =   . Since   q   0   is the image of a configuration lying neither on the boundary nor on a singularity,   f 2   is   well   defined   at   q   0 ,   and  (   ) (   ) Det   F X   f   q   q  ∂ ∂   2   0   0   0 ,  ⎛ ⎝ ⎜   ⎞ ⎠ ⎟ ≠   . If we prove that   (   )   (   ) f   q   f   q 1   0   2   0 =   .   for any point   q   in  QAb   i , theorem 1 will be proved. Let   (   )   (   ) {   } C   q   QAb   f   q   f   q i  =   ∈   = /   1   2   : we have to prove that   C = QAb   i .   C   is a not empty since it contains points  q   0 .   C   is a closed set in   QAb   i   since   C   is the set of all the roots of   (   )   (   ) f   q   f   q 2   1   0 −   =   . Now   (   ) (   ) Det   F X   f   q   q  ∂ ∂   2   0 ,  ⎛ ⎝ ⎜   ⎞ ⎠ ⎟ ≠   for any point   q   in   C , since   QAb   i   and thus   C   does not contain points which are the image under   g   of a singularity. Therefore, it stems from the implicit function theorem that   C   has a neighbourhood of each of its points, and thus is also an open set in   QAb   i . Since   QAb   i   is connected, the only subsets of   QAb   i   which are open an closed at the same time are   QAb   i   and the empty set. Since   C   is not empty,  C = QAb   i   and the theorem is proved.  Application :  The   basic   regions   are   calculated   as   the   connected components   of   the   set   obtained   by   removing   the characteristic surfaces from the aspects :  ∪   =   − WA   WA   S bi   i   ci   (16)  We   obtain   thus   28   basic   regions   for   the   planar manipulator at hand (Figure 10).  X   Y  φ  Figure 10 : Octree model of the basic regions  The basic components are computed as :  (   ) QA   g WAb i   i  =   (17)  We notice that, in accordance with proposition 1, the basic components are either coincident or disjoint. The coincident components yield domains with 2, 4 or 6 solutions for the direct kinematic problem : the upper one domains contain two coincident basic components, the   middle   five   domains   are   composed   of   four coincident domains, and the last two domains contain six coincident basic components (Figure 11).  5.2   U NIQUENESS DOMAINS  Theorem 1 yields sufficient conditions for defining domains of the workspace (the basic regions) where there is one unique solution for the direct kinematic problem.   However,   the   basic   regions   are   not   the maximal uniqueness domains. The following theorem 2 intends to define the larger uniqueness domains in the workspace.  Six coincident basic components in each domain Two coincident basic components in each domain  Four coincident basic components in each domain  ρ 1   ρ 2  ρ 3  ρ 1   ρ 2  ρ 3  ρ 1   ρ 2  ρ 3  ρ 1   ρ 2  ρ 3  ρ 1   ρ 2  ρ 3  ρ 1   ρ 2  ρ 3  ρ 1   ρ 2  ρ 3  Figure 11 : Octree model of the basic components  Theorem 2 :  The uniqueness domains   Wu   k   are the union of two sets : the set of adjacent basic regions   (   ) '  ∪   ∈ i   I   i WAb   of the same aspect   WA   whose respective preimages are disjoint basic components, and the set   S   c   (I’)   of the characteristic   surfaces   which   separate   these   basic components :  (   )   (   ) Wu   WAb   Sc I k   i   I   i  = ∪   ∪ ∈   '   '   (18)  with   I '   ⊂   I   such   as   ∀   ∈ i   I 1   2 ,i   '   ,  (   )   (   ) g WAb   g WAb i   i 1   2  ∩   = ∅   .  Proof :  Since the basic regions which define the   Wu   k   are such that their preimages in the joint space do not overlap, there is still one unique solution in each   Wu   k   .  Application :  To build the uniqueness domains, we have to consider the adjacent basic regions corresponding to disjoint, adjacent basic components. Six   uniqueness   domains   have   been   found   for   the planar manipulator (Figure 12), that is as many as the number of direct kinematic solutions. Note that in general, the number of uniqueness domains should be always more or equal to the maximal number of direct
kinematic solutions. We notice that, in the joint space, a non-singular configuration changing trajectory has to go through a domain made of only two coincident basic components, that is, a domain where there are only   two   direct   kinematic   solutions   (one   in   each aspect). In addition, it is worth noting that the domain with six coincident basic components map into six basic regions which are linked together by singular surfaces. This means that a non-singular configuration changing trajectory cannot be a straightforward motion between   two   such   basic   regions,   but   should   pass through another intermediate basic region (Figure 13).  Figure 12 : Octree model of the six uniqueness domains  ρ 1   ρ 2  ρ 3  Figure 13 : Basic regions and basic components with 6 solutions to the direct kinematic problem  6.   CONCLUSION  The problem of determining the uniqueness domains in the   workspace   of   parallel   manipulators   has   been studied   in   this   paper.   The   aspects,   originally introduced for serial manipulators, have been redefined here   as   the   largest   singularity-free   regions   in   the workspace. The aspects were shown to be divided into distinct basic regions where there is only one solution to the direct kinematic problem. These regions are separated   by   the   singular   surfaces   plus   additional surfaces   referred   to   as   characteristic   surfaces. Physically,   the   basic   regions   separate   the different solutions to the direct kinematic problem : given a point   in   the   joint   space,   the   corresponding configurations of the moving platform are distributed in the different basic regions. The maximal uniqueness domains have been defined as the union of adjacent basic regions whose preimages in the joint space are not coincident. All results have been illustrated with a 3-DOF planar RPR-parallel manipulator. An octree model of space and specific enrichment techniques have been used for the construction of all sets. This work   brings   preliminary   material   to   further investigations like trajectory planning   [Merlet 94] , which is the subject of current research work from the authors.  7.   BIBLIOGRAPHY [Borrel 86]   : Borrel Paul, «A study of manipulator inverse kinematic solutions with application to trajectory planning and workspace determination», Proc. IEEE Int. Conf. on Rob. And Aut., pp. 1180-1185, 1986.  [Chablat   96]   :   Chablat   Damien,   Wenger   Philippe, « Domaines d’unicité pour les robots parallèles », Technical report, Laboratoire d’Automatique de Nantes, n°96.13, 1996.  [Chételat 96]   : Chételat O., Myszkorowsky P., Longchamp R., « A Global Inverse Function Theorem for the Direct Kinematics », Proc. World Automation Congres-ISRAM’96, May 27-30, 1996, Montpellier, France.  [El Omri 96]   : El Omri Jaouad, « Analyse Géométrique et Cinématique   des   Mécanismes   de   Type   Manipulateur », Thèse, Nantes, 1996.  [Faverjon 84]   : Faverjon B., « Obstacle avoidance using an octree in the configuration space of a manipulator », Proc. IEEE Int. Conf. on Rob. And Aut., pp. 504-510, 1984.  [Garcia   86]   :   Garcia   G.,   Wenger   P.,   Chedmail   P., « Computing moveability areas of a robot among obstacles using octrees », Int. Conf. on Advanced Robotics (ICAR’89), June 1989, Columbus, Ohio, USA.  [Gosselin 91]   : Gosselin Clément, Sefrioui Jaouad, Richard Marc   J.,   « Solutions   polynomiales   au   problème   de   la cinématique des manipulateurs parallèles plans à trois de gré de liberté », Mech. Mach. Theory, Vol. 27, pp. 107-119, 1992.  [Innocenti   92]   :   Innocenti   C.,   Parenti-Castelli   V., « Singularity-free   evolution   from   one   configuration   to another in serial and fully-parallel manipulators », Robotics, Spatial Mechanisms and Mechanical Systems, ASME 1992.  [Kumar 92]   : Kumar V.,   « Characterization of workspaces of parallel manipulators », ASME J. Mech. Design, Vol. 114, pp 368-375, 1992.  [Khalil 96]   : Khalil W., Murareci D., « Kinematic analysis and singular configurations of a class of parallel robots », Mathematics   and   Computer   in   Simulation,   pp.   377-390, 1996.  [Meagher 81]   : D. Meagher, « Geometric Modelling using Octree Encoding », Technical Report IPL-TR-81-005, Image Processing   Laboratory,   Rensselaer   Polytechnic   Institute, Troy, 1981, New York 12181.  [Merlet   90]   :   Merlet   J-P.,   « Les   robots   parallèles », HERMES, Paris, 1990.  [Merlet 94]   : Merlet J-P., « Trajectory verification in the workspace of parallel manipulators », The Int. J. of Rob. Res, Vol 13, No 3, pp326-333, 1994.  [Pennock   93]   :   Pennock   G.R.,   Kassner.   D.J.,   « The workspace of a general geometry planar three-degree-of- freedom   platform-type   manipulator »,   ASME   J.   Mech. Design, Vol. 115, pp. 269-276, 1993.
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