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A new methodology for the determination of the workspace of six-DOF redundant parallel structures actuated by nine wires

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A new methodology for the determination of the workspace of six-DOF redundant parallel structures actuated by nine wires
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  Robotica (2007) volume 25, pp. 113–120. © 2006 Cambridge University Pressdoi:10.1017/S0263574706003055 Printed in the United Kingdom A new methodology for the determination of the workspace of six-DOF redundant parallel structures actuated by nine wires Carlo Ferraresi ∗ † , Marco Paoloni ‡  and Francesco Pescarmona † †  Dipartimento di Meccanica, Politecnico di Torino, Corso Duca degli Abruzzi, 24, Torino 10129, Italy. ‡  Robotic Section of the Nuclear Fusion Unit, ENEA C.R. Casaccia, Via Anguillarese, 301, Roma 00060, Italy. (Received in Final Form: July 27, 2006, First published online: October 12, 2006) SUMMARY The WiRo-6.3 is a six-degrees of freedom (six-DOF) roboticparallel structure actuated by nine wires, whose character-istics have been thoroughly analyzed in previous papersin reference 2 . It is thought to be a master device for tele-operation; thus, it is moved by an operator through a handleand can convey a force reflection on the operator’s hand.A completely new method for studying the workspace of this device, and of virtually any nine-wire parallel structureactuated by wire is presented and discussed, and its resultsare given in a graphical form. KEYWORDS: Parallel robot; workspace; wire actuation. 1. Introduction Mechanical structures actuated by wires (wire robots)are characterized by the presence of a mobile platform(representing the end-effector) connected by several wires toa fixed frame; the wires are fixed to the platform, rolled over pulleys and stretched by motors fixed to the frame in order to exert forces and torques. At the same time, the positionand orientation of the mobile platform can be determined bythe measured wire lengths. Wire robots are parallel deviceshaving wires as links, and belong to a set of fully parallelstructures because every wire is an independent chain withone DOF. 1, 4–6 With respect to the traditional parallel structures, wire-actuated robots have several advantages: they allow greatmanoeuvrability, thanks to a reduced mass, and also promiselowercostswithrespecttotraditionalactuators.Furthermore,thestrokelengthofeachlinearjointdoesnotfollowthesamerestrictions as with conventional structures, because wirescan be extended to much higher lengths, unwinding from aspool. This kind of a structure allows to comply with severalneeds in applications where conventional manipulationtechnology can be hardly used for technical or economicalreasons. We could mention, for example, crane robots, 1, 4 high-speed manipulation robots 5 and force feedback devicestobeusedasmastersinmaster–slaveteleoperationsystems. 6 ∗ Corresponding author. E-mail: carlo.ferraresi@polito.it Such devices offer many advantages, such as a simplifiedmechanical structure, very high speed, relatively largeworkspace and low inertia. However, it must be noted thatwires can only pull objects and not push on them: thisunilateral constraint compels the adoption of a redundantactuatingmechanism.Thiscanbeseenasananalogywiththegrasping problem for multi-finger systems with frictionlesspointcontact;theforcesexertedbythefingersonthegraspedobject are subject to the same unidirectional constraint.It has been stated 5, 6, 9, 13 that to obtain  n  degrees of freedom(DOFs)withoutexternalforces(orthegravityinthecrane case) it is necessary and sufficient to use  n  +  1 wires;Thesedevicesareusuallyreferredtoascompletelyrestrainedposition mechanism (CRPM), while devices with a higher number of wires are referred to as redundantly restrainedposition mechanism (RRPM).The study of the operative characteristics of wire-drivendevices may present more difficulties than the traditionalones, in particular for the definition of their workspace anddexterity.Theworkspaceisnotsimplythesetofnon-singular platformpositionsandorientationscompatiblewiththejointslimits, but it is also necessary that all forces and torquesexerted in such platform poses should be obtainable onlyby means of a set of wire forces directed from the platformto the frame. Furthermore, the shape and the dimensions of the workspace, and the dexterity of these devices are greatlyinfluenced by the number of wires and their geometricaldisposition.It has been stated by several authors 5, 6, 12, 13 that in acertain pose of the end-effector of a six-DOF mechanismdriven by  m  wires, it is possible to exert arbitrary force andmoment, if and only if, the transpose of the inverse Jacobian(called the  structure matrix   with six rows and  m  columns)has a rank equal to six and if it is possible to find a vector belonging to its null space with all the components strictlypositive. Practically speaking, after the evaluation of the nullspace base constituted by  m − 6 vectors with  m  components,to decide if the considered pose belongs to the workspace atleast one set of   m  −  6 coefficients must be found to forma linear combination of the  m  −  6 base vectors yielding aresultant with all the  m  components strictly positive.For the CRPM case ( m  =  7), the null space is a one-dimensional (1-D) space vector and then the examined posebelongs to the workspace if the vector chosen as base of thenull space hasall thesevencomponents ofthe samesign. For   114  Redundant parallel structures Fig. 1. The WiRo-6.1 structure. the RRPM case ( m  >  7), the null space examination is nottrivial and solutions can be found using algorithms like theones used in linear programming problems of constrainedoptimization. 13 In the Department of Mechanics of the Politecnico diTorino and in the Robotics Section of the ENEA, researcheson wire parallel structures have been carried out in order to identify appropriate analysis and design criteria for thedevelopment of devices devoted to remote manipulation andsensed teleoperation.Starting from a number of seven-wire structures, whichis the minimum number of wires able to provide six-DOFs, some analysis methodologies were implemented andadopted,suchastheuseofappropriateindexestoidentifythebestperformingstructuresthroughanobjectiveevaluationof their workspaces. 3 As a first result, one seven-wire structure,WiRo-6.1 (Fig. 1), was chosen as the starting point for futuredevelopment, although its main problems were a very smallworkspace and low dexterity.Then, a new structure was conceived in order to lessen thedisadvantages of the earlier structure. The lower, single wirewasreplacedbythreewiresconvergingin O M ,thusobtaininga highly redundant structure with a large workspace, gooddexterity and a better equilibrium in the disposal of wires.This structure was called WiRo-6.3, and its inverse andforward kinematics were both solved in a closed form. 2 In this paper, a novel analytical methodology is proposedfor identifying the workspace of a nine-wire redundantstructure. In particular, this procedure was applied to thestudy of WiRo-6.3, demonstrating that it possesses a larger workspace and a more dexterous behavior than WiRo-6.1. 2. Workspace Definition Together with the properties of the structure matrix, in thestudy of the workspace of wire robots further aspects mustbe considered: 11 the structure must be sufficiently stiff, wiresmust not intersect with each other and with the environmentand the stretching tension must lie between a minimum (in Fig. 2. The WiRo-6.3 structure. ordertopreventwiresbecomingslack)andamaximumvalue(determined by the motors, the breaking loads of the wires,etc.).Theseaspectsarenotconsideredinthispaperbecausetheypertain to the practical realization of the mechanisms, whilethe objective of our work is the development of a procedureto explore the theoretical limits of these structures. Theworkspaces obtained in this condition are called  controllableworkspaces ,whilethoseobtainedwithallaspectsconsideredare called  feasible workspaces ; usually a feasible workspaceis a subset of the correspondent controllable workspace.In the following sections, geometrical and analyticalconsiderations will lead to a procedure to calculate whether a platform pose belongs to the workspace or not. Sinceit is difficult to obtain a closed-form description of theworkspaces of such robots, they are described as grids of points, starting from a discrete subdivision of the internalvolume of the fixed frame.As can be seen in Figs. 1 and 2, the groups of sixwires of both WiRo-6.1 and WiRo-6.3 devices are disposedas the actuators of a Stewart–Gough platform, a well-known parallel mechanism with an amount of technicalbibliography. 17 The workspaces of this and similar deviceshave been widely studied 17–22 using different methodo-logies from the analytical study of the zeroes of its Jacobiandeterminant 18, 19 to the application of the Lie algebra. 20 All these studies, though, can bring a relatively poor contribution to our work because they do not deal withthe problem of the unilateral actuation of the wires, theycan be used to evaluate the rank of the structure matrix;in fact, it can be recognized that the limitations to theplatform motion imposed by the need of preventing wiretangles constraints the dispositions of the group of six wiresin non-singular configurations of a correspondent Stewart– Goughplatform.Inthisway,theblockofthestructurematrixassociated with the six wires group is a 6  ×  6 matrix withnon-null determinant, ensuring the full rank of the wholestructure matrix and allowing to concentrate on its null spaceexamination.   Redundant parallel structures  115 3. Force Closure The force closure of a parallel structure in a particular configuration can be expressed as − W  =  f   =  ˜ J · τ   (1)where, in the case of a nine-wire parallel robot,  W  is the six-component wrench acting on the platform,  f   is the wrenchprovidedbytherobot, ˜ J isthe6 × 9structurematrixevaluatedoveranyparticularconfigurationand τ   isthenine-componentvector containing the tensions of the wires.The condition representing the belonging of a given poseto the workspace can be expressed imposing that for any  f  the tensions of the wires can all be made positive (or greater than a prefixed positive value) τ   >  0 .  (2)Since ˜ J isnotsquare,amongtheinfinitesolutionsofEq.(1)for any given  f  , the minimum-norm solution can be obtainedby means of the pseudoinverse τ  min  =  ˜ J + · f   (3)where ˜ J + is the pseudoinverse of  ˜ J .The generic solution of Eq. (1) is given by τ   =  τ  min  + τ  ∗ (4)where  τ  ∗ must belong to the null space of  ˜ J ˜ J · τ  ∗ =  0 .  (5)This means that the infinite possible values of   τ   can befound by adding to  τ  min  a set of vectors that do not affect theresulting wrench.Equation (2) may be reached imposing that for eachpoint of the workspace at least a strictly positive  τ  ∗ exists,satisfying Eq. (5). In this way, knowing that all its multiplesalso belong to the null space of  ˜ J , it is possible to find anappropriatepositivemultiplier  c capableofcompensatingthenegative components of   τ  min f   =  ˜ J ( τ  min  + c · τ  ∗ ) (6)where c · τ  ∗ ∈  null(˜ J );  τ  min  + c · τ  ∗ >  0 . 4. Workspace Analysis The procedure for the analysis of the workspace consistsof studying a discretized six-dimensional (6-D) spacecomprising three displacements and three rotations, pointby point, checking whether the rank of  ˜ J  is maximum (equalto 6) and a strictly positive  τ  ∗ exists, satisfying Eq. (5).If   n  is the number of degrees of freedom of a structure, and m  is the number of its actuators, the base of the null spaceof  ˜ J  (if  ˜ J  has rank  n ) is a set of   m  −  n  vectors, each with  m components.In the case of a nine-wire robot, the base of the null spaceof  ˜ J  is composed by three nine-component vectors, a linear combination of which may provide the desired  τ  ∗ .Naming [ ker ]  ∈  R 9 × 3 the matrix whose columns generatethe null space of  ˜ J ,  τ  ∗ can be written as a linear combinationof its columns. Equation (2), considering Eq. (5) as well, canbe expressed by the following inequality: τ  ∗ =  [ ker ] ×  αβγ   >  0 (7)where  α ,  β  and  γ   are three unknown quantities.Developing Eq. (7) yields  k 11 α  + k 12 β  + k 13 γ >  0 k 21 α  + k 22 β  + k 23 γ >  0 k 31 α  + k 32 β  + k 33 γ >  0 k 41 α  + k 42 β  + k 43 γ >  0 k 51 α  + k 52 β  + k 53 γ >  0 k 61 α  + k 62 β  + k 63 γ >  0 k 71 α  + k 72 β  + k 73 γ >  0 k 81 α  + k 82 β  + k 83 γ >  0 k 91 α  + k 92 β  + k 93 γ >  0(8)where  k  ij   are the components of  [ ker ].Observing Eq. (8) shows that either it has no solution, or in the three-dimensional (3-D) space generated by  α ,  β  and γ   it determines a polyhedral angle with vertex in the srcinand polygonal cross section (Fig. 3).In case no solution of Eq. (8) exists, the correspondingpositioning point does not belong to the workspace; on thecontrary, if the polyhedral angle exists, there is an infinitenumber of sets ( α ,  β ,  γ  ) satisfying Eq. (8), and they are allcontained inside the polyhedral angle. This means that anyexternal wrench can be compensated by positive tensionsin the wires, and so the positioning point belongs to theworkspace. Sectioning the polyhedral angle with a cylinder coaxial with  α , considering cylindrical coordinates with  θ   asthe angle on the plane  βγ  , something analogous to Fig. 4 isobtained.Combining linearly the three columns of  ˜ J , it is alwayspossible to determine a new base of the null space of  ˜ J  such Fig. 3. The polyhedral angle determined by inequations given inEq. (8).  116  Redundant parallel structures Fig. 4. A generic section of the polyhedral angle. that its last three rows are  1 0 00 1 00 0 1  thus limiting the search to the portion of space in which  α ,  β and  γ   are all greater than zero.Naturally, the other elements of  [ ker ] will changeconsequently. The new matrix will be called [ ker ∗ ] tohighlight the modification performed.Thelastthreerowsof[ ker ∗ ]nowrepresenttheconditions  α >  0 β >  0 γ >  0 . (9)The first six rows of [ ker ∗ ] may be written in the form α >  − k i 2 β  + k i 3 γ k i 1 ,  if  k i 1  >  0 α <  − k i 2 β  + k i 3 γ k i 1 ,  if  k i 1  <  0 (10)for   i = 1 to 6.If   k  i 1  is null, the inequations are γ β>  − k i 2 k i 3 ,  if  k i 3  >  0 γ β<  − k i 2 k i 3 ,  if  k i 3  <  0 (11)for   i = 1 to 6.In this case, it is not important to exclude the case when k  i 3  is zero, because the ratio  γ/β  may be infinite, as will bepointed out afterwards.The first and the second inequations of Eq. (10) expressrespectively, the lower and upper limits for the values of   α ,as a function of the coefficients  k  ij   and of the value of   β  and γ  . The lower limits are counted through the letter   l = 1,  ... , l ∗ , while the upper limits are counted through the letter   u  = 1,  ... ,  u ∗ .For at least one valid  α  to exist, every lower limit mustbe smaller than every upper limit, which is equivalentto applying − k l 2 β  + k l 3 γ k l 1 <  − k u 2 β  + k u 3 γ k u 1 for   l  =  1 ,...,l ∗ , u  =  1 ,...,u ∗ (12)for each possible combination between a lower limit and aupper limit as from Eq. (10). Moreover, the first condition of Eq. (9), requiring α > 0, means that all the upper limits mustbe greater than zero − k u 2 β  + k u 3 γ k u 1 >  0 ,  for  u  =  1 ,...,u ∗ (13)for every upper limit as from Eq. (10).Remembering that  β >  0 as per Eq. (9), Eq. (12) may bewritten as γ β>  k u 2 k u 1 − k l 2 k l 1  k l 3 k l 1 − k u 3 k u 1  ,  if   k l 3 k l 1 − k u 3 k u 1  >  0 γ β<  k u 2 k u 1 − k l 2 k l 1  k l 3 k l 1 − k u 3 k u 1  ,  if   k l 3 k l 1 − k u 3 k u 1  <  0 . (14)If ( k l 3 k l 1 −  k u 3 k u 1 )  =  0 ,  no problem occurs, since  γ/β  may beinfinite as will be better explained in the following sections.The possibility that both the numerator and the denominator are zero is excluded by the check of the rank of  ˜ J .Equation (13) may be written as γ β> − k u 2 k u 3 ,  if  k u 3 k u 1 <  0 γ β< − k u 2 k u 3 ,  if  k u 3 k u 1 >  0 . (15)In the 3-D space generated by  α ,  β  and  γ  , the projectionof the polyhedral angle (8) on the plane  βγ   determines anangle delimited by an interval [ θ  1 ,  θ  2 ], as given in Fig. 5,where a generic  θ   is represented as well. Remembering that Fig. 5. Projection of the polyhedral angle on the plane  βγ  .   Redundant parallel structures  117 Fig. 6. Flowchart of the workspace-determination procedure. β > 0 and  γ > 0 as per Eq. (9)0 ◦ < θ  1  <  90 ◦ 0 ◦ < θ  2  <  90 ◦ .  (16)Now, imposing β  = cos θ   and γ   = sin θ  , the generic angle θ   can be expressed in the form θ   =  atan γ β.  (17)This explains why the ratio  γ/β  may be infinite: thecorresponding angle  θ   would be 90 ◦ .In this notation, the inequations in Eqs. (11), (14) and (15)express the lower and the upper limits for   θ   (note that thelimitations of   α  have been discussed previously). Therefore,considering the notation of Fig. 5 as well, let the greatest of the lower limits be  θ  1 , and the smallest of the upper limitsbe  θ  2 ϑ 1  =  max  atan  γ β  l = 1 ,...,l ∗ ϑ 2  =  min  atan  γ β  u = 1 ,...,u ∗ . (18)

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