Dimensional Synthesis of Wristed Binary Hands

Transcription

1 Dmensonal Synthess of Wrsted Bnary Hands Neda Hassanzadeh Department of Mechancal Engneerng, Idaho State Unversty 921 S. 8th. Ave, Pocatello, ID 83209, USA Alba Perez-Graca Assocate Professor Department of Mechancal Engneerng, Idaho State Unversty 921 S. 8th. Ave, Pocatello, ID 83209, USA The knematc synthess appled to tree topologes s a tool for the desgn of mult-fngered robotc hands, for a smultaneous task of all fngertps. Even though tradtonally wrsts and hands have been desgned separately, the wrst usually beng part of the robot manpulator arm, t makes sense to consder the wrst as a part of the hand, as many graspng and manpulaton actons are a coordnated acton of wrst and fngers. The manpulaton capabltes of robotc hands may also be enhanced by consderng more than one splttng stage, as opposed to the sngle-palm tradtonal hand. In ths work we present the dmensonal synthess for a famly of mult-fngered hands, the bnary hands, whch have a 2R wrst and several splttng stages, each of them spannng two branches consstng f a revolute jont for each edge. For these topologes, t s proved that a three-poston task can be defned for each fngertp, regardless of the number of fngers. One example s presented to show the possble desgn strateges and uses for ths famly of hands. Nomenclature S A lne or dual vector, defned usng Plucker coordnates. s A vector Ŝ A quaternon or dual quaternon ŵ A dual number [M] A matrx 1 Introducton The topology of an artculated system defnes the number and type of jonts of the system, ther adjacency, and ther ncdence on the end-effectors. In dmensonal knematc synthess, the goal s to determne the locaton and orentaton of the jont axes of a gven topology so that t can perform a gven task. It has only been recently that knematc synthess has been appled to multple end-effector systems. These systems can be smplfed, for synthess purposes, to chans wth a tree topology, wth several common jonts that branch to a number of seral chans, and possbly wth several branchng stages, endng wth several end-effectors. A typcal example of a knematc chan wth a tree topology s a wrsted, mult-fngered hand. Knematc analyss of tree topologes for applcatons n modular robots and robotc hands can be found n [1], [2], and [3], and dynamc analyss s found n [4] and [5]. Structural synthess for multple fngers wth no wrst s also consdered n [2], and n [6] for graspng and manpulaton requrements. The dmensonal synthess of the tree topologes presents partcular challenges that are dfferent from those that appear n sngle seral chans or n closed-loop systems. In partcular, the dmensonal synthess of mult-fngered hands has been explored n [7], [8] and more extensvely n [9]. In ths case, the knematc task s defned as havng the same number of postons for each of the multple endeffectors; ths means that we are dealng wth a coordnated acton of all those end-effectors, denoted as a smultaneous task. The synthess of tree topologes, when appled to wrsted mult-fngered hands, allows creatng robotc hands for n-palm manpulaton tasks, for nstance, or complete arm-plus-hand desgns. The choce of the topology so that t s more suted for the task s the object of the type, or structural, synthess. A frst attempt at the enumeraton of tree topologes for the desgn of mult-fngered hands can be found n [10], where solvablty was studed. In ths study t was concluded that there are some famles of topologes for whch the number of fngers can be ncreased arbtrarly wthout constran- JMR , Perez-Graca, page 1

2 ng ther ablty to perform a task. In ths paper we focus on one famly of tree topologes, denoted as bnary hands, whch s part of the bgger famly of fractal hands. For ths famly, the topologes are enumerated and characterzed, solvablty s studed and the dmensonal synthess s solved for the 2R wrested hand wth R- jont edges. A closed algebrac soluton s derved and the number of solutons s calculated for any number of fngers. Fnally, one example s presented that show the potental of ths famly of mult-fngered hands. 2 Mult-fngered Robotc Hands Mult-fngered robotc hands consst, n the most general case, of a seres of common jonts spannng several knematc chans, whch can splt agan n new chans, up to a set of end-effectors, or fngertps [11]. We defne a branch of the hand as the seral chan connectng the root node to one of the end-effectors, and a palm as a lnk that s ternary or above. In ths work t s consdered that the fngertps are the only elements whose moton or contact wth the envronment s beng defned by the task; ths could be generalzed by consderng also other ntermedate lnks of the hand. These tree topologes are represented as rooted tree graphs; for ths we follow the approach of Tsa [12], the root vertex beng fxed wth respect to a reference system. In tree topologes, a vertex can be connected to several edges defnng several branches. Most of the current robotc hands have one palm and several fngers, and may have a few wrst jonts. Fgure 1 shows a commercal hand, the Barret R Hand, and a modelng of the human hand consderng the wrst, wth ther tree representatons. Open hands, that s, hands not holdng an object n the fngers, are knematc chans wth a tree or hybrd topology. For our synthess formulaton, the nternal loops n the hand structure are substtuted usng a reducton process [9], so that the hand has a tree topology wth lnks that are ternary or above. Tree topologes are denoted as SC (B 1,B 2,...,B b ), where SC ndcates the common jonts and the dash ndcates a branchng or splttng, wth the branches contaned n the parenthess, each branch B characterzed by ts type and number of jonts. In the case of usng just revolute jonts, the jont type s dropped and only the number of jonts s ndcated. For Fgure 1, the tree topologes would be denoted as 0 (2,3,3) and 3 (4,4,5,5,5) respectvely. Our defnton of mult-fngered hand consders the wrst as a part of the hand, as many graspng and manpulaton actons are a coordnated acton of wrst and fngers. In addton, the hand desgn may present several splttng stages, such as the 2 (2,1 (3,3,3),2) tree topology shown n Fgure 2, wth a 2R wrst and two splttng stages. Fg. 2. A tree topology wth two splttng stages. (a) edge labelng; (b) number of jonts n each edge. Two arrays can be assocated to a hand topology. One of them s the parent-ponter representaton, n whch the value of the array for each labeled edge corresponds to the prevous edge of the graph. The other array s the jont array, whch contans the number of jonts for each edge. For nstance, for the tree graph n Fgure 2, the parent-ponter array s pt = {0,1,1,1,3,3,3} and the jont array s j = {2,2,1,2,3,3,3}, accordng to the edge notaton of the Fgure. Fg. 1. Barrett R Hand, wrsted human hand and ther tree representatons. Vertces are labeled as V, edges as E, and R denotes revolute jonts. 3 Dmensonal Synthess for Mult-fngered Hands Gven a hand topology and a knematc task consstng of a set of fnte postons for each end-effector, dmensonal knematc synthess seeks to fnd the poston of the jont axes n order for each of the end-effectors to perform the gven set of dsplacements smultaneously; see [9]. In ths secton, the formulaton of the synthess equatons and the calculaton of the solvablty of the topology are descrbed. JMR , Perez-Graca, page 2

3 3.1 Dmensonal Synthess Equatons The approach followed to create dmensonal synthess equatons conssts on equatng the forward knematcs of each root-to-fngertp branch n the hand to the set of postons defned for that partcular fngertp, whch s the endeffector. Gven a set of m task postons ˆP k, k = 1...m, for each end-effector (denoted by superscrpt ), we compute the relatve dsplacements from a selected reference poston (let us say, poston 1), and equate the relatve forward knematcs to those relatve postons. For a hand wth b fngertps, ths yelds b sets of equatons that are to be solved smultaneously, ˆP 1k = ˆθ k j e 2 S j, = 1,...,b; k = 2,...,m, (1) j {B } where the number of end-effectors, or branches as root-tofngertp chans, s ndcated by b, and {B } s the set of ordered ndces of the jonts belongng to branch. Notce that some of the jonts wll be common to several branches. The jont axes at the reference confguraton are denoted as S j. Ths yelds a total of 6(m 1)b ndependent equatons to be smultaneously solved. 3.2 Solvablty of Tree Topologes for Exact Synthess We defne a knematc chan as solvable [10] f we can fnd a postve ratonal number of postons for whch the exact dmensonal synthess yelds a fnte number of solutons. In the case of multple end-effectors, the system of equatons needs to be properly dmensoned to be solved smultaneously, whle not overconstranng any of the branches. Ths translates to checkng the relatve solvablty for all possble subgraphs startng at a root node and endng at -at least- one end effector, ncludng all possble root node changes. The number of postons for each subgraph s computed as follows: let D e j be an e 1 vector contanng the jont degrees-of-freedom for each edge of the contracted graph, and D e s be the e 1 vector contanng the number of structural parameters (four per jont n the general case) for each edge of the contracted graph. Denote as D n ee the b 1 vector contanng the degrees-of-freedom of the space of each endeffector, and D n c the b 1 vector wth the number of addtonally mposed constrants (f any) for each branch. Defne the vectors B as a b 1 vector of ones correspondng to the branches ncluded n the graph, and E as an e 1 vector of ones for the edges ncluded n the graph consdered. Here the subscrpt ndexes the set of subgraphs startng at the root node and endng n at least one end-effector, for a gven tree topology. These last two vectors can be easly computed from the end-effector path matrx and the ncdence matrx of the overall graph, see [9]. The number of postons for exact synthess of the graph or subgraph s gven by m = De s E D n c B D n ee B D e j E + 1. (2) An overall soluton can be found, for arbtrary tasks, only when consderng the solvablty of all subgraphs that start at the root node and end at end-effectors, ncludng all subgraphs obtaned when exchangng the root node wth one of the end effectors as descrbed above. In ths case, consderng m as the number of postons for exact synthess of the overall graph and m as the number of postons for exact synthess for a subgraph, wth S the set of all possble dfferent end-effector subgraphs up to somorphsm, the topology s solvable f 1. m Q + 2. m m, m Q +, S In the case of a subgraph contanng c branches and beng solvable for m = m postons, that subgraph can be solved separately, whch elmnates exactly 6c(m 1) equatons and the same number of unknowns, so that the rest of the graph can be solved a posteror. 4 Bnary Hands We denote as bnary hands those hands that are represented by a symmetrc bnary tree, that s, a tree n whch each vertex spans two edges, up to a gven level. Ths s a partcular case of the more general famly of fractal hands, n whch k branches are spanned at each splttng pont. Partal fractal hands have been mentoned n [13]; see Fgure 3 for some examples. Fg. 3. Graphs for some wrsted fractal hands: bnary and ternary. In our case, the symmetry of the bnary tree s broken at the wrst; the frst edge conssts of a seral chan wth two revolute jonts, whle the rest of edges are sngle revolute jonts. The structure of these bnary trees allows us to defne ther topology usng the number of branchng stages, or splttng stages, of the tree. If a tree has depth s + 1 (depth of a tree s defned as the number of edges from the root to the end-effector node), then the number of splttng stages s s, because there s no splttng stage for the wrst. Fgure 4 JMR , Perez-Graca, page 3

4 (1,1),1 (1,1)) bnary hand wth a planar moton at the four fngertps that can be combned to create two small loppers or a larger one, usng two end-effectors for each one. In ths case, the task defnes the workspace of the wrst, so that fnger actons can be performed wthn a gven volume and for gven orentatons. Fg. 4. For a tree wth depth s + 1, the number of splttng stages s s, as there s no splttng at the wrst. shows the relaton between the depth and the splttng stages of the bnary trees. For a wrsted, bnary hand wth s splttng stages, 1. The total number of splts s 2 s 1, 2. The total number of edges s 2 s+1 1, 3. The hand has b = 2 s end-effectors, or fngertps, 4. The parent-ponter representaton, when namng the edges as they appear sequentally from the root vertex, s pt = {0,1,1,2,2,3,3,4,4,...,2 s 1,2 s 1}. The total number of jonts of the hand s hence 2 s+1 countng the two jonts at the common wrst edge. Fgure 5 shows four examples of bnary hands, wth one to four splttng stages. Fg. 6. An example of a two-stage bnary hand 4.1 Partal Bnary Hands Partal fractal hands, n whch the symmetry s broken at a gven splttng stage by elmnatng, or prunng, some of the branches, may be also consdered as sutable topologes for the desgn of mult-fngered robotc hands. In the case of bnary hands, t s possble to elmnate up to 2 k new branchngs wth ther subsequent edges at each depth k of the tree, wth k = 1 to s. Because of solvablty ssues presented n subsequent sectons, the case studed here s that of elmnatng both edges ncdent n a gven vertex, for a certan number of vertces r 2 k 1. In partcular, for a bnary hand of depth s + 1 n whch we elmnate both edges ncdent on a sngle vertex at a gven depth k, we are elmnatng: 1. A total of 2 s+1 k 1 splts, 2. A total of 2(2 s+1 k 1) edges, 3. A total of 2 s+1 k end-effectors. Ths countng can be generalzed to the case n whch we elmnate r 1 pars of branches at a depth k 1, r 2 pars of branches at a depth k 2, etcetera, so that the number of endeffectors elmnated wll be r 1 2 s+1 k 1 + r 2 2 s+1 k A new end-effector s created for each par elmnated, so that the total number of end-effectors for a partal bnary hand of depth s + 1 n whch p prunngs are performed s Fg. 5. stages. Graphs for bnary trees wth one, two, three and four splttng b = 2 s (1 p =1 r 2 1 k ) + p. (3) Wrsted bnary hands may be used for scalable nteracton wth the envronment. Fgure 6 below shows a 1 (1 Fgure 7 shows the partal bnary hand wth one par of branches removed at depth 2 and another par removed at depth 3. JMR , Perez-Graca, page 4

5 Ths can be shown as follows: the dot products of the arrays n Equaton (2) are D e s E = 8 + 4(2 s+1 2), D n c B = 0, D n ee B = 2 s 6, D e j E = 2 + (2 s+1 2), (4) whch yeld m = 3 regardless of the depth s of the tree. In order to conclude that the hand s solvable, we also need to check all the non-somporphc subgraphs. Notce that the number of postons allowed for a subgraph wth b branches and e edges s Fg end-effectors. An bnary tree pruned at depth 2 and depth 3, wth a total of 4.2 Other Fractal Hands The characterzaton of symmetrc, fractal tree topologes can be generalzed to any number k of new branches per splt, and any depth s + 1 of the tree. The generalzaton of the propertes from prevous secton yelds the followng propertes for a k-ary wrested fractal hand: 1. The total number of splts s ks 1 k The total number of edges s ks+1 1 k The total number of end-effectors s k s. 4. The parent-ponter representaton gves the array pt = {0,1,...,1,2,...,2, ks 1 k 1,..., ks 1 k 1 }. 5. The total number of jonts, for a wrst wth p jonts and edges havng q jonts each, s j = p + k(ks 1) k 1 q. Notce that the number of end-effectors grows quckly: for nstance, for a ternary hand wth a depth of 4, 3 jonts n the wrst and 2 jonts per edge, the total number of jonts n the hand s 81, and the total number of end-effectors or fngertps s 27. The study of the solvablty and synthess of general fractal hands s beyond the scope of ths work. 5 Dmensonal Synthess of Bnary Hands The exact dmensonal synthess of a bnary hand wth s splttng stages, or 2 s end-effectors, requres checkng ts solvablty and, f solvable, defnng the number of postons n each fngertp task n order to obtan a fnte number of solutons. 5.1 Solvablty and number of solutons Usng Equaton (2), for the bnary hand topology wth jont array j = {2,1,1,...,1}, a few facts can be derved for ths famly, regardless of the number of splts: 1. The bnary hand s solvable for all topologes of the famly. 2. The maxmum number of postons per end-effector for exact knematc synthess s constant and equal to three. 4 m = + 1, (5) 6b e+1 1 whch means that the most restrctve subtrees wll be those wth the maxmum number of edges, that s, those n whch we elmnate the same number of edges as branches. For a subgraph n whch branches and edges have been elmnated, m = s s+1 + 1, (6) where < 2 s always and at most = 2 s 1. Ths number of postons tends to 3 as the number of splttng stages s ncreases, regardless of the number of branches elmnated n the subtree. Ths concludes that, for all subtrees startng at the root node, the number of postons for exact synthess s m 3. For those subtrees wth root node substtuton, the smallest of them corresponds to the RR seral chan, wth m = 3, and for the rest of them we can apply the same reasonng, to conclude that the wrsted bnary trees are always solvable. It s mportant to hghlght that some of the subtrees, n partcular those correspondng of two termnal edges (the edges ncdent on end-effectors) creatng seral RR chans, have m = 3, same as the overall number of postons. Accordng to [9], those can be solved separately. Ths s the strategy followed for the synthess of the bnary hands: 1. Solve for the relatve moton at the termnal edges of each two branches (a spatal RR chan) wth respect to the reference frame of one of the end-effectors. 2. Once all the termnal edges are dmensoned, solve for the relatve moton of the prevous branchng, agan as spatal RR chans. 3. Contnue untl the last branchng. Then the two wrst jonts can be solved for any of the branches. Usng ths methodology, the total number of solutons for a bnary hand wth s splttng stages, or branchngs, s 2 2s. The smplest hand, the 2 (1,1), has 4 solutons, and the hand topology wth four splttng stages n Fgure 5 yelds 2 16 canddate desgns for m = 3 postons. JMR , Perez-Graca, page 5

6 5.2 Solvablty for partal bnary hands The partal bnary hands obtaned by prunng some of the branches are also studed for solvablty. As t has been already mentoned n the prevous secton, the prunng at each node can ether elmnate ether a par of branches or just one of them. The trees n whch the prunng elmnates the par of branches at a gven node are always solvable. The constructve proof uses the solvng method presented n the prevous subsecton, where the hand s dmensoned by consderng moton between pars of end-effectors. A tree pruned by pars conssts of consecutve R R seral chans when consderng relatve moton, and hence they are solvable at each step. Bnary hands n whch only one of the branches s pruned at a gven node, present a heterogeneous mx of 2R and 3R seral chans when consderng moton relatve to the end-effectors. Due to ths, some of the subgraphs wll be more constraned n ther moton than others, resultng n a tree that s not solvable for smultaneous tasks. For those pruned bnary tree that are solvable, use the same strategy and equatons derved to solve for complete bnary trees. can be found smlarly usng cos θ 1 2 cos θ 2 2 = 2(ŝ 1ŝ 2 ŝ 12 ) ˆp ŝ 12 ŝ 12, sn θ 1 2 cos θ 2 2 = 2(ŝ 12ŝ 2 ) ˆp ŝ 12 ŝ 12, cos θ 1 2 sn θ 2 2 = 2(ŝ 1ŝ 12 ) ˆp ŝ 12 ŝ 12, sn θ 1 2 sn θ 2 2 = 2ŝ 12 ˆp ŝ 12 ŝ 12. (8) 6 The 2 (1,1) Hand The 2 (1,1) hand has a two-jonted wrst and two fngers, each one consstng of a sngle revolute jont. Fgure 8 shows the tree graph for the hand and Fgure 9 shows the knematc sketch. 5.3 Synthess equatons for each branchng For each of the subproblems mentoned above, one of the several methods used to synthesze the spatal RR chan can be appled, for nstance [14], [15] or [16]. It s well known that the problem yelds sx solutons, wth only two of them beng real. The desgn equatons for the spatal RR chan used n ths work are as follows: consder the task poston, expressed as a dual quaternon, ˆP = ˆp + ε ˆp 0, where ˆp s the quaternon correspondng to the task rotaton. Smlarly, consder the jont axes as Ŝ k = ŝ k + εŝ 0 k = 0 + s k + ε(s 0 k + 0), a pure dual quaternon. Then the followng equatons can be wrtten, ŝ 1 ˆp ( ˆpŝ 2 ŝ 2 ˆp) = 0, ŝ 0 1 (ŝ 12ŝ 2 ) ˆp + ŝ 0 2 (ŝ 1ŝ 12 ) ˆp (ŝ 1 ŝ ŝ0 1ŝ2)ŝ 12 ˆp = 1 2 ˆp0 ŝ 12 ŝ 12, = 2,3, (7) where ŝ 12 = ŝ 1 ŝ 2 ŝ 2 ŝ 1, and the s the usual dot product. The frst equaton n set (7) s a scalar one, and the second s a pure quaternon equaton. These eght equatons can be solved for the Plucker coordnates of the axes by mposng Plucker constrants. One of the easest solutons s obtaned by fndng the soluton when ntersectng for nstance the z=1 plane. Ths system of equatons s easy enough to be solved usng Groebner bases. The nverse knematcs soluton for the jont varables Fg. 8. Graph of the 2-(1,1) hand Solvablty calculatons [10] show, as expected, that the hand s solvable for smultaneous tasks, for m = 3 fnte postons for each fngertp. In ths secton we show how we can reduce the rgd-body gudance problem for ths hand to that of the spatal RR chan, to obtan 4 real solutons. The desgn equatons for each fnger are created by equatng the forward knematcs equatons of each branch to the desred poston, [16]. In what follows, eŝ jθ j denotes the screw dsplacement of rotaton θ j about jont axes S j, regardless of the algebra used to compose the dsplacements. In ths case, absolute dsplacements from the fxed frame assocated to the root node are used, 0eŜ1θ 1eŜ2θ 2 ˆD 1 0 = ˆP 1 0eŜ1θ 1eŜ3θ 3 ˆD 2 0 = ˆP 2, = 1,2,3, (9) where ˆD j 0 represents the absolute dsplacement of the endeffector j at a reference confguraton, for each branch j. Notce that ˆP j ( ˆD j 0 ) 1 s the relatve dsplacement of branch j JMR , Perez-Graca, page 6

7 The set n Eq.(12) corresponds to the equatons obtaned for a spatal RR chan. Solve for the axes usng Eq.(7) to obtan two soluton for axes S 2 and S 3. Once those two solutons are found, the wrst axes can be solved as another spatal RR chan n the prevous sets of equatons, Eq.(11). Each soluton of the S 2 and S 3 axes wll yeld two solutons for the S 0 and S 1 axes, for a total of four hand desgns. 7 The 2-(1-(1,1),1-(1,1)) Hand Ths hand topology has two splttng stages and four fngertps. Fgure 10 shows the tree graph and Fg 11 shows the knematc sketch. Fg. 9. Knematc sketch of the 2-(1,1) hand from the reference confguraton up to poston ˆP j, expressed n the fxed frame, so that the set of equatons can be wrtten 0eŜ1θ 1eŜ2θ 2 = ˆP 1 ( ˆD 1 0) 1 0eŜ1θ 1eŜ3θ 3 = ˆP 2 ( ˆD 2 0) 1, = 1,2,3. (10) In partcular, f the reference confguraton s taken as the frst task poston, the frst equaton s the dentty and the system s reduced to reachng two relatve postons. Ths set of equatons could be solved n order to fnd the dmensons of fngers and palm. However the system can be smplfed to obtan an algebrac soluton. In order to smplfy the set of equatons to ts mnmum, we can wrte the forward knematcs of the common jonts (the wrst) as a functon of the rest of the chan and the task poston, for each branch j, 0eŜ1θ 1 = ˆP j ( ˆP j 1 ) 1 eŝ j+1θ j+1, = 2,3; j = 1,2, (11) where j ndcates the branch and denotes the task poston. The followng system of equatons s created by mposng equalty of the common jonts at the wrst, e Ŝ 2 θ 2eŜ3θ 3 = ( ˆP 1 ( ˆP 1 1 ) 1 ) 1 ˆP 2 ( ˆP 2 1 ) 1, = 2,3. (12) Ths equatons correspond to the change of reference frame to the frst end-effector n branch 1, creatng a seral chan startng at the frst end-effector and gong to the second one. The dsplacement ˆP 1 1 ( ˆP 1 ) 1 ˆP 2 ( ˆP 2 1 ) 1 corresponds to the relatve dsplacements from the frst poston to poston, measured n the frst end-effector frame. Fg. 10. Graph of the 2-(1-(1,1),1-(1,1)) hand The synthess for m = 3 fnte postons per fngertp follows the steps descrbed n Secton 5. The relatve postons to reach, 0eŜ1θ 1eŜ2θ 2eŜ4θ 4 = ˆP 1 1, 0eŜ1θ 1eŜ2θ 2eŜ5θ 5 = ˆP 2 1, 0eŜ1θ 1eŜ3θ 3eŜ6θ 6 = ˆP 3 1, 0eŜ1θ 1eŜ3θ 3eŜ7θ 7 = ˆP 4 1, = 2,3, (13) can be transformed nto two sets of equatons n order to dmenson the termnal edges for the relatve moton between them, e Ŝ 4 θ 4eŜ5θ 5 = ( ˆP 1 1) 1 ˆP 2 1, e Ŝ 6 θ 6 e Ŝ 7 θ 7 = ( ˆP 3 1 ) 1 ˆP 4 1, = 2,3. (14) Each of these sets of equatons are solved ndependently to obtan the jont axes S 4, S 5, S 6 and S 7 and the correspondng rotaton angles, usng the spatal RR soluton method. JMR , Perez-Graca, page 7

8 After that, the prevous splttng stage s solved wth e Ŝ k θ k e Ŝ k+1 θ k+1 = ( ˆP f 1 e Ŝ m θ m) 1 ˆP g 1 e Ŝ r θ r, = 2,3, k = 2 s 1,2 s 1 + 2,...,2 s 2, (18) where k enumerates the jonts correspondng to ths splttng stage, and we have some flexblty n the selecton of the end-effectors f and g and assocated jont axes m and r to be used to create the knematc chan contanng axes k and k + 1. Ths process s repeated up to the wrst jonts, whch can be solved usng any of the seral chans leadng to an end-effector, for nstance the frst one, 0eŜ1θ 1 = ˆP 1 1e Ŝ 2 s θ 2 s...e Ŝ 2 θ 2, = 2,3. (19) Fg. 11. Knematc sketch of the 2-(1-(1,1),1-(1,1)) hand Use the solutons to dmenson the jont axes of the frst branchng stage, selectng two of the four end-effector poses, e Ŝ 2 θ 2eŜ3θ 3 = ( ˆP 1 1e Ŝ 4 θ 4) 1 ˆP 6 1 e Ŝ 6 θ 6, = 2,3. (15) After ths, any of the four end-effector chans can be used to dmenson the remanng two axes, correspondng to the wrst, for nstance 0eŜ1θ 1 = ˆP 1 1e Ŝ 4 θ 4e Ŝ 2 θ 2, = 2,3. (16) The output of the process s a set of 16 canddate solutons. 8 The Bnary Hand wth n End-effectors The synthess presented n Sectons 6 and 7 can be generalzed for any number of splttng stages s. The relatve postons for the jonts ncdent on end-effectors are, for a bnary hand {0,1,1,2,2,3,3,...2 s 1,2 s 1} wth s splttng stages and 2 s fngertps, e Ŝ k θ k e Ŝ k+1 θ k+1 = ( ˆP f 1 ) 1 ˆP f +1 1, = 2,3, k = 2 s,2 s + 2,...,2 s+1 2, f = 1,3,...,b 1 (17) wth k enumeratng the end-effector jonts, from 2 s to 2 s+1 1, correspondng to end effectors ndcated by superscrpt f. These equatons are solved for the correspondng Plucker coordnates of the jont axes and for the jont angles, usng Eqs.(7) and (8). 8.1 Canddate soluton sortng and rankng An mportant post-synthess step n the desgn process s how to rank the 2 2s canddate solutons for a bnary hand of depth s + 1 n order to select the most suted soluton for the task. In fndng an optmal soluton, t s mportant to dstngush between requrements that depend on the dmensons of the jont axes, and requrements that are defned by the placement and shape of the lnks. There s some freedom n the lnk dmensonng and placement for spatal lnkages gven the poston of the jonts, so that lnks can be optmzed separately n a post-synthess, lnk-based optmzaton process [17]. Some of the requrements that can be targeted usng the lnk optmzaton are lnk and jont lengths and force transmsson at the jont level. Some other performance requrements are dependent on the poston of the jonts, such as sngulartes, and some others depend both on the poston of the jonts and the lnks, such as overall dmensons, self-ntersectons and obstacle avodance. For an optmal soluton, a possble strategy s to perform approxmate synthess wth addtonal performance requrements. Those requrements wll n general reduce the number of sutable canddates, so an mportant frst step s the knowledge of the number of potental desgns avalable to the desgner, whch s one of the results of ths work. In the case of mult-fngered robotc hand, nterestng performance requrements may nclude graspablty and manpulablty condtons, accordng to the task beng targeted. 9 Desgn Example The synthess results presented here for bnary hands allow followng several desgn strateges. One possble strategy arses from notcng that the dfference between the moton of two consecutve end-effectors happens at the termnal edges. The bnary hand can be desgned by pre-desgnng the relatve moton between those fngertps and usng the prevous jonts to locate that acton. In ths secton we present one example that locates two grppng end-effectors for a gven task. JMR , Perez-Graca, page 8

9 For ths example, the 1 (1 (1,1),1 (1,1)) topology s used, where the last branchng stages are pre-desgned and knematc synthess s appled for desgnng the frst branchng stage as well as the wrst jonts. The two pars of fngertps, 1 2 and 3 4, are desgned wth parallel jont axes. Fgure 12 shows the selected desgn and the computaton of the moton between both endeffectors expressed n the frame attached to one of them, for the case of branches 1 and 2. Table 2. Task postons for each fngertp Fnger j + ε(7.07 j ) j k + ε( j + 4.9k + 5.7) j 0.19k + ε( j k + 7.7) Fnger j + 0.5k + ε( j 6.0k 4.0) j k + ε( j?6.3k 3.7) j k + ε( j?3.1k + 1.5) Fnger ε( j) ε( j + 1.1k 1.7) j k + ε( j + 3.3k 3.6) Fnger j + ε( j 12.50k) j k + ε( j?15.0k + 2.6) j 0.014k + ε( j 16.0k 4.9) Fg. 12. Fngertp desgn and moton wth respect to Frame 1 The desgn process yelds four canddate desgns, and one of them s selected as the fnal desgn. Ths selecton s based on a quck modelng and smulaton of the four canddate solutons. Table 3 shows the Plucker coordnates of the axes and Fgure 13 shows the CAD model of the hand at the three dfferent postons. Table 1 shows the selected relatve moton between fngertps 1 and 2, and 3 and 4, expressed as the moton of fngertp 2 seen from fngertp 1, and the moton of fngertp 4 seen from fngertp 3. Table 2 shows the three postons for each of the four fngertps, as expressed n the fxed frame. Notce that, n ths case, the fxed frame s attached to the arm holdng the 2-dof wrst. Table 1. Relatve postons between fngertps Fngers 1 and j + ε( j) j 0.1k + ε( j 0.071k 0.071) j 0.1k + ε( j k ) Fngers 3 and j + ε( j) j k + ε( j) j k + ε( j 0.071k 0.071) Table 3. Plucker coordnates of the jont axes, accordng to the jont notaton n Fgure 11 Axs Plucker coordnates S 0 (0.44, 0.54,0.71) + ε(13.0,12.0,1.2) S 1 (0.08, 0.80,0.59) + ε(5.5,7.8,9.9) S 2 ( 0.2,0.005, 0.98) + ε( 13.0, 4.3,2.7) S 3 ( 0.99,0.03,0.14) + ε(2.6,2.5,18.0) S 4 (0.0,0.0,1.0) + ε(15.0,0.0,0.0) S 5 (0.0,0.0,1.0) + ε(15.0,4.0,0.0) S 6 (1.0,0.0,0.0) + ε(0.0,0.0, 12.68) S 7 (1.0,0.0,0.0) + ε(0.0,4.0, 12.68) 10 Conclusons Fractal hands present several splttng stages, each of them splttng n the same number of branches. Here we present the desgn of wrsted bnary hands, those mult- JMR , Perez-Graca, page 9

10 Bnary hands are one of the many famles of hand topologes that can be enumerated and solved for a specfc task or applcaton. Even though t has been proved that there s an nfnte number of possble desgns that can be syntheszed, some famles of hands can be fully characterzed, as n the case of wrsted bnary hands. These bnary hands may have applcatons n multple, smultaneous mn-manpulaton and n manpulaton of objects at dfferent scales, for whch the stages can be added n synchronous moton to create more szable fngers. The use of a systematc methodology for the desgn of mult-fngered hands may lead to optmzed end-effectors for specfc sets of tasks. Acknowledgements Ths work s supported by the Natonal Scence Foundaton under Grant No The content s solely the author s responsblty. Fg. 13. Fnal hand desgn wth fngertps reachng task postons 1, 2 and 3. The task postons are ndcated wth reference frames. fngered hands that splt n two branches at each stage. Some propertes of the bnary hands are derved and the solvablty of all possble topologes n ths famly s calculated. Desgn equatons are created for the general case, based on teratvely dmensonng the relatve moton of branches, startng at the end-effectors. It s also derved that other less symmetrc hand topologes can be created by prunng the bnary hands. The dmensonal synthess equatons are derved n partcular for hands wth one and two splttng stages, and also for the general case. One desgn example llustrates the synthess process. References [1] Stramgol, S., 2001, Modelng and IPC control of nteractve mechancal systems - A coordnate-free approach, vol. LNCIS 266, Sprnger. [2] Tschler, C., Samuel, A., and Hunt, K., 1995, Knematc chans for robot hands - 1. orderly number synthess, Mechansm and Machne Theory, 30(8), pp [3] Chen, I., Yang, G., and Kang, I., 1999, Numercal nverse knematcs for modular reconfgurable robtos, Journal of Robotc Systems, 16(4), pp [4] Garca de Jalon, J. and Bayo, E., 1994, Knematc and Dynamc Smulaton of Multbody Systems: The Real- Tme challenge, Sprnger-Verlag. [5] Jan, A., 2010, Graph-theory roots of spatal operators for knematcs and dynamcs, Proc. of the 2010 Internatonal Conference on Robotcs and Automaton, Anchorage, Alaska, USA, pp [6] Lee, J.-J. and Tsa, L., 2002, Structural synthess of mult-fngered hands, ASME Journal of Mechancal Desgn, pp [7] Smo-Serra, E., Moreno-Noguer, F., and Perez- Graca, A., August 29-31, 2011, Desgn of Nonanthropomorphc Robotc Hands for Anthropomorphc Tasks, ASME Desgn Engneerng Techncal Conferences, Washngton DC, USA. [8] Smo-Serra, E., Perez-Graca, A., Moon, H., and Robson, N., June 2012, Desgn of mult fngered robotc hands for fnte and nfntesmal tasks usng knematc synthess, Advances n Robot Knematcs, Innsbruck, Austra. [9] Smo-Serra, E. and Perez-Graca, A., 2014, Knematc synthess usng tree topologes, Mechansm and Machne Theory, 72 C, pp [10] Makhal, A. and Perez-Graca, A., June 2014, Solvable mult-fngered hands for exact knematc synthess, Advances n Robot Knematcs, Ljubljana, Slovena. JMR , Perez-Graca, page 10

11 [11] Selg, J. M., 2004, Geometrc Fundamentals of Robotcs (Monographs n Computer Scence), SprngerVerlag. [12] Tsa, L. W., 2001, Mechansm Desgn: Enumeraton of Knematc Structures Accordng to Functon, CRC Press, Boca Raton. [13] Ma, R. R. and Dollar, A. M., 2011, On dexterty and dexterous manpulaton, The 15th Internatonal Conference on Advanced Robotcs. [14] Mavrods, C., Lee, E., and Alam, M., 2001, A new polynomal soluton to the geometrc desgn problem of spatal rr robot manpulators usng the denavthartenberg parameters, ASME Journal of Mechancal Desgn, 123, pp [15] Perez-Graca, A. and McCarthy, J. M., 2003, Dmensonal synthess of bennett lnkages, ASME Journal of Mechancal Desgn, 125, pp [16] Perez-Graca, A. and McCarthy, J. M., 2006, Knematc synthess of spatal seral chans usng clfford algebra exponentals, Proceedngs of the Insttuton of Mechancal Engneers, Part C: Journal of Mechancal Engneerng Scence, 220(7), pp [17] Yhun, Y., Bosworth, K., and Perez-Graca, A., 2014, Lnk-based performance optmzaton of spatal mechansms, Journal of Mechancal Desgn, 136(12). JMR , Perez-Graca, page 11