Efficient and practical determination of grasping con gurations for anthropomorphic hands

E icien and p ac ical de e mina ion o g asping con igu a ions o an h opomo phic hands ? Josep-A nau Cla e ∗Ra´ul Su´a ez ∗ ∗Ins i u e o Indus ial and Con ol Enginee ing (IOC) Technical Uni e si y o Ca alonia (UPC), Ba celona, Spain (e-mail: josep.a nau.cla [email p o ec ed]c.edu, aul.sua [email protected]). Abs ac : The pape p esen s a me hodology o apidly sol e he in e se kinema ics o an h opomo phic hands, which is pa icula ized o a mechanical hand conside ing 27 deg ees o eedom. Gi en he con ac poin s and no mal di ec ions on an objec su ace, he p oposed algo i hm inds he join alues and he w is posi ion and o ien a ion ha make he inge ips sa is y he con ac cons ain s. The app oach combines an i e a i e algo i hm wi h an o - line analysis ha allows signi ican educ ions o he execu ion ime. The app oach has been implemen ed and he pape includes applica ion examples. The e ec i eness and as execu ion o he algo i hm is demons a ed wi h s a is ical esul s. Keywo ds: obo ics, obo kinema ics, in e se kinema ic p oblem, obo ic manipula o s 1. INTRODUCTION Robo ics is a echnology applied in di e en scena ios: medical assis ance, indus y, space explo a ion, among se e al o he s. A obo can also ha e di e en objec i es in a ce ain en i onmen like o ins ance inspec ion, lo- ca ion, anspo ing o manipula ion. These wide ange o applica ions in ol e a e y la ge numbe o obo ic ac ions ha need o physically in e ac wi h he en i onmen and, in pa icula , need o g asp and manipula e di e en objec s. Robo hands, as hei e sa ili y is e y high, a e one o he mos adap able ools o g asping objec s. The ad ances in he de elopmen s o obo hands a e signi i- can (Bicchi, 2000), bu hey ha e some associa ed p ob- lems ha need be e solu ions han he cu en exis ing ones. One o hese p oblems is he g asp planning, whe e he i s decision is he selec ion o he desi ed ype o g asping (Cu kosky, 1989): powe g asp, closing he hand a ound he objec wi hou knowing he inal con ac poin s be ween he hand and he objec ; o p ecision g asp, whe e he con ac poin s a e known on he objec and ake place only on he hand inge ips. Many wo ks we e ocused on inding app op ia e con ac poin s on he objec (e.g. o 2D objec s: (Nguyen, 1988) (Pa k and S a , 1990) (Liu, 1998) (Co nell´a and Su´a ez, 2009), and o 3D objec s: (Ponce e al., 1997) (Bo s e al., 1999) (Li e al., 1989) (Polla d, 2004) (Roa and Su´a ez, 2009)), bu he e a e no gene al o mula ions o sol e p ecision g asp including he kinema ics cons ain s o a gi en hand. Sol ing he hand in e se kinema ic is an in e es ing p oblem, ha is, he sea ch o an app op ia e se o join alues o a obo hand ha sa is ies he cons ain s imposed by some con ac poin s (Rosell e al., 2005) (Rosales e al., 2011) (Su´a ez and Cla e , 2009). The main di icul y o his p oblem ?This wo k was pa ially suppo ed by he Spanish Go e nmen h ough he p ojec s DPI2010-15446 and DPI2008-02448 is o quickly ind a alid hand con igu a ion in he e y high dimensional space de ined by he hand join s. The app oach p oposed in his wo k uses he hand Jacobian o i e a i ely ind hand con igu a ions close o he desi ed cons ain s imposed by he con ac poin s on he objec s, oge he wi h a s a is ical s udy o selec ini ial hand con igu a ions ha speeds up he i e a i e p ocedu e. 2. PROBLEM DESCRIPTION The objec i e o his wo k is o ind a eachable hand con igu a ion ha sa is ies he cons ain s imposed by he desi ed con ac poin s on he objec using he inge ips, i.e. he inge ips mus be p ope ly loca ed and o ien ed, he inge join s mus be wi hin he co esponding ange, and he e mus be no collision among he hand elemen s (palm and inge s). Checking o collisions be ween he hand and he en i onmen is ou side he scope o his wo k. The mechanical hand used in his wo k is he Schunk An h opomo phic Hand (SAH) shown in Fig. 1. This hand is an h opomo phic and has ou inge s ( humb, index, medium and ing inge s). Each inge has ou join s (Fig. 2a): wo independen (join s 1 and 2, as abduc ion and lexion, espec i ely) and wo coupled (join s 3 and 4, bo h lexion), which makes h ee independen deg ees o eedom (DoF). The humb has an ex a join in he base (join 0). Then, he o al numbe o DoF o he hand is 19, 13 om he inge s plus 6 om he hand w is mo emen s. The con ac poin s on he hand mus be on he p ope egion o each inge ip. Each inge ip is conside ed sphe - ical, and he accep ed con ac egions a e shown in Fig. 2b. Two pa ame e s a e needed o iden i y a con ac poin on each inge ip, meaning he exis ence o 2 addi ional i ual DoF pe inge . Then, he o al numbe o DoF o he hand sys em is 27, 19 om he inge join s and he w is plus 8 om he inge ips. Fig. 1. Mechanical hand SAH assembled on an indus ial obo a m. Fig. 2. a) Hand and inge join s; b) Con ac egion on he inge ip su ace. A con ac be ween a inge ip and a poin on he objec bounda y imposes 5 cons ain s: a inge ip poin mus coincide wi h he poin on he objec (3 pa ame e s) and he no mal o he inge ip mus coincide wi h he su ace no mal a he objec con ac poin (2 addi ional pa ame e s). The assignmen be ween each inge and i s con ac poin on he objec is assumed o be known; i his is no he case, all he possible combina ions should be checked un il a solu ion is ound o no solu ion a all can be de e mined. 3. PROPOSED SOLUTION The p oposed app oach has wo pa s, one is based on a adi ional i e a i e algo i hm ha , s a ing om a gi en ini ial hand con igu a ion, uses he hand Jacobian o de e mine i ual mo emen s o he hand ha y o sa is y he con ac cons ain s on each inge ip. The o he pa is an o -line s udy o de e mine a sequence o hand con igu a ions ha wo ks well as ini ial con igu a ions in he i e a i e algo i hm. These wo pa s a e desc ibed in de ail in he ollowing wo subsec ions. 3.1 I e a i e Algo i hm The i e a i e algo i hm has o be execu ed each ime i is necessa y o de e mine a hand con igu a ion sa is ying some con ac cons ain s on he objec su ace. The al- go i hm has wo loops, he i s (ou e ) loop is de o ed o change he ini ial con igu a ion o he hand i no solu ion is ound wi h he cu en selec ed one, and he second (inne ) loop is de o ed o he sea ch o hand i ual mo emen s ha i e a i ely change he hand con igu a ion om he ini ial o a inal one sa is ying he con ac cons ain s. The i e a i e algo i hm is o mally desc ibed as ollows. Le : •imax be he maximum numbe o ini ial con igu a- ions. •kmax be he maximum numbe o jacobian i e a ions. •Ckbe he hand con igu a ion in he k- h jacobian i e a ion. •Pkbe he con ac cons ain s on he inge ips in i e a ion k. •P∗be he con ac cons ain s on he objec , i.e. he desi ed inal con ac cons ain s o he inge ips. • he subindices I,M,Rand Tindica e he inge s index, middle, ing and humb, espec i ely. •pi,i∈ {I, M, R, T}, be he posi ion o he con ac poin on inge i. • i,i∈ {I, M, R, T}, be he posi ion o he cen e o he inge ip i. Algo i hm 1 I e a i e Algo i hm 1: o i= 1 o imax do 2: (C1,P1)←Ob ain Ini ial Con (i, P∗) 3: k←1 4: lag collision ←False 5: lag p og ess ←T ue 6: while k≤kmax and lag collision =False and lag p og ess=T ue do 7: i Pk≃P∗ hen 8: i Check Collisions(Ck) = False hen 9: e u n (Ck) 10: else 11: lag collision ←T ue 12: end i 13: else i k≥nand Pk−n+1 ≃. . . ≃Pk hen 14: lag p og ess ←False 15: end i 16: (Ck+1,Pk+1)←Comp Nx C(Ck,Pk,P∗) 17: k←k+ 1 18: end while 19: end o 20: e u n (“No solu ion”) The main unc ions and s eps o Algo i hm 1 a e he ollowing. Func ion “Ob ain Ini ial Con (i, P∗)” (S ep 2) This unc ion e u ns an ini ial hand con igu a ion based on he numbe i−1 o ini ial con igu a ions al eady used and he gi en desi ed con ac cons ain P∗. The e e ence sys em used o desc ibe piand ihas he o igin a he humb con ac poin , pT, and he z-axis is he ec o o T−pT | T−pT|; he x- and y-axis a e chosen andomly o comple e an o hono mal basis. The se (pi, i) ully de ines he con ac cons ain s o inge i, i.e. he con ac poin posi ion on he inge ip and he di ec ion no mal o he inge ip a he con ac poin . No e ha knowing pionly wo pa ame e s a e needed o de e mine i, since kpi− ikis a cons an dis ance ( he adius o he sphe ical inge ip). Now, he se o con ac cons ain s on he inge ips can be gene ically exp essed, o a hand con igu a ion Ck, as: Pk= (pI, I,pM, M,pR, R), (1) whe e o simplici y he subindex kis no included in each componen o Pk; analogously, using he sup aindex ‘*’ o indica e desi ed alues, he desi ed con ac cons ain s on he inge ips a e: P∗= (p∗ I, ∗ I,p∗ M, ∗ M,p∗ R, ∗ R). (2) No e ha he humb con ac cons ain s a e no included in Pk(i.e. pTand T) no in P∗(i.e. p∗ Tand ∗ T); his is because he ini ial hand con igu a ion is chosen such ha he humb con ac always sa is ies i s co esponding con ac cons ain s, as i is explained immedia ely below. The 27 DoF ixing an ini ial hand con igu a ion C1a e compu ed as ollows: (1) The hand is ini ially posi ioned such ha he con- s ain s imposed by he humb con ac a e sa is ied. This is done by imposing he condi ions: pT=p∗ T, T= ∗ T. (3) This is equi alen o i e independen cons ain s, so he e a e s ill 27-5=22 DoF o be ixed in o de o de e mine comple ely he hand con igu a ion. (2) The join alues o he hand (bo h he mechanical and he i ual DoF de ailed in Sec ion 2) a e ixed ollowing a p ede e mine sequence o hand poses Q gi en by he 25-dimensional ec o Q= (φT0, . . . , φT6, φI1, . . . , φI6, φM1, . . . , φM6, φR1, . . . , φR6), (4) whe e o each componen he i s subindex iden i ies he inge and he second subindex iden i ies he inge join . In he la es case, alues 0 o 4 iden i y he inge eal join s (Fig. 2a), and alues 5 and 6 iden i y he wo i ual join s de ining he con ac poin on each inge ip (Fig. 2b); no e ha subindex 0 exis s only o he humb, acco ding o he hand s uc u e desc ibed in Sec ion 2. The gene a ion o a p ope sequence o poses Qi, i= 1, ..., imax equi ed o gene a e he imax hand con igu a ions o Algo i hm 1 is one o he key poin s o his wo k and is de ailed below in Sec ion 3.2. Each hand pose ixes he 25 join alues, bu since he 3 d and 4 h join o each inge a e coupled (Fig. 2a) he e a e 4 join s (one pe inge ) ha a e no independen and he e o e only 25-4=21 DoF a e ac ually ixed. Then, he e is 22-21=1 DoF le o de ine comple ely he hand con igu a ion. (3) The emaining deg ee o eedom co esponds o he o a ion, ψ, o he hand a ound he di ec ion no mal o humb con ac poin (i.e. no mal o pT− T), and i is ixed loca ing he hand such ha he index and ing inge s a e well o ien ed wi h espec o hei expec ed inal posi ions, which is done as ollows. Le ΠTbe he plane o hogonal o pT− Tcon aining he humb con ac poin pT, and le 1and 2be he p ojec ions on ΠTo he ec o s p∗ R−p∗ Iand pR−pIon ΠT espec i ely. Now, ψis selec ed such i minimizes he angle be ween 1and 2. In his way all he 27 DoF o he hand a e ixed. Then, he ini ial hand con igu a ion can be w i en as C1= (pT, T,C01) (5) whe e C01= (ψ, φT0, . . . , φT6, φI1, . . . , φI6, φM1, . . . , φM6, φR1, . . . , φR6). (6) C1is a ec o wi h 32 elemen s bu only 27 o hem a e independen , ep esen ing he 27 DoF o he hand in he wo kspace, and C01has 26 elemen s wi h only 22 o hem being independen . No e ha P1can be compu ed di ec ly by sol ing he di ec kinema ics o he hand a C1. Func ion “Check Collisions(Ck)” (S ep 8) This unc ion checks i he e a e collisions be ween he elemen s o he hand ( inge s and palm) o he hand con- igu a ion Ck, e u ning T ue i so o False o he wise. Func ion “Comp Nx C(Ck,Pk,P∗)” (S ep 16) This unc ion compu es a new hand con igu a ion om he cu en one, Ck, and he desi ed con ac cons ain s P∗. Once pTand Ta e gi en (no e ha hey a e cons an ∀Ck), C0khas all he in o ma ion needed o ully know he hand con igu a ion, and since i has a smalle dimension i will be used o compu e he nex hand con igu a ion in he i e a i e algo i hm. Le : •∆P0=α(P∗0−P0 k), whe e P∗0= (P∗,0,0,0,0), P0 k= (Pk,0,0,0,0) ( he eason o adding hese ze os will become e iden below), and αis a cons an alue empi ically de e mined o ob ain a good con e gence o he algo i hm. No e ha P∗0and P0 ka e ex ended ec o s o dimension 22, and so is ∆P. •∆C0=C0k−C0k−1.C0kand C0k−1a e ec o s o dimension 26, and so is ∆C0. •Jbe he hand Augmen ed Jacobian (Siciliano and Kha ib, 2008) ha is ob ained by adding o he s anda d hand Jacobian ou addi ional ows ha include he coupling cons ain s be ween join s 3 and 4, i.e. φi3=φi4wi h i={T, I, M, R}. Each o hese ows is o he ype (0, ..., 1,−1, ..., 0), i.e. a ow o ze os wi h he excep ion o he posi ions co esponding o he join s φi3and φi4in C0k o each o he ou inge s. No e ha Jis a ma ix wi h dimension 22×26. The e ec o he ows added in Jand he ze os added in ∆P0makes ha in he ela ion ∆P0=J∆C0 he elemen s 19 o 22 become 0 = φi3−φi4 o each i, i.e. φi3=φi4. Now, ∆C0can be app oxima ed as, ∆C0=J+∆P0, (7) whe e J+is a pseudoin e se o J. Then, he nex con igu a ion in he i e a i e p ocedu e is simply compu ed as, C0k+1 =C0 k+ ∆C0, (8) and Ck+1 is ob ained om C0k+1,pTand T; inally Pk+1 is compu ed om Ck+1 using he hand di ec kinema ics. Ending condi ions (S eps 1, 6, 7 and 13) One o he ollowing ending condi ions mus be sa is ied o inish he i e a i e sea ch algo i hm: (1) imax ini ial con igu a ions C1ha e been es ed wi h- ou inding a solu ion, i.e. i=imax (S ep 1). (2) The hand con igu a ion sa is ies he desi ed con ac poin cons ain s on he objec su ace (S ep 7), i.e. Pk∼ =P∗, wi h a hand con igu a ion wi hou sel - collisions (S ep 8). This is checked e i ying ha in he i e a ion k he ollowing condi ions a e sa is ied o i∈ {I, M, R}, kp∗ i−pik< dmin,k ∗ i− ik< dmin,(9) whe e dmin is a p ede ined cons an pa ame e ; and one o he ollowing ending condi ions mus be sa is- ied o exi he inne loop o he i e a i e sea ch algo i hm: (1) A numbe kmax o i e a ions ha e been compu ed wi hou inding a solu ion (S ep 6). (2) The hand con igu a ion does no p og ess enough du - ing a p ede ined numbe no consecu i e i e a ions. This is checked e i ying ha he ollowing condi ion is sa is ied du ing nconsecu i e i e a ions o k o i∈ {I, M, R}(S ep 13): 4 X 1 (kpik−pik−1k+k ik− ik−1k)< smin,(10) whe e smin is a p ede ined cons an pa ame e . 3.2 De e mina ion o he Ini ial Con igu a ions Sequence As men ioned in Subsec ion 3.1, a key poin o he ap- p oach is he de e mina ion o a p ope sequence o ini ial hand con igu a ion in S ep 2 o Algo i hm 1 ( unc ion “Ob ain Ini ial Con (i, P∗)”). A sequence o ini ial con- igu a ions Cis equi alen o a sequence o ini ial hand poses Q, which is de e mined as ollows. A la ge enough se So hand poses samples a e an- domly gene a ed and a P incipal Componen Analysis (PCA) (Jolli e, 2002) is used o ind he di ec ion in he hand wo king space wi h la ge dispe sion o samples. This is done by compu ing he eigen alue decomposi ion o he co a iance ma ix o he samples (a e a mean cen- e ing each da a a ibu e) and selec ing he eigen ec o co esponding o he la ges eigen alue. Repea ing his p ocedu e, a new base o he hand wo kspace space is ob ained, wi h he ec o s in his base o de ed acco ding o a dec easing dispe sion along each di ec ion. Taking he i s n ec o s o his base, i is possible o de ine a subspace ha app oxima es he hand wo kspace wi h a mo e ac able lowe dimension n. This p ocedu e is e y o en used o educe he dimension o mul idimen- sional da a se s, and was al eady used o educe he hand wo kspace in wo ks dealing wi h g asp sea ching (San ello e al., 1998) (Tsoli and Jenkins, 2007) (Cioca lie and Allen, 2009) (whe e he se o sampled is composed o g asping poses, and he di ec ions o he base a e called eigeng asps), wi h he syn hesis o human-like mo ions in g aphic applica ions (Sa ono a e al., 2004), and wi h mo ion planning o a hand-a m sys em (Rosell e al., 2007) (whe e he se o samples is ob ained by mapping poses o he ope a o hand du ing uncons ained mo emen s, Fig. 3. Ini ial hand poses. and he di ec ions o he base a e called p incipal mo ion di ec ions). In his wo k he i s wo ec o s o he base o he hand wo kspace ob ained wi h he PCA desc ibed abo e we e selec ed o gene a e he ini ial hand con igu a ions in Algo i hm 1. Each o hese wo ec o s indica es a di ec ion in he hand wo kspace ha co esponds o a coo dina ed mo ion o all he join s in a single DoF. Then, wi h only wo pa ame e s, λ1 iand λ2 i, i is possible o a y hese wo DoF and de e mine he ini ial hand pose Qiin a 2-dimensional space, i.e. he dimension o he subspace whe e Qis de e mined is educed om 21 o 2, ying o co e as much as possible o he hand wo kspace. Le : •mbe he mean o he se o samples Qi∈ S. •cj,j= 1,2, be he uni a y ec o s along he selec ed di ec ions o he hand wo kspace. •σj,j= 1,2, be he s anda d de ia ion o he se o samples Salong cj. •λ1 i,λ2 ibe wo eal alues be ween 0 and 1. Then, he i- h ini ial hand con igu a ion Qiin he se- quence is compu ed as he ollowing unc ion o he alues λ1 iand λ2 ias: Qi=m+ 3 σ1(2λ1 i−1) c1+ 3 σ2(2λ2 i−1) c2. (11) No e ha o λ1 i=λ2 i= 0.5 esul s Qi=m, and ha λ1 i, λ2 i∈ {0,1}p oduce ex eme poses a ±3σiin he conside ed 2-dimensional subspace de e mined by c1 and c2( his co e s 99% o he dispe sion o Son he 2- dimensional subspace). Some ini ial hand poses a e shown in Fig. 3 o di e en alues o λ1 iand λ2 i. Thus, de e mining a sequence o ini ial poses is equi alen o de e mine a sequence SEQ o λi= (λ1 i, λ2 i), which can be done o -line o a pa icula hand using Mon e Ca lo simula ions o look o a sequence ha allows a good pe o mance o Algo i hm 1. A sequence SEQ o imax poses is de e mined as ollows. (1) Disc e ize he domains o λ1 iand λ2 iin o a ini e and uni o mly dis ibu ed se o N alues λ1 ijand λ2 ik, espec i ely ( his gene a es N2po en ial duplas λijk = (λ1 ij, λ2 ik), wi h j, k = 1, ..., N). (2) SEQ =∅ Fig. 4. Eijk , i ={1, ..., 6} o he SAH hand ( he da k ed colo co esponds o he highes alues). (3) Fo i= 1 o imax do: (a) Gene a e a la ge enough se Po M andom con ac cons ain s P∗ l ha canno be sol ed wi h any λhjk ∈SEQ, i.e. he al eady selec ed duplas λhjk ∀h<i( his means ha no solu ion was ound o ∀P∗ l∈ P using he alues o λhjk al eady included in SEQ). (b) Fo each λijk ,j, k ∈ {1, ..., N}do: (i) Ob ain he ini ial hand con igu a ion Cijk using λijk in Eq. (11) and he unc ion Ob ain Ini ial Con (i, P∗) (Sec ion 3.1). (ii) Use Algo i hm 1 wi h he only ini ial con ig- u a ion Cijk o look o a hand con igu a ion sa is ying each cons ain P∗ l∈ P and sa e he a e o success E1jk , de ined as he pe - cen age o he Mcons ain s P∗ l ha we e sol ed using Cijk . (c) Selec as he i- h elemen o he sequence he alue λijk wi h associa ed highes a e o suc- cess, i.e. add λiαβ o SEQ, wi h λiαβ such ha Eiαβ ≥Eijk ∀j, k. (4) Re u n(SEQ) Fig. 4 shows he alues o E1jk o E6jk ob ained in he selec ion o λ1 o λ6 o he SAH hand wi h M= 5000 and N= 30. 4. EXAMPLES AND PERFORMANCE 4.1 Examples The app oach has been implemen ed in C++. The pa- ame e alues used in Algo i hm 1 we e imax = 12, kmax = 100, dmin = 1.5 mm and smin = 1 mm. A sequence o 12 ini ial hand poses has been compu ed using pa allel compu ing on ou een egula PC CPUs using he MPI lib a y (MPI, 2010). The ob ained alues o λ1 o λ12 a e shown in Table 1, and Fig. 5 shows he six i s ini ial con igu a ions ob ained wi h λ1 o λ6. A i s applica ion example is shown in Fig. 6. I is he g asp o a o k ha needed 1 ini ial con igu a ion iλ1 iλ2 iiλ1 iλ2 iiλ1 iλ2 i 1 0.517 0.483 5 0.690 0.448 9 0.690 0 2 0.793 0.552 6 0.138 0.241 10 0.690 0.448 3 0.310 0.690 7 0.034 0.827 11 0.724 0.276 4 0 0.517 8 0.655 0.241 12 0.276 0.827 Table 1. Values o λi= (λ1 i, λ2 i), i={1, ..., 12}. Fig. 5. Fi s six ini ial hand con igu a ions (o de ed om le o igh and om op o bo om). and 6 jacobian i e a ions o sol e ha pa icula in e se kinema ic p oblem. No e ha he e olu ion o he hand con igu a ion is e y signi ican in he i s i e a ions, while in he las i e a- ions he hand con igu a ions a e a he simila (Fig. 6b). No e also ha in he second o he in e media e hand con igu a ions he e a e collisions be ween he ing and middle inge s, his is no a p oblem a all since he only con igu a ion ha mus no ha e collisions is he inal one. Ano he h ee examples a e illus a ed in Fig. 7, showing wo iews o he objec wi h he con ac cons ain s and wo iews o he inal g asp con igu a ion sa is ying he cons ain s. The solu ion o he cup in Fig. 7a needed 1 ini ial con igu a ion and 13 jacobian i e a ions, o he s a ue e o buddha in Fig. 7b needed 3 ini ial con igu a- ions and 25 jacobian i e a ions, and o he cup o co ee in Fig. 7c, 7 ini ial con igu a ions and 38 jacobian i e a ions. A inal example wi h a can is shown in Fig. 8, wi h he con ac cons ain s in Fig. 8a (side and op iews) and he inal con igu a ion in Fig. 8b, whe e i can be seen he esul ing g asp o he SAH on a can in he IOC Robo ics Lab wi h he join alues o he mechanical hand se a he solu ion con igu a ion. The solu ion o he can needed 1 ini ial con igu a ion and 22 jacobian i e a ions. 4.2 Pe o mance The success pe cen age and he a e age execu ion ime pe ini ial con igu a ion gene a ed has ben calcula ed (in a AMD A hlon 64 X2 Dual Co e P ocesso 5400+ a 2800 MHz). The esul s can be seen in Fig. 9a. See ha wi h one ini ial con igu a ion a ound he 62% o all in e se kinema ics p oblems a e sol ed; wi h wo, a ound he 80%; un il a 97%, using wel e (o cou se, a highe pe cen age could be sol ed by gene a ing mo e han wel e ini ial Fig. 6. a) The desi ed con ac poin s wi h and wi hou he o k ( he posi ion and o ien a ion cons ain s a e gi en by he cen e o each disk and he a ached cylinde espec i ely); b) The ini ial hand con igu- a ion ( i s image) and he he hand con igu a ions ob ained in he nex i e jacobian i e a ion; c) Views o he inal g asp con igu a ion a e 6 jacobian i e a- ions ( he i s wo images ema k he con ac poin s). Fig. 7. Th ee examples wi h wo iews o he con ac con- s ain s on he objec and wo iews o he solu ion. con igu a ions, wi h he p e ious o -line compu a ion o he co esponding λi). The a e age execu ion ime o one ini ial con igu a ion is 5 ms, 7.5 ms o wo, and 15 ms i wel e. Wi h bo h measu es i is easy o see ha he a e age execu ion ime o sol e a 62% o all cases is 5 ms, and so on, as shown in Fig. 9b. Fig. 8. a) Two iews o he con ac cons ain s on a can, and, b) ob ained solu ion in simula ion, and execu ion o he solu ion wi h he eal hand SAH. In o de o ha e a ce ain mesu e o he e iciency o he p oposed app oach wi h he o -line wo k (Sec ion 3.2), he esul s shown ha e been compa ed wi h he ones ob ained by gene a ing andom ini ial con igu a ions; his means ha a any ime ha Algo i hm 1 eaches S ep 2, he ini ial con igu a ion has been compu ed by assigning andom alues o he λi. The andom esul s co espond o he blue lines in Fig. 9a and 9b. Two main conclusions can be d awn om Fig. 9a. Fi s , he success a es o he andom ini ial con igu a ions a e always in e io han he ones ob ained wi h he o -line wo k; in he wo s case he di e ence is a 16%, while in he bes , a ound a 3%. Second, he ime o he andom ini ial con igu a ions is always be ween a 50% o 67% slowe . In Fig. 9b he compa a i e be ween he wo me hods can be mo e easily done; o ob ain a 62% o exi pe cen age he o -line ini ial con igu a ions need 5 ms, while he andoms spend 12 ms; o ob ain a 94%, o -line con igu a ions ake 12 ms, and he andom, 24 ms. Tha is, he o -line algo i hm makes he p oposed solu ion om a 50% o a 58% as e ; he e o e, i doubles he compu a ion speed. 5. CONCLUSIONS AND FUTURE WORK This pape p esen s a p ocedu e o ob ain a g asp con ig- u a ion o he Schunk An opomo phic Hand when a se o con ac cons ain s is gi en ( o ins ance, by a g asp planne ). S a is ically, he app oach sol es a ound 97% o he cases in a e y easonable ime, and he pe cen age can be imp o ed i inc easing he execu ion ime is accep able. The app oach combines a classical use o he Jacobian in an i e a i e algo i hm wi h an o -line p ocedu e ha dou- bles he execu ion speed o he algo i hm. The app oach was implemen ed and he esul s a e sa is ac o y. Fu u e wo k includes conside ing he po en ial collisions o he hand wi h he objec s in he wo k en i onmen and he inclusion o he implemen ed p ocedu e in a gene al hand mo ion planne . Ano he in e es ing p oposals o speed up he app oach a e he gene a ion o a p ope ini ial hand a) b) Fig. 9. S a is ical pe o mance o he app oach. Red lines co espond o esul s ob ained wi h ini ial hand con- igu a ions ob ained wi h he p oposed o -line wo k (Sec ion 3.2), and blue lines o esul s ob ained wi h andom ini ial hand con igu a ions: a) Success pe - cen age (con inuous lines) and execu ion ime (dashed lines) s. he numbe o ini ial con igu a ions gene - a ed; b) Success pe cen age s. execu ion ime. con igu a ion as a unc ion o he cons ain s imposed by he con ac poin s on he objec , hus i would change depending on he pa icula p oblem o be sol ed, and he use o lea ning me hods o imp o e he ini ial hand con igu a ions based on he esul s o eal applica ions. ACKNOWLEDGMENTS The au ho s would like o hank Leo Palomo o his sup- po on he so wa e, and Jan Rosell, Ca los Rosales and Jose Fo ´ın o hei hough s and help in he expe imen s. REFERENCES A. Bicchi, (2000). Hands o dex e ous manipula ion and powe ul g asping: a di icul oad owa ds simplic- i y. IEEE T ans. on Robo ics and Au oma ion, 16 (6), pp. 652-662. C. Bo s , M. Fische and G. Hi zinge , (1999). A as and obus g asp planne o a bi a y 3d objec s. In Robo ics and Au oma ion. P oceedings, IEEE In e na- ional Con e ence on, pp. 1890-1896. Cioca lie MT, Allen PK, (2009). Hand pos u e subspaces o dex e ous obo ic g asping. The In e na ional Jou - nal o Robo ics Resea ch, 28 (7), pp. 851-867. Co nella, J. i R. Su´a ez. E icien de e mina ion o 4-poin o m-closu e op imal cons ain s o polygonal objec s, (2009). IEEE T ansac ions on Au oma ion Science and Enginee ing, (ISSN: 1545-5955), Volume 6, Issue 1, Jan. 2009, pp. 121-130. M.R. Cu kosky, (1989). On g asp choice, g asp models, and he design o hands o manu ac u ing asks. IEEE T ans. on Robo ics and Au oma ion, 5 (3), pp. 269-279. I.T. Jolli e, (2002). P incipal Componen Analysis. Sp inge Se ies in S a is ics, Sp inge . Z. Li, P. Hsu and S. Sas y, (1989). G asping and coo - dina ed manipula ion by a mul i inge ed obo hand. The In e na ional Jou nal o Robo ics Resea ch, 8 (4), pp. 33-50. Y.H. Liu, (1998). Compu ing n- inge o ce-closu e g asps on polygonal objec s. In Robo ics and Au oma ion. P o- ceedings, IEEE In e na ional Con e ence on, pp. 2734- 2739. MPI Lib a y. www.mcs.anl.go / esea ch/p ojec s/mpi/. June 2010. V.D. Nguyen, (1988). Cons uc ing o ce-closu e g asps. The In e na ional Jou nal o Robo ics Resea ch, 7 (3), pp. 3-16. Y.C. Pa k and G.P. S a , (1990). G asp syn hesis o polygonal objec s. In Robo ics and Au oma ion. P o- ceedings. IEEE In e na ional Con e ence on, pp. 1574- 1580. N.S. Polla d, (2004). Closu e and quali y equi alence o e icien syn hesis o g asps om examples. The In e na- ional Jou nal o Robo ics Resea ch, 23 (6), pp. 595-613. J. Ponce, S. Sulli an, A. Sudsang, J.D. Boissonna and J.P. Me le , (1997). On compu ing ou - inge equilib ium and o ce-closu e g asps o polyhed al objec s. The In- e na ional Jou nal o Robo ics Resea ch, 16 (1), pp. 11- 35. M. A. Roa i R. Su´a ez, (2009). Finding locally op imum o ce-closu e g asps. Robo ics and Compu e -In eg a ed Manu ac u ing, (ISSN: 0736-5845), Volume 25, Issue 3, June 2009, pp. 536-544. C. Rosales, Ll. Ros, J. M. Po a i R. Su´a ez, (2011). Syn hesizing g asp con igu a ions wi h speci ied con ac egions. In e na ional Jou nal o Robo ics Resea ch, 30 (4), Ap il. J. Rosell, X. Sie a, L. Palomo and R. Su´a ez, (2005). Finding g asping con igu a ions o a dex e ous hand and an indus ial obo . P oceedings o he 2005 IEEE In e na ional Con e ence on Robo ics and Au oma ion, pp. 1178-1183. J. Rosell, R. Su´a ez, C. Rosales, J.A. Ga cia and A. Pe ez, (2009). Mo ion planning o high DOF an h opomo phic hands. IEEE In e na ional Con e ence on Robo ics and Au oma ion, pp. 4025-4030. Sa ono a A., Hodgins J.K. and Polla d N.S., (2004). Syn hesizing physically ealis ic human mo ion in low- dimensional, beha io -speci ic spaces. ACM T ansac- ions G aph, 23 (3), pp. 514-521. San ello M. and Flande s M., (1998). Soech ing JF pos u- al hand syne gies o ool use. Jou nal o Neu oscience, 18 (23), pp. 10,105-10,115. B. Siciliano and O. Kha ib, (2008). Sp inge Handbook o Robo ics. Sp inge , pp. 257-258. R. Su´a ez and J.A. Cla e , (2009). Sea ching a alid hand con igu a ion o pe o m a gi en g asp. 40 h In e na- ional Symposium on Robo ics, pp. 151-156. Tsoli A. and Jenkins O.C., (2007). 2D subspaces o use - d i en obo g asping. In: P oc. o he RSS 2007 Wo k- shop on Robo Manipula ion: Sensing and Adap ing o he Real Wo ld.