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.