GRASP PLANNING & FORCE COMPUTATION FOR DEXTROUS OBJECT MANIPULATION WITH MULTI-FINGER ROBOT HANDS

GRASP PLANNING & FORCE COMPUTATION FOR DEXTROUS OBJECT MANIPULATION WITH MULTI-FINGER ROBOT HANDS Günter Wöhlke INTERNATIONAL COMPUTER SCIENCE INSTITUTE 1947 Center St., Sute 600, Berkeley, Calfrna 94704-1198, USA Techncal Reprt TR-94-018 (revsed versn) August 1994 Abstract Ths paper deals wth the prblems f grasp plannng and frce cmputatn that ccur when bjects have t be manpulated wth dextrus mult-fnger rbt hands. Assumng that the ntal cntact pnts at the bject surface are pre-selected, the jnt mtns t perfrm a desred peratn accrdng t a gven bject trajectry are determned. Ths s dne under the cnsderatn f the effect f rllng and slppng f the fngertps. Anther pnt s the cmputatn f apprprate grasp frces (nternal frces) accrdng t gven bject frces/mments needed t ensure a stable and secure grp cnfguratn frm whch the jnt trques are derved. Ths leads t an ptmzatn prblem that can be slved wth tw dfferent appraches based n search prcedures fr fndng the maxmum (best fttng value). As majr result, the frce ptmzatn prblem culd be generalzed fr the case f arbtrary rbt hands and cntact stuatns that are specfed by the fllwng parameters: number f fngers, number f jnts per fnger jn, chsen cntact mdels cmd and null space dmensn ndm. Especally, the resultng bject mtns and cntact frces are demnstrated at a smulated example f a peg-n-hle nsertn task wth the Karlsruhe Dextrus Hand. 1. Intrductn Cmpared wth a cnventnal tw-jaw grpper, mult-fnger hands have nherent advantages: frst, they have a hgher grp stablty due t three and mre cntact pnts at the bject, and secnd, t s pssble t mpart mvements nt the grpped bject by exertng adequate fnger frces. An adequate bult-n sensr equpment permts easy data prcessng and nfrmatn gatherng whch enables the use f rbt grppers as explratn tl n unstructured envrnments. Anther knd f applcatn fr dextrus hands s fund n well-structured wrkng areas ndustral manufacturng systems: t perfrm cmplex assembly tasks, nsertn peratns and bject manpulatns. Dextrus mult-fnger hands represent an nterestng mult-dscplnary research area. The develpment f such a hand s a technlgcal and scentfc challenge, and befre btanng a fully peratnal grpper fr the applcatn n manufacturng systems, a lt f prblems are t be slved. Up t the present tme a number f mult-fnger hands have been develped, but there s nly lttle wrk 1

dealng wth the prblem f determnng effcently all parameters requred fr bject handlng peratns wth artculated rbt hands. A survey f past grpper develpments was presented n [1]. Especally, at the Insttute fr Real-Tme Cmputer Cntrl Systems and Rbtcs f the Unversty f Karlsruhe a mdular three-fnger hand wth 9 degrees f freedm, the Karlsruhe Dextrus Hand (Fg. 1), has been n develpment snce 1988 [2], [3], [4]. Because the man gals f ths prject are t nvestgate dextrus graspng and manpulatn, the research effrts are fcused n the tpcs mechancal desgn, sensr ntegratn, ntellgent hand cntrl, prgrammng and grasp plannng. Fg. 1 : The Karlsruhe Dextrus Hand Therefre, the realzed framewrk cnssts f a dstrbuted real-tme cntrl system n cmbnatn wth a graphcal prgrammng and smulatn system [5] that permt relable peratn f the multfnger grpper. Als, a sphstcated apprach t autmated grasp plannng [6] has been develped. T d ths, the task parameterzatn prblem (descrbed here) - the determnatn f the mtn and especally the frce parameters requred t perfrm a specfc bject manpulatn - has t be slved. 2. Grasp plannng and fne manpulatn One f the frst develpments was a three-fngered rbt hand frm Okada [7], [8]. Okada has defned a symblc ntatn fr the fundamental fnger mtns. The pssble peratn mdes whch can be perfrmed are smple bendng r extendng, pressng nsde r utsde, adductn and abductn. Cmplex mtns can be prgrammed by jnng several fnger peratns tgether, but ths s a dffcult and cumbersme task because the exact lcatn f each fnger has t be knwn a-prr. T avd the specfcatn f steretype fnger mtn sequences wth a large number f repeatng parameters, the fur-fngered human-lke Utah/MIT Dextrus Hand [9], [10] s prgrammed by hand mtn prmtves lke clse-hand, grasp-bject, spread-fngers. These hand prmtves [11] tgether 2

wth the free, guarded and cmplant mtn types presented later n ths sectn, represent an elegant way t descrbe a desred hand actn schemata wth respect t a subset f fngers and jnts. A LISP-based prgrammng envrnment fr the three-fngered Salsbury Hand s descrbed n [12]. Attempts have als been made t synthesze grasps t perfrm specfc parts matng peratns r assembly tasks. In [13] an algrthm based n gemetrc-mechanc calculatns has been develped t generate stable, frce-clsure grasps. The authrs f [14], [15], [16] have analyzed the human bject handlng behavur t derve grasp strateges fr artculated hands. Here, the prblems arse frm the dfferent knematc hand/fnger structure and the dffcultes t supprt frce cntrl strateges. T perfrm a dextrus manpulatn wth a rbt hand, the bject has (1) t be grasped at the cntact pnts wth suffcent nternal frces and (2) t be mved under the presence f external frces and mments actng nt t. T ncrease flexblty and t decrease cmplexty, three basc mtn types are ntrduced cnsstng f a mtn, a frce and an ptnal sensr r stffness specfcatn: The smplest mtn s realzed as free mve by whch the executn f an bject mtn n the wrkng space f the hand s ttally uncnstraned. Because a cllsn wth ther cmpnents n the envrnment s nt expected, there s n need f a sensr-guded cntrl strategy. Ths mtn type s nrmally used as carse peratn fr bth hand pstnng (reachng) and bject handlng. FREE_MOVE ::= <mtn_spec> <frce_spec> The guarded mve s a mtn n the free space that s executed under sensr supervsn and nterrupted when a termnatn cndtn s fulflled whch depends n the actual sensr cnfguratn and the way the sensr nfrmatn s nterpreted. Ths mtn type s mstly used t cntrl grasp peratns, transfer mtns and dextrus manpulatns after an ntal cntact has ccurred. GUARDED_MOVE ::= <mtn_spec> <frce_spec> <sensr_spec> The mst cmplex mtn s realzed as cmplant mve by whch the executn f the bject mtn s nt pre-cmputed but depends n an actve stffness cntrl strategy under sensr-gudance t perfrm a cmplant behavur f the hand/fnger system. Ths mtn type s nrmally used t perfrm parts matng peratns lke peg-n-hle nsertn r lght-bulb/screw turnng tasks where the parameters are cnstraned by envrnmental cnstrants and uncertantes f the bject lcatn. COMPLIANT_MOVE ::= <mtn_spec> <frce_spec> <stffness_spec> Because all mtns and frces are specfed n terms f the manpulatn bject, the cntrl system has t perfrm the transfrmatn frm bject- nt grpper-based cmmands t prvde the trajectry generatn and nterplatn rutnes wth the parameter values requred fr cmmand executn. Startng frm a gven bject trajectry n hand crdnates and cnsderng rllng and slppng f the fngertps, the cntact velctes and jnt velctes are cmputed frm whch the requred jnt angles are determned. The mathematcs f manpulatn - the gemetry, knematcs and statc fr- 3

ces - s descrbed accrdng t the wrk f [16], [17], [18], [19], [20]. Startng frm external bject frces and mments, the ptmzatn prblem fr the cmputatn nternal grp frces can be slved wth tw dfferent appraches frm whch the cntact frces and the jnt trques are derved. 3. Mtn cmputatn Because the mtn planner shuld be well-suted fr ff-lne prgrammng and smulatn f dextrus manpulatns wth mult-fngered rbt hands, several assumptns have t be made t cmpute an bject mtn: (1) especally the bject and fngertp gemetry has t be gven n analytcal frm, (2) the hand/fnger knematcs has t be knwn and (3) the ntal cntact pnts f the fngertps at the bject surface have t be pre-selected. Frm all ths nfrmatn, ts pssble t determne the crrespndng fnger mtns whch perfrm the desred peratn n frm f jnt crdnates. 3.1 Descrptn f the bject-trajectry The manpulatn bject s represented by the frame C fxed at the center f mass whle the rbt hand s represented by the frame C h fxed at the hand base. The lcatn f the bject wth respect t the frame C h s descrbed by the (3 1) translatn vectr h r,h and the (3 3) rtatn matrx A,h. Thrugh an elementary mtn f the bject, lke a translatn, a rtatn r a screw mtn, the actual frame C can be transfrmed nt the frame C. The new lcatn f the bject wth respect t the frame C h s then descrbed by the translatn vectr h r,h and the rtatn matrx A,h. A,h φ ges C a A,h C s ges r h r,h h r,h C h Fg. 2 : Object mtn wth respect t hand base A screw mtn (Fg. 2) s cmpsed f a rtatn arund an arbtrary axs a: x = p + u r wth angle φ (maxmal angle φ ges ) and a translatn alng the unt drectn-vectr r f a wth length s (maxmal length s ges ). Thus, f the tempral relatnshp f s(t) and φ(t) are bth knwn, the exact lcatn f the bject can be pre-cmputed at each tmestep t. The translatnal and rtatnal part f 4

the mtn s gven n the fllwng frm whereby the parameters are determned wth the functns t and R (sφ = snφ, cφ = csφ, vφ = 1-csφ; the r's and p's are the cartesan cmpnents f r, p): h r,h = h r,h + A,h t( a,φ) + A,h ( s r) A,h = A,h R( a,φ) t( a,φ) = R( a,φ) = 3.2 Gemetry f the manpulatn system ( r z p y r y p z )sφ ( r x r z p z r 2 z p x + r x r y p y r 2 y p x )vφ ( r x p z r z p x )sφ ( r y r z p z r 2 z p y + r x r y p x r 2 x p y )vφ ( r x p y r y p x )sφ ( r y r z p y + r 2 y p z r x r z p x + r 2 x p z )vφ r 2 2 ( y + r z )cφ + r 2 x r z sφ + r x r y vφ r y sφ r x r z vφ r z sφ + r x r y vφ r 2 2 ( x + r z )cφ + r 2 y r x sφ + r y r z vφ r y sφ r x r z vφ r x sφ + r y r z vφ r 2 2 2 ( x + r y )cφ + r z The hand knematcs s needed t descrbe a gven bject trajectry n fnger crdnates. Therefre, the relatnshp between the base frames C b f the fnger chans and the hand frame C h are defned by transfrmatn matrces ( h r b,h,a b,h) whch are cnstant fr a gven hand cnfguratn (the ndex cunts the fngers, the ttal number f fngers s and the number f jnts per fnger s jn ). T descrbe the fnger mtns, the relatnshp between the fngertp frames C and the base frames C b s defned by transfrmatn matrces ( b r,b,a,b ), the s-called drect knematcs. S, the pstns h r,h and rentatns A,h f the fngertps (see Fg. 3 fr a fnger wth jn = 3 jnts) wth respect t the hand base can be determned f the actual set f jnt angles θ s knwn: h r,h =h r b,h + A b,h b r,b ( θ ) A,h = A b,h A,b ( θ ), θ = ( θ 1,, θ jn ) T = 1,, y x C 2 x θ 3 l 3 z C f = C 3 y z l 2 C 1 z y θ 1 l 1 y z θ 2 x C b = C 0 x Fg. 3 : Fngertp lcatn wth respect t fnger base 5

In the nverse drectn, the knematc prblem can be slved drectly f the abve descrbed relatns are nvertble. In ths case, the jnt angles θ can be cmputed frm the actual fnger pstns h r,h whch s very fast, but wll fal f the effect f rllng r slppng ccurs because the exact pstns h r t,h f the cntact pnts at the fngertps are nt knwn. Nevertheless, fr gven pnts f cntact, ths smple methd can be used t determne the ntal parameter values f a fnger mtn. A mre sphstcated cmputatn methd, descrbed n [19], [20], starts wth the generalzed velcty ( h v,h, h w,h ) T (lnear and angular velcty) f the bject t determne the velcty parameters ( h v,h,h w,h )T f the cntact pnts (the cmpnents f the cntact velcty v ). Accrdng t the cntact mdel, represented by the cntact matrx Q (see sectn 4.1), the respectve velcty parameters ( h v t,h,h w t,h )T f the fngertps are derved, frm whch the generalzed fnger velctes ( h v,h,h w,h )T are determned. Ther relatn t the resultng jnt velctes s gven by a Jacban: h v t,h h w t,h h v,h h w,h h v t,h h w t,h = h v,h h w = Q υ, h v,h,h h w =,h 6 h J v h J w θ θ = h J v 1 h v,h fr Q = Q PF = ( I 3 0) T h w t,h = h w,h = 0 and v = h v t,h = h v,h v t, = I 3 -A,h S r t h, 0 I 3 v,h h w,h + A,h f A,h w t, = I 3 -A,h S( r,) h 0 I 3 v,h h w,h + A,h v, A,h w,, S( r) = 0 r z r y r z 0 r x r y r x 0 The transfrmatns f generalzed velctes gven n dfferent crdnate systems are perfrmed by abve matrx equatns (usng the skew symmetrc matrx S as peratr fr the crss-prduct) whch nclude the translatn vectrs r and rtatn matrces A that descrbe the gemetrc relatnshps f the bject-fnger mtn. The cmputatn f rllng fnger mtns s shwn n the fllwng where the cntact mdel s assumed t be pnt cntact wth frctn (PF), s that nly lnear and n angular velcty cmpnents can be transferred thrugh the fngertp cntacts wth the bject. 3.3 The prblem f rllng fngertps The pstns and rentatns f the cntact pnts (tp cntacts) can be descrbed wth respect t the bject frame C as ( r,,a,) r wth respect t the fnger frames C f as ( r t,,a t, ). These vectrs and matrces (gven as rtatns alng the drectns d f the z-axes wth rtatn angles δ) can be parameterzed as functns f tw varables (see Fg. 4). Thus, due t rllng f the fngertps, nt nly the jnt angles θ but als the cntact parameter sets η and ξ have t be cmputed. r x η 1,η 2 r t, = r y ( η 1,η 2 ) = r t, η r z η 1,η 2 wth η = η 1 η 2 r x ξ 1,ξ 2, r, = r y ( ξ 1,ξ 2 ) = r, ξ r z ξ 1,ξ 2 wth ξ = ξ 1 ξ 2

A t, = R d,δ A, = R d,δ wth d = 1 snδ (0,0,1) T f ( ˆn t, ) and δ = cs -1 f ((0,0,1) ˆn t, ) wth d = 1 snδ ((0,0,1) T ˆn,) and δ = cs -1 ((0,0,1) ˆn,) One f the relatns s gven by the fact that the generalzed velctes f the cntact pnts at tme t (cnstant durng manpulatn) must be the same wth respect t the bject and the fngertps. The secnd relatnshp s gven by the fact that bth the bject and the fngertps must stay n cntact, r equvalent, that the pstn vectrs t the actual pnts f cntact must be the same. Spltted nt cmpnents, dfferentated and subtracted frm the frst relatnshp, the fllwng equatns arse: h v t,h = h v,h h w,h = h v,h ( A,h S( r,)) h w,h h v,h A,h S ( r t, ) h r t,h = h r,h A,h v t, = A,h v, The equatns fr the cntact pnts at tme t tgether wth the cntnual change f the cntact pnts durng manpulatn (expressed n frm f the tw velcty terms v t, and v,) descrbe the resultng mtns f the actual cntact pnts. Ths yelds the s-called tangental cnstrant f mtn: C ˆn t, r, ˆn, r t, h r,h hr,h C h r,h C h Fg. 4 : Pnt f cntact wth respect t bject and fngertp Anther cnstrant fr the mtn parameters s gven by the fact that the tangental planes must cncde, r equvalent, that the nrmal vectrs ( h ˆn, = A,h ˆn,) f the bject at the cntact pnts must pnt dametrcally ppste t the crrespndng nrmal vectrs ( h ˆn t, = A,h ˆn t, ) f the fngertps at the cntact pnts. Dfferentatng yelds then the s-called nrmal cnstrant f mtn: h ˆn t, = h ˆn, A,h ˆṅt, A,h S ( ˆn t, ) h w,h = A,h ˆṅ, + ( A,h S( ˆn,)) h w,h 7

The velctes and all ther mtn parameters n abve equatns can be expressed by ther crrespndng Jacban matrces that depend n the dervatns f the jnt angles and cntact parameters: v ( η ) = ṙ ( η 1,η 2 ) = J r η h v,h = h f J v θ v t, = J r η v, = J r ξ h w,h = h f J w θ f ˆṅt, = J n η ˆṅ, = J n ξ Substtutng these parameters thrugh the Jacbans f θ, η and ξ results n the fllwng system: ( h J v ( A,h S ( r t, )) h J w ) θ = h v,h A,h S( r,) f ( A,h J r ) η ( A,h J r ) ξ = ( 0, 0, 0) T ξ = A,h S ˆn, h J w θ + A,h J n A,h S ( ˆn t, ) η + A,h J n h w,h ( ) h w,h Out f ths, a matrx equatn wth the (9 7) matrx M and the (9 6) matrx R can be btaned: ( ) h J w 0 0 h J v A,h S r t,f f M = 0 A,h J r A,h A,h S ( ˆn t, ) J r h J w A,h J n A,h J n R = I 3 A,h S r, 0 0 0 A,h S ˆn, A left hand multplcatn f R wth the left generalzed nverse M f M yelds the requred velcty vectr (three cmpnents fr the jnt velctes, tw fr each f the cntact parameters) as result f the specfed bject mtn. In ths case nne equatns n seven varables have t be slved: θ M η = R ξ h v,h h w,h θ η ξ = M R h v,h h w,h The sngle parameter values fr each fnger are then calculated by slvng the equatn system wth the Gaußan (elmnatn) algrthm. T determne the fnal (ntegrated) values f θ, η and ξ wth the Runge-Kutta-algrthm abve calculatn has t be perfrmed fur tmes fr each mtn step. 3.4 The prblem f slppng fngertps Because the relatve rentatn f the cntact frames C t and C s changng durng bject handlng due t slppng f the fngertps, the cntact angle γ between them has t be cmputed cntnually. The crrespndng frames are fxed at the cntact pnts n the fngertps and the bject n such a way, that ther x- and y-axes span a tangental plane and ther z-axes pnt nt the drectn f the surface nrmal (see Fg. 5). The resultng cntact angle γ between the x-axes f the tw frames C t and C s thereafter used t determne the rtatn matrx A,t whch descrbes the relatve rentatn f the tw cntact crdnate systems and s needed n the next chapter fr frce cmputatn: 8

csγ snγ 0 A,t = snγ csγ 0 0 0 1 wth γ = cs -1 h x t h x h x t h x T T h x t = A,h A t, 1 0 0 h x = A,h A, 1 0 0 C γ r, C t C r t, h r,h C h r,h C h 4. Cmputatn f frces Fg. 5 : Relatn between the tw cntact crdnate systems Addtnal t the cmputatn f pure gemetrc mtn parameters (jnt angles) needed t manpulate a grasped bject wth the fngers f a rbt hand, apprprate mechanc frce parameters (jnt trques) t be exerted by the fngertps must be determned t hld the bject n a stable grasp cnfguratn. The fnger frces that can be appled depend manly n the cntact mdel whch has t be ncrprated nt the frce transfrmatn prcess frm generalzed bject frces ( f, m ) T ver cntact specfc frces t the requred jnt trques τ. The dmensns f the nvlved matrces relate t the prevusly ntrduced numbers, jn and the number cmd f applcable frces. 4.1 Cntact mdels and cntact frces The cmpnents f the generalzed frces (, m ) T that can be exerted by the fngers as cntact frces n the cntact pnts wth the bject are characterzed by the cntact matrx Q. Fr ths, the matrces and equatns that perfrm the cntact specfc transfrmatn are descrbed fr the case f frctnless pnt cntacts (P), pnt cntacts wth frctn (PF) and sftfnger cntacts (SF): m = Q, Q : (6 cmd ), : (cmd 1), Q {Q P,Q PF,Q SF } 9

In the case f a frctnless pnt cntact (P) nly sngle frces can be exerted n the cntact pnts nt the nrmal drectn twards the bject surface (the frce has t be negatve) t avd a lss f cntact; als, the frce must nt exceed a specfc maxmal value n rder t prevent bject damage. Q P = ( 0 0 1 0 0 0) T, cmd = 1 ( ) 0, ( ) max In the case f pnt cntact wth frctn (PF) the fngers can exert any frces pntng nt the defned frctn cnes at the cntact pnts. Ths means, the tangental frces must stay nsde f the frctn cnes (determned by the tangental ceffcents f frctn µ t ) t avd slppng f the fngertps n the tangental plane f the bject surface (they are free t rtate abut the cntact pnts). 1 0 0 0 1 0 0 0 1 Q PF = 0 0 0 0 0 0 0 0 0, cmd = 3 ( ) 0, z max z 2 2 2 ( ) + x µt y z In the case f sftfnger cntact (SF) the frctn ver the area f cntact allws the fngers t exert sngle trques n addtn t the pure frces pntng nt the defned frctn cnes. The trques can be exerted n bth drectns abut the nrmal axs such that the bject s cnstraned wth the gven cntacts (nrmal ceffcents f frctn µ n ) and can nly break the cntacts by sldng dwnward. 1 0 0 0 0 1 0 0 0 0 1 0 Q SF = 0 0 0 0 0 0 0 0 0 0 0 1, cmd = 4 ( ) 0, z max z 2 2 2 ( ) + x µt y z ( ) µ n r z 4.2 Statc frces cnsderng rllng and slppng Startng frm generalzed frces actng nt the bject durng manpulatn the cmputatn f crrespndng fnger frces s descrbed n the presence f rllng and slppng. Therefre, the cntact frces and mments (, m ) T are cnverted nt (, m ) T whch s perfrmed by transfrmatn matrces T,; the result s then summed up t cmpensate the bject frces and mments ( f, m ) T. Accrdng t the chsen cntact mdels represented by cntact matrces Q, the respectve bject cntact frces are cnverted nt the cntact frces and mments (, m ) T whch s perfrmed by transfrmatns matrces T, and leads t the defntn f grp submatrces G : f m = m = T, = [ T, Q f ] = [ T, f ] = [ G ] = G f m 10

+, n general: b f c T a,b : (6 6) wth b m = T a,b c T a,b = T a,b Q : (6 cmd ) wth a f c a m c b f c b m c, T A a,b 0 a,b = S( b r a,b )A a,b A a,b = T a,b a f c, a f c : (cmd 1) Fr a rbt hand wth (eventually ttally dfferent) fngers and chsen cntact mdels cmd, the grp matrx G fr the transfrmatn f cntact frces f that are cmpsed frm the frce cmpnents n each cntact pnt nt the bject frces and mments ( f, m ) T s gven as fllws: G = T, = T, Q : (6 cmd ) wth G = G 1 ( G ) : (6 cmd and f m = ) wth m f m = G and = : (cmd 1) = G f, f = f 1, m = [ G ] = G 1 f 1 + T : ( cmd f G f 1) A rbt hand wth = 3 dentcal fngers and cntact mdels cmd = 3 (pnt cntact wth frctn) has fllwng grp matrx G (nly frces and n mments can be exerted at the cntact pnts): G = ( G 1 G 2 G 3 ) = I 3 S r 1, I 3 S r 2, I A 3 1, 0 0 S( r 3,) 0 A 2, 0, f = 0 0 A 3, 1 f 1 2 f 2 3 f 3 T determne the jnt trques τ fr each fnger, that depend n the frces and mments actng at the cntact pnts, the vectrs (, m ) T are cmputed ut f the cntact frce cmpnents by transfrmatn matrces T,. Ths s dne accrdng t the cntact mdel specfc frce transfrmatns frm the frames C nt the frames C t (bth have the same cntact pnts as rgns, s that the pstn vectrs t r,t are equal t zer) and the frce transfrmatns nt the frames C : m T t, T,t = T,, r = T t, T,t Q = T t, T,t m = T, m, T, : (6 cmd ) Usng the prncple f vrtual wrk [18], the jnt trques τ can be cmputed frm the fnger frce cmpnents wth the help f the matrces D (see [17]) that are a cncatenatn f the transfrmatn matrces T, (dependng n ( t r,t = 0,A,t ) and ( t r t,,a t, )) and the fnger Jacbans J T. Snce the Jacbans J (used n the transpsed frm) are the dervatns f the drect knematcs ( b r,b,a,b ), ther dmensns are determned by the jn fnger parameters (jnt angles θ ): 11

G N N,λ + τ = J T T, = J T T ( v J w ) m = D, J T = 12 J T T ( v J w ), τ : (jn 1) wth D = J T T, : (jn 6) * (6 cmd ) = (jn cmd ) Befre ths relatn wll be used t deal wth the mpled frce cnstrants f the chsen cntact mdels cmd (see sectn 6.1), t s presented fr the general case f an arbtrary rbt hand wth (nt necessary dentcal) fngers. Thus, the set f jnt trques τ can be cmputed frm the ttal fnger frces f wth the help f the matrx D that descrbes all requred fnger frce transfrmatns: τ = f J T T,f f = D f and ( f, m ) T = G f f J T = dag f 1 J T f (,, J T ), T,f = dag( T 1,f 1,, T,f ), τ = τ 1,,τ wth D = f J T T,f = dag D 1, : ( jn 6,D 4.3 Decmpstn f cntact frces T : ( jn 1 ) ) * (6 cmd ) = ( jn cmd ) Gven the external frces and mments ( f ext, m ext ) T actng n a grasped bject durng manpulatn the prblem s t cmpensate the resultng bject frces ( f, m ) T n negatve drectn wth apprprate cntact frces f ; the transfrmatn between them s perfrmed by the grp matrx G. Because the space f cntact frces (the dmensn s cmd, dependng n the cntact types) s descrbed by maxmal sx ndependent equatns, there exsts n general (f the dmensn s greater than sx) mre than ne slutn fr the frce cmpensatn prblem [19]. Thus, all cntact frces can be cmbned f the partal slutn f p and ne slutn f the hmgeneus equatn G f h = 0: f ext m ext = f m wth f p = G * = G f p + f h f m and 0 = G f h f h = N λ The partal slutn f p s cmputed by the rght generalzed nverse G * f the grp matrx G, whereas the hmgeneus slutn f h cnssts f a lnear cmbnatn f the base vectrs f the n-dmensnal null space f G whch s descrbed by the null matrx N. Thus, the n-dmensnal vectr λ represents the nternal grp frces (frces actng crsswse between the cntact pnts) that must be exerted n addtn t the cntact frces f p (whch cmpensate the external frces and mments): G N λ = G 1 ( N 1 ) T λ = [ G N ] wth N = N 1 ( ) T : ( cmd λ = G 1 N 1 + ndm), λ = λ 1, λ ndm G N T : (ndm 1) Because f the lnearty f G that defnes a dependency between all cntact frce transfrmatns, the dmensn f the null matrx N has t be determned wth respect t the whle matrx G and nt ndependently fr the submatrces G. Nevertheless, the result f the null space cmputatn can be appled t parttn the matrx N nt submatrces N whch then depend n the dmensn ndm.

5. Grasp frce plannng In ths chapter, a plannng methd fr the cmputatn f all frce parameters requred fr manpulatn f a grasped bject wth pre-selected cntact pnts s presented [6]. Cmpared t the general frmulatn, the specal case = 3 ( jn = 3), cmd = 3 and ndm = 3 s handled. It s assumed that a range f external frces and mments ( f ext, m ext ) T s gven r knwn fr a certan type f peratn, lke a peg-n-hle nsertn task, fr whch apprprate nternal frces λ have t be determned. Ths s realzed as knwledge-based apprach wth rules and crtera (stablty and securty) that must be satsfed. Here, a majr step s the transfrmatn f the prblem frm the bject space wth bject frces and mments ( f, m ) T nt the cntact space wth cntact frces f (dvded nt manpulatn and grasp frces f p, f h ) whch s dne wth the grp matrx G and the null matrx N. Fr the cmpensatn f manpulatn frces all applcable grasp frces stayng n the frctn cnes f the ( cmd = 9)-dmensnal fnger frce space must be enumerated. The mappng f the frce cmpnents nt the cntact plane generates a trangle wth parameters α, β, γ that are the lnear factrs f the base vectrs f the (n-dm = 3)-dmensnal grasp frce subspace. The applcable grasp frces are cmputed frm α-parameters determnng the β- and γ 1 -ntervals, β-parameters determnng the γ 2 -ntervals and ntersectng γ-parameters that are enumerated frm a cnvex hull algrthm. Rule 1 : [Applcable grasp frces] The grasp frces must le n the ntersectn f the frctn cnes wth the cntact pnt plane. Algrthm: [Applcable grasp frces] Determne α L,α U FOR α 0 α L,α U [ ] [ ] DO (* enumerate α - nterval *) [ ]( α 0 ), [ γ 1L,γ 1U ]( α 0 ) [ ]( α 0 ) DO (* enumerate β - nterval *) [ ]( α 0,β 0 ), [ γ S,γ E ] [ γ 1L,γ 1U ]( α 0 ) [ γ 2L,γ 2U ]( α 0,β 0 ) [ ]( α 0,β 0 ) DO (* enumerate γ - nterval *) f h α 0 h 1 + β 0 h 2 + γ 0 h 3 Determne β L,β U FOR β 0 β L,β U RETURN f h Determne γ 2L,γ 2U FOR γ 0 γ L,γ U The determnatn f sutable grasp frces f h wth respect t the assembly task can be dvded nt successve steps fr the cmpensatn f a sngle manpulatn frce thrugh a stablty search wth mnmum securty r a securty search wth mnmum stablty. Startng frm the task area EA that descrbes s range f external frces and mments, the mst stable and mst secure grasp frces fr the crrespndng range MA f manpulatn frces are cmputed frm search prcedures, such that the mean value σ f the resultng range HA f grasp frces s the desred ptmal slutn f hpt. T ensure ths, tw qualty measures fr the ptmzatn f grasp frces n frm f crtera fr the bject stablty Ψ and the grp securty Ω (rbustness) [22] are ntrduced; they are based n sde 13

and bttm face prjectns Π f the bject cntact frces f t the surface f the frctn cnes FC: := Mn Ψ f Ω f h 3 := Ω max f h FC Ψ ( ) wth Ψ ( ) = Mn Π s ( ), Π b ( ) 0 FC 3 and Ω max := µ t max maxmal length 3 2 + ( max ) 2 Rule 2 : [Sngle cmpensatng grasp frce] Each grasp frce must cmpensate a crrespndng manpulatn frce f a gven frce area. Rule 3 : [Range cmpensatng grasp frce] A partcular grasp frce value must cmpensate all manpulatn frces f a gven frce area. Algrthm: [Stablty search wth mnmum securty] Input: grp matrx G, null matrx N, manpulatn frce f p, mnmum securty ω 0 Output: f p fr mnmum securty ω 0 mst stable cmpensatng grasp frce f h res ψ max 0 f h Applcable grasp frce(α 0,β 0,γ 0 ) [ > ψ max ] [ Ω ( f h) > ω 0 ] THEN f h res f h ; ψ max Ψ( f p + f h ) IF Ψ f p + f h RETURN f h res Algrthm: [Securty search wth mnmum stablty] Input: grp matrx G, null matrx N, manpulatn frce f p, mnmum stablty ψ 0 Output: f p fr mnmum stablty ψ 0 mst secure cmpensatng grasp frce f h res ω max 0 f h Applcable grasp frce(α 0,β 0,γ 0 ) [ > ω max ] Ψ f p + f h IF Ω f h RETURN f h res [ > ψ 0 ] THEN f h res f h ; ω max Ω f h Defntn: The sx-dmensnal task frce area EA mdeled as task-plyeder s the cnvex cmbnatn f frce vectrs whch are cmputed frm the range set ES f external bject frces and mments (descrbes the specfc frce spectrum as set f vectrs) by the sx-dmensnal cnvex hull peratr CH: ES EA wth ES, EA R 6 : EA Ply ( ES) := CH( ES) and ES := 14 f ext m ext The mean value σ f the range set HS f grasp frces cmpensatng the range set MS f manpulatn frces derved frm the frce spectrum ES, r fully equvalent, the center f gravty f the crrespndng grasp frce area HA cmputed by the three-dmensnal cnvex hull peratr CH, s a k eset k=1

best cmpensatng grasp frce f hbest that can be transfrmed nt the range set IS nternal frces: f h HS : Ψ f p k + f h f hbest = σ HS > 0 k 1,,eset eset { } k=1 := 1 eset k f h = σ f h k { } wth MA := CH(MS) and MS = G * ( ES) eset wth HA := CH(HS) and N * σ HS k=1 15 ( ) = σ( IS) Rule 4 : [Grasp frces fr stable grps] The grasp frces must have a certan stablty factr relatng t rllng and slppng mtns. Rule 5 : [Grasp frces fr secure grps] The grasp frces must have a certan securty factr relatng t tlerances f the frce areas. Algrthm: [Mst stable grasp frce wth respect t MA] Input: grp matrx G, null matrx N, manpulatn frce area MA, mnmum securty ω 0 Output: MA fr mnmum securty ω 0 mst stable cmpensatng grasp frce f hbest FOR k { 1,,eset} DO [ ] G,N, f p k, ω 0 f h k Stablty search wth mnmum securty f hbest σ k eset { f h } k=1 IF k { 1,,eset }: Ψ f k + f p hbest ( > 0) THEN Success TRUE RETURN ( f hbest,success) ELSE Success FALSE Algrthm: [Mst secure grasp frce wth respect t MA] Input: grp matrx G, null matrx N, manpulatn frce area MA, mnmum stablty ψ 0 Output: MA fr mnmum stablty ψ 0 mst secure cmpensatng grasp frce f hbest FOR k { 1,,eset} DO [ ] G,N, f p k,ψ 0 f h k Securty search wth mnmum stablty f hbest σ k eset { f h } k=1 IF k { 1,,eset }: Ψ f k + f p hbest ( > ψ 0 ) THEN Success TRUE RETURN ( f hbest,success) ELSE Success FALSE The man dffculty s the determnatn f the smallest mnmum stablty parameter ψ 0 fr the algrthm searchng the mst secure grasp frce n cntact space. The lwer bund s nrmally zer whereas the upper bund can be cmputed frm the algrthm searchng the mst stable grasp frce wth mnmum securty zer (pure stablty search). Buldng a stablty nterval f them, an nterval search prcedure based n the securty search wth mnmum stablty and termnatn cndtn (desred accuracy) cmputes the ptmal grasp frce f hpt cmpensatng all gven manpulatn frces.

[ ] [ ]( G,N,MA,ω 0 = 0) lwer bund: ψ mn = 0, upper bund: ψ max < µ t f max fr stablty nterval I ψ := ψ mn,ψ max wth ψ max := ψ f h and f hbest = Mst stable grasp frce wrt. MA best Algrthm (nterval search): [Optmal grasp frce wth respect t MA] Input: grp matrx G, null matrx N, manp. frce area MA, desred search accuracy ε 0 Output: MA fr stablty nterval mst stable and secure cmpensatng grasp frce f hpt ( f hpt,success) [ Mst stable grasp frce wrt. MA ]( G,N,MA,ω 0 = 0) IF Success THEN [ ] ψ max ψ f h pt ; ψ mn 0; I ψ ψ mn,ψ max REPEAT ψ 0 (ψ max + ψ mn ) / 2; ( f hbest,success0) Mst secure grasp frce wrt. MA IF Success0 THEN ψ max ψ 0 ; f hpt f hbest [ ] I ψ ψ mn,ψ max UNTIL ψ max ψ mn < ε 0 RETURN f hpt,success [ ]( G,N,MA,ψ 0 ) ELSE ψ mn ψ 0 General rule : [Grasp parameter ptmzatn] The parameters must be chsen, such that the cntact pnt dependent grasp frces f h n the cntact space are ptmal and the resultng grp s bth stable and secure wth respect t the specfc task. Algrthm: [Optmal nternal frce wth respect t EA] Input: grp matrx G, null matrx N, task frce area EA, desred search accuracy ε 0 Output: EA ptmal (regardng stablty and securty) cmpensatng nternal frce λ pt FOR k { 1,,eset} DO f k p G * f ext m ext f hpt Optmal grasp frce wrt. MA k (* mappng frward nt cntact frce space *) [ ]( G,N,MA,ε 0 ) λ pt N * f hpt RETURN λ pt (* mappng backward back nt bject frce space *) The determned ptmal grasp frce f hpt n the cntact space s transfrmed nt the ptmal nternal frce λ pt n the bject space whch s perfrmed by the rght generalzed nverse N * f the null matrx N. Summarzng, Fg. 6 shws the search prcess fr ptmal cntact frces that starts frm external frces and mments mapped t the cntact space where all manpulatn frces are cmpen- 16

sated by ne sutable grasp frce whch s mapped t the bject space as the resultng nternal frce. Space f bject frces Space f cntact frces Subspace f external frces and mments Task frce area EA f ext mext G * k f p k Area f manpulatn frces MA Subspace f manpulatn frces Subspace f nternal frces λ pt N * Area f grasp frces HA f hpt σ f h k Interval search σ cmpensates MA? + - Subspace f grasp frces 6. Grasp frce ptmzatn Fg. 6 : Schematc prcess f cntact frce searchng In ths chapter, anther plannng methd fr the determnatn f the requred grasp frce parameters s presented whch s frmulated as an ptmzatn prblem ver gven r mpsed cnstrants. Therefre, t ncrprates explctly the cntact frce cnstrants (ndrectly defned n the prevus descrbed methd by bject stablty and grp securty crtera) nt the prcess f frce cmputatn. The apprach chsen here s based n wrk fr the nrmal case = 3 ( jn = 3), cmd = 3 and ndm = 3 (see [17]) whch s generalzed fr the case f an arbtrary rbt hand and all knds f pssble cntact mdels. Fr the fnal slutn f the ptmzatn prblem several algrthms can be appled, but preference s gven t an algrthm that searches fr maxma n n-dmensnal spaces. 6.1 Cnstrants n cntact frces In rder t keep the frce cmputatn smple and fast, the nn-lnear equatns descrbng the cntact frces nsde f the cntact mdel dependng frctn cnes are apprxmated lnearly: the generalzed frctn cnes are replaced by generalzed frctn pyramds whse sdes tuch the brders f the cnes. By ths prcedure, addtnal (nt applcable) cntact frce regns are defned whch can be avded by reducng the frctn ceffcents prprtnally (fr example, as shwn n Fg. 7 frm µ t t µ t = µ t 2 ). Rewrtng f the frce cnstrants mentned n sectn 4.1 leads t an nequalty system-lke frmulatn f the pssble cntact frces whch are descrbed thrugh frctn matrces F and cnstrant vectrs c. Agan, the respectve equatns are cnsdered fr the case f frctnless pnt cntacts (P), pnt cntacts wth frctn (PF) and sftfnger cntacts (SF): F c wth F : (2 * cmd cmd ), c : (2 * cmd 1), cmd = {1,3,4} 17

In the case f frctnless pnt cntact (P) the cnstrant vectr cnsders nly the drectn f the cntact frces and the maxmal pssble cntact frces that can be appled by the fnger f the hand: ( ) 0 - ( ) 0, ( ) max F P c P wth F P = 1 1, c P = 0 max ( ) max, cmd = 1 In the case f pnt cntact wth frctn (PF), addtnally t the cnsderatn f the maxmal cntact frces, the cntact frce cmpnents wthn the tangental plane f the crrespndng frctn cnes are lnearzed and adapted t the resultng frctn pyramd by a reduced frctn ceffcent µ t : ( ) + x µ t y z ( ) x ( ) y z z µ t µ t 0 0 1 F PF ±1 0 µ c PF wth F PF = t 0 ±1 µ, c PF = t 0 0 1 µ t x ±0 z µ t y ±0 z ± ± 0 ±0 ±0 max, cmd = 3 2µ t µ t µ t µ t µ t =! µ t = 1 µ t 2µ t 2 Fg. 7 : Lnearzatn f a frctn cne (frctn pyramd) In the case f sftfnger cntact (SF), addtnally t abve descrbed restrctns n the cntact frces, the cntact trque n nrmal drectn s cnstraned by the frctn cndtns alng the z-axs: ( ) µ n r µ n z + z 0 and µ n r + z 0 r 0 0 1 0 ±1 0 µ t 0 F SF c SF wth F SF = 0 ±1 µ t 0, c SF = 0 0 ±µ n 1 0 0 1 0 18 0 ±0 ±0 ±0 max, cmd = 4

H Anther set f cnstrants s ntrduced by the fact that the jnt trque values τ f each fnger f a rbt hand must nt exceed the cnstructnal lmts t avd damage f the mtrs and/r the gears: τ mn τ τ max wth τ = D and τ mn, τ max : (jn 1) D τ mn and D τ max 6.2 Frce cmputatn: slvng nequalty systems The restrctns mpsed nt the cntact frces and jnt trques that are descrbed by frctn matrces F, cnstrant vectrs c and mtn dependent matrces D (rllng and slppng f the fngertps at the bject surface) can be expressed n matrx frm as systems f lnear nequaltes. These nequalty systems are based n the matrces H and vectrs g that are cmpsed frm abve descrbed cmpnents and lead t ptmzatn prblems fr the ptmal chce f the cntact frces. F H = D : ([2 * cmd + 2 * jn ] cmd ), g = D 19 c τ mn τ max : ([2 * cmd + 2 * jn ] 1) nequalty system n : H g wth = : (cmd 1) T cmpute apprprate slutns, the submatrces D f the matrces H have t be actualzed cntnually n each mtn step durng bject manpulatn, because they cntan all nfrmatn abut the changng parameter values f the cntact pnts and the jnt angles. Generalzed fr the case f a rbt hand wth (nt necessary dentcal) fngers, the resultng nequalty system s descrbed by the matrx H and the vectr g that are cmpsed frm the fnger specfc matrces H and g. The ptmzatn prblem s then defned as search fr the ptmal chce f the ttal cntact frce f. H = dag H 1, g = g 1, : ( [2 * cmd + 2 * jn ],g,h cmd ), frmula fr general case T : ( [2 * cmd + 2 * jn ] 1), f = f 1,, f T : ( cmd nequalty system n f : H f g H g {1, 1),} Rearrangng abve nequalty system and applyng the subdvsn f the frces n the cntact space nt manpulatn frces f p and grasp frces f h, the ptmzatn prblem can be rewrtten as prblem f fndng the ptmal nternal frces λ n the bject space usng the null matrx N. If there s a way t splt ths matrx nt submatrces N, the frce search prcess culd be dne n parallel: H f g H ( f p + f h ) g H f h g H f p H N λ g H G * ( f, m ) T wth H N = H 1 N 1 : ( [2 * cmd + 2 * jn ] N H N λ g H G * (, m ) T {1,,} ndm)

6.3 Optmzatn nternal (grp) frces T cmpute the best nternal frce λ, a dstance measure d k fr evaluatng the dstances between all defned frce cnstrants (rws f the matrces and vectrs) and the resultng hyper-planes (generalzed vlumes) that are spltted nt "allwed" and "frbdden" half-spaces s ntrduced (see [17]). The ptmal selectn f λ s the ne whch has the greatest value f the mnmal dstance t all gven cnstrants (hyper-planes). In the three-dmensnal case ths s the center f the bggest hemsphere fttng nt the "allwed" plyhedra. Therefre, the fllwng ptmzatn prblem has t be slved: ( H N) λ g + H G * f ext m ext d k = ( H N) rw=k λ g + H G * 20 f ext m ext rw=k maxmze the functn : d(λ) = mn (d k ) wth : d k 0 k = 1,, [2 * cmd + 2 * jn ] The gal functn d(λ) whch has t be maxmzed, s a cntnual, nn-lnear and nn-dfferentable functn n R ndm that s cmpsed f lnear functns d k. Hence, fr fndng the ptmal value f λ bth the smplex-methd (gal functn nt lnear) and the gradent-descend methd (gal functn nt dfferentable) wll fal. Ths means, that n rder t cmpute the ptmal frce parameters, an algrthm cnsderng the nn-lnearty and nn-dfferentablty f the gal functn s requred. T slve the ptmzatn prblem, the Hke-Jeeves algrthm [17], [23], [24] fr fndng maxma n R n can be used whch steps frm an arbtrary pnt (start pnt must nt lay wthn the allwed space) t the pnt wth the greatest dstance t each cnstrant. After an ptmal slutn s fund, the nternal frces λ tgether wth the external bject frces and mments ( f ext, m ext ) T are used t cmpute the ttal cntact frce f and fnally the crrespndng jnt trques τ fr the mtr drves: τ = D f = D f p + f h = D G * f ext m ext + N λ Thus, the ptmzatn prblem fr nternal frces culd be generalzed fr the case f a rbt hand wth (arbtrary) fngers wth jn jnts per fnger, applyng frces accrdng t the cntact mdels cmd and the n-dmensnal null space f the grp matrx. Summarzng, the frce cmputatn prcess can be fully characterzed by the parameters (dmensns), jn, cmd and ndm. 7. Example manpulatn Here, the result f a cncrete bject manpulatn (a peg-n-hle nsertn task) wth the Karlsruhe Dextrus Hand [2], [3], [4] s presented n graphcal vsualzed frm. Ths rbt hand cnssts f three dentcal fngers whse bases are fxed at the edges f an equal-sded trangle. The hand frame C h s fxed at the center f ths trangle, s that the transfrmatns between the fnger base frames C b and ths crdnate frame are gven by fllwng translatn vectrs r and rtatn matrces A: 0 50 cs30 50 cs30 h r b1,h = 50, h r b2,h = 50 sn30, h r b3,h = 50 sn30 80 80 80 0

0 1 0 0 sn 30 cs30 0 sn 30 cs30 b 1,h = 0 0 1, A b 2,h = 0 cs30 sn 30, A b 3,h = 0 cs30 sn 30 1 0 0 1 0 0 1 0 0 Each f the three fnger mdules f the hand cnssts f a tw-lnk structure munted n a revlute base and cnnected va rtatnal jnts whch s the mnmal cnfguratn needed t perfrm a desred bject manpulatn n space. The length f the lnks are (lnk ne s clsest t the fnger base): l 1 = 20 mm l 2 = 55mm l 3 = 54 mm At the end f lnk three, a half-sphere frmed fngertp wth radus r = 10 mm s fastened. By defnng frames at each jnt n a way that the x-axes pnt n drectn f the respectve lnk and the z- axes pnt n drectn f the rtatn axs (Fg. 3), the transfrmatns frm the fnger frames C fxed at the end f lnk three nt the fnger base frames are btaned by (c j = csθ j and s j = snθ j ): ( + l 2 c 2 + l 1 ) ( + l 2 c 2 + l 1 ) + l 2 s 2 c 1 l 3 c 2 c 3 s 2 s 3 b r,b = s 1 l 3 c 2 c 3 s 2 s 3 l 3 c 2 s 3 + s 2 c 3 c 1 ( c 2 s 3 + s 2 c 3 ) s 1 s 1 ( c 2 s 3 + s 2 c 3 ) c 1 c 2 c 3 s 2 s 3 0 c 1 c 2 c 3 s 2 s 3 A,b = s 1 c 2 c 3 s 2 s 3 c 2 s 3 + s 2 c 3 Frm them, all ther transfrmatns, especally the crrespndng Jacbans b J can be btaned. Nw, the cmputatn prcess f all frce parameters requred fr dextrus manpulatn f a grasped bject wth pre-selected cntact pnts s shwn n the base f the grasp plannng apprach presented n chapter 5 that deals wth the standard case f = 3 (jn = 3), cmd = 3 and ndm = 3. The plannng prcess f parameter determnatn fr a peg-n-hle nsertn task [25] uses a specfc frce area that descrbes the external frces and mments ( f ext, m ext ) T actng nt the bject durng manpulatn t determne apprprate cntact frces f fr the fngers f the hand. Accrdng t pstnal and rentatnal tlerances between peg and hle, several smulatn runs are perfrmed t cmpute the external frces n x- and z-drectn and the external mment n y-drectn. The resultng frce area EA s represented as task plyeder frmed by 2 6 = 64 (r 2 3 = 8) frce vectrs. Tlerances f peg lcatn [mm] : External frces and mments [mn] : ( x peg ), x x ( peg ), x y ( peg ) z ( x peg ), x φx ( peg ), x φy ( peg ) φz ( f ext ), f x ( ext ), f y ( ext ) z ( m ext ), m x ( ext ), m y ( ext ) z { [ ]} = [ 5,5 ],[ 0,0], 3,3 = [ 0,0 ],[ 7,7], 0,0 { [ ]} = { [ 0,550 ],[ 0,0], [ 0,320 ]} = [ 0,0 ],[ 0, 33000], 0,0 { [ ]} The determnatn f a stablty nterval s dne wth the help f a search prcedure that cmputes n a frst run the mst stable grasp frce wth mnmum securty (= 0) and the maxmum stablty para- 21

meter (= 1100). Because ths step fals n the strength f the defned frctn cndtns, an nterval search (mst secure grasp frce wth mnmum stablty) s perfrmed ver the stablty nterval [0, 1100] t cmpute the best cmpensatng grasp frce fr the manpulatn frce area MA. Ths leads after an apprprate number teratns (desred accuracy = 0.01) t the ptmal nternal frce λ. Resultng nternal (grp) frce [mn] : ( λ 1,λ 2,λ 3 ) = ( 5400, 6300, 5200) Each cmpensatng grasp frce n the cntact space that s determned wth respect t a sngle manpulatn frce r a range f manpulatn frces can be graphcally vsualzed by the smulatn system. Fr that purpse, perpendcular prjectns f the frctn cnes n the cntact pnts are dsplayed tgether wth the tangental frces n the yz-plane and the nrmal frces alng the x-drectn. Fg. 8 : Cmputatn f stable and secure grasp frces In Fg. 8, the exterr crcles represent the brders f the frctn cnes at the heght f the maxmal cntact frce whereas the nterr crcles le n the brders f the frctn cnes at the heght f the superpstn f manpulatn and cmpensatng grasp frces. The dagrams f the cmputed cntact frces pnt ut very clearly that the stablty search shwn n the left hand sde maxmzes the length f the grasp frces. Cnsequently, the dstances between the nterr and exterr crcles are relatvely small. In cntrary, the securty search shwn n the rght hand sde mnmzes the length 22

f the grasp frces, s that the dstances between the nterr and exterr crcles are relatvely large. The smulatn f reactn frces and mments ccurrng durng part matng peratns accrdng t pstnal and/r rtatnal uncertantes f bject mtns and ther effect n the wrld mdel are als cnsdered at the example f the peg-n-hle nsertn prblem. The frce smulatr s a prgram package based n a stffness cntrl mdel f the nsertn prcess [26] whch has the ablty t mdfy the cmplance parameters and t analyze the fnger-bject nteractns at the defned cntact pnts. Ths permts testng f manpulatn strateges fr specfc assembly tasks by takng nt accunt the bject frces and mments, cntact frces and stablty cndtns f the grasped bject. Fg. 9 : Vsualzatn f a peg-n-hle nsertn prcess Fg. 9 shws the graphcal vsualzatn f a peg-n-hle nsertn peratn wth the three-fngered Karlsruhe Dextrus Hand. Ths s bass fr the cmparsn f dfferent grasp cnfguratns accrdng t ther rbustness aganst external nfluences (dsturbances) and allws als the verfcatn f the planned frce parameters. Therefre, the grasp plannng prcess must ensure that the cmputed cntact frces stay nsde f the frctn cnes what results n a stable and secure grp f the bject. 8. Cnclusns und future wrk Ths paper descrbes the fundamentals f bject manpulatn and grasp plannng fr dextrus rbt hands cnsderng bth rllng and slppng f the fngertps n the bject surface. The prcess f frce cmputatn and evaluatn f sutable grasp cnfguratns s shwn fr tw dfferent appraches. As majr result, the frce ptmzatn prblem culd be generalzed fr the case f arbtrary rbt hands and cntact mdels whch are specfed by the parameters, jn, cmd and ndm. 23

The cmputatn methds wth ther cmbnatn f matrx multplcatns, matrx nversns, slutn f equatn systems and search fr ptmal frce parameters are well-suted fr the applcatn and executn n parallel machnes. Snce the presented algrthms, lke the maxmum search algrthm, are frmulated n a way that supprts rather ther realzatn n a seral r dstrbuted cmputer, ther transfrmatn and mplementatn n a parallel cmputer s an nterestng new apprach. Especally, because the parallelzatn f algrthms and ther adaptatn t massve parallel machnes wth lts f prcessr ndes s actually an underdevelped area n the feld f rbtcs research. One majr prblem arses hereby frm the nature f the cntact frces and therefre fr the nternal grp frces λ tself: they are crsswse dependng n each ther, s that they cannt be treated ndependently. Ths s expressed n the addtve nature f the grp submatrces G whch frm altgether the grp matrx G that descrbes bth the cntact pnts and cntact mdels, and characterzes the resultng cntact frces that can be appled wth respect t gven external frces and mments. Ths means, that the cmputatnal effrt requred fr slvng the ptmzatn prblem cannt easly be subdvded upn ndependent prcesses whch wuld mprve the search prcedure tremendusly. Nevertheless, t s pssble t use parallelsm t speed up the search prcedure that tres t ptmze the frce parameters by fttng them nt the bggest vlume n R ndm that has the greatest dstance t all gven cnstrants. Ths culd be dne by splttng the whle search space nt several regular shaped areas frm whch ndependent wrkng prcesses culd start ther parameter search; because there are n restrctns n the chce f the start pnts, the cnvergence f the algrthm t the resultng end pnts whch are pssble (ptmal) slutns s always ensured. Thus, research wll n the future be mre cncentrated n questns f parallelzng the search and ptmzatn algrthm. Acknwledgment Part f ths research wrk was perfrmed at the Insttute fr Real-Tme Cmputer Cntrl Systems and Rbtcs, Prf. Dr.-Ing. Ulrch Rembld and Prf. Dr.-Ing. Rüdger Dllmann, Faculty f Infrmatcs, Unversty f Karlsruhe, 76128 Karlsruhe, Federal Republc f Germany. The cntnuatn f the wrk and extensn f the thery was perfrmed durng my pst-dctral research stay at the Internatnal Cmputer Scence Insttute, Prf. Jerme Feldman, Berkeley 94704 CA, USA. References [1] Hllerbach, J.M.: "Rbt Hands and Tactle Sensng", AI n the 1980s and Beynd, Grmsn, W.E.L.; Patl, R.S.; MIT Press, 1987. [2] Wöhlke, G.: "Develpment f the Karlsruhe Dextrus Hand", Prc. f the 3rd Int. Cnf. n New Actuatrs (ACTUATOR), 1992. [3] Dll, Th.; Schneebel, H.J.: "Framewrk f the Karlsruhe Dextrus Hand", Prc. f the Symp. n Rbt Cntrl (SYROCO 88), 1988. [4] Dll, Th.: "Develpment and Prgrammng f a Rbt Hand", PhD thess (n German), Unversty f Karlsruhe, 1989. [5] Wöhlke, G.: "A Prgrammng and Smulatn Envrnment fr the Karlsruhe Dextrus Hand", Jur. f Rbtcs and Autnmus Systems, Nrth-Hlland, vl. 6(3), pp. 243-263, 1990. 24

[6] Wöhlke, G.: "Autmatc Grasp Plannng fr Multfngered Rbt Hands", Jur. f Intellgent Manufacturng, vl. 3(5), pp. 297-316, 1992. [7] Okada, T.; Tsuchya, S. : "On a Versatle Fnger System", Prc. f the 7th Int. Symp. f Industral Rbts, 1977. [8] Okada, T.: "Cmputer Cntrl f Multjned Fnger System fr Precse Object Handlng", IEEE Trans. n Systems, Man and Cybernetcs, vl. 12 (3), May/June 1982. [9] Jacbsn, S.C.; Wd, J.E.; Knutt, D.F.; Bggers, K.B.: "The Utah/MIT Dextrus Hand: Wrk n Prgress", Frst Int. Cnf. n Rbtcs Research, MIT Press, 1984. [10] Jacbsn, S.C. et.al.: "Desgn f the Utah/MIT Dextrus Hand", Prc. f the IEEE Int. Cnf. n Rbtcs and Autmatn, 1986. [11] Narasham, S.: "Dextrus Rbtc Hands: Knematcs and Cntrl", Master thess, Department f Electrcal Engneerng and Cmputer Scence, MIT, 1988. [12] Salsbury, K.; Brck, D.; Chu, S.: "Integrated Language, Sensng and Cntrl fr a Rbt Hand", Prc. f Int. Symp. n Rbtcs Research, MIT Press, 1986. [13] Nguyen, V.: "Cnstructng Stable, Frce-Clsure Grasps", Master Thess, Department f Electrcal Engneerng and Cmputer Scence, MIT, 1986. [14] Arbb, M.A.; Iberall, T.; Lyns, D.: "Crdnated Cntrl Prgrams fr Mvements f the Hand", Center f Systems Neurscence and Labratry fr Perceptual Rbtcs, COINS Techncal Reprt 83-25, Massachusetts, 1983. [15] Lyns, D.M.: "A smple Set f Grasps fr a Dextrus Hand", Prc. f the IEEE Int. Cnf. n Rbtcs and Autmatn, 1985. [16] Iberall, T.: "Grasp Plannng fr Human Prehensn", Prc. f the Int. Jnt Cnf. n Artfcal Intellgence, 1987. [17] Härtl, H.: "Analyss f Manpulatn Prcesses wth Multfnger Grpper Systems under the Cnsderatn f Rllng and Slppng f the Fngertp", Master thess (n German), Unversty f Karlsruhe, 1992. [18] Crag, J.J.: "Intrductn t Rbtcs: Mechanc and Cntrl". Addsn-Wesley, 1986. [19] Kerr, J.; Rth, B.: "Analyss f Multfngered Hands", Int. Jur. f Rbtcs Research, vl. 4(4), pp. 3-17, Wnter 1986. [20] Cle, A.; Hauser, J.; Sastry, S.: "Knematcs and Cntrl f Multfngered Hands wth Rllng Cntact". Prc. f the IEEE Int. Cnf. n Rbtcs and Autmatn, 1988. [21] Hsu, P.; L, Z.; Sastry, S.: "On Graspng and Crdnated Manpulatn by a Multfngered Rbt Hand". Prc. f the IEEE Int. Cnf. n Rbtcs and Autmatn, 1988. [22] Vgelgesang, V.: "Develpment f a plannng cmpnent fr the determnatn f stable grp cnfguratns t d parts assembly wth mult-fnger grppers", Master thess (n German), Unversty f Karlsruhe, 1990. [23] Brnshten, I.N.; Semendyayev, K.A.: "Taschenbuch der Mathematk", Harr Deutsch Verlag, 1987. [24] Brnshten, I.N.; Semendyayev, K.A.: "Ergänzende Kaptel zum Taschenbuch der Mathematk", Harr Deutsch Verlag, 1991. [25] Wöhlke, G.: "Knwledge-Based Grasp Plannng fr Mult-Fnger Rbt Hands", PhD these (n German), Unversty f Karlsruhe, 1991. [26] Whtney, D.E.: "Quas-statc Assembly f Cmplantly Supprted Rgd Parts", ASME Transactns n Dynamc Systems, Measurement, and Cntrl, vl. 104(1), 1982. 25