Distributed Trajectory Generation for Cooperative Multi-Arm Robots via Virtual Force Interactions

Transcription

1 862 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B: CYBERNETICS, VOL. 27, NO. 5, OCTOBER 1997 Distributed Trajectry Geerati fr Cperative Multi-Arm Rbts via Virtual Frce Iteractis Tshi Tsuji, Achmad Jazidie, ad Makt Kaek Abstract A trajectry geerati methd fr multi-arm rbts thrugh cperative ad cmpetitive iteractis amg multiple ed-effectrs is prpsed. The methd ca geerate the trajectries f the multiple arms i a distributed maer based a ccept f a virtual iteracti frce which represets a iteracti betwee a ed-effectr ad a evirmet. It is shw that the methd is effective t ly fr simple cperative tasks such as psitiig a cmm bject, but als fr mre cmplicated tasks icludig relative mtis amg arms. I. INTRODUCTION As regards geeratig a trajectry f a multi-arm rbt, may iterestig studies have bee reprted, up t the preset, fr a purpse f prgressig abilities f rbtic systems. Lee [1] prpsed a methd usig a maipulability measure, which ca deal with a dual-arm rbt ly. Al-Jarrah ad Zheg [2] prpsed a methd fr dual-arm rbts hadlig flexible bjects. M ad Ahmad [3] applied a trajectry time scalig ccept t the multi-arm rbts, ad later, i rder t reduce cmputati time, M ad Ahmad [4] develped a sub-time-ptimal trajectry plaig fr cperative multi-arm rbts usig a lad distributi scheme. A algrithm fr the time ptimal trajectry geerati was als prpsed by Wag ad Pu [5] based a cell-t-cell mappig methd. I the greater part f the previus researches, hwever, a desired spatial ed-effectr trajectry f a multi-arm rbt fr a task is assumed t be give befrehad, ad attetis f the ivestigatrs have bee paid fr ly simple cperative tasks such as a pick-ad-place mti f a cmm bject. O the ther had, let us csider a cperative task betwee a left had ad a right had t pare a apple. The left had tightly hlds the apple, ad the right had hlds a kife ad pares the apple. The left had shuld ctrl the psiti f the apple while the right had shuld ctrl a relative mti f the kife t the apple rather tha its abslute psiti i the task space. If the cperative task illustrated abve becmes pssible, a eed f the multi-arm rbts will icrease, i particular, i ustructured ad hazardus evirmets such as the space, udersea ad uclear pwer plats. S far, ly a few studies dedicated fr such cmplicated tasks f the multi-arm rbt have bee reprted. Fr example, Yamamt ad Mhri [6] prpsed a trajectry geerati methd fr a cperative task f a multi-arm rbt, where e arm is graspig ad mvig a bject ad thers prcessig the surface f the bject. Als, Tsuji [7] has prpsed a methd utilizig redudat degrees f freedm f a clsed lik system cmpsed by multiple rbtic arms. Hwever, sice all these methds geerate trajectries based gemetrical cstraits f a clsed lik structure cmpsed by multiple rbtic arms, plaig a trajectry fr each arm requires etire ifrmati mvemets f all ther arms. Thus, a cetralized system fr plaig Mauscript received April 26, 1995; revised August 29, T. Tsuji ad M. Kaek are with the Departmet f Idustrial ad Systems Egieerig, Hirshima Uiversity, Higashi-Hirshima, 739 Japa ( tsuji@huis.hirshima-u.ac.jp). A. Jazidie is with the Surabaya Istitute f Techlgy, Suklil-Surabaya, Idesia. Publisher Item Idetifier S (97) Fig. 1. Crdiate systems fr a -arm rbt system perfrmig a cperative task. The first t 0 th arms ctrl the mti f the task pit ad the ( 0 +1)st t th arms execute the relative mtis betwee the task pit ad the ed-pit f each arm. mvemets f all arms usig a sigle cmputer will evetually face prblems i terms f failure resistace, flexibility, expadability ad s, as the umber f arms r the degrees f freedm f the jits icreases. Oe pssible apprach that ca be take t vercme such prblems is t cstruct a autmus decetralized ctrl system which is cmpsed f a set f autmus subsystems i a distributed maer [8], [9]. I this apprach, varius characteristics may be realized as fllws: 1) partial failure i a subsystem ca be hadled lcally, sice cetral ctrller exists; 2) by chagig lcal iteractis amg subsystems, the verall system behavir ca be regulated flexibly; ad 3) the system structure cmpsed by autmus subsystems allws a easy expasi f the system withut re-plaig f the mti f the etire system. Recetly, a variety f ctrl systems based the ccept f the autmus decetralized system have bee prpsed (fr example, [10] ad [11]). Hwever, these methds attempt t ctrl a mbile rbt system r a sigle-arm rbt, t a multi-arm rbt, by decetralizig it it a set f subsystems. I the preset paper, a ew methd fr multi-arm rbts is develped, which ca geerate trajectries f the multiple arms i a parallel ad distributed maer thrugh cperative iteractis amg subsystems crrespdig t each arm. I rder t express ifrmati exchagig amg subsystems, virtual dyamics are imagied fr each maipulatr, ad virtual iteracti frces arisig frm these virtual dyamics are used. Uder the prpsed methd, it is pssible t deal with t ly simple cperative tasks such as psitiig a grasped bject but als mre cmplicated cperative tasks icludig relative mtis amg the arms. II. BASIC FORMULATION OF MULTI-ARM ROBOT A. Cperative Task f Multi-Arm Rbt Let us csider a cperative task f a multi-arm rbt, which csists f tw sub-tasks: a) hldig ad mvig a cmm bject ad b) prcessig a surface f the bject (see Fig. 1). A grup f arms (the first t the 0 th arms) ctrl the mti f the bject i the task space, while the ther grup f arms (the ( 0 +1)st t the th arms) are prcessig the surface f the bject. The secd grup f arms have t ctrl relative psitis f the ed-effectrs t the bject rather tha the abslute psitis i the task space /97$ IEEE

2 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B: CYBERNETICS, VOL. 27, NO. 5, OCTOBER Nw, we defie a referece pit f the bject (e.g., the ceter f mass f the bject) as the task pit. Usig the task pit, the mti f multiple arms perfrmig a cperative task ca be described as a) mti f the task pit ad b) relative mti betwee each arm ad the task pit. I mti plaig f the task pit, it is assumed that a virtual rigid lik [12] is cected betwee the task pit ad the edeffectr f each arm. Als, slip mti betwee the ed-effectr f each arm ad the bject is assumed. O the ther had, the relative mti betwee each arm ad the task pit is used t represet mre cmplicated tasks such as gridig ad scrapig the bject. The relative mtis amg arms are expressed by cmbiatis f relative mtis betwee the ed-effectr f each arm ad the task pit. B. Crdiate Systems The jit degrees f freedm f each arm is deted by m i (i = 1; 2; 111; ) where is the umber f arms, ad the task space dimesi is deted by l: I the preset paper, three differet Cartesia crdiate systems are defied: 1) the wrld crdiate system, 6 ; 2) the task crdiate system, 6 c, which is a mbile crdiate system accrdig t the mti f the task pit; ad 3) the ed-pit crdiate system, 6 i (i = 1;111;); the rigi f which is lcated the ed-pit f the arm i. Let the psiti ad rietati vectr f the task crdiate system, 6 c, ad f the edpit crdiate system, 6 i, with respect t the wrld crdiate system, 6, be deted as ad X c =[ p c ; 8 c ] T 2< l X i =[ p i ; 8 i ] T 2< l ;111; respectively. Let als the psiti ad rietati vectr f the ed-pit crdiate system, 6 i, represeted i the task crdiate system, 6 c; be deted as c X i =[ c p i ; c 8 i ] T 2< l ;111; as shw i Fig. 1. The, the psiti ad rietati f the task pit, X c, is uiquely cmputed frm X i ad c X i. I the three-dimesial (3-D) space (l = 6), fr example, the relatiship amg the psiti vectrs p c, p i ad c p i is give as fllws: c i p = p 0 Ri ( 8 i )[ c R i ( c 8 i )] Tc p i (1) where the rtati matrices frm 6 i t 6 ad frm 6 i t 6 c are deted as R i ( 8 i ) ad c R i ( c 8 i ), respectively. Als, the use f the Euler agle 8=[; ; ] T fr each rietati vectr leads t the fllwig expressi fr the rtatial matrix Rc ( 8 c ) frm 6 c t 6 [13] Rc ( 8 c )= R i ( 8 i )[ c R i ( c 8 i )] T = R 11 R 12 R 13 R 21 R 22 R 23 : (2) R 31 R 32 R 33 The, based the prperty f the Euler agle, the rietati vectr c 8 =[ c ; c ; c ] T ca be btaied i the fllwig way. a) Whe si c 6= 0; c = ata 2(6R23; 6R 13); (3) c = ata 2(6 R R 2 23 ;R 33); (4) c = ata 2(6R32 ; 7R 31 ): (5) b) Whe si c = 0; c = arbitrary (6) c = (1 0 R33) (7) 2 c c = ata 2(R21 ;R 22 ) 0 R 33 : (8) C. Kiematics f Multi-Arm Rbt As is well-kw, the relatiship betwee the ed-pit velcity vectr, _X i ; f the arm i ad the jit agular velcity vectr, _q i 2 < m ;is give by _X i = J i _q i (9) where J i 2< l2m is the Jacbia matrix f the arm i. O the ther had, assumig that a virtual rigid lik is cected betwee the task pit ad the ed-pit f the arm i, we ca derive the relatiships betwee the ed-pit f the arm i ad the task pit as fllws: F ci = G i H i F i = G i F tri (10) _X tri = H i _X i = G i _X c (11) where F i 2 < l expresses the frce/mmet vectr f the edpit f the arm i represeted i the wrld crdiate system, 6 ; _X c ad F ci 2 < l represet the velcity vectr f the task pit ad the frce/mmet vectr trasmitted t the task pit frm the ed-pit f the arm i, represeted i the wrld crdiate system, 6, respectively. The ctact type matrix H i 2< l2l ad the matrix G i = S i H i 2 < l2l express filterig effects that filter ut sme frces/mmets f the ed-pit f the arm i ad trasmit ther frces/mmets t the task pit depedig the ctact mechaism. Als, the ed-pit velcity vectr f the arm i trasmitted frm the task pit is deted as _X tri 2 < l, ad the vectr f the frces/mmets trasmitted frm the ed-pit f the arm i t the bject as F tri 2 < l [14], [15], where l i is the umber f the degrees f freedm f the frces/mmets that ca be trasmitted frm the ed-pit f the arm i t the bject. The matrix S i 2 < l2l expresses the gemetrical relatiship betwee the task pit ad the ed-pit f the arm i, which is give by S i = I 0 ( c p i ) 6 I (12) where I is the uit matrix; 0 is a zer matrix; ad ( c p i ) 6 is c p i represeted i 6 : I additi, is the crss peratr that satisfies (a)u = a 2 u fr ay vectr a ad u: Whe a =[a; b; c] T ; it is defied as [16] a = 0 0c b c 0 0a 0b a 0 : (13) Sice the frce/mmet vectr F c actig the task pit is the ttal sum f all frce/mmet vectrs trasmitted frm the edpits f the arms t the task pit, the et frce/mmet vectr F c is btaied by c ci F = F : (14) The kiematic relatiships are summarized i Fig. 2. Based the abve frmulatis, a trajectry geerati methd f the edeffectrs fr multiple arms is explaied i the ext secti.

3 864 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B: CYBERNETICS, VOL. 27, NO. 5, OCTOBER 1997 f each arm ad the task pit, c X i 2 < l ;which are assumed t be give as the desired mti. Sice we ca dete as l i = l ad H i = I l (a l 2 l uit matrix) fr arm i (i = 0 +1;111;); the ed-pit accelerati vectr i the trasmissi space, X tri ; ca be writte as X tri = X c + + Rc 0 c X i + 2 _R c 0 0 Rc 0 _R c _X i c R c 0 c i X : (19) 0 0 Fig. 2. Kiematic relatiship f mti ad frce f arm i: III. DISTRIBUTED TRAJECTORY GENERATION BASED ON VIRTUAL DYNAMICS The methd prpsed here ivlves a set f subsystems crrespdig t each arm ad ca geerate jit trajectries that satisfy kiematic cstraits f the ed-effectr f each arm i a parallel ad distributed maer thrugh cperative ad cmpetitive iteractis amg subsystems. I rder t express the iteractis amg subsystems apprpriately, virtual dyamics f each arm ad the task pit are itrduced. The virtual iteracti frces/mmets geerated frm the virtual dyamics ad psiti cstraits resultig frm kiematic relatiships betwee each arm ad the task pit are derived thrugh aalgy f mechaical systems t represet the iteractis amg subsystems. A. Structure ad Mti Equati f Subsystems First f all, the virtual dyamics f the arm i is defied usig the simplest secd rder differetial equati as fllws: q i = i +(H i J i ) T i (15) where i 2< m is the virtual jit ctrl trque vectr f the arm i; ad i 2< l is the frce/mmet vectr actig the ed-pit f the arm i frm the bject represeted i the wrld crdiate system, 6 : The, let us csider the mti f the task pit. Sice the virtual frce/mmet vectr i is exerted t the ed-effectr f the arm i frm the bject, a reactig virtual frce/mmet vectr 0 i is exerted cversely t the bject frm each ed-pit. As a result, a et virtual frce/mmet vectr actig the bject at the task pit is give by F c = k=1 F ci = k=1 (0G i i ): (16) O the ther had, virtual dyamics f the task pit is assumed as M c X c = F c (17) where M c 2 < l2l is iterpreted as the virtual iertia matrix f the bject; X c 2 < l is the accelerati vectr f the task pit represeted i the wrld crdiate system, 6 : Nw, let us csider psiti cstraits impsed the ed-pit f each arm. The arms i (i = 1;111; 0 ) must be cstraied by the mti f the task pit determied by the dyamics f equati (17). I ther wrds, equati (11) leads t the relatiship X tri = _ G i _X c + G i X c : (18) O the ther had, i rder t derive the kiematic cstraits fr the arm i (i = 0 +1; 111;);we have t csider t ly the mti f the task pit but als the relative mtis betwee the ed-effectr The ed-effectr accelerati vectr X tri cmputed by (18) ad (19) must agree with the ed-effectr accelerati determied by the jit mti f the arms X tri = H i _ J i _q i + H i J i q i : (20) Csequetly, the jit accelerati f the arm i ad the virtual iteracti frce/mmet vectr i ca be btaied usig (15) ad (20) q i i = I m 0(H i J i T ) H i J i 0 01 i H i J_ i _q i 0 X tri (21) where I m is a m i 2 mi uit matrix. The resulted jit accelerati i tur geerates the ed-effectr trajectry f each arm. Nw, the virtual jit ctrl trque i is cmputed usig the target psiti f the task pit, X c ; as fllws: where i =(H i J i ) T (G i ) + fk i ( X c 0 X c )g0b i _q i (22) (G i ) + =(G i G i ) 01 G i =H i (S i ) 01 2< l 2l is the pseud-iverse matrix f G i ; K i 2< l2l is a psitive defiite psiti feedback gai matrix fr the arm i; ad B i 2< m 2m is the psitive defiite velcity feedback gai matrix f the arm i. Fr the ith arm that is executig the relative mti (i = 0 +1;111;); we ca easily shw G i =0ad (G i ) + =0:The, the virtual jit ctrl trque fr the ith arm (i = 0 +1;111;) is reduced t i = 0B i _q i : (23) The trajectry geerati methd prpsed here is illustrated i Fig. 3. Each subsystem geerates a trajectry cperatively usig the virtual ed-pit frce/mmet vectr i as the ifrmati f the iteractis via the virtual dyamics f the task pit. Sice each subsystem ca perate idepedetly, it is t ecessary t mdify the mti equatis f all subsystem if the purpse f the arm chages frm the task f hldig ad mvig the bject t the task icludig a relative mti, r if a ew arm is added t the system. I the fllwig secti, stability f the system ad the kiematic prperty f the equilibrium pit is aalyzed. B. Stability Aalysis Let us csider a eergy fucti H that is cmpsed f tw types f eergy fuctis H 1 ad H 2 as fllws: H = H 1 + H 2; (24) H 1 = E i + Q c _q i _q i (25)

4 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B: CYBERNETICS, VOL. 27, NO. 5, OCTOBER (a) (a) (b) Fig. 3. Distributed trajectry geerati fr a multi-arm rbt. (b) (c) Fig. 4. Mdel f a three-arm rbt ad its iitial psture used i simulatis. The task crdiate system ad the desired psiti f the task pit are shw. H 2 = 1 2 i= +1 _q i _q (26) E i = 1 2 ( X c 0 X c ) T K i ( X c 0 X c ) (27) Q c = 1 _X c M 2 c _X c : (28) H 1 ad H 2 are eergy fuctis fr the mti f the task pit ad the relative mti betwee each arm i (i = 0 +1;111;) ad the task pit, respectively; E i represets the squared psiti errr betwee the target ad curret psitis f the task pit calculated at each subsystem i (i =1;2;111; 0 ); ad Q c expresses the virtual kietic eergy f the task pit. Nw, let us csider the eergy fucti fr the mti f the task pit, H 1. The time derivative f the eergy fucti, _H 1; ca be derived as _H 1 = _E i + _ Q c + _q i q i (29) _E i = 0( _X c ) T K i ( X c 0 X c ) (30) _Q c = _X c M c X c : (31) Fig. 5. A example f the geerated trajectry fr psitiig the task pit. (a) Stick pictures, (b) time histry f the psiti f the task pit, ad (c) time histry f the virtual iteracti frces/mmets f the arm 1. Substitutig (15) (17) ad (22) it (29) (31), the fllwig equati ca be btaied: _H 1 = 0 _X c i= +1 G i i 0 _q i B i _q i : (32) O the ther had, the time derivative f the eergy fucti, _H 2 ; ca be give by _H 2 = i= +1 [0 _q i B i _q i + _X tri i ]: (33) As a result, frm (32) ad (33) the time derivative f the eergy fucti f the whle system ca be btaied as _H = 0 _q i B i _q i : (34) Sice B i is f psitive defiite, we have H _ 0 ad the eergy fucti H decreases mtically util _H =0, i.e., _q i =0(i= 1;2;111;) ad _X c =0:This meas that the mti f the whle

5 866 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B: CYBERNETICS, VOL. 27, NO. 5, OCTOBER 1997 system is asympttically stable. C. Kiematic Prperty f the Equilibrium Pit Kiematic meaig f the equilibrium pit f the eergy fucti, H, is aalyzed uder a assumpti that the relative mtis betwee the ed-pits ad the task pit, c X i (i = 0 +1;111;), are give as the desired mti satisfyig c _X i = c X i =0at a certai time, t r, where t r is the time durati required fr relative mtis. It meas that, fr t>t r, there is relative mti betwee ed-pits ad the task pit, ad therefre, i =0(i= 0 +1;111;): Sice the task pit is at a stadstill i the equilibrium pit, usig (16) ad (17) ad i =0(i= 0 +1;111;); we have (a) 0 G i i =0: (35) Substitutig (15) ad (22) it (35) ad rememberig that at the equilibrium psiti, _q i = q i = 0 ad _X c = 0; the fllwig equati ca be btaied: G i (G i ) + K i ( X c 0 X c )=0 (36) where G i (G i ) + is a psitive semi-defiite matrix ad K i is a psitive defiite matrix frm its defiiti. Prvided that all frces/mmets actig the task pit ca be ctrlled usig the jit trque f the arm i (i =1;2;111; 0 ) ad 0 is sufficietly large, the matrix f 6 G i (G i ) + K i i (36) may be expected t be a sigular. Fr example, if we defie that K i = K (i = 1;2;111; 0 ); the (36) reduces t (b) G i (G i ) + K( X c 0 X c )=0: (37) Assumig that at least e maipulatr, fr example arm k, is cected t the bject thrugh the rigid graspig ctact type, i.e., l k = l, we have G k (G k ) + = I l (a l 2 l uit matrix). Csequetly the matrix f 6 G i (G i ) + i (37) is assured t be sigular. As a result, the sluti f (37) becmes X c = X c, ad the equilibrium pit f the task pit agrees with the crrespdig target psiti. If there is a uctrllable frce/mmet f the task pit, (36) becmes idefiite. Therefre the equilibrium pit des t always agree with the target psiti. Eve i this case, stability f the multi-arm rbt system ca be guarateed as described i the previus secti. Summig up, the distributed trajectry geerati methd fr multi-arm rbts has bee explaied i this secti. Als, the stability f the mti f the whle system ad the characteristics f the equilibrium pit f the task pit have bee aalyzed. I the ext secti, effectiveess f the prpsed methd will be verified by simulati experimets. IV. SIMULATION EXPERIMENTS Cmputer simulatis were carried ut usig a three-arm plaar rbt (Fig. 4). Each arm is a fur-jit type, ad legths f all liks are set t 0.4 (m). The task pit is defied as the ceter f gravity f the bject, which is the rigi f the task crdiate system (see Fig. 4). The parameters used fr the simulatis are fllws: the psiti feedback gais K i =diag:[100(n/m); 100(N/m); 100(N/rad)]; the velcity feedback gais B i = diag: [10; 10; 10; 10](Nm=(rad/s)) (c) Fig. 6. Results f trajectry geeratis. (a) The first jits f arms 1 ad 2 are fixed, (b) ed-effectr f each arm is cected t the bject thrugh a pit ctact, ad (c) ed-effectr f arm 3 perfrms a relative mti alg the surface f the bject. fr i = 1;2;3; ad M c = diag: [50(kg); 50(kg); 50(kgm 2 )]: Nte that diag: [1] detes a diagal matrix. Fig. 5 shws examples f the results where the psiti f the task pit is mved t the target psiti frm the iitial psiti idicated i Fig. 4. Fig. 5(a) shws stick pictures, while Fig. 5(b) expresses the time histry f the psiti f the task pit, ad Fig. 5(c) gives the time histry f the virtual iteracti frces/mmet geerated betwee the arm 1 ad the bject. I this case, all f the arms ctrl the mti f the task pit ad all arms grasp the bject rigidly, l i = 3fr i = 1;2;3: O the ther had, Fig. 6 shws ther results usig the same iitial arm psture ad the target psiti f the task pit as Fig. 5. Fig. 6(a) idicates the simulati result fr the case i which the first jits f the arm 1 ad arm 2 are fixed. I Fig. 6(b), the ed-pit f each arm is cected t the bject thrugh the pitctact, i.e., the ed-pit frces ca be trasmitted i ay directi t the bject, but the ed-pit mmets cat be trasmitted (l i = 2 fr i = 1;2;3): I Fig. 6(c), the ed-pit f the arm 3 perfrms a relative mti respect t the task pit alg the surface f the bject. Frm Fig. 6, it ca be see that i every case, the psiti f the task pit reaches the target psiti,

6 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B: CYBERNETICS, VOL. 27, NO. 5, OCTOBER whereas the itermediate trajectries ad fial pstures are differet csiderably. Sice tw jits were fixed i Fig. 6(a), bth f the arms 1 ad 2 mved usig ly three jits each. The prpsed methd ca easily deal with such cstraits, because fixig f a jit is hadled withi the subsystem lcally ad des t directly affect ther subsystems. I Fig. 5(a), the agle betwee the ed-pit f each arm ad the bject is kept cstat. O the ther had, they chages durig mti i Fig. 6(b), sice the ed-pit mmet cat be trasmitted t the bject, i.e., the ed-pit ca rtate freely. The methd ca geerate the trajectries uder varius ctact mechaism betwee the ed-pits f the arms ad the bject. I Fig. 6(c), the arms 1 ad 2 ctrl the task pit, ad the edpit f the arm 3 carried ut a relative mti with respect t the task pit. The relative mti f the ed-effectr f the arm 3 is give as a fucti f the time as fllws: c X 3 (t) = [0:1t 2 0 0:1(m); 0:1(m); 4 3 (rad)]t if 0 t 1 [00:1t 2 +0:4t00:3(m); 0:1(m); 4 3 (rad)]t : (38) if 1 t 2 [0:1(m); 0:1(m); 4 3 (rad)]t if t 2 It ca be see frm Fig. 6(c) that the cperative mti ca be realized maitaiig the clsed lik structure. V. CONCLUSION This paper has prpsed the ew trajectry geerati methd fr multi-arm rbts usig the ccept f the virtual dyamics. This methd is based a effrt t express iteractis amg the arms by usig the virtual frces/mmets trasmitted frm each arm t the task pit, s that the methd ca geerate the trajectries f the multiple arms i a parallel ad distributed maer thrugh cperati ad cmpetiti amg subsystems crrespdig t each arm. I fact, it shuld be ted that fr multiple rbts ivlved i crdiati, the prpsed methd is t the ly e which ca treat idividual rbts i a distributed maer. The cvetial dyamic aalysis will als lead t the geerati f jit trajectries f all the maipulatrs. I cmparis with ther appraches, the advatages f the prpsed methd are summarized as fllws. [2] O. Al-Jarrah ad Y. F. Zheg, Efficiet trajectry plaig fr tw maipulatrs t defrm flexible material, i Prc It. Cf. Itelliget Rbts Systems, 1994, vl. 1, pp [3] S. B. M ad S. Ahmad, Time-scalig f cperative multirbt trajectries, IEEE Tras. Syst., Ma, Cyber., vl. 21, pp , Apr [4], Sub-time-ptimal trajectry plaig fr cperative multimaipulatr systems usig the lad distributi scheme, i Prc IEEE It. Cf. Rbtics Autmati, 1993, vl. 1, pp [5] F. Y. Wag ad B. Pu, Plaig time-ptimal trajectry fr crdiated rbt arms, i Prc IEEE It. Cf. Rbtics Autmati, vl. 1, pp [6] M. Yamamt ad A. Mhri, A study fr crdiated tasks f multirbt system prblem statemet ad task plaig fr tw-rbt system, i Prc. 9th Au. Cf. Rbtics Sc. Japa, 1991, pp , (i Japaese). [7] T. Tsuji, Trajectry geerati f a multi-arm rbt utilizig kiematic redudacy, J. Rbt. Mechatrics, vl. 5,. 6, pp , [8] T. Fukuda, S. Kawauchi, ad H. Asama, Aalysis ad evaluati f cellular rbtics (cebt) as a distributed itelliget system by cmmuicati ifrmati amut, i Prc IEEE/RSJ It. Wrkshp Itelliget Rbts Systems, 1990, pp [9] R. C. Arki, T. Balch, ad E. Nitz, Cmmuicati f behaviral state i multi-aget retrieval tasks, i Prc IEEE It. Cf. Rbtics Autmati, 1993, vl. 3, pp [10] H. Asama, M. K. Habib, ad I. Ed, Fuctial distributi amg multiple mbile rbts i a autmus ad decetralized rbt systems, i Prc IEEE It. Cf. Rbtics Autmati, pp [11] T. Tsuji, S. Nakayama, ad K. It, Distributed feedback ctrl fr redudat maipulatrs based virtual arms, i Prc. It. Symp. Autmus Decetralized Systems, 1993, pp [12] M. Uchiyama ad P. Dauchez, A symmetric hybrid psiti/frce ctrl scheme fr crdiati f tw rbts, i Prc IEEE It. Cf. Rbtics Autmati, pp [13] K. S. Fu, R. C. Gzales, ad S. G. Lee, Rbtics: Ctrl, Sesig, Visi, ad Itelligece. New Yrk: McGraw-Hill, [14] M. R. Cutksky ad I. Ka, Cmputig ad ctrllig the cmpliace f a rbtics had, IEEE Tras. Rbt. Autmat., vl. 5,. 2, pp , [15] M. T. Mas ad J. K. Salisbury, Rbt Hads ad the Mechaics f Maipulati. Cambridge, MA: MIT Press, [16] R. Featherste, Rbt Dyamics Algrithm. Nrwell, MA: Kluwer, ) Usig the virtual dyamics, the mbility f the bject depedig the directi f the mti ca be regulated. Als, if a virtual iertia f each arm is icluded the basis f the ccept f the virtual dyamics, the mbility f each arm relative t ther arms ad the bject ca be set accrdig t the give task. 2) The subsystem crrespdig t each arm ca wrk idepedetly, s that the advatages f the autmus decetralized system i terms f failure resistace, expadability ad parallel cmputati are als held with the prpsed methd. 3) The relative mti betwee the ed-pit f each arm ad the task pit ca be give as the desired trajectry which are treated as the cstrait maipulatr mti. REFERENCES [1] S. Lee, Dual redudat arm cfigurati ptimizati with task rieted dual arm maipulability, IEEE Tras. Rbt. Autmat., vl. 5,. 1, pp , 1989.