Motion Planning for Cooperative Transportation of a Large Object by Multiple Mobile Robots in a 3D Environment

Transcription

1 Motion Planning for Cooprativ Tranportation of a Larg Objct by Multipl Mobil Robot in a 3D Environmnt Atuhi YAMASHITA, Maaki FUKUCHI, Jun OTA, Tamio ARAI and Hajim ASAMA Dpartmnt of Prciion Machinry Enginring, Th Univrity of Tokyo 7-3- Hongo, Bunkyo-ku, Tokyo, , JAPAN Tl: (Dirct In) Fax: (Dpartmnt Offic) Th Intitut of Phyical and Chmical Rarch (RIKEN) 2-, Hiroawa, Wako-hi, Saitama, 35-98, JAPAN addr: yamaita@princ.p.u-tokyo.ac.jp Abtract In thi papr, w propo a motion planning mthod for cooprativ tranportation of a larg objct by multipl mobil robot in a 3 dimnional nvironmnt. Thi tak ha variou kind of problm, uch a path planning, manipulation, and o on. All of th problm can t b olvd at onc, inc computational tim i xplodd. Accordingly, w divid a motion plannr into a local manipulation plannr and a global path (motion) plannr, and dign th two plannr rpctivly. And w intgrat two plannr. Namly, w aim at intgrating a gro motion plannr and a fin motion plannr. A to th local manipulation plannr, w build a manipulation tchniqu, which i uitabl for mobil robot by poition-control. W comput condition, in which th objct bcom untabl during manipulation, and gnrat th ach robot motion conidring th robot motion rror and indfinit factor from th planning tag. A to th global path plannr, w rduc th dimnion of th configuration pac (C-pac) uing th fatur of tranportation by mobil robot. W can find a olution with arching in thi mallr dimnional C-pac uing th potntial fild dfind in th C-pac. And contraint of th objct manipulation ar conidrd a th potntial function. W vrify th ffctivn of our propod motion planning mthod through imulation and xprimnt. Kyword: Multipl Mobil Robot, Motion Planning, Cooprativ Tranportation, Manipulation, Motion Error, Cll Dcompoition, Potntial Fild. Introduction It i xpctd that mobil robot undrtak variou tak in manufacturing plant, warhou, contruction it, and o on. In ordr to improv flxibility and fault tolranc of tak, th concpt of coopration by multipl mobil robot i propod []. In th futur, mobil robot hould work in a ral 3 dimnional nvironmnt. In thi complicatd ituation, a good motion planning mthod i vry important to accomplih tak fficintly. Thr i a big dmand for a tranport tak with robot in particular. Thrfor, w propo a motion planning mthod for cooprativ tranportation of a larg objct by multipl mobil robot in a 3 dimnional nvironmnt. Howvr, thi tak ha variou kind of problm. For xampl, w mut plan path of th objct to avoid obtacl, contruct a tabl manipulation mthod, and dcid robot motion. All of th problm can t olv at onc, inc computational tim i xplodd. Convntional path planning mthod conidr only gomtrical and topological condition in particular, uch a hap of obtacl and robot. Thy don t conidr tatic and dynamic of th objct whn robot manipulat it. Accordingly, w divid a motion plannr into a local manipulation plannr and a global path (motion) plannr. Th formr trat how to manipulat an objct (Fig. ()). Th lattr plan path of th objct and robot (Fig. (2)). To intgrat two plannr, contraint of th objct manipulation ar conidrd in th path (motion) plannr. In othr word, w aim at intgrating a gro motion plannr and a fin motion plannr. Obtacl (2) Objct (2) Start () Robot Fig. Tranportation Tak Th compoition of thi papr i dtaild blow. Th nxt chaptr rviw th prviou work and dcrib outlin of our motion planning mthod. In chaptr 3 and 4, th local manipulation plannr and th global motion plannr ar xplaind. In chaptr 5, w vrify our mthod with imulation and xprimnt. Th concluion ar dicud in chaptr Tranportation Tak by Mobil Robot 2. Prviou work Many rarch ar tudid about th motion planning for th claical movr problm in a 3 dimnional nvironmnt. Thi problm i vry difficult bcau of th high dimnional configuration pac (C-pac) [2]. A brut forc arch can olv problm with low dimnional C-pac (for xampl, path planning of on movr in a 2 dimnional nvironmnt). Howvr, a impl plannr cannot olv high dimnional ()

2 problm (for xampl, in a 3 dimnional nvironmnt, th dimnion of th C-pac i at lat ix). Thr ar many tudi about a path plannr with high dimnional C-pac [3]. Kondo ud multi-huritic by arching [4]. Barraquand t al. contructd a randomizd plannr [5] and Gupta t al. propod a backtracking mthod [6] by olving problm with many dgr of frdom. Hwang t al. propod rolution-complt and fficint mthod [7] and Kavraki t al. propod a probabilitic roadmap mthod [8]. Th mthod can find a path of an objct or a robot fficintly. Th plannr purpo finding a path in th C-pac to avoid obtacl. In tranportation tak, w nd to conidr motion of workr or robot that tranport and manipulat an objct. And th plannr don t tak account of workr or robot motion. Thrfor, gnratd path (motion) ar difficult to raliz, for thr i no conidration about tability of an objct and how to tranport and manipulat th objct. A to cooprativ tranportation [9-] and manipulation of objct [-2] with multipl mobil robot ytm, thr ar many tudi. Ru, t al. [9] built an objct puhing and tranporting mthod uing th nor information. Th mthod i bad on th knowldg obtaind convntionally in th rarch fild of th puhing manipulation [3-6]. Koug t al. [] aimd at tranporting an objct by lifting, and adoptd th fdback control mthod uing th information of robot forc nor. Khatib t al. [] controlld innr forc that i applid to an objct and adoptd th tabl handling tratgy to compnat th motion rror of robot. Sawaaki t al. [2] ralizd rolling po chang of an objct with two mobil manipulator. Th mthod can crat a path of an objct and robot or a manipulation way of th objct in a 2 dimnional nvironmnt by uing nor data. In a 3 dimnional nvironmnt, th dimnion of th C-pac i too larg for th mthod to plan motion of an objct and robot. 2.2 Outlin of propod motion plannr In thi mthod, w divid a motion plannr for tranportation tak into a local manipulation plannr pha and a global motion plannr pha (Fig. 2). Th local plannr output a way and information of manipulation. Th manipulation way indicat motion of an objct and robot to chang th po of th objct afly. Th information of manipulation indicat th manipulation pac that i an ara ncary to manipulat th objct and th manipulation cot that i th ncary tim to finih th manipulation. Th information of manipulation i utilizd to plan th global motion of th objct and th robot. Th global motion plannr output path of th objct and th robot whil avoiding obtacl. Th global plannr dcid whn and whr to manipulat th objct and what kind of manipulation way can b ralizd bad on th rult of th local manipulation plannr. 2.3 Th local manipulation plannr For th local manipulation plannr, on of th biggt problm whn mobil robot work i th ffct of th robot poition rror. In th prviou work, th forc nor information and th imag information from th camra ar fd back to corrct th motion rror. Th motion rror of mobil robot ar avoidd with th on-lin mthod. And xciv innr forc that i applid to an objct i avoidd by forc control approach. In othr word, fdback control approach ha bn adoptd to manipulat an objct by multipl mobil robot ytm. Howvr, it i difficult for mobil robot to maur forc corrctly and to manipulat an objct with a forc-control mthod lik fixd manipulator whil thy mov. Thn, in thi rarch, w propo th robut manipulation planning mthod for th motion rror by taking into conidration of th motion rror in a motion planning tag. W aim at contructing th planning mthod of manipulation that i uitabl for th mobil robot undr poition control [8]. 2.4 Th global motion plannr For th global motion plannr, th biggt problm i an xploion of th computation tim that rult from th high dimnional C-pac. In thi papr, w rcontruct th C-pac and rduc th dimnion of C-pac by conidring th fatur of tranportation tak by mobil robot. And w can find a olution with arching in thi mallr dimnional C-pac uing th potntial fild dfind in th C-pac. And contraint in manipulating ar conidrd a th work ara and th potntial function. Th work ara i computd a th manipulation pac, and th potntial function i computd a th manipulation cot in th local plannr. Information of Manipulation Manipulation Spac Manipulation Cot & Manipulation Way Motion of Robot Motion of Objct Start Information Environmnt: Poition of Obtacl,... Objct: Shap, Ma, Poition of COF,... Robot: Shap, Max Forc,... Local Manipulation Plannr Fig. 2 Motion of Robot and Objct Global Motion Plannr Path of Robot and Objct Poition of Robot Poition of Objct Potur of Objct Outlin of Motion Plannr 2.5 Manipulation and Tranportation Styl W propo that robot manipulat an objct by puhing with tick in multipl mobil robot ytm (Fig.

3 3). Whn th manipulating tak ar carrid out, th tability of opration i improvd bcau th contact ara to th objct bcom largr [7]. Th robot tumbl th objct to chang it fac that contact with th floor from on fac to th nxt on. By rpating thi tumbling opration, it i poibl to chang th po of th objct who arbitrary fac touch th floor. tick objct robot Fig. 3 Tumbling Manipulation Thr occur poition rror whn poition controlld mobil robot mov. Thrfor, thr xit a rik that multipl mobil robot apply xciv innr forc to an objct whn thy touch th objct at th am tim. Thn, in thi papr, mor than two tick cannot touch th objct at any tim to avoid xciv innr forc whn th robot mov. Sinc it i difficult for mobil robot to lift up a larg or havy objct, thy manipulat th objct without lifting up opration. Thrfor th objct alway contact with th nvironmnt (a floor). Th robot can mov th tick up and down. And puhing and tumbling opration ar adoptd hr. And th robot tranport th objct around it. Whil tranporting th objct, robot don t chang thir poition to th objct. In othr word, thy don t chang thir formation. Th robot dpart from th objct whn thy manipulat it and thy kp clo to th objct whn thy tranport it. 3. Local Manipulation Planning 3. Poibl Manipulation Th robot tumbl th objct whil on dg (thi dg i th cntr of rotation) alway contact with th floor. W mak th following aumption for analyi: All motion ar quai-tatic. Th gomtry of an objct and th location of it cntr of ma ar known. Cofficint of friction btwn th objct and th floor and btwn th objct and th tick ar known. All frictional intraction of th objct ar dcribd by Coulomb' law of friction. Robot can mov in omni-dirction Robot hav lift-up mchanim to control thir nd-ffctor hight (contact point hight). 7 8 〇 〇 〇〇 2 Stabl Not Stabl Stabl (a) Objct Fac (b) Poibl Manipulation Fig. 4 Objct Rprntation In th local plannr, w conidr all poibl manipulation. At firt, w comput th tat that w can ground an objct. A fac numbr rprnt th groundd tat. For xampl, in th ca of an objct hown in Fig. 4(a), whn Fac,, 2, 3, 4, 7 (ordinary fac) and Fac 8 (virtual fac) touch th floor, th objct can b groundd (In thi tat, robot can tranport an objct). And Fac 5 and 6 cannot touch th floor, th objct cannot b groundd. Nxt, w chck if th objct can b manipulatd from on tat to anothr tat by tumbling afly. W u matrix A ij to xpr whthr it i a poibl manipulation or not. If th manipulation from Fac i to Fac j i poibl, A ij =. And if it i impoibl, A ij = (Fig. 4(b)). In th ca of Fig. 4, w can gt A ij a follow (Equation ()). A ij = () 3.2 Formulation of Manipulation Undr th aumption in ction 3. and th ituation mntiond a in Fig. 5, a 2 dimnional modl can xpr thi ituation in th local manipulation plannr. W can dcrib th contraint about th objct. F + F + G (2) = p F + p G (3) c = whr F : forc at tick point S, F : forc at contact point O G = (, -mg) : forc at th cntr of th ma (G) P = (x, y) : poition of th tick P g = (x g, y g ) : cntr of th ma of th objct y Edg2 Vrtx2 Vrtx(n-) F S (x, y) αμ f d F fn F Edg G Edg S (n-) ai fn Fg Oαμ fd F Vrtx O θ x Vrtx Edg Fig. 5 2D Objct Modl Lt f b th normal contact forc at S, and lt f b th normal contact forc at O. F = f n +α µ f d (4) F = f n +α µ f d (5) whr n = (u n, v n ), d = (u d, v d ) From Equation (2-5), w obtain

4 f u α µ f u + α µ f = (6) n + d vn µ f vd + f f f + α mg = (7) ( xv yu ) + α µ f ( n n xvd yud mg( x co θ y inθ ) = g g ) (8) f > and > (9) f α < and α < () f < F max () whr f : forc at S, f : forc at O F max : maximum forc of robot μ, : cofficint of friction at S, O m : ma of th objct, θ : angl of th objct W can know whthr th objct i tabl or not by olving Equation (6-8) undr contraint of Inqualiti (9-). Inquality (9) man that th objct do not dtachd from th tick and th floor. Inquality () man that th objct do not lip and kp tabl. Inquality () man that th robot do not apply th forc to th objct byond thir ability. 3.3 Stabl Domain Graph W mak th tabl domain graph that indicat th poition of th tick and th angl of th objct whn th objct i tabl. Y-axi man th paramtr a i howing th contact poition with th objct, and X-axi man th paramtrθ howing th angl of th objct (Fig. 6). Th paramtr a i (< a i <) xpr th contact point of th tick at Edg i of th objct. For xampl, whn a =, th tick contact with th objct at Vrtx. Whn a =.5, th tick contact with th objct at th cntr of Edg. Edg2 Vrtx2 Edg Vrtx3 Edg3 Edg Vrtx O Stick Edg a Vrtx θ a Stabl domain θ[rad] π/2 (b) Stabl Domain Graph (a) Paramtr a i andθ Fig. 6 Stabl Domain of Edg Th objct i tabl, if th paramtr (θ, a i ) i in tabl domain. Thrfor, if th objct i opratd in thi domain, th objct manipulation will not fail. Bcau th numbr of th dg i four in ca of rctangl, four tabl domain graph which how tabl domain of ach dg ar gnratd (Fig. 7(a)). And th limitation on th movmnt of robot (for xampl, th nd-ffctor hight i limitd from h min to h max ) can b conidrd in th tabl domain graph. Th ffct of motion rror of th mobil robot and nvironmntal indfinitn ar takn into conidration. A th rror factor which influnc a tabl domain i; (i) th motion rror of mobil robot, and (ii) th accuracy of givn data and th chang of a cofficint of friction. A to th formr problm, w arch a domain whr th tability of th objct i maintaind if th poition rror (±Δx,±Δy) xit on th poition of th tick. W can put th bnd of th tick on thi poition rror. A to th lattr problm, w arch a domain whr th tability of th objct i maintaind if cofficint of frictionμ, chang within th rang of ± Δμ,. And th input data rror (poition of th cntr of th ma (±Δx g,±δy g ), ma of th objct ±Δm) can b conidrd. Th tabl domain graph conidring th motion rror i hown in Fig. 7(b). In Fig. 7(b), th domain rducd a compard with Fig. 7(b) i a domain in which work fail, whn thr ar motion rror or anothr factor. a a Edg θ[rad] π/2 Edg a Edg a2 Edg2 θ[rad] π/2 θ[rad] π/2 θ[rad] π/2 (a) Without Error a3 Edg θ[rad] π/2 θ[rad] π/2 θ[rad] π/2 θ[rad] π/2 Fig. 7 a Edg a2 Edg2 a3 Edg3 (b) With Error Stabl Domain Graph of Each Edg 3.4 Opration Graph W gnrat th opration graph and utiliz it to plan th motion of th robot fficintly. Th procdur of gnrating th opration graph i a follow. Firt, w lct charactritic point whr th opration mthod may chang in th tabl domain graph. Concrtly, w choo th point whoθ valu ar th largt and th mallt in tabl domain (Fig. 7(b)). And th point ar rgardd a nod in th opration graph. Thrfor, in th opration graph, all nod that hav phyical maning ar collctd (Fig. 8(a)). Information that nod hold in th opration graph i th dg numbr and th angl of th objctθ. In th opration graph, information about a i i not conidrd to plan th motion of th tick fficintly. In th opration graph, th po chang of th objct i plannd by moving from on nod to othr nod. In th rctangl ca hown in Fig. 6(a), th po chang i prformd by moving from nod 3 (θ= [rad]) to nod 8 (θ=π/2 [rad]). In th opration graph, th arc btwn two nod indicat thr typ of opration mthod, (a) th continuou opration, (b) th hand-ovr opration, and (c) th tranfr opration. Th arc that i a horizontal lin btwn two nod indicat th continuou opration (Fig. 8(a)). In thi opration, it i poibl to chang th angl of th objct without changing th dg that th tick contact with. During th continuou opration, th contact point of th tick continuouly chang. It man that a i in th tabl domain graph chang continuouly. Th prpndicular arc indicat th hand-ovr opration (Fig. 8(b)). In thi opration, it i poibl to chang th dg that th tick contact with without changing th angl of th objct. Nod 9 and ar nwly gnratd in th opration graph. Th obliqu arc indicat th tranfr opration (Fig. 8(c)). In thi

5 opration, it i poibl to chang both th angl of th objct and th dg that th tick contact with. Thi opration i prformd whn thr i no tabl domain btwn two nod. To xpr th difficulti of th opration, th ditanc of th arc, which man moving cot btwn two nod, i introducd. Thn, th problm of chooing th manipulation mthod among thr opration com back to th hortt path problm. Long ditanc from th tart nod to th goal nod man that th po changing opration of th objct i difficult. Thrfor, w choo th hortt path, bcau th po-changing tak can b prformd aily. Th ditanc (cot) of th ach arc i dfind in Equation (2) and Tabl (Δθ man th chang of th objct angl). co = A θ + B (2) ( ) t a, b, c a, b, c a, b, c Edg 2 3 Edg 4 Edg2 Edg3 Edg Angl of objctθ[rad] π/2 Edg 39 Edg2 Edg3 Edg 2 4 (a) Continuou Opration Angl of objctθ[rad] π/2 Edg 39 Edg2 Edg3 4 2 (b) Hand-ovr Opration Angl of objctθ[rad] π/2 (c) Tranfr Opration Fig. 8 Opration Graph and Opration Accordingly, th cot ar dtrmind a follow: cot (a) < cot (b) < cot (c). Th cot of th continuou opration i lowr, bcau it i th ait way to manipulat th objct. Thi opration can b accomplihd with on tick and th total tim of opration i th hortt compard to othr opration. And th cot of th tranfr opration i highr, bcau it i th mot difficult. Thi opration nd two tick and th objct i not tabl during thi opration. Whn thi opration i prformd, a gratr ffct of th uncrtain factor of th nvironmnt mut b conidrd compard with othr opration. Tabl Paramtr A and B A B a b C a b c. Aftr th cot of arc ar dtrmind, Dijktra' algorithm i ud to olv th hortt path planning problm. From th rult of th arch, th hortt path i gaind; 3(initial tat) (goal tat). Path man th continuou opration on Edg. Path 4 7 man th tranfr opration btwn th tick on Edg and Edg 3. Path 7 8 man th continuou opration on Edg 3. It i not dtrmind th poition whr th tick contact with ach dg (a i ). Th rult obtaind hr i th ordr of th opration whr th tick contact. Th orbit of a tick i not dtrmind at th prnt. 3.5 Dtrmination of Robot Motion In thi rarch, th orbit of th tick i dtrmind aftr a procdur of objct opration i dtrmind. If it i th manipulation mthod for fixd manipulator, th contact poition orbit whr th forc applid to th tick i alway minimizd may b good. For xampl, whn th tick contact with Edg, th continuou opration whr a i nar lad to th ituation whr control forc i minimum. Howvr, whn optimizing th control forc, th orbit of a tick bcom a circl-lik hap. In thi ca th prpndicular and th horizontal pd of th tick nd to b controlld in a complicatd way vry tim. Thi ha th rik of failing tak convrly, for it i inconvnint for mobil robot to mak complicatd opration. Thn, th tick orbit that mobil robot can aily control form. In thi rarch, a traight-lin orbit i adoptd whil avoiding th objct. Th tick poition whn th continuou opration bgin and nd ar dtrmind to minimiz th forc at th point, and th mantim i connctd linarly. Whn th orbit i out of th tabl domain, th tick poition of th bginning and th nding ar changd uitably. Thn olution that th orbit i alway in th tabl domain i archd. Th manipulation mthod of th objct hown in Fig. 9 i off-lin-plannd by th local manipulation plannr. Fig. 9 Plannd Manipulation Mthod 3.6 Information of Manipulation Th manipulation pac whr robot work with objct i rprntd by th cll dcompoition mthod (th octr mthod). And th manipulation cot that man th total tim to finih manipulation i calculatd in th local manipulation plannr. Th information i ud in th global plannr. 4. Global Motion Planning 4. Rprntation of th Environmnt In th global motion plannr, path of th objct and robot whil avoiding obtacl can b find. Th global plannr dcid whn and whr to manipulat th objct and what kind of manipulation way can b ralizd bad on th rult of th local manipulation plannr.

6 In our mthod, all thing (an objct, robot and obtacl) ar rprntd by an octr that i th approximat cll dcompoition mthod for 3 dimnional nvironmnt. By uing th octr mthod to rprnt thm, all thing can b dalt with in th am way. And th numbr of cll i mall compard with th xact cll dcompoition mthod. Whn w olv problm with high dimnional C-pac, thi rprntation i fficint and practical. In our mthod, w aim at th rolution-complt and fficint plannr. A 2 dimnional nvironmnt rprntd by a quadtr i hown in Fig. (a) and a 3 dimnional nvironmnt rprntd by an octr i hown in Fig. (b). Cll ar dividd into whit on (fr pac) and black on (obtacl). Th objct and robot can b rprntd a on objct, bcau th formation of th robot i not changd whil tranporting th objct (Fig. (c)). A to th objct and robot, cll whr th objct and robot xit ar calld th objct cll C obj,j (j: cll numbr). rduc th dimnion of th C-pac uing th fatur of tranportation with mobil robot. In tranportation tak with mobil robot, th poibl path and motion ar limitd and ar diffrnt from th path in th problm of fr flying objct. Robot cannot tranport th objct whil changing it po in complicatd way. And thy chang th objct po from on groundd tat that on fac of th objct contact with th floor to othr groundd tat. Th tat ar abtractd a charactritic configuration and th plannr u th configuration to find a practical path. Th rcontructd C-pac i hown in Fig. 2. In Fig. 2, () corrpond R, R 2,,R m in Fig. and xpr thm a th formation of mobil robot ( dof). (2) corrpond γ in Fig. and xpr it a th orintation of th objct and th robot ( dof). (3) corrpond α and β in Fig. and xpr thm a th fac of th objct contactd with th floor ( dof). Accordingly, (6+r) dof can b rducd to 5 dof. Start (3) (a) 2D Environmnt (Quadtr) Objct Cobj, Cobj, 2... Objct Start () (2) Cobj, i Mobil Robot Mobil Robot and Objct (b) 3D Environmnt (Octr) (c) Robot and Objct Fig. Rprntation by Cll Dcompoition For rprnting th nvironmnt by an octr, a path fr from obtacl can b find fficintly a a qunc of whit cll from a tart configuration to a goal. And for all thing ar rprntd in th am way, colliion chck btwn two thing can b aily dtctd by chcking th ovrlap of cll. Not that, in thi mthod, C-obtacl don t comput poitivly and th plannr chck th colliion in th ral world (not in C-pac). 4.2 Rcontruction of Configuration pac W dal with th problm that multipl mobil robot (th numbr of robot: r) tranport an objct around it. Th dof of th C-pac i (6+r) hown in Fig.. (Th dof of th objct i 6 and th dof of robot i r). Robot R2 cntr of qravity Objct Robot R Robot Rm z γ {Q} y β α Robot R Robot R2 θ θ R2 R x θ Rm Objct Robot Rm Fig. Dof of C-pac Bcau of th high dimnional C-pac, w Fig. 2 Rcontructd C-pac 4.3 Graph Sarch Mthod A graph arch i prformd by uing th following valuation function f. f = i= U ( q ) + i i= M ( q i, q ) i (3) whr q i : configuration of th objct and robot : total tp from a tart to a currnt configuration In Equation (3), th firt componnt i th forc rcivd from th potntial fild gnratd at th configuration q i. And th cond componnt i th manipulation cot computd in th local plannr. Aftr th cot valuation function f i dfind in Equation (3), A* algorithm i ud to find th optimal path in C-pac from th tart configuration to th goal configuration. By uing thi function, w can gain a path that i far from obtacl and total tim that robot accomplih th tranportation tak i hort. 4.4 Potntial Fild A potntial fild i contructd with a rpuliv forc from obtacl and attractiv forc from a goal configuration. Th total forc from th potntial fild whr th configuration of th objct and robot i q, can

7 b calculatd a a um of th forc actd on th objct cll C obj,j (j =,, k). Th total forc actd on q i dfind a follow. (k i th numbr of th objct cll) U ( q) = k j= P w ( C obj, j ) (4) P w ( Cobj, j ) = Prp ( x j ) + patt ( x j ) (5) whr x j : rfrnc point of objct cll C obj,j P w (C obj,j ) : Th forc actd on th objct and robot P rp (x j ) : rpuliv forc from obtacl P att (x j ) : attractiv forc from th goal In thi papr, w adopt th potntial fild from obtacl mntiond in [9], which i on of th implt potntial function. Th potntial fild in th 2 dimnional nvironmnt of Fig. (a) i hown in Fig. 3. And thi potntial function i normalizd with th vlocity of robot. Thrfor, th dimnion of U(q i ) i am a th dimnion of a tim. tranport th objct whil avoiding obtacl. And thy chang th po of th objct at point A and B. Th numbr of tim of manipulation can b rducd by uing manipulation cot a potntial function. In Fig. 6, th manipulation way of th objct at point A i computd. In thi imulation, objctiv tak ha bn ralizd without failur, uch a th objct lip or fall down undr th motion rror of robot and nvironmntal indfinitn. It tak about on hour to comput th path of th objct and robot with Ultra SPARC-II (334MHz). And thi how that our propod plannr can find appropriat olution within a practical tim fficintly dpit of complxity of th problm. A Objct Start B Robot (a) P rp (b) P att (c) P w Fig. 3 Potntial Fild 4.5Manipulation Cot and Manipulation Spac Th manipulation cot that i th total tim to finih th manipulation i computd by local manipulation plannr. If robot don t manipulat th objct and only tranport it from configuration q i- to q i, th cot M(q I-,q i ) i. Th manipulation pac computd by th local plannr and moving ara of robot (Fig. 4) computd by th global plannr ar both conidrd a th work ara. By making objct cll C obj,j bad on th work ara, th pac that i nd for manipulation and movmnt of robot i nurd in th global motion plannr. Obtacl Fig. 5 Simulation Rult I Stick Robot Stick Objct Robot (a) (b) Fig. 4 Moving Ara of Robot (Work ara) 5. Simulation and Exprimnt 5. Simulation W vrifid th propod plannr in thi papr through th imulation. Th motion plannr comput th path that two robot tranport an L-hap larg objct in a 3 dimnional nvironmnt hown in Fig. (b). Simulation rult of th global motion plannr and th local manipulation plannr ar hown in Fig. 5 and Fig. 6 (In Fig. 6, obtacl ar not diplayd bcau of th implicity of obrvation). In Fig. 5, th robot (c) () Fig. 6 (d) (f) Simulation Rult II

8 5.2 Exprimnt W applid th propod planning mthod of manipulating an objct to ral robot ytm. Th mobil robot can mov in all dirction [2] and hav th tick. (a) Mobil Robot ZEN (b)zn with Stick Fig. 7 Omni-Dirctional Mobil Robot ZEN Th xtrnal-world nor i not carrid on th mobil robot. And w t th motion vlocity of robot low nough to kp quai-tatic motion. A rult of a impl xprimnt (tumbling a rctangl with two robot) that adoptd our local manipulation planning mthod i hown in Fig. 8. Hr, it can b chckd that objctiv opration can b prformd without uing nor information, and th validity of thi mthod wa hown. (a) (c) (b) (d) () (f) Fig. 8 Exprimntal Rult 6. Concluion In thi papr, w propo a motion planning mthod for a cooprativ tranportation of a larg objct by multipl mobil robot in a 3 dimnional pac. W divid th motion plannr into a local manipulation plannr and global motion plannr. A to th local manipulation plannr, w conidr th motion rror in a planning tag bforhand, and build th manipulation tchniqu that i uitabl for poition-controlld mobil robot. And a to th global motion plannr, w rduc th dimnion of th C-pac and can find a olution with arching in thi mallr dimnional C-pac uing th potntial function. Aftr contructing two plannr rpctivly, w intgrat thm. Namly, a gro motion plannr and a fin motion plannr can b intgratd by our mthod. Simulation and xprimnt hav vrifid th ffctivn of propod planning mthod. Rfrnc [] Tamio Arai and Jun Ota: "Lt u Work Togthr -Tak Planning of Multipl Mobil Robot-", Procding of th 996 IEEE/RSJ Intrnational Confrnc on Intllignt Robot and Sytm, pp , 996. [2] Jan-Claud Latomb: "Robot Motion Planning", Kluwr Acadmic Publihr, 99. [3] Hwang Y. K. and Ahuja N.: "Gro motion planning a urvy", ACM Computing Survy, Vol.24, No.3, pp , 992. [4] Koichi Kondo: "Motion Planning with Six Dgr of Frdom by Multitratgic Bidirctional Huritic Fr-Spac Enumration", IEEE Tranaction on Robotic and Automation, Vol.7, No.3, pp , 99. [5] Jrom Barraquand and Jan-Claud Latomb: "Robot Motion Planning: A Ditributd Rprntation Approach", Th Intrnational Journal of Robotic Rarch, Vol., No.6, pp , 99. [6] Kamal Kant Gupta and Zhnping Guo: "Motion Plannning for Many Dgr of Frdom: Squntial Sarch With Backtracking", Procding of th 992 IEEE Intrnational Confrnc on Robotic and Automation, pp , 992. [7] Yong K. Hwang and Pang C. Chn: "A Huritic and Complt Plannr for th Claical Movr' Problm", Procding of th 995 IEEE Intrnational Confrnc on Robotic and Automation, pp , 995. [8] Lydia E. Kavraki, Ptr Svtka, Jan-Claud Latomb and Mark H. Ovrmar: "Probabilitic Roadmap for Path Planning in High-Dimnional Configuration Spac", IEEE Tranaction on Robotic and Automation, Vol.2, No.4, pp , 996. [9] Danila Ru, Bruc Donald and Jim Jnning: "Moving Furnitur with Tam of Autonomou Robot", Procding of th 995 IEEE/RSJ Intrnational Confrnc on Intllignt Robot and Sytm, pp , 995. [] Kazuhiro Koug, Tomohiro Ooumi and Kunihiro Chiba: "Load Sharing of Dcntralizd-Controlld Multipl Mobil Robot Handling a Singl Objct", Procding of th 997 IEEE Intrnational Confrnc on Robotic and Automation, pp , 997. [] O. Khatib, K. Yokoi, K. Chang, D. Rupini, R. Holmbrg and A.Caal: "Vhicl/Arm Coordination and Manipulat Mobil Manipulator Dcntralizd Coopration", Procding of 996 IEEE/RSJ Intrnational Confrnc on Intllignt Robot and Sytm, pp , 996. [2] Naoyuki Sawaaki and Hirochika Inou: "Cooprativ Manipulation by Autonomou Intllignt Robot", Journal of th Japan Socity of Mchanical Enginr, Sri C, Vol.59, No.564, pp , 993, in Japan. [3] Matthw T. Maon: "Mchanic and Planning of Manipulator Puhing Opration", Th Intrnational Journal of Robotic Rarch, Vol.5, No.3, pp.53-7, 986. [4] Randy C. Brot: "Automatic Grap Planning in th Prnc of Uncrtainty", Th Intrnational Journal of Robotic Rarch, Vol.7, No.7, pp.3-7, 988. [5] Michal A. Phkin and Arthur C. Sandron: "Th Motion of a Puhd, Sliding Workpic", IEEE Journal of Robotic and Automation, Vol.4, No.6, pp , 988. [6] Sriniva Aklla and Matthw T. Maon: "Poing Polygonal Objct in th Plan by Puhing", Th Intrnational Journal of Robotic Rarch, Vol.7, No., pp.7-88, 998. [7] Atuhi Yamahita, Jun Saaki, Jun Ota and Tamio Arai: "Cooprativ Manipulation of Objct by Multipl Mobil

9 Robot with Tool", Procding of th 4th Japan-Franc/2nd Aia-Europ Congr on Mchatronic, pp.3-35, 998. [8] Atuhi Yamahita, Kou Kawano, Jun Ota, Tamio Arai, Maaki Fukuchi, Jun Saaki and Yaumichi Aiyama: "Planning Mthod for Cooprativ Manipulation by Multipl Mobil Robot uing Tool with Motion Error", Procding of 999 IEEE/RSJ Intrnational Confrnc on Intllignt Robot and Sytm, pp , 999. [9] Yong K. Hwang and Narndra Ahuja: "A Potntial Fild Approach to Path Planning", IEEE Tranaction on Robotic and Automation, Vol.8, No., pp.23-32, 992. [2] Hajim Aama, Maatohi Sato, Luca Bogoni, Hayato Katu, Akihiro Matumoto and Iao Endo: "Dvlopmnt of an Omni-Dirctional Mobil Robot with 3 DOF Dcoupling Driv Mchanim", Procding of th 995 IEEE Intrnational Confrnc on Robotic and Automation, pp , 995.