Search algorithm for a Common design of a Robotic End-Effector for a Set of 3D objects

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1 Search algorhm for a Commo esg of a Roboc E-Effecor for a Se of 3D obecs Avsha Sov a Amr Shapro Deparme of Mechacal Egeerg, Be-Guro Uversy of he Negev, Beer Sheva 8405, Israel. sova@pos.bgu.ac.l, ashapro@bgu.ac.l Keywors: E-effecor; Search algorhm, Grasp. Absrac. We prese a algorhm for efg he se of 3-fger coac po cofgurao whch s suable for graspg a gve se of obecs. he coac pos cofgurao ca be use for he esg of a smple e effecor capable of graspg hs se of obecs. Gve a mesh of each obec, he search algorhm maps all possble grasps for each obec whch sasfy a qualy crero a akes o accou exeral loas apple o he obec. he mappe grasps are represee by feaure-vecors whch represe he shape of he grpper a 9-mesoal space a are regsere a aabase of all possble grasps for each obec. he we use a search algorhm for ersecg all pos over all ses a f commo feaure vecors. Each commo vecor s he grasp cofgurao of a e-effecor for graspg he group of obecs. Irouco Roboc e-effecors auomao are esge for carryg ou a specfc aco a for halg a specfc par. he esg phase of a ypcal e-effecor cosumes a coserable amou of egeerg me a as exra cos o he fal prouc. herefore, he purpose of hs work s o evelop a algorhm whch wll f a cofgurao of a smple e-effecor for graspg a se of obecs. Gve a se of 3D CAD moels of he pars, he algorhm wll f a commo cofgurao for graspg all of hem. hs cofgurao wll serve for he esg of a smple saar e-effecor capable of graspg all obecs he gve se. hs work uses he mehos of force-closure [], [] a grasp qualy measure [4] as crero for eermg a quafyg feasble grasps. o he bes of our kowlege, o pror work has bee oe for searchg commo grasps of e-effecors for a se of obecs. However, much work has bee oe he area of 3D shape smlary comparso algorhms, such as [5]. hs paper preses a algorhm for parameerzao of he force-closure grasps of each obec a usg for classfcao of he obecs o possble grasps. I he frs sage of he algorhm a Force Closure Grasp Se (FCGS) s cosruce for every obec by samplg all force-closure grasps, a represeg he possble grasps as feaure vecors he hgh-mesoal space. Smlary o base o eares-eghbor search s ha oe, for fg pars of commo feaure vecors he FCGS of all obecs. he followg seco s a backgrou overvew of graspg fuameals use hs work. he geerao of he FCGS a he smlary search for he commo feaure vecors are he presee. Fally, we prese smulaos of he propose algorhm. Backgrou Forces a orques ca be represee as a wrech vecor he wrech space. A wrech s a 6 6-mesoal vecor a s eoe as w f τ 3 3 where f s a force vecor a s a orque vecor. Furhermore, a wrech orgae from a force f apple a a coac po, c, ca be escrbe as w f p f where p s he locao of coac po c represee he obec's coorae frame.

2 Fg. - hree coac po grasp. Frco exss a he coacs bewee he fgerps a he obec's surface a ca be represee by he Coulomb frco moel. I hs moel, forces exere he coac po mus le wh a coe ceere abou he surface ormal. hs s kow as he Frco Coe (FC). he FC ca be approxmae by a s-se covex polyope a he force exere wh he FC ca be represee by a lear combao of he u vecors f ˆk (prmve forces) cosrucg he polyope. he assocae wreches ca be expresse by he prmve forces as s s fˆ k w ak w ak, () k k k ˆ p fk where wk are he prmve wreches assocae wh he prmve forces [3]. Force Closure. A grasp s sa o be force-closure f s possble o apply wreches a he coacs such ha ay exeral forces a orques acg o he obec ca be couer-balace. A sysem of wreches ca acheve force-closure whe ay exeral loa ca be balace by a o-egave combao of he wreches []. Wh sysem W of wreches w,,w, he covex-hull of W s efe as CH ( W ) aw : w W, a, a 0 () he covex hull of he sysem of coac wreches s eoe as he Grasp Wrech Se (GWS). A ecessary a suffce coo for a sysem of wreches w,,w o be force-closure s ha he org of k les he eror of he covex hull of he coac wreches [], [6]. Meag O eror CH( W) (3) Grasp qualy measure. he grasp qualy measure quafes how much a grasp ca ress a exeral wrech whou fgers loosg coac or slp [7]. he grasp qualy measure s efe as he sace from he org of he GWS o he closes face of CH(W ), a s gve by Q m w (4) ww where W s he bouary of CH(W ) [4]. FCGS geerao algorhm We geerae he se of all possble grasps for each obec. he grasps are represee as feaure vecors he Force Closure Grasp Se (FCGS). hs seco preses he propose srucure of he grasp feaure vecor a he meho for cosrucg he FCGS. Grasp feaure vecor. A 3-fger grasp of obec B ca be efe by a se of coac pos, P ( p, p, p3) a he ormals a each coac po N (,, 3). hs grasp efo ca be mappe o a -mesoal feaure vecor ecvely represeg he grasp. e u u

3 Fg. shows a 3-fger grasp. he grasp ca be represee as a ragle where he coac pos are s verces. he poso of he 3 fgers relave o each oher ca be ecvely represee as a ragle by wo agles, a he ege legh bewee hem, gve by he followg equaos: cos p p p p 3 p p p p 3 (5) cos p p p p 3 p p p p 3 (6) p p (7) he ormal o he surface a each coac po ca be represee relave o he ragle by he agle bewee he ormal of he pla of he ragle f, a he agle bewee he ormal a each ege of he ragle. hese agles are gve by he followg equaos: cos,,,3, (8) f ˆ f f f ˆ ˆ sg 3 a ˆ 3, cos 3 a3,, 3, sg f a, f cos f a,,, where aˆ, p p p p. herefore, he 3-fger grasp ca be ecvely efe by a 9-mesoal feaure vecor e, 3 3 e (0) I shoul be meoe ha here are hree ffere represeaos for a ragle, epeg o whch ege s pcke for efg. herefore, he loges ege of he ragle s always pcke o represe. hs esures ha he algorhm wll always rea he ragles from he same po of vew. Algorhm FCGS geerao Ipu: Mesh of a 3D obec B o be graspe a q - mmal grasp qualy. Oupu: FCGS E e,..., e of obec B. v : Geerae grasp efe by P ( p,, p ). : If P s force closure a (grasp qualy > q) o.. Map grasp o feaure vecor e u u.. Label e as force closure.. Sore lk bewee e a P. 3: Else label e as o-force-closure a remove from E. 4: If he grasp space E e,..., e s o fully labele, go o sep. v 5: Else Reur grasp space E e,..., e. v Grasp space geerao. Geerao of he FCGS s oe by scag he screze mesh a gog over all possble grasp feaure vecors of obec B, omg hose whch are o force-closure or her qualy s less ha a gve hreshol. Algorhm receves as a pu a 3D mesh of a 3D CAD moel. I samples all possble grasp feaure vecors of a obec a compues a se E (9)

4 represeg all combaos of 3-pos whch acheve force-closure uer he frcoal coac cosra. hs algorhm s base o he oe orgally propose [8]. Mul-ses commo vecors search Gve ses of vecors a 9-mesoal space,, E E represeg he FCGS of each obec. A smlary algorhm akes he ses a oupu a se of vecors Z whch are commo o wo or more ses of E,, E. Smlary Jo. he ma ea of he o algorhm s as follows: for each par of ses E, E, we ee o f vecors ha are commo o he wo ses. he o algorhm receves all FCGS aa ses,, E E a for each par of ses, usg eares eghbor algorhm, mach a vecor oe se o he eares vecor he oher. Afer all he pars are fou, furher speco of wheher hey are close eough o be cosere as he same grasp s oe, hus, checkg wheher hey are boh wh he preefe oleraces,, alog he maor cooraes axes. he mea vecor of he par fou s checke f exss a regsry se Z. Se Z s a cluser of vecors ha are commo o wo or more ses of E,, E. Oce a hgh-mesoal regsry se Z of vecors v,, v k Z s obae, each vecor v s couple o he bary vecor v whch eoes s exsece some of he prmve ses E,, E. he ex sep s o f a vecor wh Z whch exss all of he prmve ses E, E. Such a vecor s sa o cover he whole prmve ses E,, E. Le U be a -mesoal vecor se of sze v cossg bary vecors,.e., vecor wh compoes cossg of 0's a 's. For a vecor u Z, a s compable vecor u U, ca be sa ha a vecor u covers E, u () Each vecor fou sasfyg coo () represes a grasp cofgurao commo o all obecs. Smulaos For smulaos of he propose meho, he algorhm was mplemee MALAB o a Iel-Core 7-60M.5GHz lapop compuer wh 8GB of RAM. he followg smulaos llusrae he workg of he algorhm o 3-fger frcoal grasps of hree 3D obecs., E f, (a) (b) (c) Fg. - hree obecs o be graspe (frs row) a her mesh (seco row): (a) phoe hase, (b) bole a (c) baaa. he performace of he propose algorhm s llusrae usg he hree 3D shapes show Fg.. he shapes are escrbe wh a mesh of k = 400 ragles represeg her bouary. he frco coeffce was efe o be 0.6 a he frco coe was learze usg 5 ragles approxmag he cocal surface. Oly grasps wh qualy measure greaer ha 0. are furher cosere. Malab s a regsere raemark of Mahworks, Ic.

5 he legh oleraces use he smlary search,, 3 [ mm] were chose such way ha he eges of he ragle wll o exe or shore more ha 4% of her orgal legh ue o chages he ragle s agles. Smlarly, he agular oleraces,,,,,, [] were efe such ha each wo query ormals are wh a coe whose agle s wh 80% of he frco coe agle. he algorhm s oupu are 6 soluos of commo grasps for all obecs wh rume me of 8 hours. Fg. 3 shows oe of he 6 soluos wh he hghes qualy measure (0.55). Coclusos Fg. 3 - Oe grasp soluo for all shapes. A algorhm for fg a commo grasp for a se of obecs was propose. he algorhm uses he oo of FCGS a hgh-mesoal smlary search. he feasbly of he algorhm was llusrae hrough smulaos o 3D moels. A se of grasp feaure vecors was fou ha exs all of he prmve ses. Each of hese feaure vecors s fac a cofgurao of a e-effecor use o grasp he obec. Fuure work wll volve mprovg rume a valao expermes. Refereces [] J. Poce a B. Favero: O compug hree-fger force closure grasps of polygoal obecs. IEEE rasacos o Robocs a Auomao, Vol., No. 6, Dec [] M. A. Roa a R. Suarez: Geomercal approach for grasp syhess o screze 3D obecs. IEEE/RSJ Ieraoal Coferece o Iellge Robos a Sysems, pp , Sa Dego, 007. [3] Y. H. Lu: Compug -fger form-closure grasps o polygoal obecs, he Ieraoal oural of robocs research,vol. 9, No., pp , Feb [4] C. Ferrar a J. Cay: Plag opmal grasps, Proc, IEEE Ieraoal Coferece o Robocs a Auomao, pp , 99. [5] R. Ohbuch,. Oagr, M. Ibao a. ake: Shape-Smlary Search of hree-mesoal moels usg parameerze sascs. I Proc. Of he Pacfc Graphcs, Beg, Cha, Oc. 00. [6] B. Mshra, J.. Schwarz a M. Sharr: O he exsece a syhess of mulfger posve grps, Algorhmca, Specal Issue: Robocs, vol., o. 4, pp , Nov [7] C. Bors, M. Fsher a G. Hrzger: A fas a robus grasp plaer for arbrary 3D obecs. Proc. IEEE I. Cof. o Robocs a Auomao, Vol. 3, pp , May 999. [8] M. Roa, R. Suarez, a J. Rosell: Grasp space geerao usg samplg a compuao of epee regos. I Proc. IEEE/RSJ I. Cof. o Iellge Robos a Sysems, pp , 008.