Automatic Design Algorithm of a Robotic End-Effector for a set of Sheet-Metal Parts

Transcription

1 Automatic Desig Algorithm of a Robotic Ed-Effector for a set of Sheet-Metal Parts Avishai Sitov ad Amir Shapiro Abstract Robotic ed-effectors i automated productio lies are specially desiged ad built for a uique task ad for a particular part. A ed-effector capable of graspig differet parts with differet geometries will expad its beefit ad reduce costs. This work focuses o the grasp of sheet metal parts for the automotive idustry. I order to maximize the use of a sigle ed-effector, this paper proposes a search algorithm for a simple commo grasp cofiguratio. Such cofiguratio imply for a ed-effector desig capable of graspig a set of sheet metal parts. The algorithm maps possible grasps which are cadidate to be commo. We defie a ovel quality measure estimatig the distributio of the cotacts across the sheet metal part. Cadidate grasps which have a sufficiet quality are mapped ito high-dimesioal feature vectors. These feature vectors parameterize the geometry of the polyhedro formed by the cotacts locatios ad the directio of the surface ormals relative to the polyhedro. Thereby, they describe the desig of the ed-effector compatible to the grasp. A database is geerated for all possible grasps for each part i the feature vector space. A similarity joi based o earest-eighbor search ad classificatio algorithm are used for itersectig all possible feature vectors over all sheet metal parts ad fidig commo oes. Simulatios of a 3-figer grasp o four meshed sheet metal parts resulted i a commo grasp. Results of the simulatios validate the feasibility of the proposed algorithm. I. INTRODUCTION I automated productio lies each robotic ed-effector has its desigated usage for a specific actio ad for hadlig a specific part. Therefore, for each part ad task a robotic ed-effector is specially desiged ad built (Figure 1a). This demads a cosiderable amout of egieerig time ad cost. Thus, the demad for multiple ed effectors for each part ad task requires high costs. Therefore, it has direct impact o the price of the fial product. This work addresses the problem of desigig a ed-effector for a robotic arm capable of graspig a set of differet sheet metal parts i the automotive idustry. The purpose of this work is to develop a algorithm which will fid a desig of a simple optimal uiversal ed-effector for graspig a set of q sheet metal parts (SMP). The desig of the edeffector is extracted from the grasp cofiguratio defiig cotacts locatios ad force directios. SMP are curved thi ad flat objects formed by bedig, cuttig ad stampig sheet metal plates. The grasp ad fixture of a SMP is usually doe with clamps or suctio cups. Moreover, its geometry requires well distributed fixturig poits o the surface of the SMP to prevet distortio ad damage. Therefore, the sythesis of such a grasp should cosider some properties to Avishai Sitov ad Amir Shapiro are with the Dept. of Mechaical Eg., Be-Gurio Uiversity of the Negev, Beer-Sheva 84105, Israel sitova@post.bgu.ac.il, ashapiro@bgu.ac.il Fig. 1. (a) (b) (a) Graspig of a sheet metal part ad (b) a sheet metal clamp. be described. The proposed algorithm uses these properties to fid a commo grasp cofiguratio for a give set of SMP. Give a set of q CAD models of the SMP, the goal is to desig a ed-effector which is uiversal, that is, able to hold a wide set of compoets. We propose a ovel solutio for desigig simple ed-effectors that ca hadle a class of SMP rather tha a sigle oe. This paper is a adaptatio of our previous work preseted i [1], [2] to SMP. The proposed algorithms output is a commo grasp cofiguratio which implies for the desig of the idustrial edeffector. The desig has to be simple ad low cost, thus, it has to be with miimal degrees of freedom. We search for a feasible grasp cofiguratio which ca stably grasp the object eve uder the applicatio of exteral forces or torques due to the task beig doe. That is, we require force-closure grasp. As there are umerous possible grasps, we itroduce a ovel quality criteria termed Variace Based Quality Measure (VBQM) which quatify the grasp accordig to its distributio across the SMP. The quality measure eables filterig out iappropriate grasps. I this work we assume rigidity of the SMP ad poit cotact by the clamps. I the proposed algorithm, the CAD s of the SMP are discretized to a triagular mesh. We apply a mesh reductio algorithm to remove udesired areas which could ot be grasped. The algorithm parameterizes all grasps (up to mesh size) that are possible cadidates to be commo to all SMP ito feature vectors i a high-dimesioal space. The feature vector implies for the ed-effectors cofiguratio. These feature vectors costruct a Cadidate Grasp Set (CGS) for each SMP. Similarity joi ad classificatio are coducted, i order to fid miimal feature vectors which cover the whole set of SMP. for fidig pairs of commo feature vectors i the CGS of all SMP. The paper is orgaized as follows. Sectio III presets the graspig model used i this work. A overview of the proposed search algorithm is described i sectio IV. Sectio V presets the simulatios implemetig the proposed algorithm.

2 II. RELATED WORK Grasp plaig uses the otios of force-closure ad quality measure as criteria for determiig ad quatifyig feasible grasps. A force-closure grasp is defied as a grasp that ca resist ay exteral load. Usig the otio of wreches (forces ad torques), i [3] [5] force-closure criterio is well defied ad several algorithms for sythesis of a frictioal ad frictioless grasps were preseted. Several grasp optimizatio methods usig differet grasp quality measures have bee preseted i the literature; Ferrari ad Cay [6] ad Li ad Sastry [7] itroduced a quality measure based o the exteral wrech to be resisted. A method which is based o the covex-hull of the wreches formed by the cotact forces. As we deal with graspig of large flat objects, we focus o methods which measures the distributio of a grasp over the grasped object. Chiellato et al. [8] itroduced quality measures criteria for 3-figer plaar objects, oe based o the area of the triagle formed by the cotact poits. Kim et al. [9] preseted the stability grasp idex, which defies a polygo formed by the cotact poits ad measures the deviatio of the polygos agles from a regular polygo; this implies for the distributio of the cotacts o the object. Much work has bee doe i the area of graspig ad ed-effector desig for sheet metal parts. Park ad Millis [10] have dealt with the localizatio of sheet metal parts to acquire a precise fixture. Gopalakrisha et al. [11] proposed a algorithm for fixturig sheet metal parts usig coical grooves. Such method eables uilateral fixture avoidig tool iterferece at oe side of the sheet metal part. Ceglarek et al. [12] itroduced a method for modelig ad optimizatio of a ed-effector for SMP cosiderig task directios ad desired motio. Li et al. [13] proposed a methodology for modelig a dexterous ed-effector ad overcomig deformatios, acquirig more precise fixturig. To the best of our kowledge, o previous work has bee doe for searchig commo grasps or ed-effectors desig for a set of SMP. However, much work which has relatio to ours has bee doe i the area of 3D shape similarity compariso algorithms, such as [14] ad [15]. These algorithms are used for Iteret ad local storage search, face recogitio, image processig or parts graspig i assembly lies. Such methods deals with parameterizatio of the geometry of the objects ad caot be applied for ed-effector desig. The work of Li ad Pollard [16] is based o shape matchig for fidig the best grasp for a set of objects. The best grasp is foud by matchig had poses from a database to each object. This is doe by usig a predefied parameterizatio of the object surface ad the had poses, a method which ispired this work. III. GRASP FEASIBILITY The grasp of SMP is doe usig sheet metal clamps (Figure 1b). Clampig of a SMP provides two opposig forces at the cotact poit. I this sectio we preset the implicatios of this property o the graspig model. Frictio exists at the cotact poits betwee the clamps ad the SMP s surface. Frictio ca be represeted by the simple Coulomb frictio model. I this model, forces exerted at a cotact poit must lie withi a coe cetered about the surface ormal. This is kow as the Frictio Coe (FC). The frictio model ca be mathematically represeted by f 2 2 i,2 + f i,3 μf i,1, where f i,1 is the ormal compoet of the cotact force, f i,2 ad f i,3 are the tagetial compoets at the cotact poit. The FC i this case is o-liear ad to simplify the model the coe be approximated with a s-sided covex polyhedro ad every force exerted withi the FC ca be represeted by a liear combiatio (Figure 2) of the uit vectors ˆf ik FC (primitive forces) costructig the liearized frictio coe, s f i = a ikˆfik, a ik 0 (1) where a ik are oegative coefficiets. The ˆ. sig deotes a uit vector. Based o the model of the grasp, we ow wat to defie whether a grasp is feasible or ot. This is preseted i the followig subsectio. Fig. 2. Graspig of a sheet metal part. A. Force-Closure Forces ad torques ca be represeted as a wrech vector i the wrech space. A wrech is a 6-dimesioal vector ad is deoted as w = (f τ) T R 6 where f R 3 is the force vector ad τ R 3 is the torque vector. Furthermore, a wrech iduced by the cotact force at the cotact poit p i, ca be described as w i = (f i p i f i ) T where p i is represeted i the parts coordiate frame. A grasp is said to be force-closure if it is possible to apply wreches at the cotacts such that ay exteral forces ad torques actig o the part ca be couter-balaced. I other words, a system of wreches ca achieve forceclosure whe ay exteral load ca be balaced by a oegative combiatio of the wreches [3]. The determiatio of whether a grasp is force-closure is usually doe accordig to the followig Theorem. Theorem 1. [5], [17] A set of wreches W is said to achieve force-closure if its wreches positively spa the etire wrech space R 6. To determie whether a grasp is force-closure, the Covex- Hull aalysis [3] is usually doe. However, it is a expesive computatioal method ad i the ext Theorem we show it is uecessary. The followig Theorem is based o the grasp method where i clampig, two opposig forces are applied at the cotact.

3 Theorem 2. Give 3 frictioal cotact poits P = {p 1,..., p p i p j i j, i, j =1,..., } o the surface of the SMP, each with two opposed forces i directio of the ormals to the surface. If there are 3 o-colliear cotact poits p i, p j, p k P, the frictioal forces at P positively spa R 6 ad the grasp is force-closure. Proof: At least four frictioal cotact forces are eeded to achieve a force closure grasp [5]. For two cotact poits p i, p j, we have four frictioal forces (two opposig pairs). However, these forces caot apply torque about the axis formed by p i p j, that is, the four frictioal forces spa oly R 5. Addig two more opposig forces at poit p k = {p k p k p i +γ(p i p j ), γ} eables applyig torque about the p i p j axis. Hece, the three cotact poits p i, p j, p k with six frictioal forces positively spa R 6. Theorem 2 provides the otio that a grasp of a SMP with 3 clamps, where at least three clamps are ocolliear, is always force-closure. Therefore, there is o eed for force closure aalysis usig the covex-hull method. Moreover, there is o eed for modelig ad liearizig the frictio coes, which cosumes large computatio resources. This otio reduces the rutime of the algorithm drastically. However, despite all o-colliear grasps are force-closure, ot all of them are equally good. There is a eed for a criteria to be used to filter out udesired grasps. The ext subsectio defies such criteria. B. Variace Grasp Quality Measure It has bee show that all 3 clamp grasps are force closure. However, ot all grasps should be cosidered feasible. Oe ca thik of graspig a log sheet metal from oe edge. Although the grasp may be force closure, large forces would have to be applied to couter balace exteral loads such as gravity. Hece, the criterio for defiig the quality of a grasp should be based o the distributio of the clamps o the surface of the SMP. A good distributio would result i balaced ad relatively low loads o the clamps. We preset a ovel quality measure termed Variace Based Quality Measure (VBQM). It is a grasp quality measure defied for SMP ad is used to filter out iappropriate grasps accordig to a heuristic coditio to be preseted. The mai cocept of defiig the VBQM is by evaluatig the distributio of the cotact poits o the sheet metal objects. We use variace measure to quatify the distributio. For clamp poits P = (p 1,..., p ), where p k = (x k,y k,z k ). We compute the orm of the variaces over all coordiates of the poits as follows ( V = (x k x) 2, (y k y) 2 ) T, (z k z) 2 (2) where x, y, z are the meas of the respected coordiates. As will be show i the followig sectio, for each SMP, we go through all possible combiatio (up to mesh size) of clampig poits ad fid the largest variace possible amog all sheet metals, that is, V max =maxvj i (3) i,j where Vj i is the variace of grasp j of part i. The variace calculated is a measure of how much the grasp is distributed across the SMP. Therefore, we defie the ew quality measure for grasp j of sheet metal i to be ormalized by the maximum variace of all SMP, Q i j = Vi j. (4) V max The ormalizatio will provide a comparative measure betwee SMP grasps to fid a commo oe. IV. SM-COG ALGORITHM Give q objects to be grasped with clamps. The Sheet Metal COmmo Grasp (SM-COG) search algorithm will output a feasible commo grasp with the highest quality measure for the set of objects. The algorithm is preseted i Algorithm 1. The algorithm receives as iput a set of q CAD models of the query SMP. The first step of the algorithm is the discretizatio of each CAD model to a mesh of k triagles. Each triagle i the mesh is characterized with its ceter of gravity positio vector p i ad the ormal ˆ i (uit vector) to the surface of the triagle. Thus, the mesh of each object is defied with a set of poits o the surface P =(p 1,..., p k ) ad a set of ormals at the poits Ñ =(ˆ 1,..., ˆ k ). The ext step of the algorithm is the reductio of redudat mesh triagles as described ext. Algorithm 1 Commo grasp search Iput: CAD s of SMP B 1,..., B q ad umber of cotact poits. Output: A commo grasp for all objects or commo grasps for subsets of the objects. 1: Calculate Q max,q mi usig eq. (3) ad (4). 2: for ξ =1 q do 3: Mesh SMP B ξ to form { P ξ, Ñξ}. 4: Perform Mesh reductio. 5: Geerate possible cotact poits grasps {P 1, N 1 } ξ,..., {P λ, N λ } ξ. 6: for j =1 λ do 7: Calculate Q ξ j of grasp {P j, N j } ξ. 8: if Q mi Q i j Q max the 9: Map grasp {P j, N j } ξ to feature vector e j. 10: Add e j to set E ξ. 11: Store poiter betwee e j ad {P j, N j } ξ. 12: ed if 13: ed for 14: ed for 15: Z = JoiCGS(E 1,..., E q ) 16: H = Classif icatio(z) 17: if H = Øthe 18: retur H =(u 1,..., u σ ) 19: else 20: Report: No commo grasps. 21: ed if

4 A. Mesh Reductio As our clampig device ca oly clamp ear the boudaries of the sheet metal, the ier triagles of the mesh are excessive ad cause tremedous calculatio rutime. The redudat mesh should be removed either by the user or by a reductio algorithm. Therefore, algorithm for omittig the ier mesh was implemeted. We defie m as the umber of layers of ier triagles from the boudary we cosider as edge triagles. m is determied accordig to the mesh size ad the desired boudary width to grasp defied by the clamp. The algorithm builds a adjacecy table of the the quality measures for each SMP (Figure 4), this is doe by samplig all possible grasps o the SMP s surface. We acquire q sets of measures Q i. The maximum limit of the quality measure to search will be Q max =mi(max(q i i j)) (6) j where i is the SMP idex ad j is the grasp idex. A boudary larger tha this will be wasteful as ot all SMP has grasps i that regio. Therefore, grasps which has a quality measure larger tha Q max are ot possible cadidates to be commo. The miimum limit of the measure Q mi is Fig. 3. The triagles of the mesh marked to be edge or ier triagles whe m =3. triagular mesh. The mai cocept is markig the eighbors of each mesh triagle (Figure 3), those with oly 2 eighbors are boudary triagles (marked with e ) ad those with 3 eighbors are ier triagles (marked with i ). The, we mark as boudary triagles ( e ) those which are with distace m from the boudary triagles. Fially, the ier mesh triagles are removed from the mesh. Aother optioal step is for the user to maually remove areas i the mesh which are forbidde to grasp. Such areas are defied accordig to operatioal costraits, sesitive areas o the SMP, etc. B. CGS Geeratio The ext step is the geeratio of the Cadidate Grasp Set (CGS) for each SMP. The CGS is a set of high-dimesioal vectors which represets cadidate grasps. Cadidate grasps are those that have the possibility to be commo to all SMP. The geeratio of the CGS is doe by mappig each cadidate -clamp grasp to a set of parameters termed feature vector. The CGS cotais all (up to the mesh size) feature vectors of cadidate -clamp grasps. For example, a 3-clamp grasp ca be represeted by a triagle formed by the 3 cotact poits. With more tha 3-clamps, the cotact poits form a polyhedro. From all possible grasps, we pick oly the oes which are cadidate to be commo it term of their quality. Grasp j of part i that is defied by cotacts Pj i =(p 1,..., p ) is cosidered to be a cadidate if there are at least three ocolliear cotact poits ad if its quality measure Q i j is withi the boud Q mi Q i j Q max, (5) where Q max ad Q mi are defiedasfollows.priortothe geeratio of the CGS, we build a distributio graph of Fig. 4. Distributio of the quality measure for each SMP ad the search limitatios. defied by a maximum allowed umber β of poits i a set. Therefore, we choose Q mi to be Q mi =max(q i mi) (7) i such that each Q i mi maitais the coditio Qmax Q i mi f i (Q i )dq i β (Q max Q i mi), i =1,..., q (8) where f i (Q i ) is a polyomial fit of the distributio poits acquired previously. The choice of the umber of poits β to be i the set affects the rutime. Next, the cadidate grasps are parameterized to feature vectors i the feature space. That is, we defie trasformatio map T to map grasp j represeted with P j ad N j ito a d-dimesioal feature vector e j : T : {P j, N j } e j R d. (9) Trasformatio T forms a feature vector which ijectively represet the grasp cofiguratio ivariat of ay coordiate frame. The feature vector of a grasp is a set of parameters which costrai the size ad shape of the polyhedro formed by the cotact poits. Moreover, parameters i the feature vector costrai the ormals directios at the cotact poits relative to the polyhedro itself. For details ad examples

5 of the algorithm for parameterizatio of a grasp to a feature vector see our previous publicatio [18]. The dimesioality of the feature vector is determied accordig to the umber of cotact poits which defies the umber of vertices i the polyhedro. The feature vectors of object B i which are cosidered to be feasible are added to the CGS of the compatible SMP, deoted as E i R d. Oce all feasible grasps of all objects are mapped to the CGS sets E 1,..., E q,wewouldliketoitersectthecgs s to fid commo feature vectors which imply for commo grasps. Therefore, we defie fuctio joicgs which is a similarity joi algorithm to fid commo poits over the sets. Hece, earest-eighbor search is utilized to fid pairs of commo vectors amog the sets. Pairs of commo vectors foud are checked to satisfy tolerace demads derived from the frictio coes agle, accuracy demads ad hardware capabilities. Basically, two feature vectors over two sets are cosidered to be the same if they are both iside a hyperrectagle with predefied edge legths. Two vectors which are cosidered to be the same are further added to a registry set Z R d of commo vectors. The set Z is a d-dimesioal database of vectors take from E 1,..., E q. The vectors iserted to Z are the oes which exists i two or more sets of E 1,..., E q, i.e., those which are commo to two or more sets. For each feature vector added to Z, itis marked from which CGS sets they origiated. The fial step of the algorithm is the classificatioofthe vectors i Z. After a set of vectors commo to two or more of the sets E 1,..., E q were acquired, classificatio is eeded to fid the miimal umber of grasp cofiguratios which ca grasp subsets of the objects. We search for a miimum set H Z of vectors from the registry set which covers all of the CGS sets. Basically, we search for a sigle feature vector from Z which exists i all of the sets E 1,..., E q. Such a vector represets a ed-effector desig which ca grasp all of the objects. If a sigle vector is ot foud, we seek for a miimal umber of grasps which ca grasp subsets of the objects. That is, we divide the set of objects to subsets, where for each subset there is a compatible ed-effector. As we go through all possible -clamp grasp combiatios (up to mesh size), therefore, the algorithm will certaily fid a commo grasp or a set of commo grasps if such exist. Thus, if a sigle grasp for all SMP or a set of grasps for subsets of the SMP exist, the algorithm will fid them. If failed to do so, the algorithm reports that commo grasps do ot exist. V. SIMULATIONS For simulatios of the proposed algorithm, it was implemeted i Matlab o a Itel-Core i7-2620m 2.7GHz laptop computer with 8GB of RAM. The ruig of the algorithm was doe usig MATLAB 1 parallel computig toolbox i order to decrease rutime. The followig simulatios preset a example of the algorithm operatio with 3-clamp frictioal grasps of four sheet metal car doors. The four SMP are show i Figure 5. The objects were meshed usig COMSOL 1 Matlab is a registered trademark of The Mathworks, Ic. Multiphysics 2 to triagular meshes with average size of 10,368 triagles. However, with the mesh reductio algorithm the mesh size was reduced to a average of 2,181 triagles per SMP (Figure 6). The boudaries of the VBQM were calculated accordig to coditios (6)-(8) with β chose to be poits. Thus, the boudaries were calculated to be Q mi = ad Q max = Figure 7 presets the distributio of the grasps with respect to their VBQM ad the boudaries to be used to geerate the CGS. Fig. 6. Fig. 5. CAD s of four sheet metal parts. Four boudary meshes of the sheet metals. The toleraces for the similarity joi were chose such that the distace betwee the cotact poits will ot exted or shorte by more tha 5% of their origial legth. These toleraces are cotiuously computed durig the simulatio executio. Moreover, toleraces were defied to esure that the ormals at the cotact poits will be iside a frictio coe where the coefficiet of frictio is μ =0.7. Uder these coditios, with rutime of hours, the search algorithm provided 6 grasps which are commo to all SMP. The output of the algorithm is a sigle grasp, out of the 6, with the 2 COMSOL Multiphysics is a registered trademark of COMSOL AB.

6 degrees of freedom to the ed-effector i cases where a solutio could ot be foud because of sigificat scale variace betwee parts. Moreover, we are curretly workig o a full size experimetal setup to verify the simulatio results. ACKNOWLEDGMENTS The research was partially supported by the Helmsley Charitable Trust through the Agricultural, Biological ad Cogitive Robotics Ceter of Be-Gurio Uiversity of the Negev. Fig. 8. Fig. 7. VBQM distributio of the SMP grasps. The commo grasp cofiguratio o the four SMP. highest quality measure. Figure 8 presets the best grasp cofiguratio which has a quality measure Q = This grasp is the best commo 3-clamps grasp for the four SMP. Notice that some cotact poits are o the outer boudary of the parts ad some are o the ier boudary. Ad because of that, there is a differet approach agle for each part. This ca be solved by a rotary degree of freedom added to the certai clamp ad will be addressed i future work. VI. CONCLUSIONS The proposed algorithm discussed i this paper provides a feasible solutio for desigig a robotic ed-effector for automotive productio lies able to grasp a set of differet sheet-metal parts. The algorithm itersects possible grasps of SMP s ad provides a commo grasp. If fail to fid oe commo grasp, it will try to divide the SMP to a miimum umber of subsets where each subset has its ow grasp. The results achieved durig simulatio shows the feasibility of the algorithm. The overall complexity of the algorithm is i the order of O(k ) where k is the mesh size of the SMP. Future work will ivolve searchig for possibilities to add REFERENCES [1] A. Sitov, S. Raghothama, R. Meassa, ad A. Shapiro, A commo 3- figer grasp search algorithm for a set of plaar objects, i Proceedig of the IEEE Iteratioal Coferece o Automatio Sciece ad Egieerig (CASE), 2012, pp [2] A. Sitov, R. J. Meassa, ad A. Shapiro, OCOG: A commo grasp computatio algorithm for a set of plaar objects, Robotics ad Computer-Itegrated Maufacturig, vol. 30, o. 2, pp , [3] J. Poce ad B. Faverjo, O computig three-figer force-closure grasps of polygoal objects, IEEE Trasactios o Robotics ad Automatio, vol. 11, o. 6, pp , dec [4] M. Roa ad R. Suarez, Geometrical approach for grasp sythesis o discretized 3d objects, i Proceedigs of the IEEE/RSJ Iteratioal Coferece o Itelliget Robots ad Systems. IEEE, Oct. 2007, pp [5] R. M. Murray, Z. Li, ad S. S. Sastry, A Mathematical Itroductio to Robotic Maipulatio, 1st ed. CRC Press, Mar [6] C. Ferrari ad J. Cay, Plaig optimal grasps, i Proceedigs of the IEEE Iteratioal Coferece o Robotics ad Automatio, May 1992, pp [7] Z. Li ad S. Sastry, Task orieted optimal graspig by multifigered robot hads, i Proceedigs of the IEEE Iteratioal Coferece o Robotics ad Automatio, vol. 4, Mar. 1987, pp [8] E. Chiellato, R. Fisher, A. Morales, ad A. del Pobil, Rakig plaar grasp cofiguratios for a three-figer had, i Proceedigs of the IEEE Iteratioal Coferece o Robotics ad Automatio, vol.1, sept. 2003, pp vol.1. [9] B. H. Kim, B. J. Yi, S. R. Oh, ad I. H. Suh, No-dimesioalized performace idices based optimal graspig for multi-figered hads, Mechatroics, vol. 14, o. 3, pp , [10] E. J. Park ad J. K. Mills, Three-dimesioal localizatio of thiwalled sheet metal parts for robotic assembly, J. Field Robotics, vol. 19, o. 5, pp , [11] K. Gopalakrisha, K. Goldberg, G. Boe, M. Zaluzec, R. Kogati, R. Pearso, ad P. Deeszczuk, Uilateral fixtures for sheet-metal parts with holes, Automatio Sciece ad Egieerig, IEEE Trasactios o, vol. 1, o. 2, pp , [12] D. Ceglarek, H. F. Li, ad Y. Tag, Modelig ad optimizatio of ed effector layout for hadlig compliat sheet metal parts, Joural of Maufacturig Sciece ad Egieerig, vol. 123, pp , [13] H. F. Li, D. Ceglarek, ad J. Shi, A dexterous part-holdig model for hadlig compliat sheet metal parts, Joural of Maufacturig Sciece ad Egieerig, vol. 124, p. 109, [14] R. Ohbuchi, T. Otagiri, M. Ibato, ad T. Takei, Shape-similarity search of three-dimesioal models usig parameterized statistics, i Proceedigs of the 10th Pacific Coferece o Computer Graphics ad Applicatios, 2002, pp [15] R. Osada, T. Fukhouser, B. Chazelle, ad D. Dobki, Shape distributios, ACM Trasactios o Graphics, vol. 21, o. 4, pp , Oct [16] Y. Li ad N. S. Pollard, A shape matchig algorithm for sythesizig humalike evelopig grasps, i Proceedigs of the 5th IEEE-RAS Iteratioal Coferece o Humaoid Robots, 2005, pp [17] B. Mishra, J. T. Schwartz, ad M. Sharir, O the existece ad sythesis of multifiger positive grips, Algorithmica, vol. 2, pp , [18] A. Sitov, R. Meassa, ad A. Shapiro, 3d-ocog: A commo -figer grasp search algorithm for a set of 3d objects - techical report, [Olie]. Available: