(JPL-2557) DEFORMATION PATH PLANNING FOR BELT OBJET MANIPULATION

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1 Proeedings of 1 ISFA 1 International Symposim on Flexible Atomation Tokyo, Japan Jly 1-14, 1 (JP-557) DEFORMATION PATH PANNING FOR BET OBJET MANIPUATION Yya ASANO, Hidefmi WAKAMATSU, Eiji MORINAGA, Eiji ARAI Gradate Shool of Engineering Osaka Uniersity -1 Yamadaoka, Sita, Osaka , Japan y-asano@mapse.eng.osaka-.a.jp Shinihi HIRAI Ritsmeikan Uniersity Noji Higashi Ksats Shiga , Japan ABSTRACT A differential geometry based modeling to represent deformation path for a belt objet is proposed. Adeqate deformation path of a belt objet sh as film irit board or flexible irit board mst be generated for atomati maniplation and assembly. First, deformation of a belt objet is desribed sing the diretion of the entral axis and torsion arond it. Seond, a method to derie an adeqate transition of the objet shape from the initial state to the final state is proposed. The adeqate deformation transition an be deried by minimizing the maximm potential energy dring its maniplation proess. Flexible irit board an be easily bent along the entral axis, bt it mst not be twisted arond the entral axis bease it may ase the rak at the transerse edges leading to wiring disonnetion. So, more sitable deformation path is onsidered so that the stress in a belt objet is small. INTRODUCTION De to downsizing of arios eletroni deies sh as notebook PCs, mobile phones, digital ameras, and so on, more film irit boards or flexible irit boards illstrated in Fig.1 are sed instead of onentional hard irit board. It is diffilt to assemble sh flexible boards by a robot bease they an be easily deformed dring their maniplation proess and they mst be deform in the final state. For example, the flexible irit board shown in Fig.1-(a) mst deform to the objetie shape illstrated in Fig.1-(b) to install into the hinge part of a flip phone. Therefore, analysis and estimation of film/flexible irit boards is reqired. (a) Natral shape (b) Objetie shape Fig. 1. Example of flexible irit board In ompter graphis, a deformable objet is represented by a set of partiles onneted by mehanial elements[1]. Reently, fast algorithms hae been introded to desribe liner objet deformation sing Cosserat formlation[]. Cosserat elements possess six degrees of freedom; three for translation displaement and three for rotational displaement. Flexre, torsion, and extension of a liner objet an be desribed by se of Cosserat elements. In robotis, insertion of a wire into a hole in D spae has been analyzed sing a beam model of the wire to derie a strategy to perform the insertion sessflly[][4]. Kosge et al. hae proposed a ontrol algorithm of dal maniplators handling flexible sheet metal[5]. amirax et al. hae proposed a method of path planning for elasti objet maniplation with its deformation to aoid ontat with obstales in a stati enironment[6]. Saha and Isto proposed a motion planner for maniplating ropes and realized tying seeral knots sing two ooperating robot arms[7]. In differential geometry, red lines in D or D spae hae been stdied to desribe their shapes mathematially[8]. Moll et al. 1 Copyright 1 by ISFA

2 hae proposed a method to ompte the stable shape of a liner objet nder some geometrial onstraints qikly based on differential geometry[9]. It an be applied to path planning for flexible wires. Jian proposed the modeling method for deformable shell-liked objet[1]. As aboe, the modeling method to represent deformable objet deformation and the method to maniplate it has been proposed. Howeer, to my knowledge, there is not a researh to estimate the deformed shape and load ondition of it dring its maniplation proess. A flexible irit board is a deformable objet. So, it is possible to be deformed to ontat with itself or obstales in a stati enironment. In addition, a flexible irit board is preision mehanial eqipment. This means that it is possible to make fratre when it is deformed largely dring its maniplation proess. It an be easily bent along the entral axis, bt it mst not be twisted arond the entral axis bease it may ase the rak at the transerse edges leading to wiring disonnetion. So, it is reqired to estimate the deformation shape and load ondition of the objet dring its maniplation proess. In solid mehanis, the Kirhhoff theory for thin plates and Reissner-Mindlin theory for thik plates hae been sed[11]. For extremely thin plates, the inextensional theory was proposed[1]. In this theory, it is assmed that the middle srfae of a plate is inextensional, that is, the srfae of the plate is deelopable. Based on these theories, the deformed shape and load ondition of the objet an be allated sing the Finite Element Method (FEM). Howeer, the high aspet ratio of thin objets often ases instability in omptation of deformed shapes. Wakamats has proposed a differential geometry to represent linear objet deformations and path planning[1]. In this paper, we apply this theory to deformation and deformation path of a belt objet and propose the maniplation planning so that the objet deforms with little damage dring its maniplation proess. MODEING OF BET OBJECT DESCRIPTION OF DEFORMED SURFACE In the ase of a thin objet, one dimension is assmed to be sessflly small omparing with the other two dimensions, namely, d 1 d, d. In this paper, we define an objet with the following aspet ratio as a belt objet : d 1 d d. This implies that the thikness h of the objet is sffiiently smaller than its width b, and its width b is sffiiently smaller than its length. In maniplation of belt objets sh as film/flexible irit boards, flat ables, and so on, they bend and twist mainly and their expansion/ontration an be negligible. In differential geometry, a D srfae whih an be flattened withot its expansion or ontration is defined as a deelopable srfae. So, the deformed shape of an inextensible belt objet orresponds to a deelopable srfae. et be the distane from one end of the objet along the entral axis in its longitdinal diretion and let be the distane from the entral axis in the transerse diretion of the objet. et P(, ) be a point on the objet. In order to desribe deformation of the entral axis of a belt objet, the global spae z x O y P Figre: Coordinate of belt objet oordinate system and the loal objet oordinate systems at indiidal points on the objet are introded as shown in Fig.. et O-xyz be the oordinate system fixed in spae and P- ξηζ be the oordinate system fixed at an arbitrary point P(, ) on the entral axis of the objet. Selet the diretion of oordinates so that the ξ-, η-, and ζ-axes are parallel to the x-, y-, and z-axes, respetiely, in the natral state. Deformation of the objet is then represented by the relationship between the loal oordinate system P-ξηζ at eah point on the objet and the global oordinate system O-xyz. et ξ, η, and ζ be nit etor along the ξ-, η-, and ζ-axes, respetiely, at any point P(, ). By analogy with anglar eloities of a rigid objet, partial differentiation of these nit etor with respet to are desribed by d d (1) where ω ξ, ω η, and ω ζ are infinitesimal ratios of rotation angles arond the ξ-, η-, and ζ-axes, respetiely, at point P(, ). Note that ω ξ and ω ζ orrespond to ratres of entral axis in the ηζ-plane and in the ξη-plane respetiely, and ω η orrespond to torsional ratio arond the entral axis. Soling differential eqations desribed by eq.(1) nmerially, etor ξ, η, and ζ at point P(.) an be determined. et x(, ) = x(, ) y(, ) z(, ) T be the position etor at P(,). The position etor an be ompted by integrating etor η(, ). Namely, x (,) x η (,) d where x = x y z T is the position etor at the end point P(, ). Next, we onsider desription of the deformed srfae of a belt objet. Aording to differential geometry, the normal ratre κ in diretion d = aη + bζ of the objet srfae is represented as follow: P(,) a Ea where E, F, and G are oeffiients of first fndamental form and, M, and N are those of the seond fndamental form of the srfae. These oeffiients are defined as follows: Mab Nb Fab Gb E η η 1, F η ζ, G ζ ζ 1 () () (4) Copyright 1 by ISFA

3 η ζ ζ ξ, M ξ, N ξ. Here, we introde parameter δ, = N. It orresponds to the ratre in the transerse diretion. The normal ratre κ depends on the diretion d and its maximm ale κ 1 and its minimm ale κ are alled the prinipal ratre. Diretion d 1 of the maximm ratre κ 1 and diretion d of the minimm ratre κ are referred to as prinipal diretions. A srfae is haraterized by Gassian ratre K(, ) and the mean ratre H(, ). They are related to the prinipal ratre κ 1 and κ by 1 H (7) Ths, bending of a srfae is haraterized by infinitesimal ratio of rotation angle ω η (, ) and ω ζ (,) and the ratre in a transerse diretion δ,. If a prinipal ratre κ, i.e., the mimimm ale of the normal ratre is eqal zero, the srfae is deelopable. Namely, it an be flattened withot its expansion or ontration. Sh srfae is referred to as a deelopable srfae. In this paper, we assme that a belt objet is inextensible. Then the deformed shape of the objet orresponds to a deelopable srfae. Gassian ratre K of a deelopable srfae mst be zero at any point. So, the following onstraint is imposed on the objet. This implies that δ an be desribed by 1 The infinitesimal ratio of rotation angle arond ξ-axis ω ξ mst also be satisfied the following eqation bease of the inextensibility of a belt objet: POTENTIA ENERGY AND GEOMETRIC CONSTRAINTS (5) (6) (8) (9) (1) et s formlate the potential energy of a deformed belt objet with Kirhhoff theory[14]. In this paper, we spposed the deformation of a belt objet as follows: 1. Straight lines normal to the mid-srfae remain straight after deformation.. Straight lines normal to the mid-srfae remain normal to the mid-srfae after deformation.. The thikness of the plate does not hange dring a deformation. We an formlate the potential energy of a belt objet with Kirhhoff theory. That is desribed as follows: K K, ( ), [, ].. [, U]. 1 U b h b h E bh (1 ) 1 G bh [ ] d dddw (11) where E, G and υ represents Yong's modls, modls of rigidity and Poisson ratio, respetiely. Poisson ratio υ is zero bease the deformed shape of the objet is deelopable. κ η, κ ζ, and κ ηζ are ratre in the η-diretion, the ζ-diretion and ratios of rotation angle arond the η -axis. These orrespond to ω ζ, δ, and ω η respetiely. So, Eq. (11) is desribed by U (1) Next, let s formlate geometri onstraints imposed on a belt objet. et l = l x l y l z T be a predetermined etor desribing the relatie position between two operational points on the entral axis of a belt objet, P ( a ) and P ( b ). Reall that the spatial oordinates orresponding to distane are gien by Eq.(). Ths, the following eqation mst be satisfied: (1) The oriental onstraint at operation point P( a ) is simply desribed as follows: (14) where ξ, η and ζ are predefined nit etors at this point. Therefore, the shape of a belt objet is determined by minimizing the potential energy desribed by Eq.(1) nder geometri onstraints imposed on the objet desribed by Eqs.(1) and (14). Namely, omptation of the deformed shape of the objet reslts in ariational problem nder eqation and ineqality onstraints. COMPUTATIONA AGORITHM FOR BET OBJECT MANIPUATION Wakamats deeloped an algorithm based on Ritz s method [15] and a nonlinear programming tehniqe to ompte linear objet deformation. That algorithm an be applied to the omptation of the belt objet deformation. et s express fntion ω ζ () and ω η () by linear ombinations of basis fntion e 1 () throgh e n (): (15) where a ζ and a η are etors onsisting of oeffiients orresponding to fntions ω ζ () and ω η () respetiely, and etor e() is omposed of basis fntion e 1 () throgh e n (). Sbstitting the aboe eqation into Eq.(1), potential energy U 1 is desribed by a fntion of oeffiient etors a ζ and a η. Constraints are also desribed by onditions inoling the oeffiient etors. Conseqently, the deformed shape of a belt objet an be deried by ompting a set of oeffiient etors [ ] d E bh G bh 1 [ ( ) ] 1 d x( ) x( ) l b ξ( ) ξ, η( ) η, ζ( ) ζ ( ) a e( ), ( ) a e( ) a [ ] d. Copyright 1 by ISFA

4 a ζ and a η that minimizes the potential energy nder the onstraints. The minimizing problem an be soled by se of a nonlinear programming tehniqe sh as mltiplier method[16]. et s ompte the belt objet deformation. The natral shape of the belt objet is illstrated in Fig.-(a). It is mm long, mm wide, and.14mm thik. Positional and orientational onstraints are shown in Fig.-(b). Fig.4 shows the omptational reslts. In this approah, omptation was performed on two.ghz AMD Operation 46 CPUs with GB memory operated by Solaris 1. Program were ompiled by a Sn C Compiler 5.8 with optimization option O5. It took abot 1 mintes to ompte. O O mm (a) Initial shape mm (b) Deformed shape mm mm DEFORMATION PATH PANNING FOR BET OBJECT DEFORMATION In maniplation of a deformable belt objet, the objet is often deformed from one shape into another. et s determine appropriate deformation path from an initial shape to a goal shape. It is generally reqired to deform a belt objet with little damage. Exessie potential energy of a belt objet an be easily transformed into kineti energy by a small distrbane fore, in whih ase the shape of the objet may beome nstable and hange dynamially. Ths, the potential energy of a belt objet shold be small dring its deformation proess. It is fond that a deformable path that minimizes the ale of the maximm potential energy is preferable. Reall that the deformation of a belt objet an be desribed by oeffiient etors orresponding to ω ζ () and ω η (). et a be a olletie etor of these oeffiients. One deformation orresponds to a point in oeffiient spae. The deformation proess of a belt objet is then gien by a path in the oeffiient spae et a and a 1 be the initial and goal deformations,respetiely, and let a k ( k 1) be a path from the initial deformation to the goal deformation. Note that fntions 1 k, k, k i 1 k (i = 1, ), and k 1 k j (j = 1, ) are a set of bases of a fntion spae. Then, any path an be approximately by a linear ombination of these base fntions a( b, k) (1 k) a i j ka1 bi,k (1 k) b j,1k(1 k) i1 j1 (16) where b i,, b j,1 are expansion oeffiients. Any path an be represented by an infinite nmber of oeffiients etors: b i,, b j,1. et b be a olletie etor onsisting of these oeffiient etors, whih is referred to as the deformation path etor. The deformation path etor b determines a deformation path from the initial deformation a b, = a to the goal deformation a b, 1 = a 1. Vetors a(b, k) orresponds to an intermediate deformation along the path. et U(b, k) be the potential energy of a belt objet with deformation a(b, k). et U max (b) be the maximm of the potential energy along a deformation path represented by b: Figre: Example of belt objet deformation (a) Top iew (b) Front iew () Side iew Figre4: Comptational reslt Reall that geometri onstraints imposed on an objet an be desribed by a set of fntions of etor b. Conseqently, it is fond that the optimal deformation path an be deried by minimizing the fntion U max (b) nder the geometri onstraints. et s show a nmerial example in order to demonstrate how deformation path is ompted by or approah. The natral shape of the objet is illstrated in Fig.-(a). Both ends of the objet are fixed as shown in Fig.-(b) and the objet is deformed from being onex pward to being onex downward. Fig.5 shows the ompted optimal deformation path. Fig5-(a) shows the initial shape of the belt objet and Fig.5-(f) shows the goal shape of it. It took abot hors to ompte the deformation path. EXTENSIBE MODE OF BET OBJECT U max ( b) max U( b, k). k1 (17) 4 Copyright 1 by ISFA

5 (a) k=. () k=.4 () k=.8 (b) k=. (d) k=.6 (d) k=1. Figre5: Deformation path for a belt objet In this paper, we assmed that a belt objet bends and twists mainly and their expansion/ontration an be negligible. So, we assmed that the srfae of the belt objet is deelopable srfae and Gassian ratre of the objet is zero at any points. Howeer, generally, an elasti objet an expand and ontrat dring its bending or twisting. The srfae of the deformed objet with expansion/ontration is not deelopable. In maniplation, the fore that ases the objet expansion/ontration sally are imposed. When the objet is extensible, it expands or ontrats. Howeer, when the objet is inextensible, sh as film/flexible irit boards, this fore an lead to loally exessie stress in the objet, instead of expansion/ontration. When we ealate the damage of a belt objet with the whole potential energy as preiosly explained, the loally exessie stress less ontribtes to the damage of the objet. Howeer, loally exessie stress an lead to fratre in a belt objet. So, the loally exessie stress shold be small dring its deformation proess. In this paper, the deformation inoling expansion/ontration is referred as ndeelopable deformation. When the srfae of a belt objet is deelopable, the potential energy of a deformed belt objet is desribed as Eq.(1). Here, let s formlate the potential energy of a belt objet that deforms ndeelopably. In this paper, the model where the objet srfae is assmed to be ndeelopable is referred as extensible model while the model explained in Setion is referred as inextensible model. When a belt objet deforms ndeelopably, the srfae of the belt objet is ndeelopable and Gassian ratre of the objet isn t zero at any points. Frthermore, the entral axis an bend arondξ -axis. The potential energy by this deformation is desribed as follows: U E b h (1 ) 1 (18) In addition, when a belt objet deforms ndeelopably, it expands and ontrats dring its bending or twisting. Namely, Poisson ratio υ isn t zero in Eq.(11). So, the potential energy of the shape inlding the ndeelopable deformation is desribed as follows: U E bh (1 ) 1 E b h (1 ) 1 Then, modls of rigidity G an be desribed as follows: (19) () Expressing fntion ω ξ (), ω η (), ω ζ (), and δ() by linear ombinations of basis fntion e 1 () throgh e n () as preiosly explained, potential energy U is desribed by a fntion of oeffiient etors a ξ, a η, a ζ, and a δ. Constraints are also desribed by onditions inoling the oeffiient etors. Conseqently, the deformed shape of a belt objet an be deried by ompting a set of oeffiient etors a ξ, a η, a ζ, and a δ that minimizes the potential energy nder the onstraints. Fig.6 shows a omptational reslt of a belt objet deformation with extensible model. It seems that the belt objet shown in Fig.6 twists more largely than that in Fig.4. This is bease when Poisson ratio υ isn t zero, modls of rigidity G beomes small and the potential energy with respet to ω η () less ontribtes to the whole energy. Moreoer, let s introde the extensible model into the deformation path planning. et a be a olletie etor of oeffiients etors a ξ, a η, a ζ, and a δ, and define a(b, k) and U max (b) as represented Eqs.(16) and (17). Conseqently, the new optimal deformation path an be deried by minimizing the fntion U max (b) with the same algorithm preiosly mentioned. Fig.7 shows the omptational reslt of new optimal deformation path. It seems that the shapes in the deformation path obtained with extensible model are also more twisted. COMPARISON OF TWO MODES [ ] d. [ ] d. In this setion, the omptational reslts obtained with two models will be erified. The model that an be deelopable srfae is referred as inextensible model. First, the omptational reslts obtained with inextensible model and extensible model will be erified by measring the deformed shape of the belt objet. We measred a belt-shaped flexible polystyrol sheets with D sanner. It is mm long,mm wide, and.14mm thik. Yong s mods E is.gpa and Poisson ratio υ is.4. Fig.8 shows the experimental reslt. Fig.9-(a) and Fig.9-(b) show the omptational reslts obtained with inextensible model and extensible model respetiely. As G bh [ ] d E G (1 ) [ ] d. 5 Copyright 1 by ISFA

6 (a) Top iew Figre8: Experimental reslt 1 1 (b) Front iew () Side iew Figre6: Comptational reslt with extensible model (a) inextensible model (b)extensible model (a) k=. (b) k=. Figre9: Comptational reslt (, ) (, ) w (,, ) w (1) () k=.4 (d) k=.6 As mentioned before, is the distane from one end of the objet along the entral axis in its longitdinal diretion, is the distane from the entral axis in the transerse diretion of the objet, and w be the distane from the netral plane in the ertial diretion. The stress in a belt objet is desribed by () k=.8 (d) k=1. Figre7: Deformation path for a belt objet with extensible mode shown in these figres, both omptational reslts are well oinide with the atal shape. This means that, atally, the srfae of a belt objet is nearly deelopable srfae and a belt objet bends and twists mainly and their expansion/ontration an be negligible. Seond, in both shapes shown in Fig.9-(a) and Fig.9-(b), the loally stress is erified, and the point whih is sbjeted to maximm stress is allated. The strain in a belt objet is represented with the ratre ω ξ (), ω η (), ω ζ (), and δ() as follows: (, ) 1 G (, ) 1 (, ) 1 (1 )/ Moreoer, the prinipal stress is represented as follows: () 1 1 () 1, ( ) ( ) 4 where σ 1 is the maximm prinipal stress and σ is the minimm prinipal stress. Table 1 shows the ale of the maximm prinipal stress of the deformed objet shown in Fig.9-(a) and Fig.9-(b) and the position where the maximm prinipal stress is orred. In both ases, the position where the maximm prinipal stress is orred is the same position. Howeer, the maximm prinipal stress of the deformed shape obtained with extensible model is larger than that of the deformed shape obtained with inextensible model. This is bease a belt objet an deform ndeelopably in the 6 Copyright 1 by ISFA

7 Inextensible Model Extensible Model extensible model and ndeelopable deformation often ases the loally exessie stress in a belt objet. FUTURE WORKS It is fond that ndeelopable deformation ases the loally exessie stress in a belt objet. The loally exessie stress an make fratre. So, the ndeelopable deformation shold be small dring its deformation proess. It is assmed that when the potential energy U obtained with extensible model is nearly eqal to U 1 obtained with inextensible model, a belt objet less deforms ndeelopably and the srfae of the deformed belt objet beomes almost deelopable. So, new potential energy U new an be defined as follow: (4) where 1 and are the weighting fators of whole potential energy and ndeelopable deformation respetiely. The deformation path of a belt objet is determined by minimizing the new potential energy U new nder geometri onstraints. As a reslt, a belt objet deforms deelopably and the stress in the objet beomes small dring its maniplation proess. CONCUSION Table1. Prinipal stress U new U Prinipal stress [MPa] 1 U1 U Position (,,w) [mm] 1.7 (1, -1, -.7) 1. (1, -1, -.7) A differential geometry based modeling to represent belt objet deformation and optimal deformation path were proposed for maniplation of film/flexible irit boards. First, the deformed shape of a belt objet was assmed to be deelopable srfae and represented as fntions of two independent parameters. Then we estimated belt objet deformation by optimizing these parameters so that potential energy of the objet attains its minimm ale nder onstraints imposed on it. Frthermore, adeqate deformation path was deried by minimizing maximm potential energy dring its deformation proess. Seond, ndeelopably deformed shape and deformation path were deried. Third, the deformed shape and the maximm prinipal stress in the objets obtained with inextensible model and extensible model were ompared, and it was erified that ndeelopable deformation ases the exessie stress in a belt objet. Finally, as ftre prospets, a method to generate the path planning was proposed so that the deformed shape of a belt objet beomes deelopable. REFERENCES [1] A. Witkin and W. Welh, Fast Animation and Control of Nonrigid Strtres, Compter Graphis, Vol.4, 199, pp.4-5. [] D. k. Pai, STANDS: Interatie Simlation of Thin Solids sing Cosserat Models, Compter Graphi Form, Vol.1, No.,, pp.4-5. [] Y. E. Zheng, R. Pei, and C. Chen, Strategies for Atomati Assembly of Deformable Objets, Pro. of IEEE Int. Conf. Robotis and Atomation, 1991, pp598-6 [4] H. Nakagaki, K. Kitagaki, and H. Tskne, Stdy of Insertion Task of a Flexible Beam into a Hole, Pro. of IEEE Int. Conf. Robotis and Atomation, 1995, pp.- 5. [5] K. Kosge, M. Sakaki, K. Kanitani, H. Yoshida, and T. Fkda, Maniplation of a Flexible Objet by Dal Maniplators, Pro. of IEEE Int. Conf. Robotis and Atomation, 1995, pp [6] F. amirax and. E. Kraaki, Planning Paths for Elasti Objets nder Maniplation Constraints, Int, J. Robotis Researh, Vol., No., 1, pp [7] M. Saha and P. Isto, Maniplation Planning for Deformable inear Objets, J. IEEE Transations on Robotis, Vol., No.6, 7, pp [8] A.Gray, Modern Differential Geometry of Cres and Srfane, CRC Press, 199. [9] M. Moll and. E. Kraraki, Path Planning for ariable Resoltion Minimal-Energy Cres of Constant ength, Pro. of IEEE Int. Conf. Roboti and Atomation, 5, pp [1] J. Tian and Y. B. jia Modeling Deformable Shell-like Objets Grasped by a Robot Hand, IEEE International Conferene on Robotis and Atomation, 9, pp [11] S. Timoshenko, Theory of Plates and Shells, MGraw- Hill Book Company, In., 194. [1] E. H. Mansfield, The Inextensional Theory for Thin Flat Plates, The Qarterly J. Mehanis and Applied Mathematis, Vol.8, 1955, pp.8-5. [1] H. Wakamats and S. Hirai, Stati Modeling of inear Objet Deformation based on Differential Geometry, International Jornal of Robotis Researh, Vol., No., Marh, (4), pp [14]. E. Elsgol, Calls of Variations, Pergamon Press, [15] M. Ariel, Nonlinear Programming: Analysis and Methods, PretieHall, Copyright 1 by ISFA