INTEGRATED SYNTHESIS OF ASSEMBLY AND FIXTURE SCHEME FOR PROPERLY CONSTRAINED ASSEMBLY

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1 Proceeings of T SM esign ngineering Technical onferences an omputers an Information in ngineering onference September 8-October,, Salt Lake ity, Utah, US T INTGRT SYNTHSIS O SSMLY N IXTUR SHM OR PROPRLY ONSTRIN SSMLY yungwoo Lee an Kazuhiro Saitou * epartment of Mechanical ngineering University of Michigan nn rbor, Michigan 89-5 mail: {byungwoo, kazu}@umich.eu STRT This paper presents an integrate approach to esign an assembly, fixture schemes an an assembly sequence, such that the imensional integrity of the assembly is insensitive to the imensional variations of iniviual parts. The ajustability of critical imensions an the proper constraining of parts uring assembly process are the keys in achieving the imensional integrity of the final assembly. top own esign metho is evelope which recursively ecomposes a lump of initial prouct geometry an fixture elements matching critical imensions, into parts an fixtures. t each recursion, joints are assigne to the interfaces between two subassemblies to ensure parts an fixtures are properly constraine at every assembly step. case stuy on a simple frame structure is presente to emonstrate the metho. INTROUTION Structural enclosures of moern mechanical proucts, such as ship hulls, airplanes an automotive boies, typically are mae of hunres or thousans of parts ue to their geometric complexity an sizes. s the number of parts increases, however, achieving the imensional integrity of the final assembly becomes more ifficult ue to the inherent variations in manufacturing an assembly processes. H igure. Two box esigns (a) without an with ajustable height uring assembly []. solution is to ajust critical imensions in assembly processes when parts or subassemblies are locate an fully constraine in fixtures. This in-process imensional ajustment is typically facilitate by slip planes, mating surfaces at joints that allow a small amount of relative motions. or example, igure shows two esigns of a rectangular box. In contrast to esign in (a) with no in-process ajustability of the critical imensions (length between sections an ), esign in provies slip planes such that relative location of parts can be ajuste along the critical imension. (a) igure. Two box esigns (a) without an with properly constraine parts []. The imensional integrity of an assembly is also affecte by the post-assembly istortion ue to the internal stress inuce by joining parts with imensional mismatches. solution is to ensure the proper constraining of subassemblies at each assembly step. or example, part in igure (a) is not properly constraine an therefore the post-assembly istortion might occur, if the length of sections an are slightly ifferent ue to manufacturing variation. With two slip planes perpenicular to each other, the esign in can absorb manufacturing variations within parts an --, provie that variations in angles are negligible. In aition to the assembly esign incluing joint types at part interfaces, the assembly sequence also influences inprocess imensional ajustability an proper part constraints. In the assembly sequence in igure (a), the critical imension * orresponing author opyright by SM

2 (total length) is not ajustable since there is no slip plane parallel to it when the total length is realize with the aition of part. On the other han, the sequence shown in provies the slip plane at the assembly step where the critical imension is achieve, to absorb the variation in length. s another example, the sequence in igure, where each critical imension is inepenently ajuste at each step, is more esirable than the sequence in (a), where both imensions are ajuste at one step, inevitably requiring a compromise between two potentially conflicting critical imensions. igure 5 illustrates an effect of the assembly sequence on proper part constraints, where the sequence in (a) causes over-constraint at the secon step, whereas all parts are properly constraine at all steps in, thus avoiing potential assembly stress. critical imensions are realize inepenently on the only fixture, in two ifferent assembly sequences, (a) an. What is ifferent from igure (a) is that pins locating part an control the location of part an separately, thus enabling inepenent realization of the critical imensions. The pin locating part serves realizing both critical imensions. Inee, the fixture in igure 6 is the union of the two fixtures utilize in igure. xamining ifferent ways of arranging fixtures for multiple Ks is valuable, as using one fixture to eal with several critical imensions is quite common for large scale assemblies, especially when several parts constitute a flat subassembly. (a) igure. ssembly sequences (a) without an with inprocess ajustability (moifie from []). (a) igure 6. easible assembly sequences epen on utilization of fixtures. ompare with igure. (a) igure. ssembly sequences where two imensions are ajuste (a) at one step an inepenently at two steps (moifie from []). (a) igure 5. ssembly sequences (a) without an with proper constraints []. Let us note igure again. In igure, each critical imension is realize on a separate fixture, in which case, it is the only feasible assembly sequence to realize both critical imensions inepenently. However, to make the problem more complicate, other assembly sequences are feasible if fixtures are arrange ifferently. or example, in igure 6, both s pointe out by inustry practitioners an researchers, the proper constraint an ajustability are key elements in assembly esign to achieve high precision an accuracy with low cost parts []. Whereas it is important to carefully esign an sequence the assembly an fixtures in orer to avoi overconstraints an the loss of esire ajustability, inustry practices o not come up to systematic approaches. espite the fact that the proper constraint an ajustability shoul be ensure between subassemblies at every assembly step, not between parts, current esign practices an systems overlook this important property an mistreat joints an tolerances as the attributes of part geometry without consiering assembly sequences. or complex mechanical assemblies, this causes many imensional iscrepancies at the manufacturing stage, followe by costly reesigns an reworks. To make matters worse, typical engineering countermeasures in such situations have often been to tighten part tolerances, without examining the assembly esign an tolerance relationships as a whole []. s a remey, we have presente a top-own ecomposition-base assembly synthesis metho [] to fully enumerate all feasible sets of part ecomposition, joint assignments an an assembly sequence, for geometry. ssuming that assemblies can be built in the reverse sequence of ecomposition, the metho recursively ecomposes a given prouct geometry into two subassemblies until parts become manufacturable. t each recursion, joints are assigne to the interfaces between two subassemblies to ensure in-process imensional ajustability an properly constraint. The metho opyright by SM

3 has also been applie to beam-base structure [5], where Screw Theory [6] is utilize for the evaluation of in-process ajustability an proper constraints of subassemblies at every assembly step. However, our previous works [, 5] assume one fixture to achieve every critical imension (as shown in igure ), an hence incapable of exploring various fixture schemes. This paper extens our previous works to esign fixture scheme as an integrate part of assembly synthesis, which enumerates all feasible esigns (assembly esigns, fixture schemes, an assembly sequences) by treating fixtures as an entity of assembly. Not only oes this integration explore all feasible fixture schemes along with assembly esigns, but also reveals feasible assembly sequences that were illicit in our previous methos [, 5], such as those shown in igure 6. case stuy on a simple space frame is presente to emonstrate the metho. onsiering the number of parts, the number of fixtures, the epth of assembly tree, an the number of uner-constraints as objectives to minimize, a multi-objective graph search is performe on the enumerate feasible esigns, in orer to obtain Pareto optimal solutions. Some representative esigns in the Pareto set are examine to illustrate the trae-offs among the assembly esign, fixture scheme, an assembly sequence. RLT WORKS Since previous works in general relevance to assembly synthesis are reviewe in [], this section focuses on the literature irectly relate to the present extension of the assembly synthesis metho, namely on property constraine assembly esigns an fixture esigns. The avantages of properly constraine assemblies are well known to practitioners in precision machinery esign, an several methos have been propose in literatures incluing: Kinematic esign [7], Minimum onstraint esign [8] an xact onstraint esign [, ]. These works escribe isavantages of over-constraints an provie goo practices as well as analytical methos to compute constraints. In these works, the most commonly cite merit of properly constraint esign is repeatability that leas to high precision. owney et al. [9] analyze an classifie elements of assemblies that absorb manufacturing variations of parts. universal analytical metho for motion an constraint analysis ates back to Screw Theory, a pioneering work by all [6]. Since then, Screw Theory has been applie to areas of mechanism, robotics an machine esign. mong others, Walron [] utilize the screw theory to buil a general metho which can etermine all relative egrees of freeom (O) between any two rigi boies making contacts to each other. laning [] shows the application of screw theory to assembly esign. ams an Whitney [] also use screw theory to compute the constraints on parts an applie it to rigi boy assemblies with mating features such as pin-slot joint. saa an y [] propose kinematic analysis metho for fixture layout esign by moeling kinematic constraints of ixture scheme is efine as a plan showing which fixture will control what critical imensions where in assembly sequence. More formal efinition will follow in terminology section. fixture locators as a Jacobian matrix, which shoul have full rank to locate a given work piece uniquely at a esire position. While these works provie tools for analyzing constraints in a given assembly an esign guielines, they o not aress a systematic an integrate synthesis of an assembly an fixture scheme with esire constraint characteristics such as inprocess imensional ajustability an proper part constraints, as iscusse in this paper. lthough esign of fixture scheme shoul precee physical fixture layout esign, authors coul not fin previous works attacking this problem in a systematic way. TRMINOLOGY Since the assembly synthesis eals with objects yet to be ecompose into an assembly of separate parts, a few terms an concepts nee to be efine to avoi confusion with generic meanings use in other literatures. prouct geometry is a geometric representation of a whole prouct as one piece before ecomposition into parts. member is a section of a prouct geometry allowe to be a separate part. pair of members is connecte when they meet at a certain point in the prouct geometry. configuration is a group of members which are connecte to at least one member within the group. prouct geometry is a configuration, so as a part (as efine below). The Key haracteristics (Ks) are efine by Lee an Thornton [] as prouct features, manufacturing process parameters, an assembly features that significantly affect a prouct s performance, function an form. In this paper, a K refers to a critical imension to be achieve in assemblies. ecomposition is a transition of a configuration into two sub-configurations by removing connections between two members. part is a configuration that is not ecompose further uner given criteria, e.g., a minimum part size. part may consist of one or more members. joint library is a set of joint types available for a specific application omain (igure 7). igure 7. n example of joint library for - beam base assemblies consisting of lap, butt an lap-butt. n (synthesize) assembly is a set of parts an joints that connect every part in the set to at least one of other parts in the set. ssembly synthesis is a transformation of a prouct geometry into an assembly. fixture element is an imaginary part of a fixture to control a K. Physically, a K will be controlle by a set of locators, an the fixture element is abstract opyright by SM

4 representation of this set of locators. Thus, each K will have a fixture element corresponing to it. fixture is a group of fixture elements an contols corresponing Ks. ixture scheme is partitioning the whole set of fixture elements into groups an assigning them into assembly sequence. SRW THORY In Screw Theory, a screw is efine as a pair of a straight line (screw axis) in a artesian space an a scalar (pitch). It is commonly represente by screw coorinates, a pair of two row vectors S = (s; s ) in artesian coorinates, where s is a unit vector parallel to the screw axis an s is given as: s = r s + ps () where r is the position vector of a point on the screw axis an p is the pitch. quivalently, p can be expresse using s an s as: s s p = () s s screw with an infinite pitch oes not follow quation (), therefore it is enote by s being the zero vector an s being the unit vector parallel to the screw axis. Two types of screws, twist an wrench, are utilize in this paper. twist is a screw representing a motion of a rigi boy simultaneously rotating aroun an translating along an axis. Using screw coorinates, it is enote as T = (ω; v), where ω is the angular velocity an v is the linear velocity of a point on the boy (or its extension) locate at the origin of global reference frame. wrench is a screw representing a force along an a moment aroun an axis exerte on a rigi boy. Using screw coorinates, it is enote as W = (f; m), where f is the force an m is the moment that a point on the boy (or its extension) locate at the origin of global reference frame shoul resist. Two screws S = (s ; s ) an S = (s ; s ) are reciprocal to each other, if an only if they satisfy: s s + s s =. () If a twist T is a reciprocal of wrench W (or vise versa), W oes no work to a rigi boy moving accoring to T. When a boy can receive linear combinations of several screws (either twist or wrench), this set of screws are typically represente as a matrix where each screw in the set forms a row vector of the matrix. This matrix is calle a screw matrix. s its row space is the screw space (a space forme by the set of screws in the matrix), the rank of a screw matrix is equal to the imension of the screw space. The function reciprocal(s) returns a screw matrix whose row consists of the screws reciprocal to those in S. It can be obtaine by exchanging the former three columns an the letter three columns of the null space of S. The terminology an formalization in this section are summarize from [6], [], [], [5] an [6]. The union of screw matrices represents the sum of the screw spaces efine by the matrices, an can be obtaine by simply stacking them on top of one another: n i= S i S S S n The intersection of screw matrices is the set of screws common to the screw matrices, an can be compute through ouble reciprocals: n i i= i= n () S recip rocal( reciprocal( Si)) (5) Since a twist an a wrench are also screws, the efinitions of reciprocal, union, an intersection hol. (a) x z y igure 8. Lap (a) an lap-butt joint of a beam base moel an the local coorinate frames for twists. Woo an reuenstein [] presents kinematic properties of various joint types in screw coorinates, which are aopte to buil twist matrices of beam joint types. igure 8 (a) shows a typical lap joints foun in beam-base structures. When it is attache to another beam, the tab allows planar motion parallel to x-y plane. lso, if we assume that the length of the tab is very small compare to the length of the beam, it can be treate as a line contact along y-axis, allowing the rotation about y-axis. Thus, the lap join, with respect to the local coorinate frame shown in the figure, can be moele as a twist matrix: T lap = (6) Similarly, a butt joint in igure 8 allows the motion parallel to y-z plane, can be moele as: T butt = (7) In twist matrices in quation (6) an (7), each row represents an inepenent motion, an each non-zero number represents rotation or translation along a corresponing axis x z y opyright by SM

5 ω x, ω y, ω z, v x, v y or v z. or example, the first row in quation (6) has at the secon column, which means the lap joint allows rotational motion about y-axis. In the thir row, it has at the fourth column, meaning translation along the x-axis is allowe. Since these matrices are use only to give information on which Os are not constraine for a joint type, the magnitue of each twist (row) of these twist matrices (i.e., the magnitues of the angular an linear velocities in the twist) oes not have significant meaning in this paper. Once the twist matrix is obtaine for a joint type, the reciprocal wrench matrix can be compute as escribe above. or instance, the wrench matrices corresponing to twist matrices in (6) an (7) are: Wlap = reciprocal( T lap ) = (8) Wbutt = reciprocal( T butt ) = (9) ach non-zero number now represents force or moment along a corresponing axis f x, f y, f z, m x, m y or m z. Since a wrench that is a reciprocal of a twist oes no work to a rigi boy moving accoring to the twist, these are the forces an moments the joint supports (hence resulting no work). or example, in the first row in quation (8) has at the thir column, which means the lap joint can support a force along z-axis. Initial config. ecomposition rules onfig. onfig. ecomposition rules onfig onfig. ssembly esirable conitions Part Subassembly esirable conitions Part igure. ssembly synthesis by top-own hierarchical ecomposition. ssembly sequence is the reverse of the ecomposition sequence. SSMLY N IXTUR SHM SYNTHSIS Part ssembly synthesis via recursive ecomposition There are numerous issues relate to assembly esign. mong others, ajustability an proper constraint are the key necessary conitions for imensional integrity. issimilar to other issues such as structural stiffness an prouct function, these two conitions shoul be satisfie at every assembly step, as illustrate in igures -5. y taking avantage of this fact, one can hierarchically ecompose a given prouct geometry such that (sub)geometries at each ecomposition step satisfy the above esire conitions when they are assemble back together in the reverse orer (see igure 9). Our previous works [, 5] suggeste the framework of assembly synthesis via such hierarchical ecomposition, which was successfully applie to simple - [] an - [5] geometries. Generation of fixture elements K, in this paper, is assume to be a critical imension between parts to be achieve by the ajustment uring the assembly process. Thus, the imension note as a K will be constraine by a fixture, accoring to which parts being assemble will be locate. In this context, we know the fixture woul have to constrain at least the Os specifie by the K, regarless of its physical emboiment. Provie a K is controlle by a fixture, the assembly of two subassemblies connecte by a K can be viewe as two assembly steps, involving two subassemblies an a fixture, such as {{part, fixture}, part}. s epicte in igure, this allows each K to be replace by a fixture element connecting the same members. The graph representation shown in igure is what we call configuration graph. fter replacing Ks with fixture elements, the configuration graph is a pair: = (M, ) (), where M is the set of noes representing members an fixture elements, an is the set of eges representing connections. ach noe in M is associate with its type (members are in white an fixture element are in black in igure ), an each ege in between a member noe an a fixture element noe is associate with a wrench matrix representing the Os to be constraine by the replace K. or example, if kc in igure is the istance between members an in y-irection measure at x =.5 in the global reference frame, then the wrench matrix associate with eges {, f} an {, f} is l = (.5) W. () where subscript l inicates locator. Similarly, the wrench matrix associate with for eges {, f} an {, f} are: l = ( ) W. () While seemingly subtle, this replacement of Ks with the corresponing fixture elements is a major avance beyon our previous works [, 5], which enables an elegant integration of the synthesis of a fixture scheme into assembly synthesis process. Initially, each fixture element is connecte to all the other fixture elements, in orer to allow the exploration of all possible fixture schemes. The connections between a fixture element an a member represent minimum locators that constrain at least the Os specifie by the replace K. ny aitional Os neee to uniquely locate the part will be compute uring assembly an fixture scheme synthesis as escribe in the following section. urther, the configurations after the replacement of Ks with fixture elements are classifie to three classes: 5 opyright by SM

6 Incomplete configuration: a configuration with unconnecte members or with a fixture element connecte to less than two members. or example, the secon step of igure 6 is an incomplete configuration since, members are not connecte an the fixture element controlling the istance between members an has only one connection (to part ) ue to the absence of member. ixture: a configuration consisting of only fixture elements. omplete configuration: a configuration that is neither an incomplete configuration nor a fixture. X only one connecte subassembly an, if any, a fixture with all assigne Ks realize. Therefore, it is reay to leave the fixture for further assembly with any configuration incluing a fixture. X Y W l f W l f W l W l cut-set kc kc Y kc kc X Y W l f W l W l f W l f W l fixture element fixture element W l igure. Replacement of Ks with fixture elements whose locators constrain the same Os. easible binary ecomposition The assembly synthesis algorithm [] aopte in this paper assumes every assembly step combines a pair of subassemblies. onversely, the algorithm ecomposes a configuration into two (sub)configurations by removing some connections, which is equivalent of fining a cut-set [7] of the configuration graph. ecomposition is mae only when the reverse of it yiels a feasible assembly step, for which there are two criteria. irstly, the assembly step is binary - only two subassemblies are joine at the assembly step. This is justifie by the fact that a nonbinary assembly step (eg., assembly of multiple parts on a fixture in one step) can be broken own to an equivalent sequence of binary assemblies. Seconly, at least one of two subassemblies joine at the assembly step is a complete configuration, which is justifie in the next paragraph. When a configuration is incomplete, subassemblies shoul remain on the fixture because subassemblies are either not connecte or the fixture has at least one assigne K yet to realize (such as the status shown in igure 6 ). Since fixtures are usually heavy or groune, it woul be very rare that a subassembly attache to a fixture is assemble to another subassembly in the same situation, or to another fixture. or the same reason, assembly of two fixtures is consiere infeasible. On the other han, when a configuration is complete, it has f W l W l igure. feasible ecomposition Y igure. n infeasible ecomposition that results in two incomplete sub-configurations. or example, igure shows a feasible ecomposition yieling one complete an one incomplete sub-configurations. In igure, both sub-configurations are incomplete, thus will not be consiere as a feasible ecomposition. More formally, ecomposition from configuration a = (M a, a ) to two sub-configurations b = (M b, b ) an c = (M c, c ) is feasible if the following conitions are satisfie: M b an M c. b an c are connecte. X X Y W l W l f f W l W l f W l W l W l f W l cut-set 6 opyright by SM

7 t least one of b an c is a complete configuration. M a = M b M c. M b M c =. () The st conition states sub-configurations shoul be nonempty. The n conition states the sub-configurations must be connecte. The r an 5 th conitions specify the configuration shoul split into a pair of isjoint sub-configurations. ecomposition rule for imensional integrity Once a ecomposition satisfying conitions in quation () is foun, feasible joint types are assigne to broken connections, which is represente as mappingγ : S JL, where S is the cut-set broken by ecomposition an JL is a library of joint an locator types. With the joint assignment, (binary) ecomposition can be uniquely specifie as = (M a, γ, (M b, M c )). See igure for an example. Note that feasible joint types may epen on the local geometry near the joint location. or example, feasible joint types between two perpenicular beams woul be ifferent from those for two coaxial beams. The broken connections with the assigne joints are associate with the wrench matrices compute accoring to the assigne joint types an orientations. very connection between a member an a fixture element alreay has a wrench matrix compute in the previous step; therefore no action is taken even if it is broken by a ecomposition. j X Z Y j igure. Joints assigne to broken connections, for which wrench matrices are compute accoringly (fixture elements omitte in the left figure). Having replace all Ks with the corresponing fixture elements, the only criterion that nees consieration for assigning joint types to broken connections is the proper constraint of the mixture of subassemblies an fixtures at every assembly step. In particular, there is no nee to explicitly consier the ajustability for Ks as require in our previous work [] since the proper constraint incluing the O constraine by Ks implies the assigne joints o not interfere with the Os constraine by Ks, automatically ensuring their iniviual ajustability. In orer for subassemblies being assemble to be properly constraine, the joints shoul not constrain the same O more than once, as illustrate in igure 5. This assembly rule is inversely state as the ecomposition rule for proper constraint in our previous work [], which only allows the combination of joints yieling no over-constraint of parts. lthough when one subassembly is locate on an empty fixture, the fixture shoul constrain all six Os [], when the next subassembly is put f W l W j f W l W j together contacting the other subassembly on the fixture, it woul be constraine by both the other subassembly an the fixture. Therefore, joints must be selecte in such a way that no O is constraine twice not only among joints but also with locators. In such cases, the intersection of the wrench matrix corresponing to any subset of S an the wrench matrix of any other isjoint subset must result in the zero matrix:, S, =, which is also equivalent to: ( W ) ( W ) = O γ () e γ () e e e γ () e γ () e e S () rank( W ) = rank( W ) (5) e S urther, in orer to have all six Os constraine, rank( W ) = rank( ) 6 γ () e W = (6) γ () e e S e S When any set of joint types an fixture elements satisfies quation (5) with the total rank less than six, it is consiere to be feasible, assuming that aitional fixtures or locators on existing fixtures will be arrange. Six less the number of Os constraine is counte as the number of uner-constraints for each feasible joint assignment an recore as uc(γ ). preicate of a ecomposition = (M a, γ, (M b, M c )) for complying the rule is given as: M M M e: ( JL) ( ) { true, false} (7) where e(m a, γ, (M b, M c )) is true if an only if the conitions in () an quation (5) is satisfie. There is an important exception to quation (5), for which compensation shoul be mae before it is checke against quation (5); when there are connections in S, from multiple fixture element to one member, the wrench matrices associate to the connections shoul be unionize such that the intersection among them coul be ignore. This is base on a basic assumption that there woul be no over-constraint between a fixture an a member. Suppose there is a set of fixture elements that are connecte to a member an more than one of these connections are broken by a ecomposition. In this case, even if there is a O constraine by more than one fixture elements, the O will be constraine by one locator in actual implementation. or example, see the first step in igure 6. When part is place on the fixture, the fixture is constraining two Ks of the same O, the istances to an. However no one will use one locator for each K, which will certainly yiel over-constraint. When this step is generate through ecomposition, the ecomposition woul break connections between the member an each of the two fixture elements transforme from the two Ks. In orer to match the assumption that there woul be one locator for a O, the wrench matrices for these connections shoul be unionize. onsier the prouct geometry ecompose in igure an the joint assignment shown in igure, which has two lap 7 opyright by SM

8 joints j an j, an two locators l an l, for eges cut by the ecomposition. ecause the ecomposition is breaking multiple connections from member to fixture elements, wrench matrices for these connections shoul be unionize as escribe in the previous paragraph. rom quation () an (), we can compute:.5 Wl Wl = (8) Suppose the location of j an j in global reference frame X- Y-Z are (,, ) an (,, ). Then, base on the local coorinate frame of lap joint shown in igure 8 an orientation of j an j, W lap (quation (8)) can be transforme into W j an W j in global reference frame: W j = (9) W j = () Unionizing the three matrices in quation (8) (), W γ () e e S () Whereas the summation of the ranks of iniviual matrices is six, the rank of union is only five, which implies that this combination of joints yiel an over-constraint of one O. In fact, the intersection of W j an W j is not a zero matrix. s this joint assignment oes not satisfy the ecomposition rule, quation (5), the assembly synthesis process will iscar it. Part manufacturability The ecomposition stops when the resulting subconfigurations become manufacturable by a chosen manufacturing process. In the following case stuy on frame structures, components are assume to be extrue an bent. Therefore, a preicate of a configuration M a for stopping ecomposition is given as: M stop_ e : { true, false} (), where stop_e(m a ) is true (i.e.. ecomposition continues) if an only if none of the first four conitions are satisfie or the fifth conition is satisfie:. M a contains both member(s) an fixture member(s).. The inuce subgraph on members in M a has a close loop (cannot extrue such parts). The result has been reuce to the Row Reuce chelon orm for easy interpretation.. Three or more members in M a are connecte to each other at a single point (cannot extrue such parts).. Members in M a lie on more than one plane (ifficult to hanle/fixture). 5. M a consists of only fixture elements (assembly of two fixtures is not consiere). See igure for example. The configuration, {,, f, f}, satisfies the first conition, thus stop_e returns false, subject to further ecomposition. On the other han, the other configuration, {, }, satisfies none of the first four conitions, the ecomposition is stoppe for this configuration. N/OR graph of assembly synthesis series of ecompositions can be typically represente in a tree as shown in igure 9. However, the aim of the presente metho is to enumerate all such trees, N/OR graph [8] is aopte to facilitate the assembly synthesis, in which multiple trees share common noes. lthough the N/OR has been previously use to enumerate assembly sequences for a given assembly esign [9], it is augmente in this paper in orer to emboy joint assignments. igure shows a partial N/OR graph of assembly synthesis [] for the rectangular box shown in igure. ach noe in white backgroun contains a configuration, (M a M ), an each noe in black backgroun contains joint assignment γ i : Si JL. set of three lines which connects a configuration M a, joint assignment γ i, an two sub-configurations (M b, M c ) is a hyper-ege, represente as (M a, γ i, (M b, M c )) which is also the representation of a ecomposition efine earlier. The N/OR graph of assembly synthesis is then represente as a triple: O = (S, J, ) (), where S is a set of noes representing configurations, J is a set of noes representing joint assignments, an is a set of hypereges (M a, γ i, (M b, M c )) satisfying the following necessary conitions.. stop_e(m a ) = false.. e(m a, γ i, (M b, M c )) = true. () Then O = (S, J, ) is recursively efine as:. If stop_e(m ) = false, M S.. or M a S, if γ i, M b, M c such that f = (M a, γ i, (M b, M c )) satisfies necessary conitions (), then γ i J, M b, M c S an f.. No element is in S, J an, unless it can be obtaine by using rules an. (5) The recursive efinition in quation (5) can be easily transforme to an algorithm buil_o that generates O from initial configuration an joint library by recursively ecomposing a configuration into two sub-configurations [], whose etails are omitte ue to space limitation. Using stop_e an e as efine earlier, one can run buil_o with any configurations to enumerate all possible assemblies (ecompositions an joint assignments), fixture schemes an 8 opyright by SM

9 accompanying assembly sequences that satisfy the in-process imensional ajustability an proper part constraint. capital letter represents a part (marke with the same letter in the following assembly esign), an a noe marke with fx with a number represents a fixture. black noe represents a joint assignment an the number within the noe represents, uc(γ ), the number of uner-constraints for the joint assignment. ecompose assemble igure. part of the N/OR graph for the - rectangular box in igure. kc5 kc kc6 kc kc7 igure 5. frame structure with eight Ks. S STUY frame structure in igure 5 is ecompose base on quation (5). Only joint types in igure 6 are assigne to broken connections as require by the ecomposition rule. In orer to reuce the size of the N/OR graph, when several joint assignments satisfy ecomposition rule for a given ecomposition, one with minimum uner-constraint is inclue in the N/OR graph. Still, the ecomposition rule prouce a large N/OR graph with 996 noes representing configurations an 69 hyper-eges, which contains about 8. billion trees. However, using brute search starting from the terminal noes (either part or fixture that satisfies stop_e), non-ominate solution trees for multi-objectives can be ientifie. ase on four objectives the number of parts, the number of fixtures, the epth of the tree an the total unerconstraints, only 9 trees are foun to be non-ominate. ssociate cost vectors for these non-ominate solution trees are liste in Table. The number of fixtures an unerconstraints shows a strong correlation, because the more fixtures are use, the more Os shoul be constraine when initially placing a part on each fixture. rom igure 7 to, some of non-ominate solution trees an their corresponing assembly esigns are presente. In solution trees, a noe with a kc kc8 kc igure 6. (top) joint types for frame sturcture, an (bottom) their graphical representation use in results. Table. Non-ominate cost vectors an the number of corresponing non-ominate solution trees for the frame structure shown in igure 5. Objectives No. of epth of No. of unerconstraints fixture tree Total 9 No. of parts No. of solution trees igure 7 shows a non-ominate solution tree, an corresponing assembly esign an sequence, which has 7 parts, fixtures, the epth of 7, an 6 uner-constraints. The coorinate frame shown by each joint shows Os constraine by the joint in black an un-constraine Os in gray. In the figure, fx controls kc, kc, kc7 an kc8, an fx controls kc, kc, kc5 an kc6. The assembly sequence is as follows:. Locate G on fx. Three Ks relate to G; kc, kc an kc5 are constraine by fx. In orer to uniquely locate G, fx shoul constrain the other three Os (the number of uner-constraints) in aition to those require by the Ks.. ssemble on G-fx. Only one K, kc6 is require for, which is fixe by fx. The other five Os are constraine by the lap-butt joints with G, thus there is no unerconstraint to be controlle aitionally.. Locate G on fx. Three Ks relate to G; kc, kc an kc5 are constraine by fx. In orer to uniquely locate G, fx shoul constrain the other three Os in aition to those require by the three Ks.. ssemble on G-fx. Only one K, kc6 is require for, which is fixe by fx. The other five Os are constraine by the lap-butt joints with G, thus there is no unerconstraint to be controlle aitionally. 5. In parallel with step, place on fx. ll four Ks relate to are fixe on fx. The other two Os shoul be constraine aitionally. 9 opyright by SM

10 fx fx fx fx fx fx G G fx fx igure 7. One of non-ominate solution trees whose cost vector is (7,, 7, 6). 6. ssemble ---G on -fx, where kc an kc7 are realize by fx. The other four Os are constraine by the lap joint from to an another lap joint from to. 7. ssemble on ----G-fx, where kc is realize. The lap joint from to an the lap joint from to constrains four Os, thus one O shoul be constraine aitionally. 8. ssemble on -----G-fx, realizing kc8. The other five Os are fully constraine by one lap joint an one butt joint of. ll assembly steps are now complete. igure 8 shows a non-ominate solution tree, which has 8 parts, fixtures, the epth of 5, an uner-constraints. This tree has the minimum epth an has a few parallel steps. or this reason, the tree is suitable for parallelize an short cycle time prouction. The price it pays is the many fixtures require to realize Ks in parallel. On the other han, the tree shown in igure is completely serial, using only one fixture to control all the Ks. ecause there are less fixtures, many Os are constraine by joints between parts, thus yieling mere three uner-constraints throughout the assembly. Instea, the prouction woul require a longer cycle time. There are only four trees that have the non-ominate cost vector of (6,, 6, ). Two of these are shown as a N/OR graph in igure, which contains two ifferent assembly sequences to buil the assembly esign shown in igure (note the OR relation between the two hyper-eges from the top noe). The other two trees for the same cost vector are mirror images of ones shown in igure, which has corresponing assembly esign that is also mirror image of one shown in igure. Whereas other non-ominate solution trees have one or more ecomposition(s) solely to remove an un-planar part, a close loop or a T-joint require by stop_e, without breaking a K, all the ecompositions in these trees have been mae to remove Ks. In other wors, these trees show the most efficient way to remove Ks in terms of the number of ecompositions. s a result, these solutions have minimum number of parts, 6. SUMMRY N ISUSSION This paper presents an integrate synthesis approach of assembly an fixture scheme. Starting with initial geometry, the ecomposition rule is applie recursively, to obtain all feasible esigns (assembly esigns, fixture scheme, an assembly sequence), such that every assembly step can be free of overconstraints. n example with a simple space frame is presente to emonstrate the metho. onsiering the number of parts, the number of fixtures, the epth of assembly tree, an the opyright by SM

11 number of uner-constrains as objectives to minimize, the Pareto optimal solutions are obtaine by searching the enumerate N/OR graph of assembly synthesis that contains all feasible esigns. Non-ominate solution trees show traeoffs among the assembly esign, the number of fixtures an the assembly sequence. n assembly with many connections among parts an many Ks is likely to have less parallel assembly sequence, because it is likely that the assembling two large subassemblies at the later stage woul have many joints between the two subassemblies an realize many Ks at one step, thus more likely to have over-constraints. It has been also shown that a more parallelize assembly sequence woul require more fixtures. 5 fx fx 5 5 fx igure. One of 6 non-ominate solution trees whose cost vector is (8,, 8, ). G H fx fx igure 8. One of non-ominate solution trees whose cost vector is (8,, 5, ). G H H H G G igure. The assembly esign matching the nonominate solution tree shown in igure. igure 9. The assembly esign matching the nonominate solution tree shown in igure 8. opyright by SM

12 5 times are shown in Table **, which exhibits a rapi growth in the computation time with the increase number of members an Ks. The present metho, therefore, woul most effectively be integrate in the esign process if it is applie to subassemblies of a prouct ecompose by other means. lternatively, the repeate applications of the metho to subassemblies with incremental refinement of members (from coarse to fine) woul also be effective to manage the complexity. fx 5 fx Table. mount of computation for () the example in igure 5, () without Ks an () with 5 members an Ks remove. No. of members 7 No. of Ks 8 No. of ecompositions 69 6 No. of solution trees computation time [sec] igure. Two of four non-ominate solution trees whose cost vector is (6,, 6, ). igure. The assembly esign matching the trees in igure, which contains two assembly sequences. ue to the enumerative nature of the presente approach, the amount of computation for complex assemblies woul be inevitably large. Taking the number of prospective ecompositions that must be analyze as a measure of the amount of computation, Homem e Mello an Sanerson [] showe that computational complexity is generally O( n ) in generating N/OR graph of assembly sequence for given assembly esigns, where n is the number of parts. lthough the amount of computation for assembly synthesis woul largely epenent on the number of Ks, available joint types an the manufacturability criteria, the worst case (when every part consists of only one member) woul be comparable to that of assembly sequence generation. Some of the actual computation fx KNOWLGMNTS This work has been supporte by the National Science ounation with a RR war (MI-99866) an Toyota Motor ompany. ny opinions, finings, an conclusions or recommenations expresse in this material are those of the authors an o not necessarily reflect the views of the National Science ounation. RRNS [] Lee,. an Saitou, K.,, ecomposition-base assembly synthesis for in-process imensional ajustability, SM Journal of Mechanical esign, vol. 5, no., pp [] Whitney,.., Mantripragaa, R., ams, J.., an Rhee, S. J., 999, esigning assemblies, Research in ngineering esign, vol., pp [] laning,. L., 999, xact onstraint: Machine esign Using Kinematic Principles, SM Press, NY. [] Kriegel, J. M., 995, xact constraint esign, Mechanical ngineering, vol. 7, no. 5, pp [5] Lee,. an Saitou, K.,, Three-imensional synthesis for robust imensional integrity base on screw theory, Proceeings of the ifth International Symposium on Tools an Methos of ompetitive ngineering, Lausanne, Switzerlan, pril -7, vol., pp [6] all, R. S., 9, Treatise on the Theory of Screws, ambrige University Press. [7] Whitehea, T. N., 95, The esign an Use of Instruments an ccurate Mechanism, over Publications, New York, NY. [8] Kamm, L. J., 99, esigning ost-ffective Mechanisms, McGraw-Hill. [9] owney, K, Parkinson,. R an hase, K. W.,, n introuction to smart assemblies for robust esign, ** Test runs were conucte on a P with. GHz Intel Pentium processor with G RM. opyright by SM

13 Research in ngineering esign, vol., no., pp [] Walron, K. J., 966, The constraint analysis of mechanisms, Journal of Mechanisms, vol., no., pp. -. [] ams, J.. an Whitney,..,, pplication of screw theory to constraint analysis of assemblies joine by features, SM Journal of Mechanical esign, vol., no., pp. 6-. [] saa, H. an y,., 985, Kinematic analysis of workpart fixturing for flexible assembly with automatically reconfigurable fixtures, I Journal of Robotics an utomation, vol. R-, no., pp [] Lee,. J. an Thornton,.., 996, "The ientification an use of key characteristics in the prouct evelopment process," Proceeings of the 996 SM esign ngineering Technical onferences, Irvine,, Paper no. 96-T/TM-56. [] Woo, L. an reuenstein,., 97, pplication of line geometry to theoretical kinematics an the kinematic analysis of mechanical systems, Journal of Mechanisms, vol. 5, pp [5] Hunt, K. H., 978, Kinematic Geometry of Mechanisms, Oxfor University Press. [6] Roth,., 98, Screws, motors, an wrenches that can not be bought in a harware store, Robotics Research, The first symposium, MIT Press, ambrige, M, pp [7] ouls, L. R., 99, Graph Theory pplications, Springer- Verlag, New York, NY. [8] Nilsson, N. J., 98, Principles of rtificial Intelligence, Tioga Publishing o., Palo lto,. [9] Homem e Mello, L. S. an Sanerson,.., 99, N/OR graph representation of assembly plans, I Transactions on Robotics an utomation, vol. 6, no., pp [] Homem e Mello, L. S. an Sanerson,.., 99, correct an complete algorithm for the generation of mechanical assembly sequences, I Transactions on Robotics an utomation, vol. 7, no., pp. 8-. opyright by SM