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Research article

An assembly sequence-planning system formechanical parts using neural network

Cem Sinanog luDepartment of Mechanical Engineering, Erciyes University, Kayseri, Turkey, and

H. Rza Bo rklu Department of Machine Education, Gazi University, Ankara, Turkey

AbstractPurpose In this paper, an assembly sequence planning system, based on binary vector representations, is developed. The neural network approahas been employed for analyzing optimum assembly sequence for assembly systems.Design/methodology/approach The input to the assembly system is the assemblys connection graph that represents parts and relations betweethese parts. The output to the system is the optimum assembly sequence. In the constitution of assemblys connection graph, a different approa

employing contact matrices and Boolean operators has been used. Moreover, the neural network approach is used in the determination of optimassembly sequence. The inputs to the networks are the collection of assembly sequence data. This data is used to train the network using the bpropagation (BP) algorithm.Findings The proposed neural network model outperforms the available assembly sequence-planning model in predicting the optimum assemsequence for mechanical parts. Due to the parallel structure and fast learning of neural network, this kind of algorithm will be utilized to model anotypes of assembly systems.Research limitations/implications In the proposed neural approach, the back propagation algorithm is used. Various training algorithms can bemployed.Practical implications The simulation results suggest that the neural predictor would be used as a predictor for possible practical applications modeling assembly sequence planning system.Originality/value This paper discusses a new modelling scheme known as articial neural networks. The neural network approach has beemployed for analyzing feasible assembly sequences and optimum assembly sequence for assembly systems.

Keywords Assembly, Modelling, Neural nets

Paper type Research paper

IntroductionIn order to develop a system that addresses computer aidedassembly sequence planning, these issues should be taken intoconsideration: the connectivity structure of parts and/orsubassemblies that are used in the assembly system, if theyhave connectivity properties, then what are the theoreticalnumber of their different connections, nally the choice of theoptimum assembly process among various alternatives.

In addition to these, another related research issue is thatparts existing in the assembly should have geometry foreasiness in assembling (Boothroyd, 1994). In this respect,

product design should directly inuence the assemblysequence planning that imposes the nal product.Moreover, the selection of a special assembly sequenceshould have inuences over assembly processes. Theseprocesses are the selection of assembly equipment, thedesign of tools and xtures, construction of subassemblies,and assembly time and cost (Laperriere and El Marghy,1994). Assembly sequence planning plays a very importantrole in the manufacture of many mechanical productsconsisting of non-standard parts. The research in this eldtries to get insight into the relations among assembly sequenceplanning, design and assembled products (Seow andDevanathon, 1993). The main purpose here is to obtain anumber of possible assembly sequence plans.

An exact assembly sequence plan consists of assembly,operations, existing assembly techniques and some details of relations between parts (Garrod and Everett, 1990). Someresearchers apply assembly sequence planning in reverseorder. Recent researches on assembly sequence planning haveestablished some important issues related to concurrentengineering analyses (Laperriere and El Marghy, 1994).These issues are the representation of a product to beassembled, the generation of assembly sequence plans and the

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determination of precedence constraints, the representation of resulting assembly sequence plans and the selection of theoptimum assembly sequence plan(s).

The early assembly planning systems were developed fromthe responses of users and they were based on humaninteractions (Homem de Mello and Sanderson, 1991). Thelater work concentrated on assembly sequence planning

systems based on geometric reasoning capability and fullautomatism (Homem de Mello and Lee, 1991). Approachesused for representation of assembly sequence planning can beclassied into four groups. These groups are:. Graphbased representation. Therequired data are entered by

the user or obtained from a CAD system. A detailedassemblyanalysiscanbe performedusing these approaches.Commonly used graph based representation methods areAND/OR and directed graph (Homem de Mello andArthur, 1990; Zhang, 1989), PETRI nets (Lee and Shin,1990; Zha, 2000; Moore et al. , 2001), hierarchical partialordered graph, relational model graph and assemblysequence graph (Homem de Mello and Arthur, 1990).

. Lingual representation. This involves the use of a speciallanguage for representation of subassemblies, parts andrelations among them. PADL, AUTOPASS and GDP areexamples of this group (Homem de Mello and Arthur,1990).

. An ordered list representation. These representations can beclassied as an ordered list of task representations, anordered list of binary vectors, an ordered list of partitionsof the set of parts and an ordered list of subsets of connections. In this representation method, each assemblysequence is represented by a set of list. This can be usedto correct and complete the denition of assemblysequences, although it is not necessary to obtain themost feasible and the most useful representation of theseassembly sequences (Homem de Mello and Lee, 1991).

. Art icial intell igence based represen tat ion. Articial

intel ligence based approaches are rule-based(Chakrabarty and Wolter, 1997), heuristic search (Leeand Shin, 1990), a neural network based (Hong and Cho,1995; Huang et al. , 2000) and genetic algorithm based(Marian et al. , 2003) etc.

Since many assembly sequences share common subsequences,attempts have been made to create more compactrepresentations that can encompass all assembly sequences.Therefore, the works in this eld are graph-based approachesthat represent all feasible assembly sequences.

The basic approach to nd the most suitable assemblysequences is to represent the assembly system in a spacewhere it is possible to explore all different assemblysequences. Thus, some criteria may be used to obtain thesesequences. Then optimum assembly sequence can be selectedby the application of some other criteria. These criteria mayinclude the number of tool changes, part replacement andclamping on the jigs, concurrency in operations, reliablesubassemblies, etc. (Kandi and Makino, 1996).

In this paper, an assembly sequence planning system, whichdetermines assembly sequences of any product, is explained.This planning system is used to graph based approach in therepresentation of product and assembly sequence plans. Theneural network structure is very suitable for this kind of problem. The network is capable of predicting the assemblyplans of the assembly system. The network has parallelstructure and fast learning capacity.

System descriptionAn assembly planning system is used to assemblys connectiongraph for the representation product. Parts and relationsamong these parts are represented by this graph. Contactrelations between parts are supplied by scanning module.Scanning module scans various assembly views of any productwhose assembly view is produced by Unigraphics in thecomputer environment and determines to the contact andintersection relations between parts (Sinanoglu, 2001;Sinanoglu and Borklu , 2004). These relations are formed asan matrix. The system constitutes assemblys connectiongraph of product to be assembled, by applying Booleanoperators on the elements of the contact matrices according tocertain rules. Moreover, intersection relations are alsoaccumulated in a form of intersection matrices fordetermination of the geometric feasibility of assembly stateslater.

In the assembly planning system, assembly states arerepresented by binary vectors. Therefore, all binary vectorrepresentations, whether corresponding to assembly states ornot, are produced by the system. By evaluating assemblys

connection graph and binary vector representationsimultaneously with the scanning module, vectorrepresentations corresponding to assembly states aredetermined.

Some of the assembly states cannot take part in a feasibleassembly sequence. The determination of the assembly statesnot corresponding to feasible assembly sequence is achievedwith the analysis module. The analysis module controls allassembly states according to stability, subassembly andgeometric feasibility constraints. Determination of thegeometric feasibility is done by Boolean operators on theelements of intersection matrices. The feasible assembly statesand assembly sequences are represented by a directed graph(Homem de Mello and Arthur, 1990). Assembly statessupplying constraints are settled down in the nodes of directed graph hierarchically by the system. In the hierarchicalpositioning, the number of the established connections in theassembly state is taken as a reference.

Any path from the root node to terminal node in thedirected graph corresponds to feasible assembly sequence.The optimum one is selected from the feasible assemblysequences with optimization module. This system candetermine the least suitable sequence among the feasiblesequences.

Moreover, the neural network approach has been employedfor analyzing feasible assembly sequences and optimumassembly sequence for sample product. Owing to parallellearning structure of the network, the proposed neuralnetwork has superior performance to analyze these systems.

Block diagram of assembly system is shown in Figure 1.

Artical neural networksThe assembly sequence problem can be viewed as a variant of the traveling salesman problem (TSP), which is to nd atraveling sequence with the shortest distance that visits all thecities of the problem only once. TSP belongs to a general classof optimization problems known as NP-complete. For thiskind of problem, it is necessary to search the entire solutionspace to guarantee that an optimal solution is reached;however, this search strategy is very time-consuming and, in

An assembly sequence-planning system for mechanical parts

Cem Sinanog lu and H. Rza Bo rklu

Assembly Automation

Volume 25 Number 1 2005 3852

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Figure 1 Block diagram of assembly planning system



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addition, not very practical for many applications. Therefore,other techniques have been developed and applied to nd anear-optimal solution quickly one of which is the neuralnetwork approach (Huang et al. , 2000).

An articial neural network (ANN) is an information-processing paradigm that is inspired by the way biologicalnervous systems, such as the brain, process information. The

key element of this paradigm is the novel structure of theinformation processing system. It is composed of a largenumber of highly interconnected processing elements(neurons) working in unison to solve specic problems(Sinanoglu, 2006a). ANNs, like people, learn by example.An ANN is congured for a specic application, such aspattern recognition or data classication, through a learningprocess. Learning in biological systems involves adjustmentsto the synaptic connections that exist between the neurons.This is true of ANNs as well.

Much is still unknown about how the brain, trains itself toprocess information, so theories abound. In the human brain,a typical neuron collects signals from others through a host of ne structures called dendrites. The neuron sends out spikesof electrical activity through a long, thin stand known as anaxon, which splits into thousands of branches. At the end of each branch, a structure called a synapse converts the activityfrom the axon into electrical effects that inhibit or exciteactivity from the axon into electrical effects that inhibit orexcite activity in the connected neurons. When a neuronreceives excitatory input that is sufciently large comparedwith its inhibitory input, it sends a spike of electrical activitydown its axon. Learning occurs by changing the effectivenessof the synapses so that the inuence of one neuron on anotherchanges (Figure 2(a)).

An articial neuron is a device with many inputs and oneoutput. The neuron has two modes of operation; the trainingmode and the using mode. In the training mode, the neuroncan be trained to re (or not), for particular input patterns.

In the using mode, when a taught input pattern is detectedat the input, its associated output becomes the current output.If the input pattern does not belong in the taught list of inputpatterns, the ring rule is used to determine whether to re ornot (Figure 2(b)).

Usually, the processing units have responses like that inFigure 2(b). Where xk are the output signals of other nodes(or external system inputs), wk are the weights of theconnecting links, and f () is a simple nonlinear function suchas the sigmoid, or logistic function.

The activation (transfer) functions (AF) are possible foreach hidden layer and the output layer. In this study, thelogistic function is used to hidden layers and output layers.Linear function is taken for input layer (Sinanoglu, 2006a).Logistic function is as follows:

y f x 1

1 e2 x 1

The linear function is;

y f x x 2

A network is trained so that application of set inputs producesthe desired set of outputs. Each such input (or output) set isreferred to as a vector. Training is accomplished bysequentially applying input vectors, while adjustingnetwork weights according to a predetermined procedure.

During training, the network weights gradually converge tovalues such that each input vector produces the desiredoutput vector. A training algorithm is used to train the neuralnetwork. The best choice is dependent on the problem, andusually trial-and-error is needed to determine the bestmethod. Back propagation algorithm is used trainingalgorithm for proposed neural networks. Back propagation

is a minimization process that starts from the output andbackwardly spreads the errors (Sinanoglu, 2006b). Trainingand structural parameters of the network are given in Table I.

Figure 3 shows the currently loaded network. Theconnections can represent the current weight values for eachweight. Squares represent input nodes; circles depict theneurons, the rightmost being the output layer. Trianglesrepresent thebias for each neuron. The neural network consistsof three layers, which are input, output and hidden layers. Theinput and outputs data are used as learning and testing data.

Representation of a productIn order to model the assembly system, the assemblys

connection graph whose nodes represent assembling partsand whose edges represent connections between parts is used.Some researchers used assemblys connection graph in orderto model assembly systems. But, these researchers assumethat whenever two parts are assembled, all the contacts inbetween are established. In the assembly planning systemdeveloped, this assumption has been rejected by forming theassemblys connection graph in a different way.

For the current analyses, it is assumed that exactly twosubassemblies are joined at each assembly task, and after thoseparts have been put together,they remaintogether until theendof theassembly process. Owingto this assumption, an assemblycan be represented by a simple undirected graph kP ; C l ; inwhich P { p1 ; p2 ; . . . ; p N } is the set of nodes, and C {c1 ; c2 ; . . . ; cL } is the set of edges. In order to explain thedeveloped approach for modeling assembly systems used forthis research, we will take a sample assembly shownin Figure4.

The sample assembly is a coupling consisting of sevencomponents. These are: Flange-I, Flange-II, Shaft-I, Nut,Shaft-II, Washer and Bolt. These parts are represented,respectively, by the symbols of { a }, {b}, {c}, {d }, {e}, { f } and{ g } sets. For this particular situation, the connection graph of the assembly has the set of the nodes as P {a ; b; c; d ; e; f ; g }and the set of the connections as C {c1 ; c2 ; c4 ; c5 ; c6 ; c7 ; c8 ; c9 } :

The contact matrices are used to determine whether thereare connections between parts in the assembly state. Thesematrices are represented by a contact condition between apair of parts as an { A ; B} : The elements of these matricesconsist of Boolean values of True (1) or False (0). Forthe construction of contact matrices, the rst part is taken as areference part. Then whether this part has a contact relationin any i axis directions with other parts is examined. Thatrelation, if any, is dened as True (1); otherwise that isdened as False (0). The row and column element values of contact matrices in the denition of six main co-ordinate axisdirections are contact relations between the parts thatconstitute the coupling assembly. To determine theserelations, the assemblys parts are assigned to rows andcolumns of the contact matrices. Contact matrices are squarematrices since they have the properly of adjacency matrices,i.e. square matrices. Coupling assembly consists of sevenparts so its contact matrices should have the dimension of



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7 7 : Some example contact matrices of the couplingassembly system can be determined as follows:

The double side contact relation concept has been suggestedfor the construction of the assemblys connection graph. Forinstance, in the manner of { a ,b} ordered pair of parts, it issufcient to determine contacts related in the ordered

direction to indicate contact in any direction. For thisreason, a V: Or operator is applied to these parts. It is alsonecessary to have contact in any direction for inverselyordered pairs of part in the assemblys connection graph.If these values are 1 for every ordered pair of parts, thenthere should be edges between corresponding nodes of theassemblys connection graph. Therefore, a V: Or operator isapplied to this ordered pair of parts. Table II shows contactrelations regarding { a , b} and { b, a } pairs of parts.

In this table, { a , b} ordered pair of parts meets to at least onecontact condition in therelateddirection of 0 _ 1 _ 1 _ 1 _ 1 _1 1 : {b, a } pair of partsalso meets to at least one contact inthe related direction of 1 _ 1 _ 1 _ 0 _ 1 _ 1 1 : An ^ :And operator is applied to obtain these values. Therefore, afterapplying ^ : And logical operator, a 1 value is obtained.Therefore, the connection between these parts is represented asan edge in the graph of connections.

Figure 5 shows the coupling graph of connections that hasseven nodes and nine edges (connections). For example, thereare not any contacts between the Flange-I and Shaft-II, nut,washer. Therefore, for instance, the graph of connectionsdoes not include an edge connecting the nodes correspondingto the Flange-I and the nut. This assemblys connection graph

Table I Structural and training parameters of the neural predictor

Neural network h m n 1 n H n o N AF

0.1 0 1 10 9 500000 Logistic

Figure 2 (a) Schematic view of a real neuron; and (b) schematic representation of the articial neural network



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determines an assembly model of the coupling system. It canbe used as an input by an assembly sequence planning system.

Generation and representation of assemblysequencesIn the developed approach an L-dimensional binary vectorcan represent a state of the process x {x1 ; x2 ; . . . ; xL }:

The elements of these vectors dene the connections betweenassembly components. Assembly sequences are obtained withenumeration of the vectors, which correspond to assemblystates (Zhang, 1989).

Based upon the establishment of the connections, theelements of these vectors may have the values of either 1 or0 at any particular state of assembly task. For example,the i th component xi would have a value of True T : 1

Figure 3 Currently loaded network

Figure 4 The coupling assembly system



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if the i th connection is established in that state. Otherwise, itwould have a value of False F : 0 . Not all binary vectorrepresentations correspond to an assembly state. In order todetermine assembly states, the established connections inbinary vectors and the assemblys connection graph areevaluated together.

There are nine edges in the example assembly connection

graph. Consequently, there are nine elements of vector andnine-dimensional binary vector can represent thatc1 c2 c3 c4 c5 c6 c7 c8 c9 : For instance, the binary vector canrepresent the initial state of the assembly process for theproduct shown in Figure 2 [ FFFFFFFFF ] whereas, the nalstate can be represented by [ TTTTTTTTT ].

In the assembly sequence planning system developed, rstof all binary vector representations must be produced.Whether these binary vector representations constituteassembly states or not is determined by using assemblysconnection graph. In the determination of the assemblystates, a new approach employing Boolean operators has beendeveloped.

In order to determine whether the vector is a state or not,established connections in vector representation must betaken into consideration. Then it is required that connectionsto be established be determined through the establishedconnections in the assemblys connection graph.

For example, some of the vectors do not correspond to anassembly state. For instance, in the [111100000] vector,connections c1 between { a } and { b}, c2 between { a } and { c},c3 between { d } and { f }, c4 between { a } and { g } have beenestablished 1 : It is necessary to establish c7 connectionbetween { b} and { g } so that these connections can beestablished to c1 and c4 . But this connection has not beenestablished in [111100000] (0). Therefore, [111100000]vector is not an assembly state (Figure 6).

After the determination of the assembly states, thedetermination of the feasibility of these assembly states for

any assembly sequence is required. Assembly states notcorresponding to feasible assembly sequences must beeliminated by some assembly constraints. These constraintsare as follows:

Subassembly constraintThis constraint denes the feasibility of a subassembly of a setof partitions to established connections in assembly states. Inorder to form a subassembly set of partitions, it is not a set of partition that contains a pair of part not in a contact relationin the assemblys connection graph. For instance, there is noconnection between the Shaft-I ( c) and Shaft-II ( e) for thecoupling system. Because of that it does not meet thesubassembly constraint set partitions containing { c,e} set of

partition.

Stability constraintA subassembly is said to be stable, if assembling partsmaintains relative position and they do not break contactduring the assembly operation. It is assumed that whenever anassembly state meets a subassembly constraint, it also meetsstability condition.

Geometric feasibility constraintIn the developed assembly sequence planning system, thegeometric feasibility of the assembly states is determined withthe intersection matrices and applying Boolean operators onthese matrices. The elements of intersection matrices takeinto consideration interference conditions during the joiningparts. In the determination of geometric feasibility, ^ and _ logical operators are applied to elements of intersectionmatrices. This operation must utilize established connections,that is, joining pairs of parts. In describing intersection matrixelements, interference is taken into consideration, whichoccurs while the reference part is moving together withanother part in the related axis direction. If there isinterference during this transformation motion, intersectionmatrix elements are 0 if not, they are dened as 1.Interference relations are located in rows and columns of theintersection matrices. Therefore, intersection matrices areformed. Some of these matrices are as follows:

Figure 5 The graph of connections for seven-part coupling assembly

Table II Contact relations of {a , b } and {b , a } pairs of part

c 1 ) (a 4 b ) 1 x 1 y 1 z 2 x 2 y 2 z _ : OR ) ^ :AND +

a /b 0 1 1 1 1 1 1 1b /a 1 1 1 0 1 1 1 1

1

Figure 6 c 1 , c 4 and c 7 connections in [111100000] vector



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A x

1 1 0 1 1 1 0

0 1 0 1 1 1 0

1 1 1 1 1 1 1

0 0 1 1 1 0 0

0 0 0 1 1 1 10 0 1 1 1 1 0

1 1 1 1 1 1 1

26666666666666664

37777777777777775

A y

1 0 0 1 1 1 0

0 1 0 0 0 0 0

0 0 1 1 1 1 1

1 0 1 1 0 1 0

1 0 1 1 1 1 1

1 0 1 1 0 1 0

0 0 0 0 0 0 1

26666666666666664

37777777777777775

A2 x

1 0 1 0 0 0 1

1 1 1 0 0 0 1

0 0 1 1 0 1 1

1 1 1 1 1 1 1

1 1 1 1 1 1 1

1 1 1 0 1 1 1

0 0 1 0 1 0 1

26666666666666664

37777777777777775In order to determine whether assembly states with only one

established connection are geometrically feasible or not, it isnecessary to apply Cartesian products between ordered pairsof parts which represent established connections and partswhich are not in this ordered pair of parts. In this situation,different intersection tables are obtained and these tables areused to check geometric feasibility. For instance, thegeometric feasibility of [000010001] is determined as follows:

In this assembly state, connection of c5 between parts{b}and { e}, c9 between parts { d } and { g } have beenestablished. To determine geometric feasibility of thisassembly state, parts without established conditions arechosen, e.g. are parts { a , c, f }. The geometric feasibility of [000010001] can be determined to applying the Cartesianproduct between the ( b,e) and ( d , g ) pairs. The Cartesianproduct between ( b, e) and ( d , g ) is given as follows; b; e d ; g : b; d b; g e; d e; g :

Table III shows the interference of b; ed ; g : b; d b; g e; d e; g ordered pairs of parts.

The result of ^ and _ logical operators is 0. Thisresult explains that the [000010001] assembly state isgeometrically unfeasible. Similar operations must be appliedto other assembly states of the coupling system. Therefore,feasible assembly sequences of coupling system can bedetermined. But, if any assembly state is geometricallyunfeasible in any assembly sequence, this sequence will begeometrically unfeasible. Three hundred and seventy threeassembly sequences for the seven-part coupling system have

been determined. The feasible assembly sequences arerepresented by directed graph.

In the coupling assembly, P {a ; b; c; d ; e; f ; g ; } is theassemblys set of par ts or set of nodes, C {c1 ; c2 ; c3 ; c4 ; c5 ; c6 ; c7 ; c8 ; c9 } is the assemblys set of connections or set of edges. kx p ; T pl corresponds to directedgraph of coupling system. A path in the directed graph of feasible assembly sequences kx p ; T pl whose initial node is u I {{a } ; {b} ; {c} ; {d } ; {e} ; { f } ; { g }} and whose terminal node isu F {{a ; b; c; d ; e; f ; g }} corresponds to feasible assembly

sequence with vector representations being [000000000]and [111111111], respectively.

Optimization of assembly sequencesDeveloped assembly planning system can be determined tond the optimum assembly sequence. In this section, anoptimization approach is explained for the developedassembly sequence planning system. For this purpose, thecoupling assembly system is taken as an example. It has beenobtained from feasible assembly sequences in previoussections. In order to optimize the assembly sequence, twocriteria are developed, weight and the subassemblys degree of freedom. First certain costs are assigned to edges of directedgraph depend on these criteria, and then the total cost of eachpath from root node to terminal is calculated, the minimumcost sequence is selected as an optimum one. Thisoptimization also enables determination of the least suitablesequences from the feasible sequences.

Optimization of weight criterionIn order to determine the optimum assembly sequence, allassembly states in an assembly sequence must be taken intoconsideration. The heaviest and bulkiest part is selected as abase part and then the assembly sequence continues fromheavy to light parts. The parts with the least volume, i.e.connective parts, like bolts and nuts must be assembled last(Bunday, 1984). The weights and volumes of parts werecalculated automatically with a CAD program (Unigraphics).Therefore, determination of the costs of assembly states isnecessary to obtain an optimum feasible assembly sequence.After that these costs are used as a reference to differentassembly states. Calculated weight costs of assembly states inthe assembly sequence are compared with reference weights.The difference of weight is multiplied by unit weight value(100). The weights of parts of the coupling system are asfollows:

a : 0 : 3945 kg b : 0 : 345kg c : 0 : 25kg d : 0 : 25kg

e : 0 : 0165 kg f : 0 : 0093 kg g : 0 : 0042 kg

Table III The geometric feasibility of [000010001]

c 5 ) (b 4 e ), c 9 ) (d 4 g ) 1 x 1 y 1 z 2 x 2 y 2 z

b /d 1 0 0 0 0 0b /g 0 0 0 1 0 0e /d 1 1 1 1 0 0e /g 1 1 1 1 0 0

(b /d ^ b /g ^ e /d ^ e /g ) + 0 0 0 0 0 0 _ )0



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Using the weight criterion, the total established connectionweights of each assembly state in optimum assembly sequencecould be determined.

The total weight of assembly states in the optimumsequence according to weight criterion can be dened as

Ow m

Xn

i 1W required weight

where W is the weight of assembly states, ( n) is the number of the established connections and ( m ) is the order of assemblystates. Ow m is the required weight of assembly states in theoptimum assembly sequence. Each connection weight for thecoupling system is as follows:

Connection c1 between parts { a } and { b} has beenestablished. The number of established connections is n 1.Some of the weight of established connection is:

Ow 1 Xn1

i 1W W c1 0 : 7395 kg

Ow 3 Xn3

i 1W 1 : 7455 kg

Ow 6 Xn9

i 1W 3 : 3747 kg

Ow 4 Xn5

i 1W 2 : 4934 kg

In order to determine the optimum assembly sequence, theweights of all assembly states in assembly sequences arecalculated. This weight is expressed as

Cw m

X

n

i 1W calculated weight

where W is the weight of assembly states, ( n) is the number of the established connections and ( m ) is the order of assemblystates. After that, (Ow m ) is used as a reference for differentassembly states. Calculated weights of assembly states inassembly sequences are compared with reference weights.This weight difference (Dw) is multiplied by unit weight valueUwv 100 : The result is the weight costs of assembly states(Wc).

For example, in the assembly state of [100000000] thecalculation of weight costs of two different assembly states inthe feasible assembly sequence are given as follows: it hasbeen used to establish c1 connection between parts { a } and{b}. The number of the established connections is n 1 : Theweight of the established connection (calculated weight) is:

Cw 1 Xn1

i 1W W c1 0 : 7395 kg

.In the rst assembly state, the necessary weight is Ow 1 0 : 7395 kg : Therefore, The difference of weight Dw is asfollows:

Dw Ow 1 2 Cw 1 0 : 7395 2 0 : 7395 0 kg

If Dw is multiplied by the unit weight value Uwv 100 ; theweight cost of [100000000] will be calculated as follows:Wc DwUwv 0 : 100 0

As a result, the weight cost of [100000000] is 0. Table IVshows weight costs of some examples of feasible assemblysequences for the coupling system.

As seen in Table IV, the optimum feasible assemblysequence is,

000000000 ; 010000000 ; 010010000 ; 010110000 ; 111111111

The total weight cost is 61.Some of the sample feasible assembly sequences and the

changes of the weight costs related with these assemblysequences are shown in Figures 7 and 8, respectively. Fromthe gures, the proposed neural network exactly follows thedesired results (feasible assembly sequences and optimumassembly sequence) of the assembly system.

The error convergence graphof thecase1 isshown inFigure 9during the training of the network. As can be seen from thegures, the error is suddenly reducing to small values. Smallepochs can be employed for case 1 (51200 epoch).

Optimization of subassembly degree of freedomcriterionThe subassembly degree of freedom criterion is based on theselection of parts with low degrees of freedom. So degree of freedom between the subassembly parts is low, the assemblyof these parts can be done more easily. It is a unit cost (unitdegree of freedom value, Udofv) also used for this criterion. Itis 25 and this criterion is more important than the other.Therefore, it is selected as the lower unit cost according toweight criterion, and so that total cost of assembly sequencescan be reduced.

It determines degree of interference for pairs of partsconnections established along the six main directions of theCartesian coordinate system. The total degree of freedom(Tdof) for pairs of parts is the products unit cost.Therefore, in the directed graph costs of degree of

freedom according to this criterion are calculated as thedegree of freedom for each path from initial node toterminal node. As a result, the minimum cost of theassembly sequence can be selected as an optimum withrespect to the degree of freedom criterion. For instance, thecalculation of the degree of freedom costs of two differentassembly states in the feasible assembly sequence for thecoupling system is given as follows:

Assembly state of [100000000]In this assembly state, connection c1 between the parts { a }and { b} has been established. The degree of freedom of thispair of parts is shown in Table V.

The Tdof for [100000000] is Tdof 2 : If this value ismultiplied by the Udofv 25 unit freedom cost, the resultwill be 50. Therefore, the degree of freedom cost DOFc for[100000000] assembly state is 50. Table VI shows thedegree of freedom costs for some example feasible assemblysequences for the coupling system.

[000000000] and [111111111] are the same for allassembly sequences. Therefore, these states can beneglected when calculating the total cost of feasibleassembly sequences. Figure 10 shows some of the feasibleassembly sequences in Table VI. The results in the ANNs andtest data targets are matched.

The optimum weight cost is 61 for the coupling system.The optimum assembly sequence has a cost of 300



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according to the subassembly degree of freedom criterion.These assembly sequences are as follows:

000000000 ; 010000000 ; 010010000 ; 010110000 ; 111111111

000000000 ; 010000000 ; 010010000 ; 010010100 ; 111111111

000000000 ; 001000000 ; 000110000 ; 010110000 ; 111111111

000000000 ; 000010000 ; 010010000 ; 010110000 ; 111111111

000000000 ; 000010000 ; 010010000 ; 010010100 ; 111111111

000000000 ; 000010000 ; 010010000 ; 010010010 ; 111111111

000000000 ; 000010000 ; 000110000 ; 010110000 ; 111111111

000000000 ; 000010000 ; 000010100 ; 010010100 ; 111111111

000000000 ; 000000100 ; 000010100 ; 010010100 ; 111111111

000000000 ; 000000010 ; 010000010 ; 010010010 ; 111111111

000000000 ; 000000010 ; 010000010 ; 010000011 ; 111111111

In the optimization approach, both optimization criteria areminimal and so that assembly sequence is optimum.

The optimum assembly sequence for coupling system isWc 61 ; DOFc 300

000000000 ; 010000000 ; 010010000 ; 010110000 ; 111111111

Set representation for this sequence is as follows:

000000000 ) {{a } ; {b} ; {c} ; {d } ; {e} ; { f }{ g }}010000000 ) {{a ; c} ; {b} ; {d } ; {e} ; { f } ; { g }}

010010000 110010000 ) {{a ; c} ; {b; e} ; {d } ; { f }}

{{a ; b; c; e} ; {d } ; { f } ; { g }}

010110000 110110100 ) {{a ; b; c; e; g } ; {d } ; { f }}

111111111 {{a ; b; c; d ; e; f ; g }}

In thisrepresentation, [010010000] is the same as [110010000].Because, in the [010010000], c2 {a ; c} and c5 {b; e}

Table IV The Wc of some example feasible sequences for coupling system (ASs-assembly states)

NoASs there is no established

connectionsASs establishedone connection Wc

ASs establishedtwo connections Wc

ASs establishedthree connections Wc

ASs establishedall connections

TotalWc costs

1 000000000 010000000 12 011000000 75 011100000 96 111111111 2972 000000000 010000000 12 011000000 75 011010000 75 111111111 1623 000000000 010000000 12 010010000 14 011010000 75 111111111 101

4 000000000 010000000 12 010010000 14 010110000 35 111111111 615 000000000 001000000 75 011000000 75 011100000 96 111111111 2466 000000000 001000000 75 011000000 75 011010000 75 111111111 2257 000000000 000100000 35 001100000 98 011010000 75 111111111 2088 000000000 000100000 35 001100000 98 011000100 98 111111111 2319 000000000 000010000 14 010010000 14 011010000 75 111111111 63

10 000000000 000010000 14 010010000 14 010010100 37 111111111 65

Figure 7 (Case 1) The sample feasible assembly sequences for the coupling assembly system according to weight criterion



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connections have been established. So, a cartesian productshould be applied to elements of sets to determine a ; cb; e )a ; b; a ; e; c; b; c; e: {a ,e}, { c, b} and { c, e} sets notcorresponding to a connection (Figure 5). But, the set of { a ,b}

corresponds to c1 connection. Therefore, both vectors have thesame meaning as far as sets are concerned. To determineoptimum assembly sequence, [010010000] vector is considered.Because, the weight cost of this vector is lower than the other.

Figure 8 (Case 2) W c weight cost variations of feasible assembly sequences shown in Figure 6

Figure 9 The error convergence graph of Case 1



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In this optimum assembly sequence for the couplingassembly, the Flange-I is joined to the shaft-I in the rstassembly state; the Flange-II and the shaft-II are added insecond assembly state. In the third assembly state, the bolt isused to x this subassembly, and then the washer and nut isjoined in the last subassembly.

The total cost of feasible assembly sequence for any productis expressed as a cost function fc;

Wc Ow 1 Ow 2 . . . Ow k

2 Cw 1 Cw 2 . . . Cw k Uwv

X

l

i 1Ow k !2

X

l

i 1Cw k !" #Uwv

3

DOFc Tdof Udofv 4

fc Wc DOFc 5

where ( l ) is the number of assembly states in the feasibleassembly sequence for any product.

Table V Degree of freedom between parts {a } and {b }

c 1 ) (a /b ) 1 x 1 y 1 z 2 x 2 y 2 z

a /b 1 0 0 0 0 0b /a 0 0 0 1 0 0

Figure 10 The sample feasible assembly sequences for the coupling assembly system according to subassembly degree of freedom criterion

Table VI The DOFc of some example feasible sequences for coupling system

No

ASs there is no

established connections

ASs established

one connection DOF c

ASs established

two connections DOF c

ASs established

three connections DOF c

ASs established

all connections total DOF c costs1 000000000 010000000 50 011000000 300 011100000 350 111111111 7002 000000000 010000000 50 011000000 300 011010000 350 111111111 7003 000000000 010000000 50 011000000 300 011000100 350 111111111 7004 000000000 010000000 50 010010000 100 011010000 350 111111111 5005 000000000 010000000 50 010010000 100 010110000 150 111111111 3006 000000000 010000000 50 010010000 100 010010100 150 111111111 3007 000000000 010000000 50 010010000 100 010010010 150 111111111 3008 000000000 010000000 50 010000010 100 010000011 150 111111111 3009 000000000 001000000 250 011000000 300 011010000 350 111111111 900

10 000000000 000100000 50 000110000 100 001110000 350 111111111 500



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Some assembly case studiesTwo case studies, which are pincer and hinge system consistingof four parts, have been investigated. Figure 11 shows theseassemblys exploded views and connection graphs.

The weights of parts of pincer assembly system are: left-handle; 0.3843 kg, right-handle; 0.3843 kg, bolt; 0.0163 kg,nut; 0.0092 kg. Figure 12 shows feasible assembly sequencesand costs of them for the pincer assembly system. The rstand third assembly sequences for the pincer system accordingto the subassembly degree of freedom criterion have beenselected with an optimum total cost of 300. The weightcosts are in parentheses ( ) and the degree of freedom costsare in quotation marks .

Figure 12 shows that the third assembly sequence isoptimum 0 weight cost and 300 degree of freedom cost.Moreover, the sixth assembly sequence is the least preferablesequence in the feasible assembly sequences. In theoptimization approach, both optimization criteria indicatedthat assembly sequence is optimum. Therefore, the optimumassembly sequence for pincer system is:

00000 ; 00001 ; 10011 ; 11111

First, the left handle is joined to right handle with connectionof c5 . After that this subassembly is joined by using the bolt

with the connections of c1 and c4 . Finally, the nut xes all theparts.

The weights of parts of hinge assembly system are: Handle;4.9253kg, Plate; 2.8192kg, Bolt; 0.0638kg and Nut;0.0158kg. The assembly sequences of the hinge systemcontain four different assembly states. The rst one is [00000]and the last is [11111]. These assembly states are the same for

all sequences. Therefore, these states can be neglected tocalculate total cost of assembly sequences. The weight costsand degree of freedom costs for hinge system are given inFigure 13.

Figure 13 shows that the second assembly sequence isoptimum 0 weight cost and 300 degree of freedom cost.Moreover, the third assembly sequence is the least preferredsequence in the feasible assembly sequences. In optimizationapproach, both optimization criterion are supplied and so thatan assembly sequence is optimum.

The optimum assembly sequence for the hinge system is:

00000 ; 10000 ; 10011 ; 11111

In this assembly sequence, the plate and handle are connectedwith the connection of c1 . Then using bolt this subassembly isxed with the connections of c4 , c5 . And the assembly processis completed with the addition of nut.

Figure 11 The pincer and hinge systems and their assemblys connection graphs



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Figure 12 The weight and degree of freedom costs for pincer system

Figure 13 The weight and degree of freedom costs for hinge system



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ConclusionsIn this paper, an assembly sequence planning system, basedon binary vector representations, is explained. The input tothe system is the assemblys connection graph that representsparts and relations between these parts. The output to thesystem is the optimum assembly sequence. In the constitutionof assemblys connection graph, a different approachemploying contact matrices and Boolean operators has beenused. In this work, sample assembly systems are examined.

ANNs application was presented for analyzing assemblysequences on assembly system. As can be depicted from theresults, the neural predictor would be used as a predictor forpossible assembly system applications. Finally, due to the parallelstructureandfast learning of neuralnetwork, thiskindof algorithmwill be utilized to model another types of assembly systems.

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