On the automatic generation of product assembly sequences

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On the automaticgeneration of productassembly sequencesC. K. Choi , X.F. Zha , T.L. Ng & W.S. LauPublished online: 14 Nov 2010.

To cite this article: C. K. Choi , X.F. Zha , T.L. Ng & W.S. Lau (1998) On theautomatic generation of product assembly sequences, International Journal ofProduction Research, 36:3, 617-633, DOI: 10.1080/002075498193606

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int. j. prod. res., 1998, vol. 36, no. 3, 617± 633

On the automatic generation of product assembly sequences

C. K. CHOI ² *, X. F. ZHA ³ , T. L. NG§ and W. S. LAU¶

Automatic assembly planning is recognized as an important tool for achievingconcurrent product and process development thus reducing manufacturing costs.It plays an important role in designing and planning the assembly systems. Inorder to recommend a good sequence of assembly operations, the process plannerneeds to generate all such feasible sequences. In this paper, the representation ofthe product assembly relationships and the assembly processes is described.Considering the assembly knowledge and the assembly constraints, a methodol-ogy is presented to generate all feasible assembly sequences by determining anddecomposing the levelled feasible subassemblies. A prototype reasoning expertsystem is developed to generate all feasible assembly sequences automatically. Theresearch ® ndings are exempli® ed with a simple assembly to illustrate the method.

1. Introduction

The choice of the sequence in which components or subassemblies are puttogether in the mechanical assembly of a product can drastically a� ect the e� ciencyof the assembly process. For example, one feasible and reasonable sequence mayrequire less ® xturing, less changing of tools, and include simpler and more reliableoperations than others. Therefore, assembly sequence planning plays an importantrole in designing and planning the product assembly process.

Traditionally, the product assembly sequence was planned by an experiencedproduction engineer. However, the planning of assembly sequences is sometimes atrivial and error-prone task especially when there exists a large number of potentialassembly sequences in a complex assembly. Therefore, there is a growing need tosystematize and to computerize the generation of assembly sequences. So far, manyresearch activities have focused on various aspects of assembly sequence planningsuch as assembly modelling, assembly sequence representation and assemblysequence generation algorithms.

In this paper, focuses are put on the approach and the system for automaticgeneration of assembly sequences, which are the key issues of computer aided as-sembly process planning (CAAPP). A new representation of the product assemblyrelationships and a methodology for generating the detailed feasible subassemblysets of assembly plans for a given product will be described. Based on the assemblyknowledge or information and the method of determination and decomposition offeasible subassembly subsets, an automatic knowledge reasoning system has been

0020± 7543/98 $12.00 Ñ 1998 Taylor & Francis Ltd.

Revision received November 1996.² Department of Manufacturing Engineering, The Hong Kong Polytechnic University,

Hung Hom, Hong Kong, PR China.³ School of Mechanical and Production Engineering, Nanyang Technological University,

Singapore.§ Sonca Products Ltd., Hong Kong, PR China.¶ Hong Kong Technical College (Chai Wan), Hong Kong, PR China.* To whom correspondence should be addressed.

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developed to generate the assembly sequences. The proposed approach is a combi-nation of the assembly’ approach and the `disassembly’ approach.

This paper is structured as follows: §§ 1 and 2 present the brief introduction andliterature review about the assembly sequence generation and planning respectively.Section 3 deals with the modelling of assembly process that contains the representa-tion of assembly relationships (liaison graph, liaison matrix, knowledge assemblyliaison graph), predicate-based representation of assembly constraints, graph-basedrepresentation of the assembly or disassembly sequences. Sections 4 and 5 present anew approach to assembly sequence generation and the knowledge based automaticreasoning system; § 6 gives an example to illustrate the proposed approach; § 7 sum-marizes and gives some suggestions for the future research work.

2. Overview of previous work

2.1. Assembly sequence generationMany researchers have made attempts to generate the assembly sequences of a

product. Bourjault (1984) proposed a procedure to obtain all the precedence knowl-edge about the liaisons of an assembly by answering a set of structured questionsbased on this proposed liaison model of the assembly. Defazio and Whitney (1987)simpli® ed Bourjault’s procedure and reduced the number of the asked questions to2n against 2n of Bourjault’ s. These two methods study the assembly from the point ofview of assembling the product. For the representation of precedence knowledge ofan assembly, there have been several methodologies widely used in the past such asthe set theory, binary matrix, directed graph, establishment condition, precedencerelationships and function diagrams (Thaler 1989a, b, Homen de Mello andSanderson 1990). However, these methods can only represent the partial assemblyprecedence knowledge. To represent the set of assembly sequences completely,Homen De Mello and Sanderson (1990, 1991a, b) proposed an AND/OR graphrepresentation of all the possible con® gurations of the assembly and generate theassembly sequences of a product using a disassembly or decomposition method,based on the assumption that the disassembly sequence is the reverse of a feasibleassembly sequence. Although this representation is complete, the resultant AND/ORgraph is huge and the number of nodes grows exponentially as the part number inthe assembly increases (Ben-Arieh and Kramer 1994). Based on a mathematicalmodel of a product obtained through the de® nition of three matrices: the interfer-ence matrix, the contact matrix and the connection matrix, Dini and Santochi (1992)proposed a procedure for the detection and selection of the subassemblies and theassembly sequence of a product, which can be applied in a ¯ exible assembly system.One of the main limitations of this model is that only partial disassemblies orassemblies are considered. In the component ordering method proposed by Leeand Ko (1987) a simple sequence was generated and interference checking ensuresthat the components to be assembled do not collide during assembly. A di� erentmethod proposed by Lin and Chang (1991) uses a three layer strategy and a specialtree structure to represent the assembly and to generate one feasible sequence. Inorder to obtain the detailed assembly plans, Ben-Arieh and Kramer (1994) presenteda methodology and algorithms to consistently generate all feasible assemblysequences with consideration for the various combinations of subassembly opera-tions. The algorithms are implemented using a LISP program. In addition, BenArieh (1994) applied a fuzzy set based method to evaluate the degree of di� cultyof each assembly operation and then select the `best’ sequence of assembly opera-

618 C. K. Choi et al.

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tions. These methods and algorithms are less interactive and need much more spaceto store the representation of assembly sequences and processing time to process theassembly operations for the complex assemblies. Therefore, it is di� cult to generatethe detailed assembly plans automatically and to deal with the coordination andfeasibility of various subassemblies e� ciently.

2.2. Integrated assembly planningApart from the above mentioned, another important development o� ers an inte-

grated approach to assembly planning by linking aspects of product design, designfor assembly (DFA), assembly planning and production layout within a manufactur-ing enterprise in one procedure. All of them require some interactive input, whichcurrently seems to be the only way of generating practical results. However, the aimsof these few systems are quite di� erent. One of these systems is the standard as-sembly analysis tool (STAAT). Incorporated into a larger CAD tool, this system cangenerate and evaluate the geometric assembly sequences of complex products from20 to 40 parts and 500 to 1500 faces, and thus provides immediate feedback to a teamof product designers about the complexity of assembling the product being designed.The researchers of STAAT are working now on extending both these results and theunderlying theory to more sophisticated cases (Romney et al. 1995). However, themost comprehensive one is the project of integrated design and assembly planning(IDAP) (Seidel and Bullinger 1991, Seidel and Swift 1989, Richter 1991), which is avery long project with large scale interaction and is still under development. In thissystem, a model of OPNET (operation network) is constructed to re¯ ect the con-straints intrinsic in the product itself. An OPNET model is structured in the form ofa graph that represents the relationships among tasks and subtasks for DFA evalua-tion and assembly process planning. An interactive operation network editor hasbeen developed to integrate the procedures for network generation, modi® cation,queries and evaluation under a uniform graphical man-machine interface (Seidel1989, Seidel and Swift 1989).

3. Assembly process modelling

3.1. Notations and assumptionsNotations:

� Components of an assembly are represented as p1,p2, . . . ,pn .

� A subassembly is represented by a list of components such as [p1,p2,p3,p4].� A subassembly operation is represented by a combination of two sub-

assemblies such as [p1],[p2,p3,p4]], [[p1,p2],[p3,p4]], etc. The list should beread from left to right, i.e. the subassembly operation direction or sequence isfrom left to right, for example, [p1]® [p2,p3,p4]and [p1,p2]® [p3,p4].

� An assembly sequence is represented implicitly by nested lists of components.For example [[[[p1,p2],[p3]],[p4]],[p5,p6]]represents the following sequence ofassembly operations: p1 ® p2 ® p3 ® p4 ® p6.

Assumptions:

� For a disassembleable product, the assembly sequence is the reverse of thediassembly sequence if there is no destructive operation in the disassembly.

Automatic generation of product assembly sequences 619

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� Each component is a solid rigid object, that is, there is no changing of shapeduring assembly and disassembly. Components are interconnected wheneverthey have one or more compatible surface in contact.

� Assembly is carried out step by step and components or subassemblies areassembled one at a time.

� Once an assembly operation is completed the subassembly would remainunchanged at all assembly stages.

3.2. Representation of assembly relationships3.2.1. L iaison graph and its matrix representation

The liaison relationships between two components can be implemented by thetotal relative constraints which can be extracted from the CAD drawing and classi-® ed as ® t contact, plane contact and meshing contact. The assembly liaison relation-ships of a product can be described by the liaison graph (LG), which is de® ned as

Lg = G(V,E)

where V is the set of vertices and E is the set of connecting arcs. Each vertexrepresents a component being assembled and each arc represents the liaison betweentwo components (De Fazio and Whitney 1987, Lee 1989, Shin and Ho 1994). Theassembly liaison relationships of a product can also be represented by liaison matrixR (Dini and Santochi 1992), which is de® ned as:

R = {rij} (1)

where,

rij =1 with liaison relationship between pi and pj;0 without liaison relationship between pi and pj{

The row of matrix R represents the liaison relationships between a componentand other components. The column of matrix R represents the components con-nected by liaison relationships. The sub-matrices of R represent the local liaisonrelationships in subassemblies.

3.2.2. Knowledge assembly liaison group (KAL G)The knowledge assembly liaison group (KALG) is an enhancement of the as-

sembly liaison graph. It includes much more assembly information and constraints.In constructing the KALG, the following rules would be used:

(1) Each component assembled (and not to be disassembled anymore) is a vertexof KALG.

(2) Connecting components with the same type and functions should be regardedas one vertex.

(3) If components pi and pj are adjacent to each other, there exists an arc con-necting pi and pj. If pi is in contact with pj, the arc is expressed by a solid line.If constraints exist between pi and pj, the arc is expressed by a dotted line.

(4) The direction of an arc connecting two components or vertices (such as pi

and pj) would be determined according to the following:

� If pi is a connecting component, the direction is from pj to pi;

� If pi is inside pj, or inserted into pj, the direction is from pi to pj;

� If pj is underneath pi, the direction is from pi to pj;


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� If pj is a base component and pi is connected with pj on the side, thedirection is from pi to pj;

In the KALG, this relationship can be described by a directed solid line pi ® pj)or an oriented dotted line (pi ---> pj) . The assembly liaison relationships and KALGof a product can be generated from the geometric data available in a CAD system orfrom inquiries directed to the user.

3.3. Representation of assembly constraintsIn an assembly procedure, there exist assembly constraints among di� erent com-

ponents. The constraints consist of hard constraints and soft constraints. The hardconstraints are the geometrical and mechanical constraints, and soft constraints aremade by assembly planners. In this paper, the following types of constraints arediscussed:

3.3.1. Topological constraintsAn assembly can be represented by the topological structure of its components

and the liaison relationships between components. It means that two components aretopologically interconnected or at least one component connects with a componentin the subassembly directly. The existence of topological constraints of a sub-assembly means the coherence of its local liaison graph, which is de® ned as therebeing at least one path composed by connecting arcs from an arbitrary component ina subassembly to any other components in the subassembly (Zha 1994). Figure 1shows the local liaison graphs of an assembly composed of four components andtheir coherence.

Figure 1 (a), (b) and (c) are coherent, but Fig. 1 (d), (e) and (f ) are not. Thereforethis type of constraint can be easily identi® ed from KALG and can also be explicitlydescribed by the liaison sub-matrix. As shown in Table 1, the predicates such asconnect, connect1, not± connect, not± connect1 have been used to describe the topolo-gical constraint knowledge.

3.3.2. Geometric constraintsIn general, geometric constraints are referred to as the relative or allowable

position and orientation relations between two components or subassemblies.They can be represented by predicates such as interference, position, and orientation,as shown in Table 1. Sometimes the geometric constraints lead to assembly prece-dence (see § 3.3.3 below). The geometric constraints can be generated from the geo-metric data available in CAD system and KALG or from inquiries directed to theuser.

3.3.3. Partial precedence constraintsThe precedence constraints specify that a component or a subassembly should be

assembled in a desired direction or precedence. These constraints can be obtainedfrom geometric and non-geometric information provided by the user or available inCAD data base and KALG. Here, they are represented by predicates such as precede(inside, left, right, base, beneath, etc) as shown in Table 1. Therefore the partialprecedence relationships have the properties:

� Chain properties:precede(p1,p2) ^ precede(p2,p3) ^ precede(p3,p4) Þ precede(p1,p4)


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� Commutative properties:precede(p1,p2) º precede(p2,p1) ;

� Distributive properties:precede(p1 ^ p2,p3) = precede(p1,p3) ^ precede(p2,p3) ;precede(p3,p1, ^ p2) = precede(p3,p1) ^ precede(p3,p2) ;precede(p1 ~ p2,p3) = precede(p1,p3) ~ precede(p2,p3) ;precede(p3,p1, ~ p2) = precede(p3,p1) ~ precede(p3,p2) ;where, ` ’= AND; ~ ’= OR.

3.3.4. Stability and security constraintsA subassembly is said to be stable if its components maintain their relative

position and do not break contact spontaneously. The stability of subassemblycontains gravity stability and anti-disturbance stability under the action of gravity,assembly stability under the action of assembly force and plastic stability under theaction of inner spring force. The security of subassembly is de® ned as zero degrees offreedom for relative motion of all components in the subassembly.


Figure 1. Local liaison graph and its coherence.

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The subassembly is said to be stable if it satis® es one of the following:

� The subassembly has fastening constraints;

� The components have tight or over® t mating;

� The centre of gravity of the subassembly falls within the supporting surface;

� Each component in subassembly has stability;

The subassembly has security if it satis® es one of the following:

� The subassembly is fastened;

� The components have no direct liaison with fasteners and have zero degrees offreedom of relative motion;

� The component has direct liaison with fasteners and has motion degree offreedom of fastening constraints only.


Name of constraintsand predicates Explanations Examples

1. Topological constraintsconnect The connections exist between

two components or twosubassemblies or onecomponent and one assembly

connect (p1,p3),connect ([p1,p2],[p3,p4]) ,connect (p1,[p1,p2,p3]) , etc.

connect1 A subassembly is coherent connect1 ([p1,p2,p3,p4])not± connect The negative of the connect not± connect (p1,p3) , etc.not± connect1 The negative of the connect1 not± connect1 ([p1,p2,p3,p4])

2. Geometric constraintinterference The interference relations

between two components ortwo subassemblies or onecomponent and onesubassembly

interference (p1,p3) ,interference ([p1,p2],[p3,p4]) ,interference (p1,[p1,p2,p3]) , etc.

position Three position coordinates ofa component

position (pi,x,y,z)

orientation Three orientation coordinatesof a component

orientation (pi, a , b , u )

3. Partial precedenceconstraintprecede (base, before, after,

left, right, beneath,inside, etc.)

A component or a subassemblyis assembled with othercomponents or subassembliesin a desired direction orprecedence

precede (p1,p2) ,precede (p1,[p2,p3]) ,precede ([p1,p2],[p3,p4]),etc.

4. Stability and securityconstraint

unstable A subassembly without stability unstable ([p1,p2,p3,p4])changeable A subassembly without

changeabilitychangeable ([p1,p2,p3,p4])

5. Cost constrainttime± consuming A subassembly operation takes

much time than otherstime± consuming (p1,[p2,p3]) ,

cost± e� ective The cost of a subassembly or asubassembly operation is lower

cost± e� ective ([p1,p2,p3]) ,cost± e� ective (p1,[p2,p3,p4])

low± e� ciency The e� ciency of a subassemblyoperation is lower

low± e� ciency p1,[p2,p3]) ,low± e� ciency ([p1,p2],[p3,p4])

Table 1. Predicate representation of assembly constraint.

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A subassembly without stability or security constraints means that it is unstableand changeable. So the stability and security constraints would be represented bypredicates such as unstable and changeable as shown in Table 1.

3.3.5. Cost constraintsThe cost of an assembly operation can be described by many indexes such as

assembly time and assembly e� ciency. So the cost constraint of an assembly opera-tion can be described by predicates such as time± consuming and low± e� ciency asshown in Table 1.

In general, a subassembly or a subassembly operation is considered to be unfea-sible if the following conditions exist, such as without topological constraints, with-out stability or security constraint or with interference constraints. The precedenceconstraint is the key to determination of feasibility of the subassembly and thesubassembly operation and the generation of component based and subassemblybased sequence. However, the subassembly or the subassembly operation withcost constraints should also be considered in detail.

3.4. Graph-based representation of assemblyThere are three main approaches to the assembly representation, namely lan-

guage-based representation, graph-based representation, and advanced data struc-ture representation with three di� erent underlying goals (Ben-Arieh and Kramer1994). The graph-based approach uses an information source such as a CAD data-base, or information supplied by the user. Its forms are numerous, which includedirected graph, AND/OR graph (Homen de Mello and Sanderson 1990), connectiv-ity graph (Shpitalni et al. 1989), Petri nets (Zhang 1989), hierarchical partial ordergraphs (Lee and Shin 1988), liaison diagram (De Fazio and Whitney 1987), prece-dence diagram (Bullinger and Ammer 1984), assembly constraint graph (Wolter1989) and interference graph (De Floriani and Nagy 1989).

In this paper, the AND/OR graph is used to represent the assembly sequence.When the AND/OR graph is applied to describe the product assembly or dis-assembly sequence, the vertex in the highest level n - 1 represents the product.The vertices in lower levels (n - 2,n - 3, . . . ,1,0) represent the feasible sub-assemblies. The edges represent the feasible assembly operations. Assembly sequenceAND/OR graph represents the feasible subassemblies and the paths of disassemblyor assembly. It can be described by a mathematical expression as:

Sfsop,Sfs{ }where Sfsop represents the set of feasible subassembly operations, and Sfs representsthe set of feasible subassemblies.

If P represents the set of components, Ssp(P) and Sfsp(P) represents the set ofcomponents of subassembly and feasible subassembly respectively, and S(P) repre-sents the total subsets of components, then the AND vertices can be described by:

A ^ = Pk, Pi,Pj{ }( ), Pk = Pi ~ Pj (2)

A feasible subassembly Sfs can be represented by:

Sfs = P Î S(P),Ssp(P),Sfsp(P){ } (3)


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If a feasible subassembly composed of two kinds of component sets Pi and Pj isrepresented by FS (Pi,Pj) and PS (Pi,Pj) = Pi ~ Pj is the juxtaposed set of Pi andPj , then the set of feasible subassembly operations Sfsop can be written as:

Sfsop = Pk, Pi,Pj{ }( ) Pi,Pj,Pk Î S(P)( )[ ] PS(Pi,Pj, ) ^ FS(Pi,Pj) (4)

An example of an assembly AND/OR graph is shown in Fig. 2.

4. Assembly sequence generation

4.1. Determination of levelled feasible subassemblies4.1.1. Theoretical levelled subassembly con® gurations

From the assumption of assembling one component at a time, we can classify thewhole assembly procedure of n-component product into n - 1 levels of assemblies.Level 0 assembly is a primary subassembly each consists of a primitive component.Level n - 1 assembly is the last subassembly which forms the product. Level i - 1( i - 2, . . . ,n - 1) assembly is a subassembly consisting of at least two up to n - 1components. The number of theoretical subassembly con® gurations in level i - 1assembly is Ci

n( i = 1,2, . . . ,n) which is determined by compound principles inmathematics. All theoretical levelled subassembly con® gurations are shown inTables 2 and 3.

4.1.2. Removal of levelled unfeasible subassembliesIn spite of the large number of theoretical sequences, only a few of them are

feasible. The above mentioned types of constraints limit the number of feasiblesequences. The key to determination of infeasibility of a subassembly is to identifyall the constraint conditions and the corresponding predicates in assembly knowl-edge base (AKB) for all subassemblies. This can be implemented automatically bythe method of the backtracking mechanism in arti® cial intelligence (AI).

4.1.3. Identi® cation of levelled feasible subassembliesBased on the constraint knowledge in AKB, all unfeasible subassemblies in level

i - 1 assembly can be eliminated. The feasible subassembly subsets of this level canthen be formed. The number of feasible subassemblies in level i - 1 can be expressed


Figure 2. Example assembly AND/OR graph.

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asNi

fs = Cin - Ni

infs, i = 1,2,3, . . . ,n (5)

where Nifs is the number of feasible subassemblies in level i - 1;

Cin is the number of theoretical assembly con® gurations in level i - 1; Ni

infs is thenumber of unfeasible subassemblies in level i - 1.

The set of feasible subassemblies in level i - 1 can be determined by the followingformula

Sifs{ } = Si

ts{ } - S iinfs{ } , i = 1,2,3, . . . ,n (6)

where {Sifs} is a set of feasible subassemblies in level i - 1, Si

ts is a set of theoreticalsubassembly con® gurations in level i - 1, and {Si

infs} is a set of total unfeasiblesubassemblies in level i - 1.

4.1.4. Decomposition of feasible subassembliesA feasible subassembly in level i - 1 assembly is likely to be decomposed into

many corresponding combinations of the feasible subassemblies in lower levelj ( j = 0,1,2, . . . , i - 2) , which are de® ned as subassembly operations here. For


No. of components Assembly levels Total no. of subassemblies

1 0 12 0,1 33 0,1,2 74 0,1,2,3 155 0,1,2,3,4 316 0,1,2,3,4,5 637 0,1,2,3,4,5,6 1278 0,1,2,3,4,5,6,7 255. . . . . . . . .n 0,1, . . . ,n - 1 C1

n + C2n + . . . + Cn- 1

n = 2n - 1

Table 2. Theoretical no. of subassemblies.

Level Theoretical subassembly con® gurations

0 [p1],[p2],[p3], . . . ,[pn]1 [p1,p2],[p1,p3],[p1,p4],[p1,p5], . . . ,[p1,pn]

[p2,p3],[p2,p4],[p2,p5], . . . ,[p2,pn][p3,p4],[p3,p5], . . . ,[p3,pn]. . .[pn- 2,Pn- 1],[Pn- 1,Pn][pn- 1,Pn]

. . . . . .i - 1(i = 3,4, . . . ,n - 2) [p1,p2, . . . ,pi- 1,pi],[p1,p2, . . . ,pi- 1,pi+1], . . . ,[p1,p2, . . . ,pi- 1,pn]

[p1,p2, . . . ,pi,pi+1],[p1,p2, . . . ,pi,pi+2], . . . ,[p1,p2, . . . ,pi,pn]. . .

n - 1 [p1,p2,p3, . . . ,pn]Table 3. Theoretical subassembly con® gurations of n-component product.

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example, the levelled feasible subassemblies and the subassembly operations of afour component product are shown in Table 4.

4.2. Generation of assembly sequencesAfter the feasible subassemblies and their subassembly operations are deter-

mined, the assembly sequences can be easily generated by AND/OR logical relationsand their graph representation. As the stability and security constraints and costconstraints are introduced in the formation of feasible subassembly subsets, theevaluation and choice of assembly sequences can be accomplished by these con-straints.

5. Automatic reasoning

In § 4, all assembly or disassembly sequences of a product can be generated bydetermining the levelled feasible subassemblies and their decompositions into corre-sponding subassembly operations. However, this method needs much more reason-ing experience or intervention and the user has to verify many feasible sequences (n!maximum for an n-component product). In order to generate assembly or dis-assembly sequences automatically, an automatic knowledge reasoning systemwould be required to solve this problem based on principles of arti® cial intelligence(AI). A prototype assembly planning expert system (APES) has been developed fordetermining and decomposing the feasible subassemblies and generating the assem-bly sequence and the assembly sequence graph. The system is written in arti® cialintelligence language Visual prolog 4.0. The procedure ¯ ow chart of the system isshown in Fig. 3.


Level Subassembly Subassembly operation

3 [p1,p2,p3,p4] [[p4],[p1,p2,p3]]or [[p1,p2,p3],[p4]][[p3],[p1,p2,p4]]or [[p1,p2,p4],[p3]][[p1],[p2,p3,p4]]or [[p2,p3,p4],[p1]][[p2],[p1,p3,p4]]or [[p1,p3,p4],[p2]][[p1,p2],[p3,p4]]or [[p3,p4],[p1,p2]][[p1,P3],[p2,p4]]or [[p2,p4],[p1,p3]]

2 [p1,p2,p3],[p1,p2,p4],[p2,p3,p4],[p1,p3,p4] [[p3],[p1,p2]]or [[p1,p2],[p3]][[p2],[p1,p3]]or [[p1,p3],[p2]][[p1],[p2,p4]]or [[p2,p4],[p1]][[p4],[p1,p2]]or [[p1,p2],[p4]][[p2],[p3,p4]]or [[p3,p4],[p2]][[p3],[p2,p4]]or [[p2,p4],[p3]][[p1],[p3,p4]]or [[p3,p4],[p1]][[p4],[p1,p3]]or [[p1,p3],[p4]]

1 [p1,p2],[p1,p3],[p2,p3],[p2,p4],[p3,p4] [[p1],[p2]]or [[p2],[p1]][[p1],[p3]]or [[p3],[p1]][[p2],[p3]]or [[p3],[p2]][[p2],[p4]]or [[p4],[p2]][[p3],[p4]]or [[p4],[p3]]

0 [p1],[p2],[p3],[p4] [p1],[p2],[p3],[p4]Table 4. Example of subassembly and subassembly operation.

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Figure 3. Program ¯ ow chart.

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The key features of the knowledge based reasoning system are a knowledge basecomposed of assembly knowledge and a reasoning mechanism formed by the infer-ring rules, which decide how to utilize the knowledge base. The assembly knowledgebase (AKB) can be constructed by many predicate logical clauses. To constructinference rules, the predicates such as test± constraint, exist and not± exist are usedto check whether the subassemblies and subassembly operations are feasible. Inaddition, there are rules adopted in the procedure of automatic inference, where,Û ’ means equivalent’ and Þ ’ means implies’ , as follows:

� Equivalent rule: A subassembly is a compound of at least one component. As aresult, it is not associated with the location of components in the subassemblyrepresentation. For example,

p1,p2,p3,p4]Û p1,p2,p4,p3[ ]Û p1,p4,p3,p2[ ][� Topological constraint existence rule: Based on the above mentioned, the local

liaison graphs for a subassembly with topological constraints must be coher-ent.

� Subassembly operation rule: If a subassembly is unfeasible then all its sub-assembly operations decomposed are not unfeasible.

� Superset rule: If there is no mating between two components or subassembliesdue to interference in the approach path, then adding a component to either acomponent or a subassembly, which is not associated with the mating liaisons,will not change this situation. For example, not± connect ( p1,p2) ^ interferencep1,p2) ^ not± connect ( p3,p2) Þ not± connect ([p1,p3],p2) .

� Subset rule: If a mating can occur between two subassemblies, then removing acomponent from either a subassembly, which is not associated with the matingliaison(s), will not change this situation. For example, connect ([p1,p2],[p3,p4]) ^ connect (p2,p3) ^ not± connect (p1,p3) Þ connect ([p2],[p3,p4]) .

6. Example

In this paper, a four-component assembly, which was described by Homen deMello and Sanderson (1990) is taken as an example to illustrate the feasibility ande� ciency of the proposed approach and the reasoning system. Figure 4 (a) describesthe example model product composed of four components namely cap (1), stick (2),receptacle (3) and handle (4) respectively. Its KALG is shown in Fig. 4 (b). Based onthe above mentioned approach and automatic reasoning system, the assemblysequences can be generated as follows:

(1) Theoretical subassembly con® gurations:

level 0: [1],[2],[3],[4]level 1: [1,2],[1,3],[1,4],[2,3],[2,4],[3,4]level 2: [1,2,3],[1,2,4],[2,3,4],[1,3,4]level 3: [1,2,3,4]

(2) Subassemblies with constraints:

(a) Topological constraintsnot± connect([1],[4]) or not± connect([4],[1]) ® not± exist([1,4])

(b) interference([1],[4]) or interference([4],[1]) ® not± exist([1,4]) Þ


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® not± exist([1,2,4]) ® not± exist([[1],[2,4]]) and not± exist([[2],[1,4]]) andnot± exist([[4],[1,2]])

® not± exist([1,3,4]) ® not± exist([[1],[3,4]]) and not± exist([[3],[1,4]]) andnot± exist([[4],[1,3]])

(c) Precedence constraints

before([2],[1,3,4]) and inside([2],[3]) ® not± exist([1,3,4])


Figure 4(a). Example product modelling.

Figure 4(b). Example product KALG.

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(d) Stability and security constraints

unstable([1,4]) ® not± exist([1,4]) andunstable([1,2,4]) ® not± exist([1,2,4])

(e) Cost constraints

time± consuming([[3],[1,2]],[[2],[1,3]]) andtime± consuming([[3],[4,2]],[[2],[4,3]])

(3) Unfeasible subassemblies:

[1,4],[1,2,4],[1,3,4](4) Feasible subassemblies and their decompositions:

The reasoning system ® rst searches and outputs the feasible subsets and thendecomposes them in di� erent level assemblies as follows:

Level 0: [1],[2],[3],[4]Level 1: [1,2]Þ {[[1],[2]]or [[2],[1]]}

[1,3]Þ {[[1],[3]]or [[3],[1]]}[2,3]Þ {[[2],[3]]or [[3],[2]]}[2,4]Þ {[[2],[4]]or [[4],[2]]}[3,4]Þ {[[3],[4]]or [[4],[3]]}

Level 2: [1,2,3]Þ {[[3],[1,2]]or [[1,2],[3]]and[[2],[1,3]]or [[1,3],[2]]and[[1],[2,3]]or [[2,3],[1]]}

[2,3,4]Þ {[[2],[3,4]]or [[3,4],[2]]and[[3],[2,4]]or [[2,4],[3]]and[[4],[2,3]]or [[2,3],[4]]}

Level 3: [1,2,3,4]Þ {[[1],[2,3,4]]or [[2,3,4],[1]]and[[4], [1, 2, 3]] or [[1, 2, 3], [4]]

(5) AND/OR graph and feasible assembly sequences: The system outputs thefollowing 12 feasible assembly sequences:

1-2-3-4, 1-3-2-4, 3-4-2-1, 2-1-3-4, 3-1-2-4, 4-3-2-12-3-1-4, 2-3-4-1, 2-4-3-1, 3-2-1-4, 3-2-4-1, 4-2-3-1

If cap (4) or handle (2) is considered as the base part of the assembly, thefollowing precedence constraints must be added:

base(1,[2,3,4]) or base(4,[1,2,3])So, four assembly sequences are remaining as follows:

1-2-3-4, 1-3-2-4, 4-3-2-1, 4-2-3-1

If cost constraints are described as above then in the end the two assemblysequences are:

1-3-2-4, 4-3-2-1

The assembly sequence AND/OR graph is shown in Fig. 4.


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7. Conclusions

The application results show that the representation for product assembly pro-cess presented in this paper is e� cient and the combination method for determina-tion of feasible subassembly based on the principles of compound and classi® cationand decomposition of feasible subassembly in assembly knowledge reasoning isuseful in assembly sequence planning. A prototype knowledge reasoning system(APES) is developed in the Visual prolog 4.0 programming language to generatethe assembly sequences automatically. This system is convenient to use and makesthe speed of reasoning fast and the assembly sequences to be selected less subjective.It will ® nd wide applications in knowledge based assembly planning systems. Furtherresearch work is required to enhance its interactivity and the integration with DFAand CAD systems, especially those about assembly representation (e.g. assemblyconstraints) that are based on assembly features and predicates. Also, the approachand system for concurrent design and planning for assembly in an integrated envir-onment is expected to be developed. The authors are working now on these problemsunder the support of a project entitled concurrent integrated design and planning for¯ exible assembly.

Acknowledgments

The second author would like to thank the National Natural Science Foundationof China and `863’ High-tech. R&D Programme of China for their support for thisresearch under grant No. 59485005.

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Documents

On the automatic generation of product assembly sequences