Symbolic OBDD representations for mechanical assembly sequences

doi:10.1016/j.cad.2007.12.001

Computer-Aided Design

Volume 40, Issue 4, April 2008, Pages 411-421

https://doi.org/10.1016/j.cad.2007.12.001 Get rights and content

Abstract

Assembly sequence planning is one typical combinatorial optimization problem, where the size of parts involved is a significant and often prohibitive difficulty. The compact storage and efficient evaluation of all the feasible assembly sequences is one crucial concern. Ordered binary decision diagram (OBDD) is a canonical form to represent and manipulate the Boolean functions efficiently, and appears to give improved results for large-scale combinatorial optimization problems. In this paper, subassemblies, assembly states and assembly tasks are represented as Boolean characteristic functions, and the symbolic OBDD representation of assembly sequences is proposed. In this framework, the procedures to transform directed graph and AND/OR graph into OBDDs are presented. The great advantage of OBDD-based scheme is that the storage space of OBDD-based representation of all the feasible assembly sequences does not increase with the part count of assembly dramatically so quickly as that of both directed graph and AND/OR graph do. We undertake many experimental tests using Visual C++ and CUDD package. It was shown that the OBDD scheme represented all the feasible assembly sequences correctly and completely, and outperforms either directed graph or AND/OR graph in storage efficiency.

Introduction

The highly competitive nature of the global market of manufacturing products requires frequent changes of the product design and manufacturing strategies. In order to shorten the time and reduce the costs required for the development of the product and its manufacturing process, it is desirable to automate and computerize the assembly sequence planning activity. Typically, a product can have a very large number of feasible assembly sequences even at a small parts count, and this number rises exponentially with increasing parts count, which renders it staggeringly difficult and even impossible for one to represent all the sequences individually. Thus, the choices of representation for assembly sequences can be crucial in assembly sequence planning, and there has been a need to develop systematic and efficient methods to represent all the available alternatives.

The representation for assembly sequences has received much attention due to the requirement of less storage and easy user comprehending. Various representations of assembly sequences used in the literature on assembly sequence planning can be classified into two groups [1], [2]: ordered lists and graphical representations. The ordered list could be a list of tasks, list of assembly states, or list of subsets of connections. In the ordered lists each assembly sequence is represented by a set of lists. Although this set of lists might represent a complete and correct description of all feasible assembly sequences, it is not necessarily the most compact or most useful representation of sequences. The graphical schemes map the assembly operations and assembly states into specified diagrammatic elements, and share common subsequences and common states graphically in many assembly sequences, which create more compact and useful representations that can encompass all feasible assembly sequences. The graphical representations include precedence diagrams [3], state transition diagrams [4], inverted trees [5], liaison sequence graphs [6], assembly sequence graphs [7] directed graphs [2] and AND/OR graphs [8] etc.

In the precedence diagram [3], assembly operations are represented by numbered circles. The circles are connected by arrows showing the precedence relations. The shortcoming of this formalism is the lack of algorithmic nature in the development of the diagram. In the state transition diagrams [4], subassemblies are represented by a separate diagram with a corresponding node in the main diagram. Hence, it cannot provide a compact representation of all assembly sequences. Bourjault represented all valid assembly sequences in the form of an inverted tree [5] that describes the possible orders of assembly. The inverted tree gives the liaison sequences only, not the actual assembly sequences. Also, it does not contain any information about subassemblies. The liaison sequence graph [6] representation is similar to that of state transition diagram, but this scheme gives liaison sequences instead of assembly sequences. However, here the states do not represent a set of parts, but a set of relations between parts. The assembly sequence graph described in [7] represents all feasible sequences through a series of assembly states and assembly tasks. Although assembly sequence graph provides a detailed hierarchical representation, it becomes quite clumsy for products with large number of parts. In a directed graph of assembly sequences [2], the nodes correspond to the sets of stable subassemblies, and the edges correspond to the assembly tasks. A directed graph might contain infeasible assembly sequences. Homem de Mello and Sanderson described an AND/OR graph [8] representation of assembly sequences. Though the AND/OR graphs usually provide a compact representation, they give the disassembly sequences only. These traditional representations have the same shortcoming that increasing parts count in an assembly makes it staggeringly difficult and even impossible to represent all the sequences.

In recent years, implicitly symbolic representation and manipulation technique, called as symbolic graph algorithm or symbolic algorithm [9], [10], has emerged in order to combat or ease combinatorial state explosion. Typically, ordered binary decision diagram (OBDD) or variants thereof are used to represent the discrete objects [11], [12]. Efficient symbolic algorithms have been devised for hardware verification, model checking, testing and optimization of circuits. Hachtel and Somenzi developed OBDD-based symbolic algorithm for maximum flow in 0–1 networks that can be applied to very large graphs (more than 10³⁶ edges) [13]. Gu and Xu presented the symbolic ADD (Algebraic Decision Diagram) formulation and algorithms for maximum flow problems in general networks [14]. Zhong, Huang and Gu discussed the operations of directed graphs using OBDDs on the view of a kind of general data structures [15]. Symbolic algorithms appear to be a promising way to improve the computation of large-scale combinatorial computing problems through encoding and searching nodes and edges implicitly.

In this regard, we present the symbolic OBDD formulation of all the assembly sequences. The subassemblies, assembly states, assembly tasks and assembly sequences are represented by Boolean characteristic functions, and the procedures to translate directed graph and AND/OR graph into OBDDs are presented. Experiment tests show that the OBDD formulation outperforms either directed graph or AND/OR graph in storage efficiency.

The rest of this paper is organized as follows. In Section 2, we introduce some concepts and properties regarding ordered binary decision diagram. The symbolic formulations for subassemblies, assembly states, assembly tasks and assembly sequences are described in Section 3; Section 4 and Section 5 develop the procedures to translate directed graph and AND/OR graph into OBDDs respectively; Some experimental results are presented in Section 6; The last section gives some conclusions.

Section snippets

Ordered binary decision diagram

An ordered binary decision diagram (OBDD) [9], [10] provides compact, canonical and efficiently manipulative representation for Boolean functions. Given a Boolean algebra $B$ , the Boolean formulae on the variables $x_{1}, \dots, x_{n}$ are obtained by recursively applying negation, conjunction, and disjunction to the elements of $B$ and the variables. We denote conjunction, disjunction and negative by “ $\cdot$ ”, “ +” and “ $'$ ” respectively. Unless otherwise noted, we assume $B = {0, 1}$ . Thus, $x_{1} \cdot x_{3} + x_{2} \cdot x_{3}$ is a formulae.

Symbolic representation of assembly sequences

A mechanical assembly is a composition of interconnected parts forming a stable unit. Each part is a solid rigid object, that is, its shape remains unchanged. Parts are interconnected whenever they have one or more compatible surfaces in contact. Surface contacts between parts reduce the degree of freedom for relative motion. These contacts and relative motions are embedded in various logical and physical relations among the parts of the assembly, called as assembly knowledge, and can be

Formulating directed graph via OBDDs

The set of all assembly sequences for a assembly, whose set of parts is $P = {p_{1}, p_{2}, \dots, p_{n}}$ , can be represented by a directed graph, in which the nodes correspond to the sets of stable subassemblies, or the partitions of part set $P$ , and the edges correspond to the assembly tasks, or ordered pairs of nodes. It is noted that directed graph here is different from general one in having domain-specific knowledge as following: for any edge, there are only two subset $θ_{i 1}$ and $θ_{i 2}$ in the state partition

Formulating AND/OR graph via OBDDs

In an AND/OR graph representation of assembly sequences, the nodes are the subsets of $P$ that characterize stable subassemblies. It is noted that AND/OR graph here contains domain-specific knowledge as following: each hyperarc is associated to a decomposition of the subassembly $θ_{i 3}$ that corresponds to its incoming node, and is an ordered pair in which the first element is a node that corresponds to a stable subassembly $θ_{i 3}$ , the second element is a set of two nodes ${θ_{i 1}, θ_{i 2}}$ such that $θ_{i 3} = θ_{i 1} \cup θ_{i 2}$ .

Experiments

We implement the symbolic OBDD scheme in windows XP and software package CUDD [16]. The storage efficiency of symbolic OBDD formulation has been tested, and compared with directed graph and AND/OR graph.

We choose many practical assemblies with different part numbers. For example, we undertake the test on a gear pump shown in Fig. 7. All of its feasible assembly sequences are generated using Gottipolu and Ghosh’s technique [1], and represented as directed graph and OBDDs respectively. The

Conclusions

The choice of representation for assembly sequences has been crucial in assembly sequence planning. AND/OR graphs and directed graphs are two popular representations. However, they face the same challenge that increasing parts count renders it staggeringly difficult and even impossible to represent all the sequences individually. We present the novel symbolic OBDD scheme for representing all the feasible assembly sequences, and explain the equivalences between them. Experiments give the proof

Acknowledgments

The authors are very grateful to the anonymous reviewers for their helpful comments. This work has been supported by National Natural Science Foundation of China (Grant No. 60563005) and Natural Science Foundation of Guangxi Province.

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Symbolic OBDD representations for mechanical assembly sequences

Abstract

Introduction

Section snippets

Ordered binary decision diagram

Symbolic representation of assembly sequences

Formulating directed graph via OBDDs

Formulating AND/OR graph via OBDDs

Experiments

Conclusions

Acknowledgments

Computers in Industry

Information Processing Letters

Theoretical Computer Science

Representation of mechanical assembly sequences

IEEE Transactions on Robotics and Automation

The precedence diagram: A tool for analysis in assembly line balancing

Journal of Industrial Engineering

Simplified generation of all mechanical assembly sequences

IEEE Journal of Robotics and Automation