Ontology and Assembly Joint Topology Representation

The broadest application domains of ontologies are upper-level ontologies that describe common sense-level knowledge (e.g., thing, individual, etc.) in a machine-interpretable manner. Cycorp's CYC is a commercial ontology containing over 200,000 terms and assertions. Its goal is to define high-level, common sense-type of concepts in a machine-interpretable manner. CYC's potential applications include online brokering of goods and services, enhanced virtual reality, improved machine translation, improved speech recognition, data mining, true language processing, etc. (Cycorp). However, since CYC is still a high-level ontology, it has not had a strong impact in the mechanical design domain. Nonetheless, in 1999, the National Institute of Standards and Technology (NIST) chose CYC as an ontology for further investigation in the manufacturing domain [20]. The results from this investigation led to the development of Process Specification Language (PSL), a language that is generic enough to represent discrete manufacturing and construction process data [10]. Narrower in scope than upper-level ontologies, enterprise-level ontologies attempt to formalize the practices and processes that occur within an organization. The level of the concepts is enterprise-specific, and the concepts are meant to promote knowledge reuse with regard to business decisions and transactions [23], [6], [7].

Some ontological research has been applied at both the conceptual and detailed design levels. Kitamura et al. [14] and Horváth et al. [11] attempt to develop ontologies to represent functional design and design features. In particular, the work of Kitamura et al. captures the flow of fluids or the flow of parts in manufacturing. However, these research works have limitations on representing mechanical assemblies and their joining constraints. Kim et al. [12][13] showed feasibility of ontological representation of assembly relations and associated constraints. This paper further enhances the scope of the explicit representation to various joining methods to properly represent topological information associated to assembly joints.

Mereology is the name of the formal theory of parts and associated concepts developed by Leśniewski [15]. To distinguish his theory from other mereological theories, the theory of Leśniewski is referred to as "Mereology" written with a capital M. Just as mereology is a theory for the part-whole relation in general, there is also a theory of the "is-connected-to" relation in general. This theory is called topology. Region-based theories of mereotopology [19], [5] suggest that it can be used to represent entities that exist in other spaces besides the usual physical one. A region is a portion of a space, typically the portion occupied by some entity, whether material (e.g., a physical part) or otherwise (e.g., a hole). The overarching goal of any mereotopology theory is to describe the nature of regions and the entities that occupy them, and the interrelations between regions. Regions themselves, however, need not be primitive entities within the theories. There are many mereotopology theories that do not take regions as primitive entities, such as Casati and Varzi [3] and Smith [21]; rather they build them up from more primitive entities. Mereology claims to be not only a theory about the part-of relationship between components, but also between a large class of "things" for which parts and wholes are relevant. To represent assembly joints, connectivity of individuals to which the part-whole relation applied is obviously important. There are two approaches to develop such a theory. One is to extend an existing theory of mereology with the topological is-connected-to relation. The other is to integrate mereological and topological concepts and relations into one mereotopological theory. While the first approach is arguably more intuitive; however the second usually leads to more compact theories because axioms can be combined. This research uses Smith's mereotopology [22], which can represent regions in physical domains (e.g., parts and products), as well as non-physical domains (e.g., form feature) that are associated to assembly design formalisms.

4.1.1 belongTo relations: A belongTo relation defines relations between a part and a form feature. The inferred constraints: 1) every form feature must belong to a part (C1-1); and 2) a form feature must not belong to two parts (C1-2), can be transformed to two asserted conditions (in Protégé 3.1). For the C1-1, an asserted condition, “belongTo Part” is used and an asserted condition, “belongTo = 1” is used to represent C1-2. This cardinality condition means the property belongTo has exactly one value.

4.1.2 inter-featureAssociation relations: The inter-featureAssociation relation represents the relations between form features. The relational constraint stands for the relationship between two form features in the form feature hierarchy. For example, a block may have a blind hole at a certain location. The distance between the coordinates of the block and the blind hole is a dimensional constraint. Since the block form feature contains the hole form feature (since the block is a parent feature of the hole in the feature decomposition hierarchy), their relational constraint is 0. If a form feature has some form feature, then the relational constraint is 1. When two form features do not belong to each other (e.g., two holes in a block), the relational constraint is 2. The associated implied constraints: 1) the associated form features must not be identical (C2-1); 2) the associated form features must belong to same part (C2-2); and the relational constraint must be represented and included (C2-3), can be modeled in SWRL rules shown below.

SWRL rules:

4.1.3 Assembly/joining relations: The assembly/joining relations represent the relations between form features that belong to different parts. It implies two constraints: 1) the associated form features must belong to two non-equivalent parts (C3-1); and 2) the associated form features must be a joining pair (C3-2), and the implied constraints are represented in SWRL rule as shown below. This assembly/joining relation is extended in the following section to accommodate the complexity of joining relations.

SWRL rule:

To represent assembly joints, connectivity of individuals for which the part-whole relation applied is obviously important. There are two approaches to make such a theory. One is to extend an existing theory of mereology with the topological is-connected-to relation. The other is to integrate mereological and topological concepts and relations into one mereotopological theory. While the first approach is more intuitive, the second usually leads to more compact theories because axioms can be combined. This paper follows the second approach.

In order to define various assembly joints, some of mereotopological definitions, axioms, and theorems are adopted. Adopting the mereotopology definitions, we need to understand the meaning of operators used in the mereotopology. According to mereological primitive the relation of parthood or constituency, Smith [22] addressed that $x$ is a part of $y$, and write '$xPy$', when $x$ is any sort of part of $y$, including an improper part. From the part relationship, the author derived the following further purely mereological notions. The first definition is that $x$ overlaps $y$, and write '$xOy\; :=\; \backslash exists\; z(zPx\; \backslash wedge\; zPy)$', when $z$ is any sort of part $x$ and $y$. The second one is that $x$ discrete from $y$, and write '$xDy\; :=\; \backslash neg\; xOy$,' when $x$ does not overlap $y$.

Smith addressed that a condition '$\backslash phi$' in a single free variable '$x$' is satisfied iff the sentence '$\backslash phi\; x$' is true for at least one value of '$x$'. The sum of $\backslash phi$s can be defined as that entity $y$ which is such that, given any entity $w$, $w$ overlaps with $y$ iff $w$ overlaps with something that $\backslash phi$s, that is '$\backslash phi\; x(\backslash phi\; x)\; :=\; \backslash iota\; y(\backslash forall\; w(wOy\; \backslash equiv\; \backslash exists\; v(\backslash phi\; v\; \backslash wedge\; wOv)))$'. And then the following theorem can be proved: $y\; =\; \backslash phi\; x(\backslash phi\; x)\; \backslash rightarrow\; \backslash forall\; x(\backslash phi\; x\; \backslash rightarrow\; xPy)$. With this proven theorem, we say that $j$ is defined as any geometric entity that can be mating boundary including line, point, and face, of two joining entities of $x$ and $y$ ($j$ is often virtual geometric entity, such as datum plane in CAD), and write '$J\; :=\; \backslash phi\; z(\backslash phi\; z\; \backslash rightarrow\; \backslash forall\; z(\backslash phi\; z\; \backslash rightarrow\; zBx\; \backslash wedge\; zBy))$'.

In order to show what it is boundary, '$B$,' some understandings are required. When $x$ is a part of $y$ that is off the boundary of $y$ (i.e., it is neither tangential nor itself a boundary), $x$ is an interior part of $y$ and write '$xIPy$'. As the first step towards defining what it is for $x$ to be a boundary of $y$, we define '$xXy$' ($x$ crosses $y$) by: '$xXy\; :=\; \backslash neg\; xPy\; \backslash wedge\; \backslash neg\; xDy$' or equivalently, for $y\; \backslash neq\; 1$, '$xXy\; :=\; xOy\; \backslash wedge\; xO(1-y)$,' that $x$ overlaps both $y$ and its complement. The relation of straddles is defined as $x$ straddles $y$ by '$xSty\; :=\; \backslash forall\; z(xIPz\; \backslash rightarrow\; zXy)$'. From the sense, the entities straddling a given entity can be divided, intuitively, into two classes. There are those, which include among their parts a boundary of the straddled entity; there are those – characteristically non-connected – which include no such boundary. The former group is called tangents defined by '$xTy\; :=\; \backslash exists\; z(zPx\; \backslash wedge\; zBy)$' and the later boundary by '$xBy\; :=\; \backslash forall\; z(zPx\; \backslash rightarrow\; zSty)$'.

We derived the definition of assembly joints based on mereotopology of Smith. In this section, two geometrically and topologically similar joint pairs (i.e., fusion welding and adhesive bonding, and mechanical fastenings with threaded fasteners and rivets) are presented to describe how the mereotopological representation can be defined.

4.2.1 Fusion Welding: The thermal energy required for these welding operations is usually supplied by chemical or electrical means. Filler metals, which are metals added to the weld area during welding of the joint, may or may not be used. When filler metals are not used, the melted material still generates a weld zone. Solid state welding (e.g., friction welding) also generate a weld zone. In mereotopology, fusion welding can be defined as $xJ\_\{fw\}y\; :=\; \backslash exists\; w(wOx\; \backslash wedge\; wOy)\; \backslash wedge\; \backslash exists\; w(wStj\; \backslash vee\; wXj)$. This definition indicates that if there is an entity, $w$, associated to joining, we observe that the entity $w$ (i.e., a weld) overlaps $x$ and $y$, which are fusion welded. In addition, the $w$ should either straddle or cross to $j$. When a groove (e.g., Butt-V, Butt-double V, Tee square groove) is used, we can say $kDx\; \backslash wedge\; kDy$, where $k$ is a groove.

4.2.2 Adhesive Bonding: Adhesive bonding joins components using an adhesive. Many types of adhesives provide adequate joint strength, including fatigue strength. The mereotopological definition of adhesive bonding is $xJ\_\{ab\}y\; :=\; \backslash exists\; g(gStj\; \backslash wedge\; gTx\; \backslash wedge\; gTy)$. As described in this definition, the characteristics of adhesive bonding are different from fusion welding. In other words, the entity associated to joining is a glue; it should straddle to mating boundary, $j$ and overlap both of $x$ and $y$, which are to be assembled. Note that after the joining process, the joint is identical to the mereotopological definition. Note also that surface reaction by filler metal or adhesive treatment is not considered in this mereotopological representation.

4.2.3 Mechanical Fastening by Threaded Fasteners: Threaded fasteners are the most common method of mechanical fastening. Examples include bolts, nuts, screws, pins, and a variety of other fasteners. The mereotopological definition of mechanical fastening by threaded fasteners is $xJ\_\{jf\}y\; :=\; \backslash exists\; u,\; v\backslash \{(uXx\; \backslash wedge\; uXy)\; \backslash wedge\; (vTx\; \backslash vee\; vTy)\; \backslash wedge\; (uPfs\; \backslash wedge\; vPfs)\backslash \}\; \backslash wedge\; fsXj$. To represent this relationship in a mereotopology, fasteners are classified into two sections: $u$ = threaded body; $v$ = head of fastener and/or nut. This definition indicates that the body of any threaded fastener should cross joining parts, and the head/nut should be tangent with one of the joining parts (Figure 2). Sometimes, the threaded fastener is used to clamp a part, rather than joining two separate parts. Even though the threaded fastener for clamping is not joining two parts, it provides similar functionality with joining, since it combines two section of the parts. The mechanical fastening for clamping has a similar definition with the definition above except that '$x$' and '$x$' should be part of $x$, $x\text{'}J\_\{j\backslash kappa\}x\text{'}\text{'}\; :=$

$\backslash exists\; u,\; v\; \backslash \{\; (u\backslash mathbf\{X\}x\text{'}\; \backslash wedge\; u\backslash mathbf\{X\}x\text{'}\text{'})\; \backslash wedge\; (v\backslash mathbf\{T\}x\text{'}\; \backslash vee\; v\backslash mathbf\{T\}x\text{'}\text{'})\; \backslash wedge\; (u\backslash mathbf\{P\}fs\; \backslash wedge\; v\backslash mathbf\{P\}fs)\; \backslash \}\; \backslash wedge\; fs\backslash mathbf\{X\}j\; \backslash wedge\; (x\text{'}\backslash mathbf\{P\}x\; \backslash wedge\; x\text{'}\text{'}\backslash mathbf\{P\}x)$. This definition needs to be understood for the case study in Chapter 5.

4.2.4 Riveting or Mechanical Fastening by Rivets: Rivets are also another common mechanical fastening device. Installing a rivet consists of placing the rivet in the hole and deforming the end of its shank by upsetting or heading. The mereotopological definition of riveting is $x\backslash mathbf\{d\}\_m\; y\; :=\; \backslash exists\; u,\; v\; \backslash \{\; (u\backslash mathbf\{X\}x\; \backslash wedge\; u\backslash mathbf\{X\}y)\; \backslash wedge\; (v\backslash mathbf\{T\}x\; \backslash wedge\; v\backslash mathbf\{T\}y)\; \backslash wedge\; (u\backslash mathbf\{P\}fs\; \backslash wedge\; v\backslash mathbf\{P\}fs)\; \backslash \}\; \backslash wedge\; fs\backslash mathbf\{X\}j$. Similar to the threaded fastener, a rivet is divided into two sections for the definition: $u$ = body of rivet; $v$ = head and upset tail of rivet. By the definition, the rivet's body crosses the joining components, and the head and upset tail should be tangent to the joining components (Figure 3). It indicates a difference between the mereotopological definitions of riveting versus mechanical fastening with threaded fasteners. This definition is valid for mathematically differentiating joints; however the definitions should be modeled in a standard ontology to increase the universality of the semantic definitions.

While a mereotopological ontology for assembly design provides a theoretical foundation, it lacks the universality of semantic definitions. Therefore, mereotopological notions, definitions, and primitives should be represented in a standard manner. The Semantic Web Rule Language (SWRL) is used in this research. In this paper, implicit assembly constraints of joining are derived from the mereotopological definitions described in the previous sub-section, and the constraints are explicitly represented using SWRL rules.

Mereotopological operators and definitions of joints are converted into SWRL rules based on the conversion rules: 1) OR relation is converted to a disjunctive normal form (DNF) and generate separated rules; 2) logical implication is converted to $\backslash neg\; p\; \backslash vee\; q$; and 3) existential quantifier is constrained with specific rules. For example, to define the straddle relation, $xSty\; :=\; \backslash forall\; z(xIPz\; \backslash rightarrow\; zXy)$, a logical implication ($p\; \backslash rightarrow\; q$) is equivalent to $\backslash neg\; p\; \backslash vee\; q$ and the logical OR can be represented with two separated rules as shown in Table 1. For fusion welding, the mereotopological definition of a joint is $xJ\_\{fw\}y\; :=\; \backslash exists\; w(wOx\; \backslash wedge\; wOy)\; \backslash wedge\; \backslash exists\; w(wStj\; \backslash vee\; wXj)$. To represent this in a SWRL form, the definition should be converted to a disjunctive normal form (DNF), that is, $xJ\_\{fw\}y\; :=\; \backslash exists\; w(wOx\; \backslash wedge\; wOy\; \backslash wedge\; wStj)\; \backslash vee\; \backslash exists\; w(wOx\; \backslash wedge\; wOy\; \backslash wedge\; wXj)$. Table 1 presents two examples of SWRL modeling from the mereotopological definitions. In SWRL, all variables are assumed to be universal and an existential variable is explicitly represented with the rules. For example, the existential condition of variable $z$ in overlap is restricted by using strict rules (i.e., $is\backslash \_part\backslash \_of(z,x)$ and $is\backslash \_part\backslash \_of(z,y)$).