Towards an Ontology for Generative Design of Mechanical Assemblies

Introduction

This posts is written after reading the study on an Ontology for Generative Design of Mechanical Assemblies by Bahar Aameri, et al. (2019)¹ for the project under Prof. Aameri during 2024 Summer. The paper studies the use of ontologies to model and reason about designs. The paper specifically provides an ontology to specify connection, parthood, and shapes in mechanical assemblies. The idea extends the Ground Mereotopology of Casati and Varzi (1999)² to multi-dimention, and combines it with a qualitative shape ontology based on the Hilbert’s axiomatic theory of geometry (1902)³.

Why Ontology?

There were many approaches to deal with configuration design problems. They includes setting constraints, using shape or generative grammers to modify a given design, and generic search algorithms. However, they were either not proven to be complete or the feasible solution space is so small compared to infeasible solution space. Therefore, the use of logical ontologies is investigated to serve two main purposes:

1. model qualitative constraints; and
2. prune infeasible designs during search.

The ontology presented in this paper is called the Assembly Ontology, which is a module of an overarching ontology called the PhysicalWorld Ontology. It is developed to specify contraints for a generative design software tool.

PhysicalWorld Ontology

The main aim of PhysicalWorld Ontology is to axiomatize concepts and properties required for representation and reasoning about physical domains. It consists of three main modules:

1. Assembly Ontology: will be discussed in detail below.
2. Occupation Ontology: specifies the relationship between a physical object and the space it occupies.
3. Kinematics Ontology: is an axiomatization of fundamental concepts required for qualitative representation of kinematics of physicsl systems

Each module has their own sub-modules and is axiomatized in first-order logic.

Past Related Work

The paper categorizes the existing related work in ontologies into three groups:

1. General consepts in assembly design and process: Common Feature Ontology by Imran and Young (2015, 2016), upper-level OWL by Mosca et al. (2009), OntoSTEP by Barbau et al. (2012) as the OWL translation of STEP, and Open Assembly Model by Fiorentini et al,. (2007) as standardiztion effort extending NIST’s CPM ontology.
2. Feature Ontologies: The first attempt from Borst et al., (1997) as a module of PhysSys ontology specifies the relationships between components of a physical system, and incomplete axiomatizations by Horváth et al. (1998). Horváth divides design features in three fundamental concepts: the set of components of the product (entities), the arrangements of the components (situations), and the physical environment the product is within (phenomena).
3. Ontologies of specific properties: Kim et al. (2008), Demoly et al. (2012), and Gruhier et al. (2015) take a similar approach to axiomatize assembly joining methods, but their ontologies are incomplete as they make implicit assumptions about shapes and dimensionality of assembly components.

Ontological Choices for the Assembly Ontology

An assembly is a collection of components that are attached together by some sorts of mechanical joints.

The paper focus on axiomatizing part-whole and connection relationships; shape, relative position, and dimensionality of componenets; and the boundaries of components.

Part-Whole and Connection Relationships

For formalization of connection and part-whole relations, multi-dimensional mereotopologies are used. Especially physical domains are typically finite, and this finiteness implies two restrictions on the mereotopological theory:

1. The theory must not be atomless as such models are infinite
2. The theory must not be extensional since finite mereotopological configurations are not necessarily extensional.

To read more about the theory of mereotopologies, see Aameri and Gruninger (2017)⁴.

Hence, the paper choose to extends the General Mereotopology (MT), the weakest theory among the mereotopologies presented by Casati and Varzi (1999)²

Shape, Relative Position, and Dimensionality

The paper choose to extend an ontology presented by Gruninger and Bouafoud (2011)⁵ to capture three-dimensionality of assembly components and their shapes. The ontology is based on a subtheory of Hilbert’s axiomatic theory of geometry (1902)³ which entails

1. Incidence relations: captures the relationship between entities with different dimensions.
2. Betweenness relations: captures the relative position of components.

Shape Boundaries

For the representation of boundaries, we should adopts two ontological viewpoints: boundary is a region which has empty interior (not consider as lower-dimensional entities) (Smith, 1996); and boundary is a lower-dimensional entity which is part of the bounded entity adopted by Baumann et al. (2016), and Hahmann (2013).

Ontological Commitments

The list of key ontological commitments in axiomatizing the Assembly Ontology:

1. There are four dimensions: there are primitive predicate representing zero-, one-, two-, and three-dimensional objects.
2. A collection of three-dimensional entities is a physical entity: there is a primitive predicate representing collections of three-dimensional objects.
3. Each physical entity is of exactly one of the four dimensions, or is a collection of three-dimensional objects
4. Individuals with different dimensions are only related by incidence relations: each class of objects is mereotopologically independent of the other classes, and there is no betweenness relationships between objects with different dimensions.
5. Zero- and one-dimensional entities do not exist independently of two-dimensional entities; two-dimensional entities do not exist independently of three-dimensional entities.
6. Relative positions of equi-dimensional individuals are captured by betweenness relations.
7. Spatial relationships between equi-dimensional individuals are captured by mereological and/or topological relations.
8. A boundary always bounds an entity with a higher dimension.
9. Every three-dimensional entity has at least one boundary, where the dimension of the boundary is exactly two-dimensional

The Assembly Ontology

The Assembly Ontology (AO) consists of three primary modules:

1. The Shape Ontology: based on the CardWorld and BoxWorld Ontologies by Gruninger and Bouafound (2011)⁵ which itself is based on Hilbert’s axiomatic theory of geometry by Hilbert (1902)³. The ontology specifies properties and relationships among five disjoint categories of entities that represent zero-, one-, two-, three-, and four-dimensional objects where four-dimensional objects are collections of three-dimensional objects.
2. The MT Multidimensional Object Mereotopology (MT MOM): captures mereological and topological relationships between individuals with zero-, one-, two-, and three-dimensions.
3. The Boundary Ontology*: extends the Shape Ontology with axioms that describe properties of physical boundaries.

The Shape Ontology

The Shape Ontology consists of a number of modules that extends from the CardWorld and BoxWorld ontologies. The main modules are CardWorld, BoxWorld, and PolyWorld ontologies which axiomatize properties of a stand-alone two-dimensional object, a stand-alone three-dimensional object, and relationships between sets of three-dimensional objects, respectively. Altogether, the ontology characterizes five disjoint categories: point, edge, surface, box, and poly.

The axioms of the Shape Ontology are decomposed into nine modules, specifying properties of two-dimensional shapes, three-dimensional shapes, and collections of three-dimensional shapes. I will not list all the axioms here, but encourage the reader to read the paper from the page 11¹. The axioms ensure disjointness between different class of objects and an incidence relation.

The MT Multidimensional Object Mereotopology

The MT MOM extends the Shape Ontology with two parthood predicates: surface_part and box_part; and four connection predicates: point_C, edge_C, surface_C, and box_C. It extends the Ground Mereotopology (MT) by Casati and Varzi (1999)², and characterizes mereotopological properties between objects in the same category.

The Boundary Ontology

The Boundary Ontology is a conservative extension of the Shape Ontology with binary predicates: point_bound(x,y), edge_bound(x,y), surface_bound(x,y) respectively denoting that $x$ is a point boundary, an edge boundary, or a surface boundary.

Example Applications of Assembly Ontology

The example given below is based on the suspension systems depicted in the original paper¹.

Axiomatic Description of Components

Torus

Idea: a torus is a box that has exactly one boundary surface, and no edges or vertices:

\[\begin{equation} \forall x( torus(x) := (box(x) \land \exists s(bound\_surface(s) \land in(s,x)) \land \forall s_1 (bound\_surface(s_1) \land in(s_1,x) \implies s_1 = s))) \end{equation}\]

Cylinder

Idea: a cylinder is a box with exactly three boundary surfaces, where two of them have a circular shape, and the third one shares a common edge with each of the other two surfaces. Also, a circular shape is a surface with exactly one boundary edge with no vertices:

\[\begin{equation} \forall s(circular(s) := (surface(s) \land \exists e (bound_edge(e) \land in(e,s)\land \forall e_1 (bound\_edge(e_1) \land in(e_1,s) \implies (e_1 = e))))) \end{equation}\]

Cube

Idea: a cube is a box with exactly eight quadrilateral boundary surfaces and at least twelve boundary edges, where a quadrilateral shape is defined as a surface with four boundary edges and four vertices.

Suspension Components

Axiomatic definitions of the shapes of the components:

\[\begin{align} \forall x\ (beam(x) &\implies cube(x))\\ \forall x\ (wheel(x) &\implies cylinder(x))\\ \forall x\ (shock(x) &\implies torus(x))\\ \end{align}\]

Disjointness of classes of components:

\[\begin{align} \forall x\ (beam(x) &\implies \lnot wheel(x) \land \lnot shock(x))\\ \forall x\ (wheel(x) &\implies \lnot shock(x)) \end{align}\]

Axiomatic Description of Joints

A fixed joint is a shared surface between two three-dimensional components (boxes): \(\begin{equation} \forall x,y (fixed\_joined(x,y) := (box(x)\land box(y)\land \exists z (surface(z) \land in(z,x) \land in(z,y)))) \end{equation}\)

A ball joint is a shared point between two three-dimensional components (boxes): \(\begin{equation} \forall x,y(ball\_joined(x,y) := (box(x)\land box(y)\land \exists z (point(z) \land in(z,x) \land in(z,y)))) \end{equation}\)

Axiomatic Description of Suspension Systems

TBU

1. Aameri, B., Cheong, H., & Beck, J. (2019). Towards an ontology for generative design of mechanical assemblies. Appl. Ontology, 14, 127-153. ↩︎ ↩︎² ↩︎³

2. Casati, R. and Varzi, A. (1999). Parts and places: The structures of spatial representation. MIT Press ↩︎ ↩︎² ↩︎³

3. Hilbert, D. (1902). The foundations of geometry. Open court publishing Company. ↩︎ ↩︎² ↩︎³

4. Gruninger, M. and Aameri, B. (2017). A new perspective on the mereotopology of RCC8. In COSIT 2017. ↩︎

5. Gruninger, M. and Bouafoud, S. (2011). Thinking outside (and inside) the box. In Proceedings of SHAPES 1.0: The Shape of Things. Workshop at CONTEXT-11, volume 812. CEUR-WS. ↩︎ ↩︎²