# Integrated product relationships management: a model to enable concurrent product design and assembly sequence planning

Frédéric Demoly, Xiu-Tian Yan, Benoît Eynard, Samuel Gomes, Dimitris Kiritsis

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Frédéric Demoly, Xiu-Tian Yan, Benoît Eynard, Samuel Gomes, Dimitris Kiritsis. Integrated product relationships management: a model to enable concurrent product design and assembly sequence planning. Journal of Engineering Design, 2012, 23 (7), pp.544-561. 10.1080/09544828.2011.629317. hal-05455053

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A theoretical framework for product relationships
description over space and time in integrated design

<table><tr><td>Journal:</td><td>Journal of Engineering Design</td></tr><tr><td>Manuscript ID</td><td>CJEN-2015-0157.R1</td></tr><tr><td>Manuscript Type:</td><td>Article</td></tr><tr><td>Keywords:</td><td>design representations < Product and Systems Modelling, Design Theory and Research Methodology, information representation < Knowledge and Information Management, design understanding < Knowledge and Information Management, mechanical product design < Engineering Design in Industry</td></tr></table>

SCHOLARONE™
Manuscripts
Journal of Engineering Design

Vol. 00, No. 00, Month 200x, 1–31

ORIGINAL ARTICLE

A theoretical framework for product relationships
description over space and time in integrated design

(Received 10 June 2015; revised version received 20 September 2015)

This paper presents a novel qualitative description theory in the context of in-
tegrated design, which here incorporates assembly sequence planning in the
early product design stages (also called assembly-oriented design - AOD).
Based on a literature review of current AOD approaches, product models and
mereotopology-based theories, the authors introduces a promising mereotopo-
logical theory which enables the formal product relationships description in
integrated design by introducing an emerging framework, four dimensional-
ism (i.e. perdurantism in philosophy). The proposed efforts aim at providing
a concrete basis for describing the evolution of spatial entities (i.e. product
parts) and their relationships over time and space during the design phases by
considering their related assembly processes. In product design, the interpreta-
tion of engineering reality from a perdurantist point of view leads to a seamless
consideration of relationships between spatial, temporal and spatiotemporal di-
mensions. Indeed, with such a mereotopological theory, formal product design
description is enabled by proactively considering its assembly sequence from
a temporal point of view in the early stages of the product development pro-
cess, and ensures information and knowledge consistency over space and time.
Finally, an ontological implementation is presented within Protégé through a
mechanical assembly as a case study, so as to illustrate the applicability and
the relevance of the theory.

**Keywords:** Mereotopology; Qualitative description ; Product design ; Assembly sequence planning ; Design formalism ; Product-process description

**1. Introduction**

The current industrial context is faced to severe competition and leads companies to
make numerous tradeoffs in terms of internal and external development strategies and
contraints. Industry is at a turning point where production strategy (i.e. global versus
local), product strategy (e.g. short versus long-life cycle), information/knowledge cap-
ture/accessibility stragegy (e.g. ontology-based, cloud-based, etc.), to name a few, need
to be addressed in a well-balanced manner. From an internal point of view, engineering
processes become more and more knowledge-intensive and then require adapted intel-
ligent environment, especially in product design and manufacturing. Indeed during the
product development process, product architects and designers decisions have a major impact on downstream processes and then demand better understanding and awareness in their activities. Currently, the knowledge-intensive and scattered information systems covering the beginning of product lifecycle also requires advanced interaction, information flows consistency and interpretable description, especially in the early design phases where proactive integration of knowledge from downstream processes and qualitative specification are needed (Demoly et al. 2013). This can be done through the fact of delivering product models in line with realistic situations over time. Such stake opens new research challenges in engineering design, which covers different research areas, such as philosophical investigations, mathematics, artificial intelligence and engineering to name a few. In general, products models are considered as a representation of our perception of the reality from a three-dimensionalism (i.e. endurantism in philosophy) point of view. Literature has proved that researchers have built product models regarding their functionalities and their extension in space. Endurantism actually supports that objects have three spatial dimensions and move through time. In others words, this conception argue that material objects are persisting things wholly present at every moment of their existence.

Nowadays this kind of framework is widely accepted in industry and conditions current engineering applications related to 3D product definition. At first glance, this trend is understandable since it does not exist yet ubiquitous system able to represent information of all things at all times. This has provided real advances in engineering and manufacturing but also causes difficulties in engineering definitions and wrong design interpretations. By following three-dimensionalism, design activity is mainly dedicated on the development of the spatial aspect of the product without considering intrinsically its evolution over time. A possible reason is due to the functional, geometric, physical complexity and multiple temporal configurations inferred in the design process, and also when the product information passes from design to manufacturing engineering and later stages (Chandrasegaran et al. 2013). This recurring issue denotes a lack of association between product and process models at various abstraction levels, particularly at a logical and semantical point of view (Demoly et al. 2012, 2013).

These aforementioned stakes fall within a broader research challenge which consists in the knowledge integration of products lifecycle phases (i.e. process planning, assembly planning, etc.) in an appropriate manner at the earliest product design stages (Demoly et al. 2012). To tackle this challenge, Demoly et al. (Demoly et al. 2012) have highlighted the current stakes in engineering design and emerging needs in proactive engineering based on qualitative description of lifecycle knowledge, so as to improve designers' awareness and understanding. This statement has also been claimed by Kusiak and Salustri (Kusiak and Salustri 2007), who have stated that design engineering is currently changing from an informal approach based on experience to a science-based approach (Zeng and Gu 1999).

Furthermore the dynamic aspect of the product definition has not been yet described in a seamless fashion (Zeng and Gu 1999). Indeed, current CAD applications are mainly dedicated the spatial and geometric definition and does not capture the design story (that is quite different from a CSG tree for example) from a temporal point of view. Such design practices actually lead to static product definition and time-consuming efforts from manufacturing side in order to interpret assembly intents for example. A similar issue can be identified in engineering change management, where changes are represented through objects' effectiveness. This results in the capture of multiple product configurations and increase complexity in the management of product design information. But evolution
and changes ((i.e. related to part, assembly, relationship, etc.) that may occur during the product definition are not adequately described, captured and represented. Hence, it is needed to refer to philosophical concepts which aid to identify things to be described here. A promising one is about four dimensionalism or perdurantism, on which an object has distinct temporal parts throughout its existence.

Based on these research stakes, new research efforts have to be addressed on formalism and theory in engineering design in order to represent product information and knowledge in a qualitative and machine-interpretable manner. The main objective is therefore to propose and describe a region-based theory (called mereotopology) for assembly-oriented design (AOD), which is built on four dimensionalism. The paper firstly presents, in section 2, a comprehensive literature review on AOD approaches and mereotopology-based theories so as to highlight the research issue to be tackled. Then section 3 describes the proposed theory, which provides product-process associations through semantics and logics. Section 4 introduces an ontological implementation within Protégé, in which a mechanical assembly is defined in order to illustrate the relevance and the application of the proposed theory. In section 5 the limits and the advantages of such a theory are discussed. Finally, in section 6, conclusions and future work are given.

## 2. Literature review

### 2.1. Assembly-Oriented Design and related product models

The concept of integrating assembly sequence planning (ASP) in product design was introduced at the beginning of the previous decade in order to overcome the current limitations of design for assembly (DFA) and ASP approaches. Based on detailed product geometry, and a part-to-part oriented evaluation, DFA approaches generally lead to a redesign. In such a context, the issue of concurrent product design and ASP (Zha and Du 2002), also called assembly-oriented design (AOD) (Mantripragada 1998), has received much attention in research work during the last decade (Wang et al. 2008). Product design and assembly sequence planning phases are normally undertaken separately and sequentially, which results in missing an efficient integration between both phases. Here, the assembly-oriented practice of product development can be considered as a top-down approach by proactively considering the assembly related product design and its relationship issues in the early phases of the product development process.

Among the models and ontology in the literature, relevant work has been done by focusing on assembly formalisms and semantics, in a collaborative product development context (Kim et al. 2008). In addition, NIST has proposed generic models, such as Core Product Model (CPM) (Fenves et al. 2008b), CPM2 (Fenves et al. 2008a) and Open Assembly Model (OAM) (Sudarsan et al. 2005) that are particularly relevant to represent the product in an integrated manner, including assembly features and so on. Indeed, Swain et al. (Swain et al. 2014) has extended OAM model by using extended liaison so as to associate assembly and design processes. Here liaison supports technological notions such as riveting, welding and so on. As a result of these efforts, the introduction of semantics and knowledge in assembly models has enabled a better understanding and interpretation of the product architects' and assembly planners' intents. These representations facilitate reasoning about assembly relationships for assembly process planning and assembly line balancing issues as well as, in a more efficient manner, at a high abstraction level of the product definition. More close to the product relationships description issue, Horvath and van der Vegte (Horvath and van Der Vegte 2003) have proposed a
nucleus-based modelling theory, in which three groups of reflexive relations have been defined (i.e. existence, reference, substance) and eight kinds of binary relations are also defined (i.e. connectivity, positioning, morphological, kinematical, deformation, kinetic, physical phenomenon nad physical field).

Although these research efforts have provided strong foundation and support for designers to understand physical effects in product design, it remains a need to provide a unified foundation which enables the entire understanding of the establishment of product relationships over time in order to bridge product with process models.

## 2.2. *Mereotopology-based theories*

Lesniewski (Lesniewki 1929b) was the first to describe a mereology-based theory including theorems and axioms, in order to develop the parthood relationship in a formal manner (Demoly et al. 2012). This initial effort has provided an interesting basis and also led to limits and basic problems in mereology (Varzi 1998, Salustri 2002). As an automotive example, the speedometer is *Part of* the dashboard, the dashboard is *Part of* the car and the car is *Part of* the fleet. This seems to be logical sentences, but in no case the speedometer is *Part of* the fleet. The problem encountered here is that parthood relationship is transitive (i.e. parthood means only one thing at once) in the first two statements and intransitive (i.e. parthood means something else) in the last one (Salustri and Lockledge 1999).

To overcome this paradox, the notion of topology has been incorporated and considered together with mereology so as to initiate mereotopology. This theory enables the qualitative formalisation of two fundamental predicates: parthood (i.e. one entity is *Part of* another) and connection (i.e., an entity is *connected to* another, denoted *C*) (Salustri 2002). By developing such a mereotopological theory, two main notions have to be introduced, so as to describe product evolution, such as regions (e.g. a spatial region is a portion of space occupied by some entities such as a physical part (Salustri 2002)) and primitives (i.e. used to describe region-to-region relationships). The current challenge of the mereotopology-based theory in engineering design is to consider the product as it is perceived in the real world (Demoly et al. 2012). In other words, mereotopology describes relationships between parts with an “engineering sense”. Here, the engineering sense requires a more accurate and more structured prospect (Salustri 2002) needed in AOD.

Relevant existing theories in product design is presented in Table 1 according to authors, their first spatial primitives on which theories are based, related primitives in spatial, temporal and even spatiotemporal dimensions and their application domain (Gruhier et al. 2014). Based on this primitives inventory, missing parts can be highlighted to fully cover the dynamic aspect of the product design domain. Besides, it seems that most of the listed theories does not cover entirely spatial and temporal dimensions, but rather provide an accurate description of spatial dimension for a specific area.

A novel dimension in mereotopological theory is needed, insofar as objects evolution has to be described over space and time. Moreover, temporal description is not applied to product design information, which means that evolution in product definition assembly cannot be considered. Table 1 enables the identification of primitives used in the mechanical engineering domain. It seems that primitives such as *Part of*, *Interior Part of*, *Overlap*, *Tangent* or their equivalents are used by every author in the research area.

By focusing on mechanical engineering application, few research efforts have been done, especially at different abstraction levels (i.e. functional, kinematic, technological, etc.).
<table><caption>Table 1. Relevant region-based theories</caption><thead><tr><th rowspan="2">Author</th><th colspan="2">Dimension</th><th rowspan="2">Temporal</th><th rowspan="2">Spatial-temporal</th><th rowspan="2">Application domain</th></tr><tr><th>Spatial</th><th>Developed primitive</th></tr></thead><tbody><tr><td>Allen (1983)</td><td></td><td></td><td>&lt;=</td><td></td><td>Artificial intelligence</td></tr><tr><td>Asher and Vieu (1995)</td><td>C</td><td>P; PP; O; EC; TP; NTP</td><td></td><td></td><td>Artificial intelligence</td></tr><tr><td>Bennett et al. (2000)</td><td>P; S</td><td>PP; D; O; PO; B;COB; EQD; MID; NEARER</td><td></td><td></td><td>Artificial intelligence</td></tr><tr><td>Borgo et al. (1996)</td><td>P; SR; CG</td><td></td><td></td><td></td><td>Knowledge representation and reasoning</td></tr><tr><td>Clarke (1981)</td><td>C</td><td></td><td>&lt;</td><td>⊂t</td><td>Mathematics</td></tr><tr><td>De Laguna (1922)</td><td>C</td><td></td><td></td><td></td><td>Philosophy</td></tr><tr><td>Demoly et al. (2012)</td><td>P</td><td>O; IP; D; Point; X; St; B; T; IB</td><td>&lt;</td><td>Ot; ⊂t; *</td><td>Mechanical assembly</td></tr><tr><td>Duntsch et al. (2001)</td><td>C</td><td>P; PP; O; PO; TPP; NTPP; EC; DC</td><td></td><td></td><td>Computer Science</td></tr><tr><td>Eschenbach (1999)</td><td>P; DC</td><td>PP; O; PO; D; C; EC; TPP; NTPP; component</td><td></td><td></td><td>Cognitive science</td></tr><tr><td>Galton (1997)</td><td>C</td><td>DC; P; O; D; C; EC; PO; EQ; PP; TPP; NTPP</td><td>&lt;=</td><td></td><td>Knowledge representation and reasoning</td></tr><tr><td>Goodman (1951)</td><td>D</td><td>P; O</td><td></td><td></td><td>Mathematics</td></tr><tr><td>Kim et al. (2008)</td><td>P; IP</td><td>O; D; Point; X; St; B; T</td><td></td><td></td><td>Mechanical assembly joint</td></tr><tr><td>Lemon and Pratt (1998)</td><td>C</td><td></td><td></td><td></td><td>Visual languages and computing</td></tr><tr><td>Lesniewki (1929a)</td><td>I</td><td></td><td></td><td></td><td>Mathematics</td></tr><tr><td>Muller (2002)</td><td>C</td><td>P; PP; O; PO; EC; TP; NTP; TPP; NTPP; TPPI; NTPPI</td><td>&lt;</td><td>TS; ⊂t; *; MOTION</td><td>Knowledge representation and reasoning</td></tr><tr><td>Randell and Cohn (1989)</td><td>C</td><td>DC; P; O; PP; D; EC; TP; NTP; TPP; NTPP; TPI; NTPI; PO</td><td>B</td><td></td><td>Knowledge representation and reasoning</td></tr><tr><td>Salustri (2002)</td><td>P; C</td><td>PP; O; EC; TPP; NTPP; SC; E</td><td></td><td></td><td>Mechanical engineering</td></tr><tr><td>Smith (1996)</td><td>P; IP</td><td>O; D; Point; X; St; B</td><td></td><td></td><td>Mechanical engineering</td></tr><tr><td>Stell and West (2004)</td><td>C</td><td>P; PP; O; EC; NTPP</td><td>&lt;</td><td>Pt; GAL; Hc</td><td>Information system</td></tr><tr><td>Tarski (1956)</td><td>P; S</td><td></td><td></td><td></td><td>Mathematics</td></tr><tr><td>Varzi (1996)</td><td>P; C</td><td>PP; O; U; OX; UX; PO; PU; EC; TP; IP; E; IE; TE; SU; SX; PS</td><td></td><td></td><td>Philosophy</td></tr><tr><td>Whitehead (1929)</td><td>R; C</td><td></td><td></td><td></td><td>Mathematics and philosophy</td></tr></tbody></table>

< (Precedence), ⊂t (temporal inclusion), * (temporal connection), B (Wholly before), C (Connected), D (Discrete), E (Enclosure), EC (Externally Connected), IB (Internal Boundary), IP (Interior Part), NTPP (Non Tangential Proper Part), O (Overlap), Ot (temporal overlap), P (Part of), Point, X (Cross), PP (Proper Part), SC (Self-Connection), St (Straddle), T (Tangent), TPP (Tangential Proper Part).

More precisely Demoly et al. (Demoly et al. 2012) have described product relationships by mainly focusing on kinematic pairs. Kim et al. (Kim et al. 2008) have proposed a region-based theory to differentiate assembly joints obtained by welding or adhesive bonding. They both used Smith's mereotopological primitives (Smith 1996). In addition, Salustri (Salustri 2002) has also introduced similar primitives which have the same properties but not the same names (e.g. InteriorPart IP is equivalent to Non Tangential Proper Part NTPP or Tangential Proper Part TPP), so as to cover product modelling issue. Moreover he did not use the Crosse X primitive, which seems to be relevant in the above mentioned context, and which is intensively used during assembly sequence planning. Finally, Randell and Cohn (Randell and Cohn 1989) research works have enabled the representation of relationships between fluid and solid objects at different states.
## 2.3. Motivation

The outcome of the above literature review highlights need for a new qualitative description theory in the context of integrated design, so as to extend current engineering capabilities and overcome current philosophical framework (i.e. endurantism) in design systems. Indeed, most of the research works have been based upon an endurantist point of view in product design so far, by only considering product definition in the spatial dimension. Such engineering practices led to understand design changes in terms of things, that is not enough suitable to be aware of design and assembly evolution. The fact of considering assembly sequence planning issue in product design in a proactive fashion requires a novel description able to associate product and process models from a logical and semantical point of view. This therefore causes a revision of the commonly used philosophical foundation. The whole thing is to describe the entire evolution of the product definition over space and time. Such issue introduces a promising philosophy, perdurantism, in order to represent spatial, temporal and spatiotemporal objectifs in a unified framework (Al-Debei et al. 2012). In this way, engineering practices will interpret things in terms of changes.

<figure><img src="image_4.png" /><figcaption>Figure 1. Proposed vision of the research</figcaption></figure>

As a consequence, the main objective of the paper is to describe product-process relationships in a formal manner by considering a perdurantist philosophy. The development of a first-order logic mereotopological theory is addressed to cover three dimensions (i.e. spatial, temporal and spatiotemporal). The theory enables to reason and provides a complete awareness and understanding of product evolution (i.e. design changes and assembly plan) to product architects and designers. Built on this, Figure 1 presents the overall research vision, in which only Step 1 (gray disc of Figure 1) is addressed in this article. Compared to recent research works (Demoly et al. 2012, Kim et al. 2008), the
current efforts are focused on the description of product evolution in the context of AOD. As such, this paper extends existing spatial mereotopology-based theories in the temporal and spatiotemporal dimensions in order to describe persistent objects over time and space. In addition, this step requires an ontological implementation of the theory by using OWL-DL (Ontology Web Language - Description Logic) and SWRL (Semantic Web Rule Language) (Gruhier et al. 2015). Therefore, its machine interpretation and usage within PLM (Product Lifecycle Management) systems (Step 2 of Figure 1) and CAD (Computer-Aided Design) application (Step 3 of Figure 1) are supported. Ontology also enables having a common semantic between all stakeholders (i.e. product architect, assembly planner and designer) and improves understanding with its reasoning capabilities.

The second step of Figure 1 will set up complex rules to reason, check information consistency and understand designer's intents within Product Data Management (PDM) and Manufacturing Process Management (MPM) systems. Thus, information related to the product evolution will be available to product architects and designers. In addition, top-down design approach will be considered so as to facilitate the designer's understanding and awareness. The third step of Figure 1 will introduce a spatiotemporal layer in CAD systems by supporting the swept volumes visualisation related to design changes and assembly process. The understanding of the product architect and the designer and their awareness about their decisions will be then facilitated with such spatiotemporal descriptions. Steps 2 and 3 of Figure 1 will be approached in a future paper.

### 3. Spatiotemporal mereotopology for product-process design description

The current section describes the developed theory, called JANUS (Joined Awareness and Understanding in assembly-oriented design with mereotopology), in the field of AOD (Gruhier et al. 2014). The theory foundation is based on regions and primitives covering spatial, temporal and spatiotemporal dimensions.

#### 3.1. Overall description of JANUS theory

##### 3.1.1. General philosophy

Figure 2 introduces the main objective of JANUS theory. Indeed, the formal representation of the relational information of the product with its parts and its lifecycle operations (i.e. here assembly operations) at various abstraction levels, is a critical point in the product design process. At this stage, an additional layer is needed so as to ensure a seamless integration of assembly sequence planning in product design with semantics and logics. In this article, spatial objects as well as temporal objects are introduced in order to formally describe product-process relationships in a machine-interpretable manner. The fact of introducing spatiotemporal objects (such as swept volumes) and spatiotemporal relationships enables the appropriate understanding of spatial objects evolution (i.e. over time) and then their associations. By introducing such a formal representation, AOD tackles current engineering issues related to the integration of product lifecycle constraints and knowledge in a proactive manner. The perdurantist paradigm leads product architects and designers to play a role through spatial, temporal and spatiotemporal dimensions (also considered as four-dimensionalism) (Sider 2001).
<figure><img src="image_5.png" /><figcaption>Figure 2. Introduction of a semantics and logics layer for ensuring product-process integration</figcaption></figure>

Moreover, the JANUS theory is derived from ISO 159262 Standard (Standard 2003) – used in oil and gas industry – to describe a foundation based on industrial needs for the support of information changes over time (Demoly et al. 2012). This standard is based on four-dimensionalism, which means that object can be extended over space and time (Batres et al. 2007). In such a context, the object can be described in four dimensions so as to enable the existence of the past, present and future. The object is then considered as being perdurant over time and is defined according to its relationships with others. In this approach change is expressed using mereotopology through the object lifecycle (Sider 2001).

### 3.1.2. Object change definition

The theory frame, is based on Le Moigne's framework (Le Moigne 1994). By stating that product is composed of spatial parts assembled over time, the author has considered that objects differ from others according to:

* Their position in space;

* Their position in time;

* Their form (i.e. the object structure).

As soon as an attribute (i.e. time, space and form) is modified, a change during the assembly design occurs. This implies that time can be considered with spatial coordinates. This point is related to the concept of *spatialised time* or *spatial time*. As such, Bergson (Bergson 1923) stated "I simultaneously get that I think in duration and that I am in duration". This means that "I am defining the product and I am aware of its evolution (assembly sequence and design changes)" in the AOD context. The definition of spatial time from Heidegger (Heidegger 1962) can be adapted to the moving process in assembly, which starts from the object initial position and finishes as soon as the object reaches its final destination (and changes its spatial primitives). By adding temporal part, it enables the description of the object changes during product design stages. The object description – *when* (defined by the assembly planner), *where* (defined by the product architect), *what* form (defined by the designer) – can be considered as complete with such kind of modelling technique (i.e. theory frame). Afterwards, the stakeholders can
<table><caption>Table 2. Fundamental mereotopological operators (Demoly et al. 2012)</caption><thead><tr><td>Symbol</td><td>Name</td></tr></thead><tbody><tr><td>∧</td><td>Logical conjunction</td></tr><tr><td>∨</td><td>Logical Disjunction</td></tr><tr><td>:=</td><td>Definition</td></tr><tr><td>→</td><td>Logical implication</td></tr><tr><td>∃</td><td>Existential quantifier</td></tr><tr><td>¬</td><td>Logical negation</td></tr><tr><td>∀</td><td>Universal quantifier</td></tr><tr><td>∅</td><td>Empty region</td></tr><tr><td>A</td><td>Overall assembly</td></tr><tr><td>φ</td><td>Condition</td></tr><tr><td>σ</td><td>Sum (fusion or join)</td></tr><tr><td>≡</td><td>Equivalence</td></tr><tr><td>ι</td><td>Definite descriptor</td></tr></tbody></table>

understand the transformation process (how and why questions) (Rodier et al. 2010). As such, the object adaptation is seen as understandable (e.g. why a planar pair has been used?). Indeed, the product lifecycle is a series of adaptations and decisions that the designer has to make. Since an object is related to numerous other objects, decisions will impact the entire assembly (and not only the object itself).

## 3.2. Description of the spatial dimension of JANUS

The spatial dimension of the theory describes the spatial mereotopological relationships between spatial objects (e.g. mechanical parts) at a specific instant (e.g. at a particular brief moment where a part touches another part) of the assembly process. Each spatial object is considered, as a spatial region at a specific instant. Here a discrete space is considered as the description of the spatial position of an object according to others, which depends on their spatial mereotopological primitives. An empty region (denoted ∅), where no object is located, is also considered as a spatial region. The overall assembly (denoted A) is the region considering all regions forming the final product. Spatial relationships capture the way of which objects are related at a certain time (Del Mondo et al. 2010). So the relative position of objects compared to others can be fully described (Rodier et al. 2010). Each region is in relation with at least another region. If there is no contact between two objects, then the Discrete D primitive is used. From that moment on, the relationship is valid for every spatial primitive (except discrete primitive) and the letter R, which means Relationship, is used. Consider that at time 't' there are two different parts: x and y (see Equation (1)).

$$ \forall t \forall x \exists y / x_t R y_t \quad (1) $$

Here spatial primitives are introduced to formally describe part-to-part relationships. They are based on Smith's mereotopological primitives (Smith 1996), and have already been addressed and adapted in Demoly et al. (Demoly et al. 2012) and Kim et al. (Kim et al. 2008), who have both focused their research efforts in assembly design and sequence planning. As for Salustri (Salustri 2002), similar primitives – same properties but not the
<figure><img src="image_6.png" /><figcaption>Figure 3. Spatial mereotopological primitives representation (Demoly et al. 2012)</figcaption></figure>

<table><caption>Table 3. Spatial mereotopological primitives description (Demoly et al. 2012)</caption><thead><tr><th>Name</th><th>Mereotopological description</th></tr></thead><tbody><tr><td>x is Part of y</td><td>x<sup>P</sup>y := &forall;z(z<sup>O</sup>x &rarr; z<sup>P</sup>y)</td></tr><tr><td>x is InteriorPart of y</td><td>x<sup>IP</sup>y := x<sup>P</sup>y &and; &not;x<sup>T</sup>y</td></tr><tr><td>x Overlaps y</td><td>x<sup>O</sup>y := &exist;z(z<sup>P</sup>x &and; z<sup>P</sup>y)</td></tr><tr><td>x is Discrete from y</td><td>x<sup>D</sup>y := &not;x<sup>O</sup>y</td></tr><tr><td>x Crosses y</td><td>x<sup>X</sup>y := &not;x<sup>P</sup>y &and; &not;x<sup>D</sup>y</td></tr><tr><td>x Tangent y</td><td>x<sup>T</sup>y := &exist;z(z<sup>P</sup>x &and; z<sup>B</sup>y)</td></tr><tr><td>x is Boundary of y</td><td>x<sup>B</sup>y := &forall;z(z<sup>P</sup>x &rarr; z<sup>Sty</sup>y)</td></tr><tr><td>x Straddles y</td><td>x<sup>Sty</sup> := &forall;z(x<sup>IP</sup>z &rarr; z<sup>X</sup>y)</td></tr><tr><td>Sum of φ ers</td><td>&sigma;x(φx) := &iota;y(&forall;w(w<sup>O</sup>y &equiv; &exist;v(φv &and; w<sup>O</sup>v)))</td></tr></tbody></table>

same names (e.g. *InteriorPart* *IP* is equivalent to *NonTangentialProperPart* *NTPP* or *TangentialProperPart* *TPP*) – have been introduced. Besides the same author did not use the *Crosse* *X* primitive, which seems to be relevant in the above mentioned context and is intensively employed in ASP. By using mereotopological operators (see Table 2), eight spatial primitives have been represented in Figure 3 and described in Table 3. Here *z* is considered as a spatial region. In addition, “not *x*” is a logical negation and an operation that can be interpreted as being true when its operand (i.e. quantity on which a mathematical operation is performed) is false, and false when its operand is true.

At the beginning of the design phase, part-to-part relationships are considered as critical input for AOD. This information is then described into geometric skeletons in order to control geometric assembly models in a central manner (Demoly et al. 2011). At a more detailed level, the condition *φ* is used and states that a condition *φ* in a single variable *x* is satisfied if and only if the sentence *φx* is true for at least one value of *x*. Based on this assumption, it is possible to introduce and define two entities (Figure 4 and Figure 5), on which the object form depends (Demoly et al. 2012):

* *Assembly skeleton*, which ensures assembly positioning, is represented by *k* (e.g. straight line representing the rotation and/or translation axis): $k := \sigma z(\phi z) \rightarrow \forall z(\phi z \rightarrow (z \mathbf{T} x \wedge z \mathbf{T} y) \wedge (z \mathbf{P} x \wedge z \mathbf{P} y))$ (See Figure 4);

* *Interface skeleton*, which describes geometric boundaries (e.g. circle, square) used to build a functional surface and supported by an assembly skeleton, is represented by *f*: $f := \sigma z(\phi z) \rightarrow \forall z(\phi z \rightarrow z \mathbf{B} x \wedge z \mathbf{B} y)$ (See Figure 4).

Multiple assembly or interface skeletons on the same assembly, are respectively denoted
$k_1$ and $k_2$ or $f_1$ and $f_2$. Moreover, the interface skeleton type depends on the shape of the parts on which a physical contact exist.

<figure><img src="image_7.png" /><figcaption>Figure 4. Geometric representation of *k* and *f* entities (Demoly et al. 2012)</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 5. Example of cylindrical pair assembly and related skeletons (i.e. assembly and interface skeletons)</figcaption></figure>

### 3.3. Description of the temporal dimension of JANUS

Based on the spatial part of JANUS, the temporal aspect is then defined with temporal relationships and temporal objects. The lifetime of an object is broken down into several smaller temporal objects (Hadjieleftheriou et al. 2002), which are created each time a part change occurs (Rodier et al. 2010). The object change is generally associated to an
event or a process. The purpose here is to describe changes that affect objects during design and assembly process. As soon as an event occurs, changes occur at time 't'. An event indicates a discontinuity in a stable referential. There is a break between the "before" and the "after" of the event. In that sense, Sider (Sider 2001) provides the following definition: x is an *Instantaneous Temporal Part of y* if, at instant *t*, x exists at, but only at, *t*. Therefore, the region is only spatial and there is no swept volume (i.e. no spatiotemporal region). On the other hand swept volumes, which are generated during assembly process, are used to understand and represent product evolution over time. So process expresses a change during the intern evolution of the object and is composed of an initial event, a duration between both events and a final one.

Here each temporal object is considered as a Temporal Region (TR). To understand the notion of temporal region, mechanical assembly design history has to be considered as a story. It describes when the base part (i.e. first part to be assembled) is placed, how the second is assembled to the previous one and so on. As commonly admitted, this story has temporal regions created at each new assembled spatial region. Temporal regions can be either intervals or instants (Sider 2001). Assembly operations are a kind of temporal regions. Thus there exist two types of temporality, such as the temporality of the object itself (i.e. how it evolves over time) and the temporality of the object regarding to the others (i.e. which is shown with spatial primitives).

The relative chronology between assembly operations can be understood by using temporal relationships (Rodier et al. 2010). Table 4 shows the temporal relationships between two intervals (third column), one interval and one instant (fourth column), and two instants (with **Bt** the temporal boundary primitive) (fifth column). The proposed description is similar to Allen's intervals (Allen 1983), who formalized topological relationships between temporal intervals. Here, *a*, *b* and *c* are used to describe three temporal regions. Since the dimensions of the regions and primitives in the section 3.3 (i.e. temporal dimension) are different from the ones in section 3.2 (i.e. spatial dimension), the authors have decided to use other letters for variables in order to remove any ambiguity.

Temporal primitives are inspired from spatial primitives and adapted to represent every temporal phenomenon that may be identified during an assembly process. Phenomena have been classified according to the type of mechanical assemblies. Just two of them can describe all types of assembly *presented* in Table 5: *Temporally Precede* (denoted *tp*) and *Temporally Tangent* (denoted *Tt*). The different types of assembly are considered as follows (Demoly et al. 2011): interconnected serial, serial, constrained serial and parallel (see Table 5). Both "interconnected serial" and "constrained" are constraint dependency relationships. The only difference is that in the latter, some parts can be temporally constrained by other parts without being in physical contact with them (i.e. precedence constraint). For instance, in the "constrained" assembly type, *Part 2* must be assembled before *Part 3* and there is no contact between both parts. As a consequence, time is considered as linear (Renolen 1999) and discrete.

## 3.4. Description of the spatiotemporal dimension of JANUS

### 3.4.1. Basic problem with spatiotemporal visualisation

The description of assembly design over space and time (evolution) requires an additional dimension, the spatiotemporal dimension. Each spatiotemporal object is considered as a spatiotemporal region such as swept volume during deformation, modification or transformation. The latter notion is the spatial region occupied by an object during
<table><caption>Table 4. Temporal mereotopological primitives description and representation</caption><thead><tr><th>Name</th><th>Mereotopological description</th><th>Repres. with two intervals</th><th>Repres. with one interval and one instant</th><th>Repres. with two instants</th></tr></thead><tbody><tr><td>b is TemporallyPart of a</td><td>bPta := &forall;c(cOtb &rarr; cPta)</td><td><u>a</u> <u>b</u></td><td><u>a</u> <u>b</u></td><td></td></tr><tr><td>b is StartTemporallyPart of a</td><td>bPtsa := &exist;c(bPta &and; cTtb &and; ctpa)</td><td><u>b</u></td><td><u>b</u></td><td></td></tr><tr><td>b is FinishTemporallyPart of b</td><td>bPtfa := &exist;c(bPta &and; bTtc &and; atpc)</td><td><u>a</u> <u>b</u></td><td><u>a</u> <u>b</u></td><td></td></tr><tr><td>b is TemporallyInteriorPart of a</td><td>bIPta := bPta &and; &not; bPtsa &or; &not; bPtfa</td><td><u>a</u> <u>b</u></td><td><u>a</u> <u>b</u></td><td></td></tr><tr><td>b TemporallyOverlaps a</td><td>bOta := &exist;c(cPtb &and; cPta)</td><td><u>b</u> <u>a</u></td><td></td><td></td></tr><tr><td>a TemporallyPrecedes b</td><td>atpb := &exist;c(aTtc &and; bPtfc)</td><td><u>a</u> <u>b</u></td><td><u>a</u> <u>b</u></td><td><u>a</u> <u>b</u></td></tr><tr><td>b is TemporallyTangent of a</td><td>bTta := &exist;c(cPtsa &and; cBtb)</td><td><u>b</u></td><td><u>b</u></td><td><u>a</u> <u>b</u></td></tr><tr><td>b is TemporallyEqual to a</td><td>btea := bIPta &and; aIPtb</td><td><u>a</u></td><td></td><td><u>a</u></td></tr><tr><td></td><td></td><td><u>b</u></td><td></td><td><u>b</u></td></tr></tbody></table>

<table><caption>Table 5. Assembly types of three components (Part 1, Part 2 and Part 3) (Demoly et al. 2011)</caption><thead><tr><th>Assembly type</th><th>Interconnected serial</th><th>Serial</th><th>Parallel</th><th>Constrained</th></tr></thead><tbody><tr><td>Directed graph</td><td><img src="image_0_0.png" alt="A directed graph with three nodes labeled t12, t23, and t13."></td><td><img src="image_0_1.png" alt="A serial representation of the directed graph."></td><td><img src="image_0_2.png" alt="A parallel representation of the directed graph."></td><td><img src="image_0_3.png" alt="A constrained representation of the directed graph."></td></tr><tr><td>Example</td><td><img src="image_0_4.png" alt="A 3-D object model."></td><td><img src="image_0_5.png" alt="A 3-D object model with a solid, shaded region."></td><td><img src="image_0_6.png" alt="A 3-D object model with a solid, shaded region and a hole."></td><td><img src="image_0_7.png" alt="A 3-D object model with a solid, shaded region and a hole, and a 3-D object model with a hole."></td></tr></tbody></table>

<figure><img src="image_9.png" /></figure>

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<figure><img src="image_11.png" /></figure>

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<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /></figure>

a specific duration. By considering swept volume, the past coexists with the present. In our context, geometric skeletons do not have spatiotemporal parts, since spatiotemporal regions consist of spatial and temporal components (Bittner et al. 2004). For instance, it is possible to see during a change (e.g. Move of Part 5 in Figure 6) the path of an object and the related needed space to reach its final position.

Several problems concerning spatiotemporal visualisation can be encountered by human. Actually human can only see the initial spatial region and the final spatial region
<figure><img src="image_17.png" /><figcaption>Figure 6. Decomposition of *Part 5* insertion</figcaption></figure>

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during a move. The actor cannot percept the swept volume. For instance, during an assembly sequence, if a part is being moved in order to cross the shaft, the designer will not be able to see generated spatiotemporal part. Indeed the move need to be broken down into several instants. This is due to human intelligence which is only able to see motionless object and fixes the time. Humans cannot think in terms of continuous evolution (Bergson 1923). As a consequence, the spatiotemporal dimension of the theory has been defined as a sequence of pictures.

If an object must undergo two consecutive changes, then the final spatial region of the first change corresponds to the first spatial region of the second change. This kind of description does not provide all information about what happens during intervals. A quantitative study has to be carried out with specific laws to know what happens in the interior of an interval.

### 3.4.2. Filiation relationships

Descartes (Descartes 1996) made an experiment with wax. When he saw it, the wax was solid. But when he put it near fire, it melts. His conclusion is that with human understanding, the wax is wax and will always be wax whatever happens to it (e.g. transformation). Moreover, Sider (Sider 2001) and Hawley (Hawley 2004) describes the object as persisting over time. As soon as an object is modified, its properties change (but not the object itself). The object will have attributes showing for instance if the object is deformed or not. For few changes (e.g. union or split), a new object is created from the "initial" one. The same identity basis is described with filiation relationships. It is a dependence relationship where the kinship is transmitted from the parent (i.e. assembly) to the child (i.e. part). It therefore enables the identification of every object and its uniqueness. Based on the identity concept, it is possible to distinguish an object from others. Two types of filiation relationships can be found during AOD phase (Del Mondo et al. 2010), such as:

* Continuation relationship, denoted $\gamma$ (i.e. exactly the same object but at two different times);

* Derivation relationship, denoted $\delta$ (i.e. a part of the identity from the original object is present in the changed object).

In the JANUS theory, the continuation relationship is considered as implicit. The latter always occurs between two objects with the same name at different instants. Derivation relationships exist if two objects are in relation with spatiotemporal primitives and do not have the same name.

### 3.4.3. Spatiotemporal primitives classification

Changes have spatiotemporal aspects and need spatiotemporal relationships to link spatial objects over time. The latter, which are introduced to link spatial and temporal dimensions (Del Mondo et al. 2010), can be described as follows:

* Between two spatial regions at two different instants (e.g. *Move*);

* From a spatial to an empty spatial region at two different instants (e.g. *Deletion*);

* From an empty spatial to a spatial region at two different instants (e.g. *Addition*).

As a consequence, spatiotemporal primitives express the change over space and time (Haddad 2009), which provides an effective mechanism to visualise and communicate design and assembly intents (McKinney et al. 1996). In that case, the evolution of an object can be seen as a succession of events and processes. Let us consider two different
<table><caption>Table 6. Part-to-part relationships definition for spatiotemporal primitives (Demoly et al. 2012)</caption><thead><tr><td>Relationship type</td><td>Definition</td></tr></thead><tbody><tr><td>Contact</td><td>Physical contact relationship between two parts</td></tr><tr><td>Kinematic pair</td><td>Additional information on contact relationship which enables the description of constrained degrees of freedom (rotation and translation) for each part</td></tr><tr><td>Technological pair</td><td>Additional information on contact relationship which enables the definition of the assemblability of the product, and therefore on the joining relationship between two parts</td></tr></tbody></table>

times $t_i$ and $t_j$ and two different parts $x$ and $y$ (see Equation (2)):

$$ \forall t_i \neq t_j \quad \forall x \quad \exists y / x_{t_i} \mathbf{R} y_{t_j} \qquad (2) $$

In rigid mechanical assembly, two parts during the assembly process cannot physically overlap, on the contrary two spatiotemporal regions can overlap themselves at two different instants. For instance, several parts can be positioned on a shaft in a serial process manner and so their spatiotemporal regions are overlapping but are assembled at different times in order to have no physical overlap of their spatial regions. That is the reason why spatiotemporal primitives are needed to model the actual world. By describing new spatiotemporal primitives such as kinematic and technological pairs positioning (as described in Table 6), evolution information in product design can be managed and understood. In Table 7, all spatiotemporal primitives are described and classified in terms of their changes over time, space and form (Haddad 2009). For growing and move, the change occurs during the process: SpatioTemporal Region (denoted STR) form changes, but also during the instant just after, i.e. change of Spatial Region Final (denoted SRF). Here, STR simply represents spatiotemporal region and SRF as final spatial region. On the contrary for Union, changes in form just occur at the final spatial region. For instance, kinematic pairs undergo spatial and spatiotemporal changes, but part number remains identical. Moreover, the form of the part, which is moving/progressing, is not the same during the move (i.e. swept volume) and at the final stage of the evolution (i.e. part).

### 3.4.4. Kinematic pairs positioning and design changes description

Each spatiotemporal primitive is described in details using spatial and temporal mereotopological primitives (see Table 2 and Table 4). This kind of spatiotemporal primitives has been developed in order to enable the description of the product evolution over time in the context of AOD. The spatiotemporal primitive called CylindricalOP (here OP stands for operation) is described in details in a specific graph (Figure 7) with its related legend (Table 8). The definition says that at the beginning Part 1 and Part 5 are Discrete and that 1 is Overlapped by its assembly skeleton $k_1$, which is Discrete with 5. After that, 5 Moves and $k_1$ Overlaps 1 and 5. As such, both 1 and 5 are aligned. At the end of the assembly positioning, 5 Moves again, Crosses 1 and are Tangent to the interface skeleton $f_5$. Processes are considered continuous, non-instantaneous (e.g. object cannot move from one point to another in one instant) and linear. 5 CylindricalOP 1 means that 1 is the base part and 5 is moving to be positioned in a cylindrical manner. For next descriptions, $TRi$ always precedes $TRi+1$. The mereotopological descriptions of kinematic pairs positioning and design changes (such as
<table><caption>Table 7. Classification of changes and associated spatiotemporal primitives</caption><thead><tr><th>Changes types</th><th>Spatiotemporal primitives</th><th>Temporal change</th><th>Spatial change</th><th>Form change</th><th>Parts number change</th><th>Exemple of Event (E) or Process (P)</th></tr></thead><tbody><tr><td rowspan="2">Move</td><td>Kinematic pairs</td><td>●</td><td>●</td><td>STR+SRF</td><td>○</td><td>P: Place an object</td></tr><tr><td>Rotation</td><td>●</td><td>●</td><td>STR+SRF</td><td>○</td><td>P: Orient an object</td></tr><tr><td>Replacement process</td><td>Permutation</td><td>●</td><td>●</td><td>STR+SRF</td><td>○</td><td>P: Designer oversight</td></tr><tr><td rowspan="2">Basic change</td><td>Addition</td><td>●</td><td>○</td><td>SRF</td><td>●</td><td>E: Missing object</td></tr><tr><td>Deletion</td><td>●</td><td>○</td><td>SRF</td><td>●</td><td>E: Useless object</td></tr><tr><td rowspan="2">Restructuring Processes</td><td>Split</td><td>●</td><td>○</td><td>SRF</td><td>●</td><td>E: Complex assembly</td></tr><tr><td>Union</td><td>●</td><td>○</td><td>SRF</td><td>●</td><td>E: Subassembly creation</td></tr><tr><td rowspan="4">Transformation</td><td>Growing</td><td>●</td><td>○</td><td>STR+SRF</td><td>○</td><td>P: Object too small</td></tr><tr><td>Decrease</td><td>●</td><td>○</td><td>STR+SRF</td><td>○</td><td>P: Object too large</td></tr><tr><td>Change Of Form</td><td>●</td><td>○</td><td>STR+SRF</td><td>○</td><td>P: Change of idea</td></tr><tr><td>Deformation</td><td>●</td><td>○</td><td>STR+SRF</td><td>○</td><td>P: Rivet positioning</td></tr><tr><td rowspan="3">Technological pairs</td><td>Permanent with added part</td><td>●</td><td>●</td><td>STR+SRF</td><td>○</td><td>P: Place permanently an object</td></tr><tr><td>Non-permanent with added part</td><td>●</td><td>●</td><td>STR+SRF</td><td>○</td><td>P: Place non-permanently an object</td></tr><tr><td>Non-permanent without added part</td><td>●</td><td>●</td><td>STR+SRF</td><td>○</td><td>P: Place non-permanently an object</td></tr></tbody></table>

*Addition, Growing* and so on) are defined in Appendix (cf. Table A1 and Table B1).

<figure><img src="image_18.png" /><figcaption>Figure 7. Representation of *CylindricalOP* in a spatiotemporal graph</figcaption></figure>

### 3.4.5. Technological pairs positioning description

At a lower granularity, several technological pairs can be used when the product architect decides to design a rigid assembly (e.g. adhesive bonding, welding and so on). These
<table><caption>Table 8. Legend of the spatiotemporal graph</caption><thead><tr><th>Representation</th><th>Meaning</th></tr></thead><tbody><tr><td>⊥</td><td>Spatial relationship (e.g. Tangent)</td></tr><tr><td>►T<sub>t</sub></td><td>Temporal relationship (e.g. TemporallyTangent)</td></tr><tr><td>Move</td><td>Spatiotemporal relationship (e.g. Move)</td></tr><tr><td>TR<sub>i</sub></td><td>Temporal region number i</td></tr><tr><td>5</td><td>Spatial region (e.g. Part 5)</td></tr><tr><td>5</td><td>Spatiotemporal region (e.g. Swept volume 5)</td></tr><tr><td>k</td><td>Assembly skeleton (e.g. k)</td></tr><tr><td>f</td><td>Interface skeleton (e.g. f)</td></tr></tbody></table>

<table><caption>Table 9. Classification of technological assembly operations</caption><thead><tr><th>Assembly Type</th><th>With added part</th><th>Without added part</th></tr></thead><tbody><tr><td>Non-permanent</td><td>Mechanical Fastening</td><td>None</td></tr><tr><td rowspan="2">Permanent</td><td>Parts destruction</td><td>Gas Welding<br>Brazing<br>Adhesive Bonding<br>Arc Welding<br>Spot Welding<br>Clinching<br>Crimping<br>Interference Fit</td></tr><tr><td>Fastening destruction</td><td>Nail Assembly<br>Simple Rivet<br>Double Rivet<br>Blind Rivet<br>Snap In<br>Metal Stitching</td><td>None</td></tr></tbody></table>

relationships have been classified (see Table 9) into four groups according to different criteria (e.g. added entity needed, non-permanent assembly, etc.). Thus it is not possible to have a non-permanent technological pair without added entities. Some rules have been set up, such as:

* p and s denote tools;

* u and v denote added entities;

* x and y denote both entities to be assembled.

According to the classification in Table 9, technological pairs positioning are described in Appendix (cf. Table C1, Table D1, Table D2, Table D3 and Table E1). Spatiotemporal descriptions expressed at TRi are not written again at TRi+1 in order to lighten the equations and to facilitate understanding. Here, the authors have considered Kim et al.'s propositions related to the technological relationships description (Kim et al. 2008). With such descriptions, the aim is to trace the objects being assembled and also their tools. Hence, spatial and temporal primitives are needed.

## 4. Ontological implementation with a case study

This section introduces an ontological implementation of the JANUS theory with an illustrative case study, so as to highlight the applicability of the proposal.
## 4.1. Case study and its mereotopological description

For this case study, the eBOM (engineering Bill Of Material) is composed of six parts (Table 10) and the part-to-part relationships have been defined in a directed graph (Figure 8) by the product architect. Based on such information, the ASDA algorithm (Assembly Sequence Definition Algorithm) (Demoly et al. 2012) has generated a relevant assembly sequence that is expressed as follows [1, [3, 2, 4], 5, 6]. So the spatial regions (i.e. product parts) 2, 3 and 4 are embedded in a Sub-Assembly (denoted SA). The assembly sequence is then refined by the assembly planner.

<table><caption>Table 10. Parts list of the case study</caption><thead><tr><td>No.</td><td>Part name</td><td>No.</td><td>Part name</td></tr></thead><tbody><tr><td>1</td><td>Shaft</td><td>4</td><td>Bearing 2</td></tr><tr><td>2</td><td>Bearing 1</td><td>5</td><td>Spacer</td></tr><tr><td>3</td><td>Hub</td><td>6</td><td>Nut</td></tr></tbody></table>

<figure><img src="image_19.png" /><figcaption>Figure 8. Graph of part-to-part relationships</figcaption></figure>

Based on the previous directed graph, the temporal information (see Table 11) and the kinematic pairs positioning can be converted into mereotopological descriptions with

<table><caption>Table 11. Temporal information: assembly operations and their associated primitives</caption><thead><tr><td>Operations description (Boothroyd et al. 2002)</td><td>Temporal relationships between operations</td></tr></thead><tbody><tr><td>OP10: Insertion of 2 on 3</td><td>OP10 Ot OP20</td></tr><tr><td>OP20: Insertion of 4 on 3</td><td>OP20 Tt OP30</td></tr><tr><td>OP30: Insertion of SA on 1</td><td>OP10 tp OP30</td></tr><tr><td>OP40: Insertion of 5 on 1 and 4</td><td>OP30 Tt OP40</td></tr><tr><td>OP50: Insertion of 6 on 1 and 5</td><td>OP40 Tt OP50</td></tr></tbody></table>
the JANUS theory (see Table 12). All information is then gathered to describe product evolution (See Figure 9) (Gruhier et al. 2014). Hence, spatial (i.e. mechanical part), temporal (i.e. assembly operation) and spatiotemporal (i.e. swept volume in the context of AOD) regions are represented and linked with primitives. Only regions, which undergo a change at *OPi*, are presented at *OPi* in order to facilitate understanding of the reader. Changes are here considered once novel spatiotemporal relationships appear. As such, product architect and designer can really understand what happens to each operation, especially where a part is situated and linked regarding to others, and which part is moving or evolving and so on. For instance, Theses actors can assess that at *OP50*, Swept volume 6 is moving to be positioned in a *ScrewOP* manner on *Part 1* and in a *PlanarOP* manner on *Part 5*. Therefore, this representation provides an overview of the product evolution along its assembly sequence and so a suitable design support to the product architect and designer before starting the product modelling phase in CAD.
<figure><img src="image_20.png" /><figcaption>Figure 9. Representation of the whole assembly process in a spatiotemporal graph</figcaption></figure>

Based on Figure 8, the final stage of the assembly design process is represented in Figure 10. Kinematic pairs are expressed with mereotopological primitives. For instance *PlanarOP* is equivalent to *Tangent* primitive at the end of the process, as well as *RevoluteOP* is equivalent to *Cross* and *Tangent* primitives. This kind of representation allows the designer to fully understand the relationships between parts at the end of the assembly process. For example, such graph informs that *Part 2* and *Part 6* are never in contact and that all parts are linked to *Part 1*. This information will aid the designer during the CAD modelling phase.

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<figure><img src="image_21.png" /><figcaption>Figure 10. Spatial representation of the whole assembly process</figcaption></figure>
<figure><img src="image_22.png" /><figcaption>Figure 11. Assembly sequence planning of the case study</figcaption></figure>

Since JANUS theory is used at the early design stages, qualitative information is favored (i.e. detailed geometry does not exist yet). Build on part-to-part relationships, the assembly sequence can be defined and represented such as in Figure 11. Moreover, spatial and temporal relationships are simultaneously represented in Figure 12 within a space-time graph in order to understand product relationships design over time. This graph is obtained with the previous information on spatial, temporal and spatiotemporal dimensions. The horizontal axis represents temporal connections between regions, and the vertical axis shows spatial relationships. For instance if two objects are touching and if the tangency between them can be projected in the horizontal axis, it means that they are *Temporally Tangent* (and vice versa for the vertical axis and the *Tangent* relationship). Figure 10 and Figure 12 enable the introduction of a complementary vision to the current 3D definition of the product. By considering such a description, product architects and designers will know how and when parts are related via the early-defined assembly sequence. Therefore, the entire understanding about the assembly process and product design evolution in the early design phase is ensured.

## 4.2. System execution

Built on this formal description, the JANUS theory has been implemented in an ontology called PRONOIA2 (i.e. PRODUCT relationships description based on mereotopology ICAI theory 2) in Protégé in order to be machine-interpretable by product lifecycle management (PLM) systems. PRONOIA2 is based on PRONOIA (Demoly et al. 2012) and extends it in the temporal and spatiotemporal dimensions. The OWL language has been used to formalise the ontology. The terms from JANUS have been reused, especially the two top-level classes, Region and Primitive (See Figure 13). In addition, description logic restrictions and SWRL rules have been incorporated to be able to reason on information and check its consistency from a logical and semantical point of view. The ontology in its current shape proposes some available queries to designers so as to display and represent evolutions and changes in product design stages.

For this case study, PRONOIA 2 has been populated with previous information. The individuals are shown in Figure 14, in which on the right individuals have been linked together with properties, and on the left part the caption of properties. This figure is composed of five states represented in the middle. Each state is composed of spatial, temporal and/or spatiotemporal regions and primitives. The different states allow visualising what happens in the context of AOD and show the product evolution over time
<figure><img src="image_23.png" /><figcaption>Figure 12. Representation of the product relationships over space and time</figcaption></figure>

and space.

After checking information consistency related to product architect' and the assembly
planner's intents (Gruhier et al. 2015), the designer has a suitable design support based
on a perdurantist vision to define consistent mechanical assemblies (See Figure 15). In-
deed, designer is aware of product architect's and assembly planner's knowledge and can
take into consideration lifecycle constraints at the early design stages so as to enable
concurrent and proactive engineering. This mechanical assembly case can then be visu-
alised within a CAD application (e.g. CATIA v5) as illustrated in Figure 16. This last
step of the framework - regarding the awareness and understanding support in design -
will be addressed in detailed in Step 3 of Figure 1.

**5. Discussions**

The proposed JANUS theory enables the description of product-process relationships
by introducing spatial, temporal and spatiotemporal regions and related primitives in
the context of perdurantism. Thus, the relationships between parts of the product along
the assembly process are fully described and exploitable as well as their design changes,
whatever the part number and the product complexity. Product evolution over time is
therefore captured at each change, enabling a better understanding of the product. As
a result, the first step (Step 1 of Figure 1) has been fully covered in this article, by
describing new temporal and spatiotemporal mereotopological primitives. This research
<figure><img src="image_24.png" /><figcaption>Figure 13. Overall view of the PRONOIA2 ontology (Gruhier et al. 2015)</figcaption></figure>

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<figure><img src="image_25.png" /><figcaption>Figure 14. Instances of the ontology for the presented case study</figcaption></figure>
<figure><img src="image_26.png" /><figcaption>Figure 15. Cross-section view of the considered mechanical assembly</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>Figure 16. Implementation of the case study within CATIA v5 (CAD application).</figcaption></figure>

effort is considered here as the basis for future studies where additional spatiotemporal layers will be specified to assist designers in the detailed design stage.
Actually information is not always available or understandable by designers. Since only one assembly sequence has been considered in the case study, only one product evolution has been carried out. In the case where another one is integrated, the product evolution will be slightly different. However the proposed theory has some limitations since it does not describe the exact shape, position (e.g. within a coordinates system) and orientation of spatial objects. At this stage, the purpose is to develop a qualitative theory, which could be applied to any kind of *mechanical* assembly (i.e. independent of shape and size), insofar the 3D model is not yet well-defined in the early design stages. In its current shape, JANUS does not describe all types of evolution that can *exist* during the product design process, but enables the description of the most important and *recurrent* ones.

In addition JANUS could be extended to describe not only assembly *process* knowledge, but also knowledge from manufacturing process (e.g. additive manufacturing, tapping, etc.). In such a case, new spatiotemporal regions will be described by considering material flow as well as new specific primitives. As a consequence, the theory seems to be enough extensible to cover all the lifecycle stages in product design. Moreover, such a theory could be applied to any domain where description of relationships between objects is needed. If the theory is planned to be adapted to another domain (e.g. geographical information system - GIS), the basis will remain the same but new regions (e.g. city) and related primitives will have to be developed.

## 6. Conclusions and future work

The article has introduced a literature review on AOD approaches and related product models, and mereotopology-based theories, on which a research problem formulation has been highlighted. Built on this, a novel mereotopological theory (called JANUS) following a four dimensionalism philosophy has been presented. Such a theory covers three dimensions (i.e. spatial, temporal and spatiotemporal) so as to describe product-process relationships over space and time. Spatial and temporal objects and primitives have been defined, as well as their associations between both dimensions. Indeed, new spatiotemporal objects and primitives have been introduced in order to describe product-process knowledge and information. By considering the spatiotemporal dimension and the associated mereotopology-based theory, information has been added to design objects in order to express their history in a consistent and understandable manner. This description aids to understand phenomena during AOD process. The actual stake is to get a long term dynamic vision of the space over time in order to facilitate the understanding of assembly and design changes.

Moreover, the JANUS theory has been implemented into an ontology by using OWL-DL and SWRL languages. As such, by considering these novel rules, a reasoning layer has been implemented so as to reason on spatiotemporal associations. Such efforts enable the introduction of novel procedures for consistency checking of product-process information and knowledge in PLM, and create a robust epistemological and methodological platform.

In future work, the PRONOIA2 ontology model will be linked to information systems covering product design and assembly planning phases such as planned in Figure 1. As such, PDM and MPM systems will be associated through an ontology, which will bring a novel semantic layer in PLM. Hence, designers will be supported to design a consistent product over space and time. Next research efforts will also address the representation of spatiotemporal information within CAD application, so as to support concurrent and proactive engineering by defining product evolution at the early design stages.
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Appendix A. Mereotopological description of the “kinematic pairs positioning” primitives

Appendix B. Mereotopological description of the “design changes” primitives

Appendix C. Mereotopological description of non-permanent assembly with added entities

Appendix D. Mereotopological description of permanent assembly with added entities

Appendix E. Mereotopological description of permanent assembly without added entities
<table><caption>Table 12. Mereotopological description of the mechanical case study</caption><thead><tr><th>OP No.</th><th>Spatiotemporal primitive</th><th>Mereotopological description</th><th>Skeleton</th></tr></thead><tbody><tr><td>10</td><td>2 RevoluteOP 3</td><td>((3D2) ∧ (3Ok1) ∧ (2Dk1))TR1 ∧ (2Move3)TR2 ∧ (3D2) ∧ (3Ok1) ∧ (2Ok1))TR3 ∧ (2Move3)TR4 ∧ (3X2) ∧ (3Ok1) ∧ (2Ok1))TR5 ∧ (2Move3)TR6 ∧ (3T2) ∧ (3X2) ∧ (3Ok1) ∧ (3Tk2) ∧ (3Tf2) ∧ (2Ok1) ∧ (2Tk2) ∧ (2Tf2))TR7</td><td>k1 line, k2 plane, f2 surface</td></tr><tr><td>20</td><td>4 RevoluteOP 3</td><td>((3D4) ∧ (3Ok1) ∧ (4Dk1))TR1 ∧ (4Move3)TR2 ∧ (3D4) ∧ (3Ok1) ∧ (4Ok1))TR3 ∧ (4Move3)TR4 ∧ (3X4) ∧ (3Ok1) ∧ (4Ok1))TR5 ∧ (4Move3)TR6 ∧ (3T4) ∧ (3X4) ∧ (3Ok1) ∧ (3Tk4) ∧ (3Tf4) ∧ (4Ok1) ∧ (4Tk4) ∧ (4Tf4))TR7</td><td>k1 line, k4 plane, f4 surface</td></tr><tr><td>30</td><td>SA RevoluteOP 1</td><td>((1DSA) ∧ (1Ok1) ∧ (SADk1))TR1 ∧ (SAMove1)TR2 ∧ (1DSA) ∧ (1Ok1) ∧ (SAOk1))TR3 ∧ (SAMove1)TR4 ∧ (1XSA) ∧ (1Ok1) ∧ (SAOk1))TR5 ∧ (SAMove1)TR6 ∧ (1TSA) ∧ (1XSA) ∧ (1Ok1) ∧ (1Tk3) ∧ (1Tf3) ∧ (SAOk1) ∧ (4Tk3) ∧ (4Tf3))TR7</td><td>k1 line, k3 plane, f3 surface</td></tr><tr><td>40</td><td>5 CylindricalOP 1 and 5 PlanarOP 4</td><td>((1D5) ∧ (1Ok1) ∧ (5Dk1))TR1 ∧ (5Movex)TR2 ∧ (1D5) ∧ (1Ok1) ∧ (5Ok1))TR3 ∧ (5Move1)TR4 ∧ (1X5) ∧ (1Ok1) ∧ (5Ok1))TR5 ∧ (5Move4)TR6 ∧ (4T5) ∧ (4Tf5) ∧ (5Tk5) ∧ (k5Tf5))TR7</td><td>k1 line, k5 plane, f5 surface</td></tr><tr><td>50</td><td>6 ScrewOP 1 and 6 PlanarOP 5</td><td>((1D6) ∧ (1Ok1) ∧ (6Dk1))TR1 ∧ (6Move1)TR2 ∧ ((1D6) ∧ (1Ok1) ∧ (6Ok1))TR3 ∧ (6Move1)TR4 ∧ (1X6) ∧ (1Ok1) ∧ (6Ok1))TR5 ∧ (6Move5)TR6 ∧ (5T6) ∧ (5Tf6) ∧ (6Tk6) ∧ (f6Tk6))TR7</td><td>k1 line, k6 plane, f6 surface</td></tr></tbody></table>

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<table><caption>Table A1. Mereotopological description of the “kinematic pairs positioning” primitives</caption><thead><tr><th>Name</th><th>Mereotopological description</th><th>Representation</th><th>Comment</th></tr></thead><tbody><tr><td>Moves</td><td>(xT<sub>k1</sub>)TR1 ∧ (xT<sub>k2</sub>)TR2</td><td>k<sub>1</sub> / k<sub>2</sub></td><td>k<sub>1</sub> and k<sub>2</sub> point, line or surface</td></tr><tr><td>CylindricalOP</td><td>((xDy) ∧ (xOk) ∧ (yDk))TR1 ∧ (yMovex)TR2 ∧ ((xDy) ∧ (xOk) ∧ (yOk))TR3 ∧ (yMovex)TR4 ∧ ((xXy) ∧ (xOk) ∧ (yOk))TR5 ∧ (yMovex)TR6 ∧ ((xXy) ∧ (xOk) ∧ (yOk) ∧ (yTf))TR7</td><td>f x y</td><td>k line, f surface</td></tr><tr><td>PrismaticOP</td><td>((xDy)TR1 ∧ (yMovex)TR2 ∧ ((xSty) ∧ (xIPy) ∧ (xTf) ∧ (yTf) ∧ (xOk) ∧ (yOk))TR3)</td><td>k y x</td><td>k line, f surface</td></tr><tr><td>SphericalOP</td><td>((xDy) ∧ (xOk1) ∧ (yDk1))TR1 ∧ (yMovex)TR2 ∧ ((xDy) ∧ (xOk1) ∧ (yOk1))TR3 ∧ (yMovex)TR4 ∧ ((xXy) ∧ (xOk1) ∧ (yOk1))TR5 ∧ (yMovex)TR6 ∧ ((xTy) ∧ (xXy) ∧ (xOk1) ∧ (xTk2) ∧ (xTf) ∧ (yOk1) ∧ (yTk2) ∧ (yTf))TR7</td><td>f x y</td><td>k line, f surface</td></tr><tr><td>RevoluteOP</td><td>((xDy) ∧ (xOk1) ∧ (yDk1))TR1 ∧ (yMovex)TR2 ∧ ((xDy) ∧ (xOk1) ∧ (yOk1))TR3 ∧ (yMovex)TR4 ∧ ((xXy) ∧ (xOk1) ∧ (yOk1))TR5 ∧ (yMovex)TR6 ∧ ((xTy) ∧ (xXy) ∧ (xOk1) ∧ (xTk2) ∧ (xTf) ∧ (yOk1) ∧ (yTk2) ∧ (yTf))TR7</td><td>f x y</td><td>k<sub>1</sub> line and k<sub>2</sub> plane, f surface</td></tr><tr><td>PlanarOP</td><td>((xDy) ∧ (xTk))TR1 ∧ (yMovex)TR2</td><td>f x</td><td>k plane, f surface</td></tr><tr><td>PointContactOP</td><td>∧((xTy) ∧ (xTf) ∧ (yTk) ∧ (fTk))TR3</td><td>x</td><td>k point, f point</td></tr><tr><td>LineContactOP</td><td>((xDy) ∧ (xOk1) ∧ (yDk1))TR1 ∧ (yMovex)TR2 ∧ ((xDy) ∧ (xOk1) ∧ (yOk1))TR3 ∧ (yMovex)TR4 ∧ ((xXy) ∧ (xOk1) ∧ (yOk1))TR5 ∧ (yMovex)TR6 ∧ ((xXy) ∧ (xOk1) ∧ (yOk1) ∧ (yTf1) ∧ (f<sub>2</sub>O<sub>f3</sub>) ∧ (yTf2)) ∧ (xTf3))TR7</td><td>f<sub>1</sub> f<sub>2</sub> f<sub>3</sub> y</td><td>k line, f<sub>1</sub> area, f<sub>2</sub> and f<sub>3</sub> elliptical curve</td></tr><tr><td>ScrewOP</td><td>((xDy) ∧ (xOk2) ∧ (k<sub>1</sub>Ok<sub>3</sub>) ∧ (k<sub>2</sub>Dk<sub>3</sub>))TR1 ∧ (xMovex)TR2 ∧ ((xDy) ∧ (xOk2) ∧ (k<sub>1</sub>Ok<sub>2</sub>) ∧ (k<sub>1</sub>Ok<sub>3</sub>) ∧ (k<sub>2</sub>Ok<sub>3</sub>) ∧ (xOk<sub>3</sub>))TR3</td><td>f<sub>1</sub> f<sub>2</sub> y</td><td>k<sub>1</sub> line, f<sub>1</sub> area, f<sub>2</sub> and f<sub>3</sub> elliptical curve</td></tr><tr><td>Rotation</td><td>((xDy) ∧ (xOk2) ∧ (k<sub>1</sub>Ok<sub>3</sub>) ∧ (k<sub>2</sub>Dk<sub>3</sub>))TR1 ∧ (xMovex)TR2 ∧ ((xDy) ∧ (xOk2) ∧ (k<sub>1</sub>Ok<sub>2</sub>) ∧ (k<sub>1</sub>Ok<sub>3</sub>) ∧ (k<sub>2</sub>Ok<sub>3</sub>) ∧ (xOk<sub>3</sub>))TR3</td><td>f<sub>1</sub> f<sub>2</sub> y</td><td>k<sub>1</sub> point, k<sub>2</sub> and k<sub>3</sub> line</td></tr></tbody></table>
<table><caption>Table B1. Mereotopological description of the “design changes” primitives</caption><thead><tr><th>Name</th><th>Mereotopological description</th><th>Representation</th><th>Comment</th></tr></thead><tbody><tr><td rowspan="10">Serial-Permutation</td><td>((xRz) ∧ (yRz) ∧ (xRy) ∧ (xTf) ∧ (xRk) ∧ (zTf1) ∧ (zTf2) ∧ (yTf2) ∧ (yRk))<sub>TR1</sub> ∧ (xMovez)<sub>TR2</sub> ∧ ((xDz) ∧ (yRz) ∧ (xDy) ∧ (xRk) ∧ (zTf2) ∧ (yTf2) ∧ (yRk))<sub>TR3</sub> ∧ (yMovez)<sub>TR4</sub> ∧ ((xDz) ∧ (yDz) ∧ (xDy))<sub>TR5</sub> ∧ (yMovez)<sub>TR6</sub> ∧ ((xDz) ∧ (yRz) ∧ (xDy) ∧ (yTf1) ∧ (yRk) ∧ (zTf1))<sub>TR7</sub> ∧ (xMovez)<sub>TR8</sub> ∧ ((xRz) ∧ (yRz) ∧ (xRy) ∧ (xTf2) ∧ (xRk) ∧ (zTf1) ∧ (zTf2) ∧ (yTf1) ∧ (yRk))<sub>TR9</sub></td><td>k line, f<sub>1</sub> and f<sub>2</sub> surface</td><td rowspan="10">k<sub>1</sub> and k<sub>2</sub> line, f<sub>1</sub> and f<sub>2</sub> surface</td></tr><tr><td>((xRz) ∧ (yRz) ∧ (xDy) ∧ (xTf1) ∧ (xRk1) ∧ (zTf1) ∧ (zTf2) ∧ (yTf2) ∧ (yRk2) ∧ (f<sub>1</sub>Df<sub>2</sub>) ∧ (k<sub>1</sub>Dk<sub>2</sub>))<sub>TR1</sub> ∧ ((xMovez) ∧ (yMovez))<sub>TR2</sub> ∧ ((xDz) ∧ (yDz) ∧ (xDy))<sub>TR3</sub> ∧ ((xMovez) ∧ (yMovez))<sub>TR4</sub> ∧ ((xRz) ∧ (yRz) ∧ (xDy) ∧ (xTf2) ∧ (xRk2) ∧ (zTf1) ∧ (zTf2) ∧ (yTf1) ∧ (yRk1) ∧ (f<sub>1</sub>Df<sub>2</sub>) ∧ (k<sub>1</sub>Dk<sub>2</sub>))<sub>TR5</sub></td><td></td></tr><tr><td>(xP<sup>0</sup>)<sub>TR1</sub> ∧ ((xPA) ∧ (xTf))<sub>TR2</sub></td><td>Ø</td><td>f surface</td></tr><tr><td>Deletion</td><td>((xPA) ∧ (xTf))<sub>TR1</sub> ∧ (xP<sup>0</sup>)<sub>TR2</sub></td><td>x</td><td>f surface</td></tr><tr><td>Split</td><td>((<sup>0</sup>Additiony) ∧ (<sup>0</sup>Additionz) ∧ (xδy) ∧ (xδz) ∧ (xTf<sub>1</sub>))<sub>TR1</sub> ∧ ((yPx) ∧ (zPx) ∧ (zTf<sub>1</sub>) ∧ (yTf<sub>2</sub>))<sub>TR2</sub> ∧ ((<sup>0</sup>Deletiony) ∧ (zTf<sub>1</sub>) ∧ (yTf<sub>2</sub>))<sub>TR3</sub></td><td>f<sub>1</sub> and f<sub>2</sub> surface</td><td></td></tr><tr><td>Union</td><td>((<sup>0</sup>Additiony<sup>0</sup>) ∧ (zTf<sub>1</sub>) ∧ (yTf<sub>2</sub>))<sub>TR1</sub> ∧ ((yPx) ∧ (zPx) ∧ (xP<sup>0</sup>) ∧ (zTf<sub>1</sub>) ∧ (yTf<sub>2</sub>))<sub>TR2</sub> ∧ (<sup>0</sup>Deletiony<sup>0</sup>) ∧ (<sup>0</sup>Deletionz<sup>0</sup>) ∧ (xδy) ∧ (xδz)(xTf<sub>1</sub>))<sub>TR3</sub></td><td>f<sub>1</sub> and f<sub>2</sub> surface</td><td></td></tr><tr><td>ChangeOfForm</td><td></td><td></td><td></td></tr><tr><td>Deformation</td><td>(xTf<sub>1</sub>)<sub>TR1</sub> ∧ (xTf<sub>2</sub>)<sub>TR2</sub></td><td></td><td>f<sub>1</sub> and f<sub>2</sub> surface</td></tr><tr><td>Growing</td><td></td><td></td><td></td></tr><tr><td>Decrease</td><td></td><td></td><td></td></tr></tbody></table>
<table><caption>Table C1. Mereotopological description of non-permanent assembly with added entities</caption><thead><tr><th>Name</th><th>Mereotopological description</th><th>Representation Comment</th></tr></thead><tbody><tr><td></td><td>((xTy) ∧ (uDx) ∧ (pTu) ∧ (pTf<sub>2</sub>) ∧<br>(uTf<sub>2</sub>) ∧ (pTk<sub>4</sub>) ∧ (uTk<sub>4</sub>) ∧<br>(xTk<sub>3</sub>) ∧ (yTk<sub>3</sub>) ∧ (xOk<sub>1</sub>) ∧<br>(yOk<sub>1</sub>) ∧ (uOk<sub>1</sub>) ∧ (pOk<sub>1</sub>))<sub>TR1</sub> ∧<br>((pCylindricalOPx)<br>Mechanical-((uScrewOPx))<sub>TR2</sub> ∧ (uXx)<sub>TR3</sub> ∧<br>Fastening-((pCylindricalOPx)<br>ByThreaded((uScrewOPx))<sub>TR4</sub> ∧ (uXy)<sub>TR5</sub> ∧<br>Fasteners ((pCylindricalOPx)<br>(uScrewOPx))<sub>TR6</sub> ∧<br>((uTx) ∧ (uTf<sub>1</sub>) ∧ (xTf<sub>1</sub>) ∧<br>(uTk<sub>2</sub>) ∧ (xTk<sub>2</sub>))<sub>TR7</sub> ∧<br>(pPrismaticOPx)<sub>TR8</sub> ∧<br>(pDeletion<sub>0</sub>)<sub>TR9</sub></td><td>(k<sub>1</sub> line,<br>k<sub>2</sub>, k<sub>3</sub> and<br>k<sub>4</sub> plane,<br>f<sub>1</sub> and f<sub>2</sub><br>surface)</td></tr></tbody></table>

<figure><img src="image_28.png" /></figure>

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<table><caption>Table D1. Mereotopological description of permanent assembly with added entities (1)</caption><thead><tr><th>Name</th><th>Mereotopological description</th><th>Representation</th><th>Comment</th></tr></thead><tbody><tr><td>Gas Welding</td><td>((xTy) ∧ (xTk) ∧ (yTk))<sub>TR1</sub> ∧ ((xDeformationp) ∧ (yDeformationp) ∧ (uDeformationp) ∧ (uMovex) ∧ (pMovex))<sub>TR2</sub> ∧ ((xSty) ∧ (uStx) ∧ (uSty) ∧ (uDp) ∧ (xDp) ∧ (uTf) ∧ (xTf) ∧ (yTf) ∧ (yDp))<sub>TR3</sub> ∧ (pDeletion<sup>0</sup>)<sub>TR4</sub></td><td>k line and f segment</td><td></td></tr><tr><td>Soldering</td><td>((xTy) ∧ (xTk) ∧ (yTk))<sub>TR1</sub> ∧ ((uMovex) ∧ (uDeformationp) ∧ (pMovex))<sub>TR2</sub> ∧ ((uStx) ∧ (uSty) ∧ (uDp) ∧ (xDp) ∧ (uTf) ∧ (xTf) ∧ (yTf) ∧ (yDp))<sub>TR3</sub> ∧ (pDeletion<sup>0</sup>)<sub>TR4</sub></td><td>k line and f segment</td><td></td></tr><tr><td>Brazing</td><td>((xTu) ∧ (xTk1) ∧ (uTk1) ∧ (uTf1) ∧ (xTf1))<sub>TR1</sub> ∧ (yPrismaticOPx)<sub>TR2</sub> ∧ ((yTk2) ∧ (yTu) ∧ (uTk2) ∧ (uTf2) ∧ (yTf2))<sub>TR3</sub> ∧ ((pPrismaticOPy) ∧ (uDeformationp))<sub>TR4</sub> ∧ ((pTy) ∧ (uStx) ∧ (uSty) ∧ (yTk3) ∧ (pTk3) ∧ (pTf3) ∧ (yTf3))<sub>TR5</sub> ∧ (pPrismaticOPy)<sub>TR6</sub> ∧ (pDeletion<sup>0</sup>)<sub>TR7</sub></td><td>k1, k2 and k3 line, f1, f2 and f3 surface</td><td></td></tr><tr><td>Adhesive-Bonding</td><td>((xDu) ∧ (uOk1) ∧ (xOk1)<sub>TR1</sub> ∧ ((uPrismaticOPx) ∧ (uDeformationx))<sub>TR2</sub> ∧ ((uTx) ∧ (uXx) ∧ (uTk3) ∧ (xTk3) ∧ (uTk2) ∧ (xTk2))<sub>TR3</sub> ∧ (uPrismaticOPx) ∧ (uDeformationx))<sub>TR4</sub> ∧ (uStx)<sub>TR5</sub></td><td>k1 line, k2 and k3 plane</td><td></td></tr><tr><td>SnapIn</td><td>((xDu) ∧ (uOk1) ∧ (xOk1)<sub>TR1</sub> ∧ ((uPrismaticOPx) ∧ (uDeformationx))<sub>TR2</sub> ∧ ((uTx) ∧ (uXx) ∧ (uTk3) ∧ (xTk3) ∧ (uTk2) ∧ (xTk2))<sub>TR3</sub> ∧ (uPrismaticOPx) ∧ (uDeformationx))<sub>TR4</sub> ∧ (uStx)<sub>TR5</sub></td><td>k1 line, k2 and k3 plane</td><td></td></tr></tbody></table>
<table><caption>Table D2. Mereotopological description of permanent assembly with added entities (2)</caption><thead><tr><th>Name</th><th>Mereotopological description</th><th>Representation</th><th>Comment</th></tr></thead><tbody><tr><td rowspan="10">Nail-Assembly</td><td>((xTy) ∧ (pTu) ∧ (uDx) ∧ (xOk<sub>1</sub>) ∧ (yOk<sub>1</sub>) ∧ (xTk<sub>2</sub>) ∧ (yTk<sub>2</sub>) ∧ (uOk<sub>1</sub>) ∧ (pOk<sub>1</sub>) ∧ (pTk<sub>4</sub>) ∧ (uTk<sub>4</sub>) ∧ (pTf<sub>2</sub>) ∧ (uTf<sub>2</sub>))<sub>TR1</sub> ∧ ((pPrismaticOPx) ∧ (uPrismaticOPx)<sub>TR2</sub> ∧ (uXx)<sub>TR3</sub> ∧ ((pPrismaticOPx) ∧ (uPrismaticOPx) ∧ (xDeformationu) ∧ (yDeformationu))<sub>TR4</sub> ∧ ((uXx) ∧ (uXy)<sub>TR5</sub> ∧ ((pPrismaticOPx) ∧ (uPrismaticOPx) ∧ (xDeformationu) ∧ (yDeformationu))<sub>TR6</sub> ∧ ((uXx) ∧ (uXy) ∧ (uTx) ∧ (uTk<sub>3</sub>) ∧ (xTk<sub>3</sub>) ∧ (uTf<sub>1</sub>) ∧ (xTf<sub>1</sub>))<sub>TR7</sub> ∧ (pPrismaticOPx)<sub>TR8</sub> ∧ (pDu)<sub>TR9</sub> ∧ (pPrismaticOPx)<sub>TR10</sub> ∧ (pDeletion<sub>0</sub>)<sub>TR11</sub>)</td><td>k<sub>3</sub><sub>k1</sub> line, k<sub>2</sub>, k<sub>3</sub> and k<sub>4</sub> plane, f<sub>1</sub> and f<sub>2</sub> surface</td><td></td></tr><tr><td>((xTy) ∧ (pXx) ∧ (pXy) ∧ (uIPp) ∧ (xTk<sub>2</sub>)<sub>TR1</sub> ∧ (pPrismaticOPx) ∧ (uPrismaticOPx))<sub>TR2</sub> ∧ ((pXx) ∧ (pXy) ∧ (pTx) ∧ (pTy))<sub>TR3</sub> ∧ ((uPrismaticOPx) ∧ (xDeformationu)<sub>TR4</sub> ∧ (uXx)<sub>TR5</sub> ∧ ((uPrismaticOPx) ∧ (yDeformationu))<sub>TR6</sub> ∧ ((uTp) ∧ (uXy))<sub>TR7</sub> ∧ (uDeformationx)<sub>TR8</sub> ∧ ((uXy) ∧ (uXy) ∧ (uTx) ∧ (pXx) ∧ (pXy) ∧ (uTf) ∧ (xTf) ∧ (xTk<sub>1</sub>) ∧ (uTk<sub>1</sub>))<sub>TR9</sub> ∧ (uDeformationp)<sub>TR10</sub> ∧ ((uTy) ∧ (uTk<sub>3</sub>) ∧ (yTk<sub>3</sub>))<sub>TR11</sub> ∧ (pPrismaticOPx)<sub>TR12</sub> ∧ (pDeletion<sub>0</sub>)<sub>TR13</sub></td><td>k<sub>1</sub>, k<sub>2</sub> and k<sub>3</sub> x plane and f surface</td></tr></tbody></table>

<figure><img src="image_29.png" /></figure>

<figure><img src="image_30.png" /></figure>

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<table><caption>Table D3. Mereotopological description of permanent assembly with added entities (3)</caption><thead><tr><th>Name</th><th>Mereotopological description</th><th>Representation</th><th>Comment</th></tr></thead><tbody><tr><td rowspan="10">SimpleRivet</td><td>((uBs) ∧ (φAdditiony) ∧ (uBf2) ∧ (uOk1) ∧ (sOk1))TR1 ∧</td><td rowspan="10">k1 line, k2, k3 and k4 plane, f1 and f2 surface</td><td rowspan="10"></td></tr><tr>((yPrismaticOPs)TR2 ∧ ((uTy) ∧ (uXy) ∧ (φAdditionx) ∧</td></tr><tr>(uTk2) ∧ (yTk2) ∧ (yOk1))TR3 ∧ (xPrismaticOPy)TR4 ∧</td></tr><tr>((uTy) ∧ (uXx) ∧ (φAdditionp) ∧ (xOk1) ∧ (xTk3) ∧</td></tr><tr>(yTk3))TR5 ∧ (pPrismaticOPx)TR6 ∧ ((pBu) ∧</td></tr><tr>(pOk1) ∧ (uBf1) ∧ (pBf1))TR7 ∧ ((pPrismaticOPx) ∧</td></tr><tr>(uDeformationp)TR8 ∧ ((uTx) ∧ (pTx) ∧ (uTk4) ∧</td></tr><tr>(pTk4))TR9 ∧ (pPrismaticOPx)TR10 ∧ (pDeletion)TR11</td></tr><tr>((vTs) ∧ (φAdditiony) ∧ (vOk1) ∧ (vTk5) ∧ (sOk1))TR1 ∧</td><td rowspan="10"></td></tr><tr>((yPrismaticOPs)TR2 ∧ ((φAdditionx) ∧ (vTk2) ∧</td></tr><tr>(yTk2) ∧ (vXy) ∧ (yOk1))TR3 ∧ (xPrismaticOPy)TR4 ∧</td></tr><tr>((φAdditionu) ∧ (xOk1) ∧ (xTk3) ∧ (yTk3) ∧ (vXx))TR5 ∧</td></tr><tr>(uPrismaticOPx)TR6 ∧ ((φAdditionp) ∧ (uTk5) ∧ (vStv) ∧</td></tr><tr>(uXx) ∧ (uXy))TR7 ∧ (pPrismaticOPx)TR8 ∧ ((pBu) ∧</td></tr><tr>(pOk1) ∧ (uBf1) ∧ (pBf1)TR9 ∧ ((pPrismaticOPx) ∧</td></tr><tr>(uDeformationp) ∧ (vDeformations))TR10 ∧ ((uTx) ∧</td></tr><tr>(pTx) ∧ (uTk4) ∧ (pTk4) ∧ (vBf2) ∧ (vTy) ∧ (vBs))TR11 ∧</td></tr><tr>(pPrismaticOPx)TR12 ∧ (pDeletion)TR13</td></tr><tr><td rowspan="7">DoubleRivet</td><td>((xTy) ∧ (pXv) ∧ (pTu) ∧ (uDx) ∧ (uXv) ∧</td><td rowspan="7"></td></tr><tr>((xTk2) ∧ (yTk2) ∧ (xOk1) ∧ (yOk1) ∧ (vOk1) ∧</td></tr><tr>(uOk1) ∧ (pOk1))TR1 ∧ (pPrismaticOPx)TR2</td></tr><tr>(uXx)TR3 ∧ (pPrismaticOPx)TR4 ∧ (uXy)TR5</td></tr><tr>(pPrismaticOPx)TR6 ∧ ((uTx) ∧ (uTk3) ∧ (uTf) ∧</td></tr><tr>((xTf) ∧ (xTk3))TR7 ∧ ((vPrismaticOPx) ∧</td></tr><tr>(uDeformationp))TR8 ∧ (uTy)TR9 ∧ ((vSplita) ∧ (vSplitb) ∧</td></tr><tr><td rowspan="2">Compression</td><td>(pPrismaticOPx)TR10 ∧ ((pDx) ∧ (aIPu) ∧ (bIPp) ∧</td><td rowspan="2"></td></tr><tr><td>(pDeletion) ∧ (bDeletion)TR11</td></tr><tr><td rowspan="2">BlindRivet</td><td>((xTy) ∧ (pXv) ∧ (pTu) ∧ (uDx) ∧ (uXv) ∧</td><td rowspan="2"></td></tr><tr><td>((xTk2) ∧ (yTk2) ∧ (xOk1) ∧ (yOk1) ∧ (vOk1) ∧</td></tr><tr><td rowspan="10">BlindRivet</td><td>(uOk1) ∧ (pOk1))TR1 ∧ (pPrismaticOPx)TR2</td><td rowspan="10"></td></tr><tr>((xXx)TR3 ∧ (pPrismaticOPx)TR4 ∧ (uXy)TR5</td></tr><tr>(pPrismaticOPx)TR6 ∧ ((uTx) ∧ (uTk3) ∧ (uTf) ∧</td></tr><tr>((xTf) ∧ (xTk3))TR7 ∧ ((vPrismaticOPx) ∧</td></tr><tr>(uDeformationp))TR8 ∧ (uTy)TR9 ∧ ((vSplita) ∧ (vSplitb) ∧</td></tr><tr>(pPrismaticOPx)TR10 ∧ ((pDx) ∧ (aIPu) ∧ (bIPp) ∧</td></tr><tr>(pDeletion) ∧ (bDeletion)TR11</td></tr><tr>((xTy) ∧ (pXv) ∧ (pTu) ∧ (uDx) ∧ (uXv) ∧</td><td rowspan="10"></td></tr><tr>((xTk2) ∧ (yTk2) ∧ (xOk1) ∧ (yOk1) ∧ (vOk1) ∧</td></tr><tr>(uOk1) ∧ (pOk1))TR1 ∧ (pPrismaticOPx)TR2</td></tr><tr>(uXx)TR3 ∧ (pPrismaticOPx)TR4 ∧ (uXy)TR5</td></tr><tr>(pPrismaticOPx)TR6 ∧ ((uTx) ∧ (uTk3) ∧ (uTf) ∧</td></tr><tr>((xTf) ∧ (xTk3))TR7 ∧ ((vPrismaticOPx) ∧</td></tr><tr>(uDeformationp))TR8 ∧ (uTy)TR9 ∧ ((vSplita) ∧ (vSplitb) ∧</td></tr><tr>(pPrismaticOPx)TR10 ∧ ((pDx) ∧ (aIPu) ∧ (bIPp) ∧</td></tr><tr>(pDeletion) ∧ (bDeletion)TR11</td></tr></tbody></table>

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<table><caption>Table E1. Mereotopological description of permanent assembly without added entities</caption><thead><tr><th>Name</th><th>Mereotopological description</th><th>Representation</th><th>Comment</th></tr></thead><tbody><tr><td>ArcWelding</td><td>((xTy) ∧ (pDx) ∧ (pDy) ∧ (xTk) ∧ (yTk))<sub>TR1</sub> ∧ ((xDeformationp) ∧ (yDeformationp))<sub>TR2</sub> ∧ ((xSty) ∧ (xStk) ∧ yStk) ∧ (pDeletionØ)<sub>TR3</sub></td><td>k plane</td><td>k plane</td></tr><tr><td>SpotWelding</td><td>((xTy) ∧ (pDx) ∧ (sTy) ∧ (pOk1) ∧ (sOk1) ∧ (xOk1) ∧ (yOk1) ∧ (sTf2) ∧ (sTk2) ∧ (yTk2) ∧ (yTf2) ∧ (xTk3) ∧ (yTk3))<sub>TR1</sub> ∧ (pPrismaticOPx)<sub>TR2</sub> ∧ ((pTx) ∧ (pTf1) ∧ (xTk4) ∧ (pTf1))<sub>TR3</sub> ∧ ((xDeformationp) ∧ (yDeformations))<sub>TR4</sub> ∧ ((xSty) ∧ (xStk3) ∧ (yStk3))<sub>TR5</sub> ∧ ((pPrismaticOPx) ∧ (sPrismaticOPx))<sub>TR6</sub> ∧ ((pDeletionØ) ∧ (sDeletionØ))<sub>TR7</sub></td><td>k1 line, k2, k3 and k4 plane, f1 and f2 surface</td><td>k1 line, k2, k3 and k4 plane, f1 and f2 surface</td></tr><tr><td>Clinching</td><td>((xTs) ∧ (xTk2) ∧ (xTk1) ∧ (xTf1) ∧ (pTk2) ∧ (pTf2) ∧ (sTk1) ∧ (sTf1) ∧ (xTy) ∧ (yTp) ∧ (yTk4) ∧ (yTk2) ∧ (yTf2))<sub>TR1</sub> ∧ ((pPrismaticOPx) ∧ (xDeformations) ∧ (yDeformations))<sub>TR2</sub> ∧ ((sBx) ∧ (pBy) ∧ (xSty) ∧ (xStk4) ∧ (yStk4))<sub>TR3</sub> ∧ (pPrismaticOPx)<sub>TR4</sub> ∧ ((pDeletionØ) ∧ (sDeletionØ))<sub>TR5</sub></td><td>k3 line, k1, k2 and k4 plane, f1 and f2 surface</td><td>k3 line, k1, k2 and k4 plane, f1 and f2 surface</td></tr><tr><td>Crimping</td><td>((xXy) ∧ (xTk3) ∧ (yTk3) ∧ (pTy) ∧ (pXy) ∧ (xOk1) ∧ (yOk1) ∧ (yTk2) ∧ (pTk2))<sub>TR1</sub> ∧ ((pDecreasex) ∧ (yDeformationp) ∧ (xDeformationy))<sub>TR2</sub> ∧ ((pBy) ∧ (yStx) ∧ (xStk3) ∧ (yStk3))<sub>TR3</sub> ∧ (pPrismaticOPx)<sub>TR4</sub> ∧ (pDeletionØ)<sub>TR5</sub></td><td>k1 line and k2, k3 plane</td><td>k1 line and k2, k3 plane</td></tr><tr><td>Interference-Fit</td><td>((xDy) ∧ (xOk1) ∧ (yOk))<sub>TR1</sub> ∧ ((yGrowingx) ∧ (yPrismaticOPx))<sub>TR2</sub> ∧ (xXy)<sub>TR3</sub> ∧ (yDecreasex)<sub>TR4</sub> ∧ ((xStk2) ∧ (yStk2) ∧ (xSty))<sub>TR5</sub></td><td>k1 line and k2 plane</td><td>k1 line and k2 plane</td></tr></tbody></table>

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Dear Editor-in-Chief and Reviewers,

We would like to thank reviewers for their valuable comments on the proposal. We have considered their comments carefully. As a consequence substantial changes have been made. Please find the list of changes related to each point raised by the reviewers below. We have divided reviewers' comments into the following table. For each point, we have provided a response and given justification and the place in the text where the changes have been made. All these comments have been taken into account. As a result, the paper has been improved from a scientific point of view, and English language has been verified by a native speaker. Further details have been provided in the paper in order to improve understanding.

<table><thead><tr><td>Comment number</td><td>Reviewer's comment</td><td>Response</td><td>Changes made in the manuscript</td></tr></thead><tbody><tr><td>R1.1</td><td>This paper deals with formal relations that can exist between a product design and its assembly process planning. This is a very significant problem targeting the « ultimate goal »: provide a product design to a computer and ask the computer to provide you its assembly process... The good points of the paper: <br> 1 - Serious study and writing. <br> The authors really try to introduce in details all the elements that support the Janus theory. <br> 2 - Large scope from conceptual contributions to software proofs (also a weak points) <br> Once theory is described, they suggest a software implementation and a case study. <br> 3 - Results that clearly sound interesting and that deserve to be published.</td><td>The authors would like to thank this positive comment about the general vision of the article. Indeed, the authors have decided to introduce an implementation and a case study to demonstrate the operational side of the described theory.</td><td>/</td></tr><tr><td>R1.2</td><td>1 - Paper presentation and readability. <br> For a reader not familiar with this subject the paper is really hard to follow. Of course a theory presentation is always delicate to achieve. And many times, the style looks a little too much like "blabla". In some cases, the references used to justify concepts are not strong and rather old.</td><td>The authors have made some efforts to increase clarity in the description of the theory. Nevertheless, since this theory has a mathematical and philosophical foundation, it has been necessary to keep some explanations. In addition, few references (i.e. Bittner et al., Huang & Gu and Muller) have been replaced by more recent ones. However, some references (i.e. Goodman, Heidegger, Lesniewski and so on) have been retained since being</td><td>References</td></tr></tbody></table>
<table><tr><td></td><td>2 – Paper ambition.<br/>This was also a good point (!) but... Of course gathering:<br/>(i) a theory proposition (ii) a software implementation (iii)<br/>a case study in a single paper, is a very good idea. But the<br/>result is a very dense paper too hard to follow. I would<br/>suggest dividing this paper in two papers, 1: theory +<br/>example, 2: Implementation + case study.</td><td>strongly associated to the basis of the theory.<br/>Sure, the fact of incorporating a theory, an ontological<br/>implementation with a case study gives the article quite<br/>dense but the authors think that without such parts, the<br/>article risks to be too theoretical. The authors would like<br/>to have the indulgence of the reviewer on this aspect.<br/>For your information, the implementation part of the<br/>theory has already been described in detailed and<br/>published in another article with the following reference:<br/>Gruhier E., Demoly F. Dutartre O., Abboudi S.,<br/>Gomes S. "A formal ontology-based<br/>spatiotemporal mereotopology for integrated<br/>product design and assembly sequence planning".<br/>Advanced Engineering Informatics, Vol. 29, No. 3,<br/>2015, pp. 495-512.</td><td>/</td></tr><tr><td>R1.3</td><td>3 – Paper goal.<br/>The paper goal or the clear interests of the JANUS theory<br/>should also clearly more highlighted in the beginning of the<br/>paper. Can't you : (i) describe a small example (ii) explain<br/>the problem that cannot be solved with existing theories<br/>(iii) deduce the goal of Janus.</td><td>The authors have well understood this comment. As a<br/>consequence, the introduction has been improved so as to<br/>highlight current problem of existing theories and the<br/>objectives/interests of JANUS theory.</td><td>Introduction<br/>section</td></tr><tr><td>R1.4</td><td>More in details about weak points 2 and 3<br/>"In general, products models are considered as a<br/>representation of our perception of the reality from an<br/>endurantist point of view"<br/>Very quickly, reader not familiar with the subject can be<br/>lost.</td><td>The authors agree that a definition of endurantism is<br/>missing. As such, a specific explanation has been<br/>introduced in order to facilitate understanding of the<br/>reader.</td><td>Introduction<br/>section</td></tr><tr><td>R1.5</td><td>"This research challenge emphasizes a lack of awareness<br/>and understanding of product lifecycle information and<br/>knowledge flows."<br/>An example of style which sounds too much general or<br/>"blabla"</td><td>True. This sentence has been removed by an appropriate<br/>explanation.</td><td>Introduction<br/>section</td></tr><tr><td>R1.6</td><td>"A remaining challenge consists in integrating knowledge of<br/>products lifecycle"</td><td>This issue has been removed from the paper since it does<br/>not fit appropriately with the scope of the paper.</td><td>Introduction<br/>section</td></tr><tr><td>R1.7</td><td></td><td></td><td></td></tr></table>
<table><tr><td></td><td>Why ??</td><td></td><td></td></tr><tr><td>R1.8</td><td>"the dynamic aspect of the design process has not been yet taken into account in an appropriate manner"<br/>Style : meaning of appropriate manner ? thanks to explain.</td><td>This sentence has been improved and justified.</td><td>Introduction section</td></tr><tr><td>R1.9</td><td>"Based on these research stakes, new research efforts have to be addressed on formalism..."<br/>"Based on this literature review on AOD approaches and mereotopology-based theories, the authors propose to develop their own theory...."<br/>Too weak justifications - Theory need unclear - Paper goal too fuzzy<br/>The simple example (weak point 3) in section 2.3 (motivation) is necessary.</td><td>The introduction has been improved so as to highlight current problem of existing theories and the objectives/interests of JANUS theory. A simple example has also been incorporated.</td><td>Introduction section.<br/>Section 2.3</td></tr><tr><td>R1.10</td><td>Globally a serious and detailed work but a paper with a too large scope.<br/>A resulting paper too hard to follow.<br/>I suggest reconsidering the global architecture of your contribution.</td><td>The paper has been reinforced and improved in each section in order to facilitate understanding. The structure of the paper has been slightly changed but not globally. The authors would like to have the indulgence of the reviewer on this aspect.</td><td>All sections.</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td>R2.1</td><td>There are many unclear sentences in the paper, practically in every section. For instance, the title of the paper may be interpreted so that the intention is to use spatiotemporal mereotopology to describe the relationships between product design engineering and assembly planning in a formal manner, rather than mereotopological relationships among the components of a complex artefact. Then, in the Abstract: "a promising mereotopological theory which enables the formal relationships description of product-process design information and knowledge". (What is product-process design information and knowledge?) Or, still in the Abstract: "This region-based theory links together spatial, temporal and spatiotemporal dimensions, therefore leading to a perdurantist philosophy in product</td><td>True. The paper has been enhanced in each section in order to increase clarity. The title of the article has also been changed in order to fit better with the research efforts.<br/>The authors agree with the comment related to perdurantism. As a consequence, the introduction section has been changed and improved from a clearness point of view, by introducing specific definitions of endurantism and perdurantism and how these philosophical concepts have been considered in the context of information modelling in integrated design.<br/>Finally, a native speaker has checked the whole article.</td><td>Title.<br/>Introduction section.<br/>All section.</td></tr></table>
<table><tr><td></td><td>design". (How does a spatiotemporal mereotopological theory lead to a perdurantist philosophy - actually, what it is and why it is needed in product design?) Or, in the Introduction: "In general, products models are considered as a representation of our perception of the reality from an endurantist point of view. This paradigm often leads to misunderstanding in engineering definitions and causes wrong design interpretations." (What is an endurantist point of view in the context of the paper, and what paradigm the authors talk about?) This question stands even after considering what is discussed in 2.3. These, as well as the other ambiguous sentences should be fixed. The whole of the text needs a careful brush up before the manuscript can be considered for publication.</td><td></td><td></td></tr><tr><td>R2.2</td><td>The Introduction is somewhat general (not structured and focused enough). It does not explain why the proposed spatiotemporal mereotopological theory is needed. Is it just an opportunity to adopt the results of recent mereotopological studies, or are there specific bottlenecks in the current early product development and assembly modelling practice that could be eliminated by applying them?</td><td>True. The introduction section was not enough structured in its current shape. As a consequence, the authors have enhanced this section. As such, needs of such a theory have been better addressed.</td><td>Introduction section.</td></tr><tr><td>R2.3</td><td>Table 1 provides an overview of the research efforts to develop region-based mereological/mereotopological theories, involving an overview of the proposed formal relationships between regions. The caption of the Table says that it is a classification of the relevant region-based theories, but I cannot understand in which sense it is classification. It seems to be an inventory of the theories and the introduced (primitive) relationships.</td><td>Right. This is actually a list of relevant mereotopology-based theories. Caption of the table has been changed.</td><td>Table 1</td></tr><tr><td>R2.4</td><td>Mereotopological relationships can be conceptualized on multiple abstraction levels. The issue of specifying and using spatiotemporal mereotopological relations (both containment and associative relations) on multiple</td><td>Right. The state-of-the-art section has been improved by introducing reference ([Horvath and van der Vegte, 2003]) related to nucleus-based modeling in product modelling.</td><td>Sections 2.1. and 2.2. References section</td></tr></table>
<table><tr><td></td><td>architectural abstraction levels (system, subsystem, module, unit, component, etc.) and on different conceptualization levels (functional, structural, behavioural, technological, etc.) is not sufficiently addressed in the literature review, and in the rest of the text. The findings of subsections 2.1 and 2.2 are not blended and concluded.</td><td>Brief conclusions have been included.</td><td></td></tr><tr><td>R2.5</td><td>The authors wrote: "As such, this paper extends existing spatial mereotopology-based theories in the temporal and spatiotemporal dimensions in order to describe persistent objects over time and space." The literature shows that there have been efforts to include time aspects in mereotopology before this work of the authors.</td><td>Indeed, the authors have highlighted existing research effort towards the integration of temporal dimension in mereotopology, but although few temporal primitives have been defined, none of them fully covers all possibilities (related to intervals and instants) addressed by the authors. Besides, primitives are generally described to a specific domain, and in mechanical engineering, new ones have been introduced to represent mechanical pairs over time.</td><td>Section 2.3</td></tr><tr><td>R2.6</td><td>The authors wrote: "In addition, this step requires an ontological implementation of the theory by using OWL-DL (Ontology Web Language - Description Logic) and SWRL (Semantic Web Rule Language) (Gruhier et al. 2015), therefore enabling its machine interpretation and usage within PLM (Product Lifecycle Management) systems (Step 2 of Figure 1) and CAD (Computer-Aided Design) application (Step 3 of Figure 1)." Why is an ontological implementation of the proposed theory indeed needed (the theory could be used in an implicit form too to underpin an artefact and process inclusive assembly sequence planning methodology), and why OWL-DL and SWRL are specifically needed (used) for the development of the ontology based on the JANUS theory?</td><td>The authors have decided to implement this theoretical framework in a formal ontology. Such ontology requires a strong mathematical basis and the consideration of a philosophical trend in order to have a fully independent construction from technological constraints and oriented towards the understanding of the reality even if it remains simple. The use of OWL-DL and SWRL languages have been needed, given that they are the most common restrictions / rules languages used in Protégé Editor. Such kind of language depends on the authors' purpose as well as the rules complexity to be described.</td><td>Introduction section and Section 4.2.</td></tr><tr><td>R2.7</td><td>Related to 3.1.2: In principle, the state of an object can also change in the spatiotemporal domain. For instance, an object can become deformed (or should be deformed for assembly), or ice melt and water evaporated. Should these</td><td>Here the JANUS theory only considers physical objects, which are defined in the design of mechanical assemblies. Nevertheless, deformed objects can be represented with such one. Actually, the authors did not focus their</td><td></td></tr></table>
<table><tr><td></td><td>state changes be considered together with spatial and temporal positions and form?</td><td>research work on physical phenomenon but rather on is based on contact/relationships between physical objects. As such, if the change of states of an object implies a change of relationships or the move of an object, then object like ice melt and water evaporated can be described. Inner change (at a very low granularity) of an object cannot be directly described here. This description should consider the atoms composing the objects and the relationships between atoms.</td></tr><tr><td>R2.8</td><td>Subsection 3.2 repeats what has been presented in (Demoly et al. 2012). The description of forms considers only rigid objects, but not deformed (deformable) of flexible objects. Furthermore, the need for combining the spatial and the temporal dimensions in assembly process planning is obvious. But assembly also needs a compositional logic. How has it been considered in the Janus theory? How can multistate objects (e.g., snap-fits) be represented? How are possible collisions handled?</td><td>JANUS is enough suitable to represent deformed or flexible objects over time, as soon as the interface with another object is deformed. Since the theory is adapted to the early design stages, i.e. before the product geometry is committed, such a description can be only based on spatial and temporal position as well as form. In addition, multistate object is always considered as the same object from the perdurantist framework. Such objects (e.g. snap-fits) have been defined in Table D1 in Appendix Section, where each state is described with spatial, temporal and spatiotemporal mereotopological primitives. Regarding the last question, the authors do not focus their theory on the feasibility checking of the assembly and the collision between parts that the product can undergo.</td></tr><tr><td>R2.9</td><td>In Section 4, the proposed theory is transferred to an ontological implementation using Protégé, “to illustrate the relevance of the proposed theory”. Talking about ‘relevance’ only is not enough, the authors should show how it helps integrating assembly sequence planning into early product design and the benefits too. More importantly, this section discusses a process of assembly sequence planning, rather than the representation of the specification of the spatiotemporal mereotopology theory (4.1. Case study and its mereotopological description). Neither the next subsection (4.2. System execution) does present a formal specification of the content of the</td><td>Right actually the proposed ontological implementation has enabled the development of a formal ontology (called PRONOIA 2). This ontology makes explicit and machine-interpretable the JANUS theory from a logical and semantical manner. The authors have introduced such implementation to prove that the theory works. For more information, a detailed development of PRONOIA2 has already been published.<br/><br/>Gruhier E., Demoly F., Dutartre O., Abboudi S., Gomes S. “A formal ontology-based spatiotemporal mereotopology for integrated</td></tr></table>
<table><tr><td></td><td>PRONOIA2 ontology - instead, a prototype system. The use of this ontology in the assembly sequence planning remained is not explained with sufficient details.</td><td>product design and assembly sequence planning". Advanced Engineering Informatics, Vol. 29, No. 3, 2015, pp. 495-512.<br/>Since the content of the paper seems quite dense, the authors have preferred to not explain all the details associated to the ontology. However, the authors have understood this comment and then have incorporated addition explanations in the appropriate section.</td><td></td></tr><tr><td>R2.10</td><td>The section Conclusions and future work provides a summary of the work and some findings, but fails to offer explicitly tested propositions. The specification of the referenced papers should be made complete and uniform. At least a spellchecker should be used to eliminate the grammatical errors and typos.</td><td>Right. This section has been improved so as to highlight the added value of the research efforts. Grammatical errors and typos have been eliminated with the aid of a native speaker.<br/>The specifications of the referenced papers have also been checked.</td><td>Conclusions section.<br/>All sections.<br/>References section.</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td>R3.1</td><td>This is a very good paper on the topic of the application of the mereotopological theory in Assembly Design and Process Planning. It provides a rigorous and robust presentation of the theoretical background with appropriate examples and a case study. The presentation is overall correct but it requires a serious revision of the writing style by a native English spoken person.</td><td>The authors kindly accept this positive comment. A native English spoken person has made a serious revision of the writing style.</td><td>All sections.</td></tr></table>