# Formalising Autonomous Construction Sites with the Help of Abstract Mathematics

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## Abstract

**:**

## 1. Introduction

- (i)
- Ologs follow the ideas of “lattice of theories” presented in [24], implying, simply speaking, that the same system can be represented by ologs with different levels of details, thus constituting a lattice of representations. This point of view can be adapted to the conceptual modelling of autonomous construction sites or engineering systems in general. A system could be modelled on a very general level at first and, after that, by using specific movements along the lattice, such as contraction, expansion, revision, and analogy, specific parts of the system can be “zoomed in”.
- (ii)
- The categorical foundation of ologs provides a clear formal procedure for relating two different ologs. This procedure is based on the concept of common ground, which is represented by a third olog related to the two other ologs. Practically, it implies that different ologs can be created for individual parts of an engineering system and then coupled together in one system of ologs.

## 2. Fundamentals of Category Theory and Ologs

#### 2.1. Basics of Category Theory

**Definition**

**1**

- (i)
- Objects:$A,B,C,\dots $
- (ii)
- Arrows:$f,g,h,\dots $
- (iii)
- For each arrow f, there are given objects$\mathrm{dom}\left(f\right)$,$\mathrm{cod}\left(f\right)$called the domain and codomain of f. We write$f:A\u27f6B$to indicate that$A=\mathrm{dom}\left(f\right)$and$B=\mathrm{cod}\left(f\right)$.
- (iv)
- Given arrows$f:A\u27f6B$and$g:B\u27f6C$, i.e., with$\mathrm{cod}\left(f\right)=\mathrm{dom}\left(g\right)$, there is given an arrow$g\circ f:A\u27f6C$called the composite of f and g.
- (v)
- For each object A, there is given an arrow${1}_{A}:A\u27f6A$called the identity arrow of A.

- These data are required to satisfy the following laws:$h\circ \left(g\circ f\right)=\left(h\circ g\right)\circ f$and$f\circ {1}_{A}=f={1}_{B}\circ f$.

**Definition**

**2**

**(**[25]

**).**A functor$F:\mathbf{C}\u27f6\mathbf{D}$between categories$\mathbf{C}$and$\mathbf{D}$is a mapping of objects to objects and arrows to arrows in such a way that:

- (i)
- $F(f:A\u27f6B)=F\left(f\right):F\left(A\right)\u27f6F\left(B\right)$;
- (ii)
- $F\left({1}_{A}\right)={1}_{F\left(A\right)}$;
- (iii)
- $F(g\circ f)=F\left(g\right)\circ F\left(f\right)$.

- That is, F respects domains and codomains, identity arrows, and composition. In other words, functors are structure-preserving mappings between categories.

#### 2.2. Introduction to Ologs

**o**ntology

**logs**are closely related to ontologies, which focus on defining what entities exist, thus consequently categorising entities and defining relationships between these categories. In engineering applications, ontologies are used to develop models of reality. Subsequently, ologs are intended to structure and represent the results of defining entities and modelling relationships between categories by recording them in a structure based on category theory.

**Definition**

**3.**

- A
**type**is an abstract concept represented as a box containing a singular indefinite noun phrase. Types are allowed to have compound structure, i.e., being composed of smaller units. The following boxes are types:

- Aspects are functional relationships between the types represented by labelled arrows in ologs. Consider a functional relationship called f between types X and Y, which can be denoted$f:X\to Y$, then X is called the domain of definition for the aspect f, and Y is called the set of result values of f. Here are two examples of using aspects:

- Facts are commutative diagrams, i.e., graphs with some declared path equivalences, in ologs. Facts are constructed by composing several aspects and types.

## 3. Abstract Description of Autonomous Construction Sites

**Definition**

**4**

**(**Autonomous construction site

**).**An

**autonomous construction site**or a

**robotic environment**is the object$\mathfrak{A}=\langle \mathfrak{T},\mathfrak{R},\mathfrak{H},\mathfrak{E},\mathfrak{G},\mathfrak{B}\rangle $, where

- $\mathfrak{T}$is a task to be solved by robots on a construction site;
- $\mathfrak{R}=\left({R}_{1},{R}_{2},\dots ,n\right)$is an n-tuple of robots;
- $\mathfrak{H}$is the object describing human–robot interaction on a construction site;
- $\mathfrak{E}$is a set of pairs representing environmental conditions on a construction site;
- $\mathfrak{G}$is a 4-tuple of GPS information for important parts of a construction site;
- $\mathfrak{B}$is a base station controlling the autonomous construction site.

- The $\mathfrak{T}$ is represented by a tuple $\mathfrak{T}=(\mathcal{V},\mathcal{A})$, where $\mathcal{V}$ is a natural language sentence formulating the task, and $\mathcal{A}$ is a formalisation of $\mathcal{V}$ in terms of a sequence of control signals controlling cyber components of the autonomous construction site.
- The n-tuple of robots $\mathfrak{R}$ evidently contains information about all robots used on the construction site. A precise definition of a robot in the framework of the abstract approach presented in this paper is provided in Definition 5.
- Considering that autonomous construction sites naturally combine human workers and robots, it is necessary to address the question of human–robot interaction [29]. However, an abstract definition of such an interaction goes beyond the scope of the current paper. Therefore, we address the point of human–robot interaction simply by placing a specific object $\mathfrak{H}$ for it, which can still be defined later without the need to change any other definition presented in this paper.
- The role of set $\mathfrak{E}=({E}_{1},{E}_{2})$ is to provide information about environmental conditions on a construction site. In this way, this information is formalised in terms of a denotation ${E}_{1}$ and the corresponding value ${E}_{2}$.
- For integration of a robotic system in the construction progress, it is necessary to provide information on the positioning of the robotic system, as well as all essential parts of the construction site. For that purpose, the 4-tuple $\mathfrak{G}=(O,{x}_{1},{x}_{2},{x}_{3})$ is introduced, where O denotes the object, and ${x}_{1},{x}_{2},{x}_{3}$ are object coordinates.
- Finally, cyber parts of the autonomous construction site must be controlled, and therefore, the base station $\mathfrak{B}$ needs to be included in the definition.

**Definition**

**5**

**(**Robot

**).**A

**robot**is the object$\mathfrak{R}$ = $\langle \mathfrak{C},\mathfrak{K},\mathfrak{P},\mathfrak{S},\mathfrak{Ac}\rangle $, where

- $\mathfrak{C}$is a robotic controller generating control signals;
- $\mathfrak{K}$is a finite set of kinematic properties of a robot;
- $\mathfrak{P}$is a k-tuple of physical properties of a robot;
- $\mathfrak{S}=\left({\mathcal{S}}_{1},{\mathcal{S}}_{2},\dots ,{\mathcal{S}}_{n}\right)$is an n-tuple of sensors installed on a robot;
- $\mathfrak{Ac}=\left({\mathcal{A}}_{1},{\mathcal{A}}_{2},\dots ,{\mathcal{A}}_{m}\right)$is an m-tuple of actuators installed on a robot.

- The robotic controller $\mathfrak{C}$ is needed for a communication with a base station $\mathfrak{B}$ introduced in Definition 4, and in general, it sends a sequence of control signals for operating the robot.
- A set of kinematic properties $\mathfrak{K}$ represents physical constraints limiting the possible movements of a robot. In practice, $\mathfrak{K}$ is determined by the kinematic chain that is formed by the series of manipulators, connected by joints, and may differ in specifications and movement, providing the (internal) axes of the robot. Furthermore, a robot system may consist of external axes, e.g., track systems. The degrees of freedom of the robotic system is the combination of internal and external axes determined by the kinematic chain. Based on the kinematic properties representing specific constraints, the robotic controller $\mathfrak{C}$ is able to generate control signals for the robot to reach target coordinates in the determined work area. Additionally, it is important to notice that for making $\mathfrak{K}$ consistent from the point of view of set theory, it is assumed that all kinematic constraints are formalised in terms of equations and inequalities, i.e., mathematical expressions.
- The tuple $\mathfrak{P}$ contains robot specification information (e.g., type, manufacturer, or information about a motor driving the system), which includes information. Physical properties have to be also known for generating control signals by the robotic controller $\mathfrak{C}$.
- Evidently, various sensors might be installed on a robot for measuring environmental conditions, as well as important physical quantities of a robot itself, e.g., the temperature of individual parts. These sensors are combined in an n-tuple $\mathfrak{S}$.
- Similar to sensors, various actuators need to be installed on a robot and are activated via control signals. These actuators are combined an m-tuple $\mathfrak{Ac}$.

**Definition**

**6**

**(**Sensor

**,**[13]

**).**A

**sensor**is the object $\mathcal{S}=\langle \mathcal{I},\mathcal{Y},\mathcal{T}\rangle $, where

- $\mathcal{I}=\left({I}_{1},{I}_{2},\dots ,{I}_{n}\right)$is an n-tuple of finite index sets;
- $\mathcal{Y}=\left({Y}_{1},{Y}_{2},\dots ,{Y}_{n}\right)$is an n-tuple of measurements with${Y}_{i}\in {\mathbb{R}}^{{N}_{i}}$,$i=1,\dots ,n$;
- $\mathcal{T}$is a k-tuple of specifications (type information).

**Definition**

**7**

**(**Sensor cluster

**,**[13]

**).**A sensor cluster is the object ${\mathcal{S}}_{\mathcal{C}}=\langle \mathfrak{B},\mathfrak{S},\mathfrak{R}\rangle $, where

- $\mathfrak{B}$is a sensor node or a base station controlling the sensor cluster;
- $\mathfrak{S}=\left({\mathcal{S}}_{1},{\mathcal{S}}_{2},\dots ,{\mathcal{S}}_{n}\right)$is an n-tuple of sensors, introduced in Definition 6;
- $\mathfrak{R}=\left({R}_{1},{R}_{2},\dots ,{R}_{m}\right)$is an m-tuple of relations.

**Definition**

**8**

**(**Actuator

**).**An actuator is the object$\mathcal{A}=\langle \mathfrak{B},{\mathcal{A}}_{\mathcal{S}},\mathcal{T}\rangle $, where

- $\mathfrak{B}$is a sensor node or a base station controlling the actuator;
- ${\mathcal{A}}_{\mathcal{S}}$is an actuation signal;
- $\mathcal{T}$is an k-tuple of specifications (typing information).

## 4. Olog Representations of Robotic Construction Sites

- where the dashed arrow indicates that we cannot arrive at olog $\mathbf{C}\left(\mathbf{R}{\mathcal{O}}_{1}\right)$ from the olog ${\mathcal{O}}_{1}$ in one step, and a combination of contractions and expansions is required. Hence, we obtain a lattice of representations containing several ologs that are convertible between each other and represent different levels of details about an autonomous construction site.

- (i)
- Olog ${\mathcal{O}}_{1}$ can be expanded to include olog ${\mathcal{O}}_{2}$ as a sub-part.
- (ii)
- Olog ${\mathcal{O}}_{1}$ can be expanded to include olog ${\mathcal{O}}_{2}$, and then the resulting olog should be contracted to olog ${\mathcal{O}}_{2}$.

- ${\mathcal{O}}_{1}\cup {\mathcal{O}}_{2}$ denotes the olog obtained by expanding the autonomous construction site olog by adding the robot olog ${\mathcal{O}}_{2}$ to it;
- ${\mathcal{O}}_{1}\cup {\mathcal{O}}_{2}\cup {\mathcal{O}}_{3}$ denotes the olog obtained by expanding the olog ${\mathcal{O}}_{1}\cup {\mathcal{O}}_{2}$ by adding the sensor olog ${\mathcal{O}}_{3}$ to it;
- ${\mathcal{O}}_{1}\cup {\mathcal{O}}_{2}\cup {\mathcal{O}}_{3}\cup {\mathcal{O}}_{5}$ denotes the olog obtained by expanding the olog ${\mathcal{O}}_{1}\cup {\mathcal{O}}_{2}\cup {\mathcal{O}}_{3}$ by adding the actuator olog ${\mathcal{O}}_{5}$ to it.

- (i)
- The resulting ontological description can be easily extended by adding new definitions and ologs without a need for changing previous results;
- (ii)
- The lattice of representation can even be created at first, serving as a guideline for creating ologs and definitions;
- (iii)
- Each olog can be directly converted into a database, see again [23], and, thus, used as a basis for practical implementations of ontologies and formal representations.

## 5. Discussion and Conclusions

**Abstract description of autonomous construction sites**Several abstract definitions formalising autonomous construction sites have been introduced in Section 3. The idea of these definitions is to provide a common ground for an olog-based description of autonomous construction. A top-to-bottom approach for conceptual modelling of autonomous construction sites has been chosen. Hence, starting with an autonomous construction site, definitions of its more detailed components have been added step-by-step. The main advantage of this approach is that the resulting conceptual modelling framework is scaleable and extendable with more details, if necessary. Any of the Definitions 4–8 can be revised or updated without the need for a general restructuring of the complete framework presented in this paper.It is also important to underline that the field of robotic construction still misses generally accepted “standard” definitions. Therefore, the results presented in Section 3 should not be understood in the way of the definitions to become an industrial standard but rather as an approach on how to address practical engineering problems on a more abstract level sieving out all concrete details.**Olog-based representations of autonomous construction**An olog-based representation of autonomous construction sites has been presented in Section 4. As described in Section 2.2, ologs are designed to handle the subjectivism of the creator of the abstract model. This point has been further strengthened by coupling ologs with abstract definitions introduced in Section 3. This coupling makes the relation and comparison, as well as the translation of ologs, even more mathematically sound and formal. Hence, the ologs presented in this paper can be straightforwardly implemented in the form of databases, as well as the extension/contraction rules. Further, if more details are desired in a concrete application, these details can be easily added via revision of existing ologs, as has been demonstrated in the paper.**Lattice of representations**Finally, Section 4 presents a lattice of representations, which is developed by extending and revising existing ologs. Arguably, the concept of the lattice of representations is the most powerful tool of olog-based description of engineering systems. First, the lattice can be easily extended without the need for changing previous results. In this case, a new olog is simply added to the lattice, and the corresponding extension is then formally defined. Second, the lattice of representation can even be created first and, hence, provide a guideline for creating ologs and missing definitions.

- The first step should be the formal creation of a lattice of representations, where, of course, instead of ologs, only names of important parts to be described are written. In this step, it is important to decide what should be the least detailed olog and how many different parts need to be modelled.
- Collect/create definitions of all parts to be described by ologs. In this step, it is important to keep the balance between the number of details and the level of abstractions. This balance is generally to be defined by the modeller and the objective of the work. Evidently, existing definitions, for example, industry standards, can be used, or new definitions can be developed, as has been done in this paper.
- Create ologs for each part and fit them into the lattice of representations defined in Step 1. Further, if necessary, ologs can be converted into databases and connected to other conceptual models, if available.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Olog | Ontology log |

UAV | Unmanned aerial vehicle |

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**Figure 3.**Olog representation of an autonomous construction site after the application of an expansion mapping $\mathbf{E}$.

**Figure 4.**Olog representation of an autonomous construction site after the application of a contraction mapping $\mathbf{C}$ to the expanded olog $\mathbf{E}{\mathcal{O}}_{1}$.

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**MDPI and ACS Style**

Legatiuk, D.; Luckey, D.
Formalising Autonomous Construction Sites with the Help of Abstract Mathematics. *Eng* **2023**, *4*, 799-815.
https://doi.org/10.3390/eng4010048

**AMA Style**

Legatiuk D, Luckey D.
Formalising Autonomous Construction Sites with the Help of Abstract Mathematics. *Eng*. 2023; 4(1):799-815.
https://doi.org/10.3390/eng4010048

**Chicago/Turabian Style**

Legatiuk, Dmitrii, and Daniel Luckey.
2023. "Formalising Autonomous Construction Sites with the Help of Abstract Mathematics" *Eng* 4, no. 1: 799-815.
https://doi.org/10.3390/eng4010048