Optimal Morphologies of n-Omino-Based Reconfigurable Robot for Area Coverage Task Using Metaheuristic Optimization

Abstract

Reconfigurable robots design based on polyominos or n-Omino is increasingly being explored in cleaning and maintenance (CnM) tasks due to their ability to change shape using intra- and inter-reconfiguration, resulting in various footprints of the robot. On one hand, reconfiguration during a CnM task in a given environment or map results in enhanced area coverage over fixed-form robots. However, it also consumes more energy due to the additional effort required to continuously change shape while covering a given map, leading to a deterioration in overall performance. This paper proposes a new strategy for n-Omino-based robots to select a range of optimal morphologies that maximizes area coverage and minimizes energy consumption. The optimal “morphology” is based on two factors: the shape or footprint obtained by varying the angles between the n-Omino blocks and the number of n-Omino blocks, i.e., “n”. The proposed approach combines a Footprint-Based Complete coverage Path planner (FBCP) with a metaheuristic optimization algorithm to identify an n-Omino-based reconfigurable robot’s optimal configuration, assuming energy consumption is proportional to the path length taken by the robot. The proposed approach is demonstrated using an n-Omino-based robot named Smorphi, which has square-shaped omino blocks with holonomic locomotion and the ability to change from monomino to tetromino. Three different simulated environments are used to find the optimal morphologies of $Smorphi$ using three metaheuristic optimization techniques, namely, MOEA/D, OMOPSO, and HypE. The results of the study show that the morphology produced by this approach is energy efficient, minimizing energy consumption and maximizing area coverage. Furthermore, the HypE algorithm is identified as more efficient for generating optimal morphology as it took less time to converge than the other two algorithms.

1. Introduction

The cleaning and maintenance (CnM) of floor area and other infrastructure is a necessary part of everyday life, but it is often considered tedious and challenging. In recent years, researchers have made significant efforts to develop autonomous cleaning robots to make these tasks easier and more efficient. It is estimated that the market for these robots will reach a value of $25.9 billion by 2027. Some well-known brands of these robots include Roomba, Xiaomi, Samsung, Dyson, and Electrolux. One common problem with many of these robots is that they have a fixed morphology and footprint, which can limit their ability to access a given floor map efficiently. To overcome this problem, researchers have developed cleaning robots with the ability to change their shape or configuration. These reconfigurable robots are able to produce better area coverage than fixed morphology robots. However, because they constantly change their shape, they tend to consume more power than fixed morphology robots. Hence, it is essential to devise more energy-efficient strategies for reconfigurable robots performing area coverage tasks in a given environment or map.

Reconfigurable systems can change their morphology to either introduce a new function or improve an existing function. The concept of reconfigurable systems was first introduced in 1980 in reconfigurable manufacturing systems [1]. Since then, reconfigurable systems have been widely adopted in various fields of innovation, such as surveillance [2], space missions [3], and reconfigurable toys [4]. However, one of the most prominent applications of a reconfigurable system is in self-reconfigurable robots, which are designed to vary their morphology by themselves [5]. These robots have multiple states and perform different functions for each unique state based on environmental conditions. Self-reconfigurable robots are widely used in situations where the performance of fixed-morphology robots is limited. A reconfigurable robot can fold, bend, expand/collapse, attach/detach to change its morphology to access narrow spaces, overcome obstacles, perform multiple functions, carry heavy payloads by connecting more modules, and split into multiple robots to execute tasks in parallel [6]. In general, reconfigurable robots can be classified into three categories: inter, intra, and nested, based on their mechanism for changing configuration [7]. Intra-reconfigurable robots act as a single unit that can change its shape and functionality without adding external components, whereas inter-configuration is achieved by attaching or detaching homogenous or heterogeneous blocks. Nested reconfiguration combines the behaviors of inter and intra-reconfiguration [7]. However, to the best of authors’ knowledge, no research has focused on identifying the optimal number of blocks (inter-reconfiguration) or the optimal configuration within a number of blocks (intra-reconfiguration) which maximizes the task efficiency.

The scope of reconfigurable robots in the domain of cleaning and maintenance is extensive and ever-expanding. Reconfigurable robots are highly suitable for area coverage applications as they can adapt their shape to overcome environmental constraints. For example, the polyomino-based tiling robot hTetro [8] can change its morphology to clean the environment efficiently by transforming into seven unique morphologies: I, J, L, O, S, T, and Z. Similarly, the modified version of the hTetro robot, reported in [9], can transform into an infinite number of configurations to perform cleaning tasks. The robot, named sTetro [10], is designed with three independent reconfigurable modules to traverse stairs by overcoming level changes to perform cleaning tasks. Additionally, the robot named vTetro navigates over facades to perform mopping while overcoming obstructions in the facade [11]. These applications clearly demonstrate the significance of reconfigurable robots in area coverage tasks. Although these robots produce higher efficiency compared to fixed morphology systems, their energy consumption may be higher because they may require additional actuators to shift their morphology.

Energy efficiency during a reconfigurable robot’s operation is highlighted in a few studies in the literature. In [12], the author proposed an energy-efficient multimode locomotion-based reconfigurable robot that can move in different terrains. Similarly, the robot reported in [13], designed to perform crucial tasks in space exploration, uses an energy-efficient locomotion type based on the terrain or environmental conditions. Additionally, in the reported work [14], the author developed a reconfigurable end effector by adapting shape memory alloys with energy-free rest locations to improve the system’s energy efficiency. In [15], the author introduced an energy-efficient path-planning strategy for reconfigurable robots based on batch-informed trees (BIT), which consider the energy cost for every reconfigurable action of the robot. In [16], the author proposed an evolutionary algorithm-based optimal coverage path planning for a differential drive-based reconfigurable robot. Additionally, in the reported work [17], the authors studied the energy consumption of reconfigurable robots by considering different configurations, which is useful for choosing the optimal reconfiguration process with minimal energy consumption.

The existing research on reconfigurable robots for area coverage has primarily focused on using multiple modular units that can be connected in a serial or parallel configuration. These modules have the ability to change their functionality by altering their shape, but they typically maintain the same number of modules throughout. However, this approach may not be the most efficient solution for all scenarios. For example, an environment with many obstacles may require a smaller, more agile reconfigurable robot with a relatively lower number of modules. In contrast, a reconfigurable robot with multiple modules may need to constantly shift its morphology to overcome the environmental constraints, resulting in lower overall efficiency. To address this research gap, in this work, we are taking the initial step toward identifying the optimal number of modules and configurations that will maximize area coverage while minimizing the energy consumption based on the environment. Furthermore, this will allow a robot to adapt to a wide range of environments while maximizing task efficiency.

Path planning is an inherent part of any robot deployed for area coverage. Defining a path over a sequence of waypoints that covers the entire region of the map while avoiding obstacles is often referred to as complete coverage path planning (CCPP) [18]. Most coverage path-planning algorithms decompose the given map into a uniform space based on the decomposition approach. Generally, this decomposition-based path planning is classified into exact, approximate, and semi-approximate cellular decompositions according to the taxonomy reported in [19]. The grid-based path-planning approach is one of the most prominent and widely used coverage planning strategies. In the grid-based path-planning approach, cells are decomposed into a uniform grid where every grid carries information stating whether the grid is free or occupied with obstacles. Spanning tree-based coverage and wavefront algorithms are the two most widely adapted grid-based path-planning approaches, as they minimize the coverage time compared to other algorithms [20,21]. Other path-planning methods include neural network-based coverage, graph-based coverage, 3D coverage, bio-inspired coverage, etc. [22,23,24,25]. However, most of these path-planning approaches do not account for the varying footprint sizes of the reconfigurable robot.

In conventional path planning and area coverage, global planners treat each grid as equal in size, regardless of the robot type, application, or footprint. Recently, some works have explored footprint-based path planning, taking into account the robot’s operating limits, such as the arm angle in hydroblasting robots [26]. Another example is the camera field of view (FoV)-based coverage planning in drones, where the camera’s FoV serves as the robot’s footprint [27], resulting in more efficient system performance through a decrease in path length. However, no prior research has been conducted on footprint-based global coverage planning for reconfigurable robots with changing morphologies. In this work, we consider the footprint size of the reconfigurable robot and propose a new coverage planner considering the variable footprint of a robot.

Several approaches have been studied for identifying the optimal solution for path planning, and area coverage tasks, such as Reinforcement Learning (RL), neural networks, metaheuristics, fuzzy logic, and linear optimization [28,29,30]. Metaheuristic-based algorithms have proven superior to the other approaches due to their simplicity, faster computation time, and ability to find optimal solutions without prior information [31]. As a result, they are widely adopted in fields such as machine learning, computer vision, scheduling, and robot motion planning. Some of the most commonly used metaheuristics algorithms for path planning and area coverage tasks in mobile robots include Particle Swarm Optimization (PSO), Genetic Algorithm (GA), Hypervolume-based Many Objective Optimization (HypE), and Multiobjective Evolutionary Algorithm based on Decomposition (MOEA/D) [32,33,34]. However, a framework for identifying the optimal morphologies of a reconfigurable robot using metaheuristic optimization has not been reported yet, which is highlighted in this work.

In this paper, we present a novel approach to identify the optimal morphology of reconfigurable n-Omino-based tiling robots which maximize the area coverage and minimize energy consumption. To demonstrate the effectiveness of our approach, we introduce $Smorphi$, which is a holonomic and nested reconfigurable robot. $Smorphi$ can assemble and disassemble with other $Smorphi$ blocks, resulting in large configurations due to varying the hinge angle and the number of blocks connected to it. Our approach uses three metaheuristic optimization techniques to identify a range of optimal morphologies that maximizes area coverage and minimizes energy consumption. The motivation for using three metaheuristic algorithms is to validate the proposed framework and identify the best optimization method for identifying the optimal morphology. The objectives and contributions of this work are listed below:

• Design of an n-Omino inspired reconfigurable robot named $Smorphi$, which has a square-shaped and holonomic mobile base as a building block of n-Omino. This paper takes the maximum number of omino blocks as four (n = 4). Furthermore, each block can attach and detach at a specific joint location resulting in nested reconfiguration, and the design principle associated with $Smorphi$ is also highlighted.
• Proposing Footprint-Based Complete coverage Path planner (FBCP) based on the shape or footprint obtained by varying the angles between the n-Omino blocks.
• Finding the optimal “morphology”, i.e., the shape or footprint obtained by varying the angles between the n-Omino blocks and the number of n-Omino blocks, i.e., “n” for a given environment using the metaheuristic optimization algorithm.
• Validation of the proposed framework for finding the optimal morphology for three simulated environmental maps with the objective of minimizing energy consumption and maximizing area coverage. Three different optimization techniques, namely, Multiobjective Evolutionary Algorithm based on Decomposition (MOEA/D), Optimized Multi-Objective Particle Swarm Optimization (OMOPSO), and Hypervolume-Based Many Objective Optimization (HypE) are used to find the optimal morphology from the pool of solutions.

The remainder of the article is organized as follows: a bird’s-eye view of the proposed approach is introduced in the Section 2. The mechanical design of the nested-reocnfigurable robot $Smorphi$ is discussed in Section 3 along with the tiling theory and transformational design principles. Section 4 discusses the kinematic model of the holonomic $Smorphi$ robot. Section 5 explains the novel Footprint-Based Complete coverage Path planner (FBCP). Section 6 explains the objective functions along with the optimization approach used. Results and discussion of the proposed framework are presented in Section 7. Finally, Section 8 concludes the paper by summarizing the key findings and highlighting future work.

2. System Overview

A bird’s-eye view of the proposed framework is shown in Figure 1. This work addresses the selection of the optimal morphologies for an n-Omino-based reconfigurable robot, aiming to maximize area coverage and minimize energy consumption. The n-Omino robot has numerous configurations obtained by varying hinge angles and the number of blocks. This work proposes a solution combining a Footprint-Based Complete coverage Path planner (FBCP) with metaheuristic optimization algorithms to select the optimal morphology from a vast range of configurations. The output of this approach provides a range of optimal morphologies which can be used for area-coverage tasks depending on the scenario, as shown in Figure 1. Furthermore, this approach enables us to determine the optimal configuration that balances area coverage and energy consumption. The framework is divided into three layers: the Design space, the Footprint-Based Complete coverage Path planner (FBCP), and the Optimization space. The Design space represents the mechanical design of the robot, which forms its physical structure. It is generalized for the design of any reconfigurable system by leveraging transformational design principles [35,36,37,38,39,40,41]. These principles classify reconfigurable systems based on their means of reconfiguration, and they aid in creating any reconfigurable system through a systematic design approach. In this work, we used these principles to design the reconfigurable tiling robot $Smorphi$. The detailed mechanical design of $Smorphi$ and the transformational design principles are discussed in Section 3.

In the Footprint-Based Complete coverage Path planning (FBCP) space, the footprint model is generated based on the type of robot, and the footprint varies for each robot based on its functionality. For example, the footprint is considered based on the effective cleaning area for a floor-cleaning robot, and the footprint is the camera’s field of view for a surveillance robot. The following are the assumptions and considerations in this paper:

• The robot’s, i.e., $Smorphi$, footprint is modeled as the occupied area by the configuration at a given instance. The number of blocks varies depending on the application and scenario. In this work, we are dealing with an indoor cleaning application, and the maximum number of blocks accounted for is two to four of an n-Omino (i.e., n = 2, 3, and 4) due to the size restrictions of the operating scenario.
• The energy consumption of a reconfigurable robot during area coverage is proportional to its path length as the robot expends energy to move and overcome forces encountered. There are several factors that can affect the energy consumption of the reconfigurable robot, such as path length, robot weight, drive type, payload, and terrain, among others. In this work, to simplify the analysis, we assume that the energy consumption of the reconfigurable robot during area coverage is directly proportional to the path length of the robot traversed. The longer the path, the more time the robot must spend traversing it, further increasing energy consumption.
• In this work, we adopted the spiral-spanning tree coverage algorithm as the complete coverage path planner, which is discussed in Section 5. The complete coverage planner then calculates the path length and the percentage of area covered in a given map based on the footprint size.
• We considered the path length and the percentage of area covered as the two critical factors for identifying the optimal morphology. However, the outcome of the footprint-based coverage can be several variables, depending on the requirement, such as the path length, overlap, number of turns, number of repetitions, percentage of area covered, etc.
• Three different metaheuristic algorithms, namely, the Multi-objective Evolutionary Algorithm based on Decomposition (MOEA/D), the Optimized Multi-Objective Particle Swarm Optimization (OMOPSO), and the Hypervolume-based Many Objective Optimization (HypE).

3. Smorphi Robot Design

This section outlines the mechanical design process of the $Smorphi$ robot as well as the tiling theory and transformational design principles used.

3.1. Tiling Theory

Tiling theory is a branch of mathematics that explores the arrangements of shapes that cover an area without gaps or overlaps. Tiling theory has been widely used in the field of computer graphics, arts, textural synthesis, etc. Based on the morphology, the tiling can be classified into three major chunks: polyominoes, heptiamonds, and hexiamonds. Polyominoes are plane geometrical figures constructed by connecting one or more squares on their side faces or edges; similarly, heptiamonds and hexiamonds are formed by connecting one or more equilateral triangles and hexagons. Based on their orientation, transformation, and chirality, these shapes can be further categorized into three configurations: free, fixed, and one-sided. Figure 2a–c shows some of the possible configurations of polyominoes, heptiamonds, and hexiamonds depending on the number of blocks, spatial arrangement, and orientation.

Self-reconfigurable robots, which are inspired by tiling theory, are often referred to as tiling robots. These robots are designed as individual blocks that are connected by hinges, allowing them to change their shape or morphology. The usefulness of tiling theory in robotics has been demonstrated in several articles [42,43,44,45]. In this work, we propose a novel framework that can be adapted to identify the optimal morphology for reconfigurable robots based on n-Ominoes as shown in Figure 2d. An n-Omino is a geometric figure made up of n squares that are joined together along their edges. The most well-known examples of n-Ominoes are tetrominoes, which are made up of four squares, and pentominoes, which are made up of five squares.

Table 1. Number of unique morphologies based on the number of block and angle interval.

Angle Interval (in Degrees)	No. of Blocks (n)
	$\mathit{n}$ = 2 (${\mathbf{\alpha }}_{\mathbf{12}}$)	$\mathit{n}$ = 3 (${\mathbf{\alpha }}_{\mathbf{12}},{\mathbf{\alpha }}_{\mathbf{23}}$)	$\mathit{n}$ = 4 (${\mathbf{\alpha }}_{\mathbf{12}},{\mathbf{\alpha }}_{\mathbf{23}},{\mathbf{\alpha }}_{\mathbf{34}}$)
180	2	3	6
90	3	8	21
45	5	20	95
30	7	42	259
15	13	144	1651
5	37	968	37,999
1	181	24,362	5.1 × 105

shows the number of unique morphologies that can be generated using the $Smorphi$ robot with different angle intervals. ${\alpha }_{12}$, ${\alpha }_{23}$, and ${\alpha }_{34}$ represent the hinge angles, and they are allowed to rotate from 0 to 180 degrees based on the angle interval. When the angle interval is higher, the number of morphologies is lower. On the other hand, as the interval becomes smaller, the number of morphologies increases exponentially. For example, with a number of blocks as four ($n_B$ = 4) configuration and an angle interval of 180, there are six unique morphologies with angles $({\alpha }_{12},{\alpha }_{23},{\alpha }_{34})\equiv$ (0, 0, 0), (0, 0, 180), (0, 180, 0), (0, 180, 180), (180, 0, 0), and (180, 0, 180). At the same time, when the angle interval is 1, the number of unique morphologies is 5.1 × 10${}^5$. Therefore, based on the objective, our framework is designed to pick the optimal morphologies from the vast number of configurations that maximizes the area coverage and minimizes the path length. The detailed mechanical design of the $Smorphi$ robot is discussed in the following section.

3.2. Mechanical Design

Creating a reconfigurable system is influenced by various factors, including the type of applications, environmental constraints, the type of reconfiguration behavior needed to satisfy the requirements, and geometrical constraints. Among these considerations, the type of reconfiguration behavior needed for the reconfigurable system is crucial because it determines the novelty of the overall system and aids in overcoming environmental constraints. In this work, we followed a systematic design approach by adopting transformational design principles as an enabler for designing the self-reconfigurable robot named $Smorphi$. In the reported work [35], the authors empirically analyzed a wide range of reconfigurable products and classified them into three different principles and twenty facilitators based on their means of reconfiguration. In this work, the transformational design principle acts as a base layer for defining the required reconfiguration behavior of the robot. For a detailed definition of the transformational design principles and facilitator, please refer to the article [35]; below is a brief description of the three transformational principles.

• Expand/collapse: Changing the physical dimensions of an object to bring about an increase/decrease in an occupied volume primarily along an axis (1D), in a plane (2D), or 3D (three dimensions).
• Expose/cover: Concealing or revealing a new surface to alter functionality.
• Fuse/divide: Make a single functional device become two or more devices or vice versa where at least one of the multiple devices has a distinct functionality separate from the function of the single device.

The transformational design principle that has been adopted to design a reconfigurable pavement sweeping robot is demonstrated in this work [46], where the subsystems are designed by mapping with the transformational design principles and facilitators. By adopting this approach, the $Smorphi$ robot behavior is designed based on the two transformational principles, namely expose/cover (the designed robot $Smorphi$ can reconfigure into different shapes by revealing/concealing their side surface) and fuse/divide (identical modules can be added to increase the footprint of the robot).

The mechanical design of the $Smorphi$ robot is shown in Figure 3. Each unit of the $Smorphi$ consists of four Meccanum wheels coupled with the DC motor to enable the holonomic locomotion. Each $Smorphi$ block is modular as it can be attached/detached through a hinge placed at the corner of each module; depending upon the configuration, each module can be attached to the right/left side of the module (LLR/RRL). Using this configuration, ‘n’ number of modules can be attached; however, in this paper, we only used four independent modules to demonstrate the developed framework in LLR configuration. Furthermore, block 2 is considered the core unit, and it carries all the processing unit which controls the other modules when they are attached.

4. n-Omino Kinematic Model

The geometry of the single block of n-Omino with four Mecanum wheels is shown in Figure 4a. The assumptions for the kinematic analysis of the n-Omino wheeled mobile robot are namely: (a) the hinged joints are in a locked state in a given configuration, (b) the relative angular changes between each block is attained by keeping the first attached block, i.e., Block #2 assigned with the robot frame and the rest of the attaching blocks (Figure 4b) can change the angles between consecutive blocks. The kinematic equation for the four Mecanum wheel mobile base is given below:

$${\mathbf{\varphi }}_{m1}=\frac{1}{r_w}{\mathbf{J}}_{m1}{\zeta }_{m1}$$

(1)

where ${\mathbf{\varphi }}_{m1}$ is the vector of the wheel velocities of module 1, i.e., ${\left[{\dot{\phi }}_1{\dot{\phi }}_2{\dot{\phi }}_3{\dot{\phi }}_4\right]}^T$ and ${\zeta }_{m1}\equiv {\left[\dot{x}\dot{y}\dot{\theta }\right]}^T$ is the vector of robot velocity in the robot frame. To transfer this information to the world frame, the rotation matrix about the Z-axis is included as $\mathbf{R}\left(\theta \right){\zeta }_{m1}$. ${\mathbf{J}}_1$ is the Jacobian matrix for a single robot with four Mecanum wheels with size $4\times 4$ with elements as $[-11K_1;11-K_1;-11-k_1;11k_1]$ where the constant $k_1=(a+b)$.

Figure 4c shows the n-Omino multiple units of a Mecanum-wheeled system with $m=1,2,\cdots ,n$. In order to establish the relation between the robot frame and the frame associated with individual blocks, the second block, i.e., #2 origin is selected as the robot frame with $O_R{X}_R{Y}_R$. Relative to this frame, the locations of other individual blocks are assigned with the vector $d_m$ with $m=1,\cdots ,n$ and the angle subtended by it with robot frame as ${\psi }_m$, as shown in Figure 4c. Then, the kinematic relation mapping the wheel velocity to the designated robot coordinate system located with the second block #2 is given by:

$${\mathbf{\varphi }}_{m_n}=\frac{1}{r_w}{\mathbf{R}}_n\left({\alpha }_{(n-1),n}\right){\mathbf{R}}_n\left({\psi }_{(n-1),n}\right){\mathbf{T}}_n\left({\mathbf{d}}_n\right)$$

(2)

where ${\mathbf{R}}_n\left({\psi }_{(n-1),n}\right)$ is the rotation about the Z-axis by an angle $\left({\psi }_{(n-1),n}\right)$, and ${\mathbf{T}}_n$ is the translation matrix in a 2D plane, i.e., XY plane by the magnitudes of vector $\mathbf{d}$ expressed as:

$${\mathbf{R}}_n\equiv \left[\begin{array}{ccc}cos{\alpha }_{(n-1),n}& sin{\alpha }_{(n-1),n}& 0\\ -sin{\alpha }_{(n-1),n}& cos{\alpha }_{(n-1),n}& 0\\ 0& 0& 1\end{array}\right];{\mathbf{T}}_n\equiv \left[\begin{array}{ccc}1& 0& -d_{n}sin{\psi }_n\\ 0& 1& d_{n}cos{\psi }_n\\ 0& 0& 1\end{array}\right]$$

(3)

Moreover, in the inertial frame, $X_i{Y}_I{O}_i$, the inverse kinematics equation of the n-Omino system, is derived as:

$$\left[\begin{array}{c}{\dot{\mathrm{\Phi }}}_{1m_1}\\ {\dot{\mathrm{\Phi }}}_{2m_2}\\ ⋮\\ {\dot{\mathrm{\Phi }}}_{n{m}_n}\end{array}\right]=\left[\begin{array}{c}{J}_1{K}_1\\ J_2{K}_2\\ ⋮\\ J_n{K}_n\end{array}\right]\mathbf{R}\left(\theta \right)\left[\begin{array}{c}{\dot{x}}_R\\ {\dot{y}}_R\\ \dot{\theta }\end{array}\right]$$

(4)

where ${\mathbf{J}}_n$ is the Jacobian for the $n^{\mathrm{th}}$ block of an n-Omino. The connection constraints from the hinge joint between each block are basically the rotation and translation of each block with respect to block #2. Matrix ${\mathbf{K}}_1=R_{Z,1}{T}_{n,1}$. The n-Omino based robot configuration is mainly defined by the angle between each block, i.e., ${\alpha }_{1,2}$, ${\alpha }_{2,3}$, ⋯, ${\alpha }_{n-1,n}$. The variation in these angles will result in a varied footrpint of the robot, which is accounted for in the next section for the path planner.

5. Footprint-Based Complete Coverage Path Planner (FBCP)

This section details the footprint-based path planning approach used in the $Smorphi$ robot to identify the optimal configurations. We developed a novel footprint-based path planning approach that is adaptable for all possible configurations of the reconfigurable robot. This approach is divided into two layers: (1) footprint generation, and (2) complete coverage path planning based on the generated footprint.

The concept of footprint generation refers to the identification of the functional area of a robot. The footprint of a robot is determined by its functionality, and it can vary from one robot to another. In this case, the footprint is defined as the surface area occupied by the robot at any instance of time. The $Smorphi$ robot’s footprint is modeled as square blocks connected in a either “Left (L)” or “Right (R)” hinge configuration, as shown in Figure 4c. Each block is assumed to have four vertices, and the vertices of each block are projected onto the $X_I$ and $Y_I$ axes to determine the occupied area, as described in Equations (5) and  (6). This information is then used to create a matrix of 0s and 1s, where 0 represents unoccupied space, and 1 represents the occupied space.

$$\begin{array}{cc}\hfill {\mathit{a}}_x^n& ={[{\mathit{a}}_1^n·\mathbf{x}{\mathit{a}}_2^n·\mathbf{x}\cdots {\mathit{a}}_n^n·x]}^T\end{array}$$

(5)

$$\begin{array}{cc}\hfill {\mathit{a}}_y^n& ={[{\mathit{a}}_1^n·\mathbf{y}{\mathit{a}}_2^n·\mathbf{y}\cdots {\mathit{a}}_n^n·\mathbf{y}]}^T\end{array}$$

(6)

To determine the dimensions of the footprint bounding box, we need to find the difference between the maximum and minimum values of the coordinate points along the X and Y axes. This can be expressed in Equations (7) and  (8). The length of the bounding box is denoted by $l_{fp}$, and the width of the bounding box is denoted by $w_{fp}$.

$$\begin{array}{c}\hfill w_{fp}=|min\left({\mathit{a}}_{\mathbf{x}}^{\mathbf{n}}\right)-max\left({\mathit{a}}_{\mathbf{x}}^{\mathbf{n}}\right)|\end{array}$$

(7)

$$\begin{array}{c}\hfill l_{fp}=|min\left({\mathit{a}}_{\mathbf{y}}^{\mathbf{n}}\right)-max\left({\mathit{a}}_{\mathbf{y}}^{\mathbf{n}}\right)|\end{array}$$

(8)

In this work, we adapted the Spiral Spanning Tree Coverage (SSTC) algorithm as the complete coverage path-planning approach. The Spiral Spanning Tree Coverage (SSTC) algorithm is a method for performing a complete area coverage in a given map. It involves creating a spanning tree of the network, with the root node placed at the center of the area to be covered. The tree is then grown outward in a spiral pattern, with each new node added to the tree being placed at the furthest point from the center that is still within the area to be covered. This allows the algorithm to efficiently cover the area while minimizing the amount of overlap. The resulting network of a spanning tree provides full coverage of the area with a minimal number of nodes. One advantage of using a spiral spanning tree coverage algorithm is that it allows the robot to cover the area in a more efficient and systematic way than other coverage algorithms. Because the robot is following a predetermined set of rules, it is able to avoid areas that have already been covered and focus on exploring new areas. This can help the robot to map the environment quickly and effectively. Figure 5 shows the implemented SSTC algorithm in a given map. The pseudocode for the implemented spiral path-planning approach is detailed in Algorithm 1. The final step in the process is to resize the initial map based on the dimensions of the footprint, as specified by Equation (9).

$$M_r=\left(\frac{x}{w_{fp}},\frac{y}{l_{fp}}\right)$$

(9)

Algorithm 1: Spiral Spanning Tree Coverage (SSTC) Planner

Next, the Spiral Spanning Tree Coverage (SSTC) algorithm is used to compute a path from the start node to the goal node, covering all the grids on the resized map. Once the path has been computed, the waypoints are transformed back to the original map. The robot then follows these waypoints, sweeping the area to compute the percentage of the area covered for a given shape. This footprint-based coverage planner computes the path length and the percentage of area covered for any given shape. Figure 6 demonstrates the footprint-based coverage planner for the $Smorphi$ robot. For example, the footprint-based coverage planning for a configuration of ${\alpha }_{12},{\alpha }_{23},{\alpha }_{34}\equiv 5,165,5$ is shown in Figure 6a–c. In these figures, the original map is resized based on the footprint value, and the complete area coverage is performed on the resized map as shown in Figure 6c. Afterwards, the waypoints are transformed back to the original map to compute the percentage of area covered. The number of blocks in a $Smorphi$ configuration has a significant impact on both the path length and the percentage of area covered. This is due to the fact that the footprint of each individual block affects the overall performance of the system. For example, a $Smorphi$ configuration with a smaller number of blocks can cover a larger area but may also have a larger path length. On the other hand, a $Smorphi$ configuration with a larger number of blocks may have a shorter path length, but it may also cover a larger area due to the combined footprint of all the blocks. Overall, the footprint-based coverage planner allows us to optimize the performance of their system by selecting the most appropriate configuration based on the specific needs and requirements. The pseudocode for the complete coverage path planning based on the footprint is detailed in Algorithm 2.

Algorithm 2: Footprint-Based Complete Coverage Path planner (FBCP)

6. Optimization Using Metaheuristic Algorithms

This section describes the modeling of the problem and the optimization approach used in this work. Since this work aims to find the optimal morphology that optimizes the % of area coverage and path length, the problem is modeled as a multi-objective optimization problem, which allows for the consideration of multiple conflicting objectives in order to find a solution that balances them. This approach is often used in problems where there is no single optimal solution but rather a range of solutions that trade off different objectives. The total cost function $\left(C_t\right)$ of the optimization problem is shown in Equation (10). The problem is modeled in such a way that the optimization approach should aim to minimize the path length ($f_1$) while maximizing the area coverage ($f_2$).

This optimization problem considered two design variables: namely, the number of blocks ($B_n$) and the hinge angle between the two consecutive blocks. In this paper, we limit the number of blocks up to 4, and the hinge angles were constrained to vary between 0 and 180 degrees. The angular constraint accounts for the physical limitations of the robot’s movement and ensures the optimization algorithms consider only feasible configurations of the robot.

The number of blocks determine the number of hinge angles in the configuration. For example, a configuration with four blocks has three hinge angles (${\alpha }_{12}$, ${\alpha }_{23}$, and ${\alpha }_{34}$), while a configuration with three blocks has two hinge angles (${\alpha }_{12}$, ${\alpha }_{23}$), and a configuration with two blocks has only one hinge angle (${\alpha }_{12}$).

$$C_t=min\left(f_1\left({P_l}_{{\alpha }_{12},{\alpha }_{23},{\alpha }_{34}=0,n=2}^{{\alpha }_{12},{\alpha }_{23},{\alpha }_{34}=180,n=4}\right)\right)+max\left(f_2\left({A_c}_{{\alpha }_{12},{\alpha }_{23},{\alpha }_{34}=0,n=2}^{{\alpha }_{12},{\alpha }_{23},{\alpha }_{34}=180,n=4}\right)\right)$$

(10)

To solve the problem, we employed three metaheuristic optimization algorithms: MOEA/D, HypE, and OMOPSO. These algorithms are specifically designed to solve complex combinatorial optimization problems and have been shown to be effective in a wide range of applications. The following subsection details the three optimization techniques used.

6.1. MOEA/D: A Multiobjective Evolutionary Algorithm Based on Decomposition

The MOEA/D optimization technique generates the solution by splitting the Multi-Objective Problems (MOP) into smaller scalar problems and then optimizing them concurrently. MOEA/D follows the four-step approach to solve any problem: initialization, reproduction, decomposition, and environmental selection. First, the number of populations, generations, neighborhood selection probability, and population size are determined during the initialization process. Next, the offspring values are generated during reproduction based on the selection, crossover, and mutation operators. Then, the problems are subdivided depending on the scalar function in the decomposition step to determine the appropriate environment. There are three decomposition approaches: Tchebycheff Approach, Weighted Sum Approach, and Boundary Intersection (BI) Approach [47]. We used the Tchebycheff Approach in this optimization problem as the scalar optimization problem stated in Equation (11).

$$min{g}^i\left(x\right)=g\left(x\right|{\lambda }^i,z^{*})=\underset{1\le j\le m}{max}{\lambda }_j^i|f_j\left(x\right)-z_j^{*}|subjecttox\in \mathrm{\Omega }$$

(11)

where ${\lambda }^i=({{\lambda }_1}_i,\cdots ,{{\lambda }_m}_i)$ is a weight vector and $z^{*}=:(z_1^{*},\cdots ,z_2^{*})$ is a reference point. Finally, the algorithm chooses the contributing solution and corresponding population for every iteration until the termination criteria are met. The pseudocode for the implemented MOEA/D is shown in the Algorithm 3.

Algorithm 3: MOEA/D for Generating Optimal Morphology

6.2. OMPOSO: Optimized Multi-Objective Particle Swarm Optimization

Particle swarm optimization is the bio-inspired metaheuristics-based optimization technique inspired by the social behavior of the birds. In PSO, a particle is considered as each potential solution to the problem, and the swarm is referred to as the population of solutions. PSO updates its particle position as mentioned in Equation (12).

$${\overrightarrow{\mathit{x}}}_i\left(q\right)={\overrightarrow{\mathit{x}}}_i(q-1)+{\overrightarrow{\mathit{v}}}_i\left(q\right)$$

(12)

where the factor ${\overrightarrow{\mathit{v}}}_i\left(q\right)$ is referred to as the velocity, and it is given by,

$${\overrightarrow{\mathit{v}}}_i\left(q\right)=\omega ·{\overrightarrow{\mathit{v}}}_i(q-1)+C_1·r_1({\overrightarrow{\mathit{x}}}_{pi}-{\overrightarrow{\mathit{x}}}_i)+C_2·r_2({\overrightarrow{\mathit{x}}}_{gi}-{\overrightarrow{\mathit{x}}}_i)$$

(13)

In this formula, ${\overrightarrow{\mathit{x}}}_{pi}$ is the best solution that $x_i$ has viewed, ${\overrightarrow{\mathit{x}}}_{gi}$ is the best particle in the swarm, and $\omega$ is the inertial weight of the particle which controls the local and global experience. $r_1$ and $r_2$ are uniformly distributed random numbers within the range [0, 1]. Optimized MOPSO (OMOPSO) is the modified version of the PSO to solve multi-objective optimization problems. The OMOPSO algorithm combines the feature of NSGA-II’s leader solution filtering with mutational operators to speed up convergence [48,49]. Additionally, it exploits the concept of e-dominance to limit the number of output solutions. The pseudocode of the implemented OMOPSO is shown in Algorithm 4.

Algorithm 4: OMOPSO for Generating Optimal Morphology

6.3. HypE: An Algorithm for Fast Hypervolume-Based Many Objective Optimization (HypE)

The hypervolume estimation algorithm for multi-objective optimization belongs to the indicator approach that orders solutions based on their hypervolume (HV) contributions in the population. In general, hypervolume-based approaches are computationally expensive. In HypE, the Monte Carlo method is used to efficiently solve the multi-objective problem by trading off the accuracy of the estimates and available computer resources [50]. The working flow of the HypE algorithm is similar to the NSGA-II algorithm, which consists of selection, crossover, and mutation. Initially, binary tournament selection is used to choose the N Parent solution from the initial set of populations. By exploiting the mutation and crossover operator, new offspring solutions are generated. Finally, non-dominated solutions are produced through environment selection.

Algorithm 5 outlines the working flow of HypE, where S represents the set of reference points, N is the number of solutions, $i_{max}$ is the maximum number of iterations, L is the sampling point, P is the parent solution, and $P^{{}^{\prime }}$ is the solution obtained after mating selection.

Algorithm 5: HypE for Generating Optimal Morphology

7. Simulation Results and Discussion

The analysis of the three meta-heuristic algorithms in the proposed framework is presented in this section. The algorithms are validated on three different manually created environments. Figure 7 shows the chosen environments with distinct obstacle density and obstacle distribution patterns. Figure 7a–c have 7.56%, 13.76%, and 28.92% obstacle density, respectively.

All spatial environments have an area of $5\times 5$, and they are represented as an occupancy grid having a resolution of 0.01 m${}^2$ per grid size. The occupancy grid represents the traversable region (free space) with a value of 0 and non-traversable region (obstacle) with a value of 1. The obstacles are inserted into the occupancy grid manually as rectangular primitives. Algorithm 2 is used to insert the robot footprint into the environment. In all environments, MOEA/D (Algorithm 3), OMOPSO (Algorithm 4) and HypE (Algorithm 5) are applied to estimate the morphology parameters. The algorithms were executed using the jMetalPy library for optimization [51]. This library provides a comprehensive set of optimization algorithms, including MOEA/D, OMOPSO, and HypE, that were used to find the optimal morphologies of $Smorphi$ in three simulated environments. In addition, the algorithms were executed on an Ubuntu 20.04 operating system in a Python environment, utilizing a workstation with 32 GB of RAM. This configuration ensured that the algorithms had sufficient computational resources to run effectively and efficiently and allowed us to obtain reliable results.

7.1. Analysis of Convergence Time

For the MOEA/D algorithm, the polynomial mutation and differential evolution crossover was used as the crossover and mutation operators, and the aggregative function was set to Tschebycheff. For the OMOPSO algorithm, the swarm size was set to 100, and the epsilon value was set to 0.0075 to bound the size of the e-archive ($ϵ$). For the HypE algorithm, the polynomial mutation and SBX crossover were used as the crossover and mutation operators. The Pareto front obtained using different algorithms on the simulation environment is illustrated in Figure 8 along with the optimal morphologies generated.

The convergence times taken by all the algorithms are listed in

Table 2. Time taken by the three algorithms.

	MOEA/D			OMOPSO			HypE
Environment	I	II	III	I	II	III	I	II	III
Time (s)	264.15	259.11	242.70	493.93	500.98	454.19	264.15	258.22	244.07

. The time taken to obtain the Pareto front by all three algorithms shows the same pattern in all three environments. All algorithms took comparatively more time for environment-1 and environment-2 than environment-3, despite its complexity due to obstacle density. This observation is attributed to the feasibility of shorter path lengths in environment-3 compared to environment-1. Among the simulations, it is observed that MOEA/D in environment-3 has converged faster than the other two algorithms. The HypE algorithm in environment-3 took 242.70 s to converge. Among all the algorithms, OMOPSO in environment-2 took the longest time interval, with 500.98 s to yield the Pareto front. Analyzing the convergence time of all the algorithms in all the environments, MOEA/D and HypE have the upper hand over OMOPSO.

7.2. Analysis of Morphology Candidate in the Pareto Front

Table 3. Comparison on the $f_1$ and $f_2$ based on the footprint size $\left(C_{s,fp}\right)$ in $C_s$ and PF $C_s$ for the three environments.

N	Environment	Algorithm	${\mathit{B}}_{\mathit{n}}$	PF ${\mathit{C}}_{\mathit{s}}$ $({\mathit{\alpha }}_{12},{\mathit{\alpha }}_{23},{\mathit{\alpha }}_{34})$ (degree)	${\mathit{w}}_{\mathbf{fp}}$ (cm)	${\mathit{l}}_{\mathbf{fp}}$ (cm)	${\mathit{C}}_{\mathit{s},\mathbf{fp}}$ (cm)${}^2$	${\mathit{f}}_1$ (cm)	${\mathit{f}}_2$ (%)
1	Environment-1	OMOPSO	4	(41.18,73.42,107.40)	50	25	1250	176	81.83
2	Environment-1	MOEA/D	3	(89.11,8.65)	31	19	589	372	84.16
3	Environment-1	OMOPSO	3	(83.78,4.06)	31	19	589	372	83.97
4	Environment-2	MOEA/D	4	(64.96,51.05,112.88)	50	19	950	228	82.8
5	Environment-2	OMOPSO	4	(174.99,79.31,48.24)	31	31	961	210	72.03
6	Environment-2	HypE	2	(17.96)	22	13	286	690	73.8
7	Environment-2	MOEA/D	3	(81.10,104.65)	31	22	682	294	72.37
8	Environment-2	OMOPSO	3	(110.68,76.70)	31	22	682	294	72.19
9	Environment-3	HypE	2	(158.00)	20	19	380	440	61.15
10	Environment-3	HypE	3	(56.16,78.02)	35	19	665	244	57.66
11	Environment-3	OMOPSO	3	(67.40,72.35)	35	19	665	244	57.67
12	Environment-3	MOEA/D	3	(66.11,71.52)	35	19	665	244	57.77

C_s: is the configuration space defined by angles α₁₂, α₂₃, α₃₄ in degrees (refer Figure 4); C_s,fp: Footprint area by the configuration space, PF C_s: Pareto front configuration space.

lists the optimal values obtained from the Pareto front obtained by MOEA/D, OMOPSO, and HypE in all environments. In environment-1, the OMOPSO algorithm yields the optimal morphology composed of four blocks that cover $81.83\%$ of the area ($f_2$) with a path length ($f_1$) of 176 cm (Table 3, row 1). Even though the MOEA/D algorithm provides a morphology candidate with the highest area coverage ($f_2$) (Table 3, row 2) in environment-1, the path length for this area coverage is considerably higher than all other Pareto-optimal solutions in environment-1. For the case of environment-2, the OMOPSO algorithm yielded the optimal morphology that can provide the highest area coverage ($f_2$) of $72.03\%$ with a path length ($f_1$) of 210 cm (Table 3, row 5). This four-block configuration has a larger footprint size and requires a shorter path length to maximize the area coverage.

The morphology candidate generated by the HypE algorithm showed a maximum area coverage ($f_2$) of $73.80\%$ in environment-2 (Table 3, row 6); however, the path length ($f_1$) is higher due to the fact that the footprint size of the two blocks is smaller. In the case of environment-3, the best morphology candidate is given by the OMOPSO algorithm, where a three-block configuration provides $57.77\%$ of the area coverage ($f_2$) with a path length of 244 cm (Table 3, row 12). The HypE algorithm produced a two-block morphology with the highest area coverage but a higher path length than the three-block morphology (Table 3, row 9). However, the morphology selection from the Pareto front is highly dependent on the task and environment and can vary from one situation to another.

7.3. Validation of Optimal Morphology

To evaluate the effectiveness of the proposed method, a comparison was conducted using nine randomly selected morphologies from the set of design variables.

Table 4. Random configuration taken from the design space.

N	Environment	${\mathit{C}}_{\mathit{s}}$(${\mathit{B}}_{\mathit{n}}$)	${\mathit{C}}_{\mathit{s}}$$({\mathit{\alpha }}_{12},{\mathit{\alpha }}_{23},{\mathit{\alpha }}_{34})$ (degree)	${\mathit{w}}_{\mathbf{fp}}$ (cm)	${\mathit{l}}_{\mathbf{fp}}$ (cm)	${\mathit{C}}_{\mathit{s},\mathbf{fp}}$ (cm)${}^2$	${\mathit{f}}_1$ (cm)	${\mathit{f}}_2$ (%)
1	Environment-1	4	(24.52,179.22,74.15)	50	25	1250	320	76.71
2	Environment-1	3	(159.58,61.73)	27	22	594	332	73.66
3	Environment-1	2	(100.21)	19	20	380	556	80.27
4	Environment-2	4	(170.82,38.13,49.33)	37	21	777	192	53.39
5	Environment-2	3	(161.09,107.02)	22	23	506	344	63.00
6	Environment-2	2	(7.30)	21	11	231	820	71.72
7	Environment-3	4	(175.75,83.85,178.30)	30	22	660	150	35.14
8	Environment-3	3	(47.42,177.88)	21	20	420	310	47.44
9	Environment-3	2	(166.52)	20	19	380	568	55.11

C_s: is the configuration space defined by angles α₁₂, α₂₃, and α₃₄ in degrees and the number of blocks B_n; C_s,fp: Footprint area by the configuration space.

lists the area coverage ($f_2$) and path length ($f_1$) obtained from a random selection of 4, 3, and 2-block configurations for each environment, respectively. Among the morphologies obtained from the random selection, the four-block configurations (${\alpha }_{12}=24.52deg,{\alpha }_{23}=179.22deg,{\alpha }_{34}=74.15deg$ (Table 4, row 1)) showed the highest area coverage of $76.71\%$ with a path length ($f_1$) of 320 cm in environment-1. However, the optimal morphology obtained by OMOPSO in environment-1 shows a clear edge over the other randomly selected morphology candidates in terms of the path length ($f_1$) and the % of the area covered ($f_2$) (Table 3, row 1). A similar observation can be made for environment-2, where the randomly selected morphology showed less area coverage compared to the optimal morphology generated, irrespective of the number of blocks used. In environment-3, the best area coverage obtained from a randomly selected morphology candidate is $55.11\%$ (Table 4, row 9). However, the optimal morphology candidate shows consistent performance in terms of area coverage in all trials, with more than $57.00\%$ with significantly smaller path length.

For further analysis of the edge of the proposed system, the area coverage of the optimal morphology candidate is compared with the standard morphology of the $Smorphi$ robot.

Table 5. Standard polyomino morphologies in environment-1.

N	${\mathit{B}}_{\mathit{n}}$	${\mathit{C}}_{\mathit{s}}$$({\mathit{\alpha }}_{12},{\mathit{\alpha }}_{23},{\mathit{\alpha }}_{34})$ (degree)	${\mathit{w}}_{\mathbf{fp}}$ (cm)	${\mathit{l}}_{\mathbf{fp}}$ (cm)	${\mathit{C}}_{\mathit{s},\mathbf{fp}}$ (cm)${}^2$	${\mathit{f}}_1$ (cm)	${\mathit{f}}_2$ (%)
1	2	(0)	20	10	200	1062	88.44
2	2	(180)	20	10	200	1062	88.43
3	3	(0,0)	30	10	300	722	87.16
4	3	(0,180)	20	20	400	510	78.29
5	3	(180,0)	20	20	400	510	78.34
6	4	(0,0,0) ≡ I	40	10	400	502	84.61
7	4	(0,0,180) ≡ J	30	20	600	344	79.56
8	4	(0,180,0) ≡ O	20	20	400	510	85.43
9	4	(180,90,0) ≡ S	20	30	600	328	73.85
10	4	(90,180,180) ≡ T	20	30	600	328	74.04
11	4	(180,0,180) ≡ Z	20	30	600	328	74.17

tabulates the standard morphology that can be obtained using 2, 3, and 4-block configurations of the $Smorphi$ robot. For the standard morphology, the design variables ${\alpha }_{12}$, ${\alpha }_{23}$, and ${\alpha }_{34}$ are taken as combinations of $0.0deg$, $180.00deg$, and $90.00deg$, respectively. The area coverage ($f_2$) simulations have been run on environment-1. In environment-1, the standard two-block configuration shows an upper hand over the optimal morphology candidates in terms of area coverage (Table 5, row 1). However, the standard two-block configuration took a longer path length ($f_1$) than the optimal morphology candidates obtained from the Pareto front. The analysis shows that the selection of an optimal morphology generated by the proposed method has a significant advantage over the standard morphology and other random configurations from the design space. Even though the standard configurations could yield a better area coverage due to their regular geometry, the higher area coverage is achieved at the expense of a higher path length. This comparison further shows the significance of using infinite morphologies in area coverage tasks.

7.4. Path Length and Area Coverage with Optimal Morphologies Footprints

Figure 9 illustrates the footprint-based coverage planner for the optimal morphology generated from the Pareto front. The three optimal shapes depicted in Figure 9 are taken from Table 3, Row 4 (Figure 9a), Table 3, Row 8 (Figure 9b), and Table 3, Row 9 (Figure 9c). To illustrate the effect of the footprint size, we depicted the path planning for four, three, and two blocks in environments-1, 2, and 3, respectively. The green color represents the area covered by the robot in the given environments, and the black line with an arrow mark represents the path taken by the robot to cover the environment. For example, as we stated earlier, the shape generated for environment-1 is the four-block configuration; since there is a lower density of obstacles, the four blocks can easily cover the area with a higher percentage of coverage.

Similarly, in environment-2 and 3, blocks 3 and 2 are chosen as the optimal morphologies as they can cover more area in the complex maps. Additionally, we can see that as the complexity of the environment increases, fewer blocks are more suitable for area coverage, even though they may have a slightly longer path length than higher numbers of blocks. In addition, the path length increases as the number of blocks decreases. This is mainly due to the combined effect of the footprint: the larger footprint results in the shorter path length. Similarly, as we highlighted earlier, we can achieve a higher percentage of area covered using standard morphologies with even geometry. However, it takes a higher path length compared to the optimal morphologies generated from the Pareto front. Therefore, even though the shape of the optimal morphology generated by the Pareto front is not uniform, they tend to take significantly less path length than the standard morphologies with a slightly lesser percentage of area covered, which is mainly due to their footprint size and surface area. The comparison between Table 3 and Table 5 clearly supports this statement.

In summary, the three algorithms’ results demonstrate our framework’s effectiveness in producing optimal morphologies, regardless of the optimization technique used. For example, Table 3 shows the convergence of hinge angles. For example, as listed in Table 3 row 11 and 12, the OMOPSO and MOEA/D algorithms produced nearly the same shape, similarly in row 2 and 3. This indicates the reliability of the proposed framework with different optimization techniques. The limitation of the current approach is that some of the optimal morphologies suggested by the algorithms may not be feasible for area coverage, considering the design constraints.

Overall, the proposed framework is efficient at identifying the optimal morphologies, which results in a higher percentage of area coverage with a shorter path length. Additionally, this work has revealed the potential for the use of infinite morphologies in area coverage tasks.

8. Conclusions

This paper proposes a novel framework for identifying the optimal morphology of a reconfigurable robot which can maximize area coverage and minimize energy consumption while performing area coverage tasks. We present the design of a new n-Omino-based tiling robot, $Smorphi$, which can reconfigure into an infinite number of configurations by varying the hinge angle and the number of blocks connected to it. The designed $Smorphi$ robot can be assembled from domino (n = 2) to tetromino (n = 4) by adding a monomino building block. We validate our proposed framework in three simulated environments using three metaheuristics algorithms: MOEA/D, OMOPSO, and HypE. The simulation results show that our approach could generate optimal morphology, which maximizes the area coverage and minimizes the energy consumption. The simulation results were compared with the random configurations of the $Smorphi$ robot and the standard polyomino configurations. In addition, Pareto front results generated optimal morphology with different numbers of blocks and different configurations.

Overall, the Pareto front results suggest that the module-based robot with inter and intra-reconfiguration capabilities is suitable for generating optimal morphology that is energy efficient. The three algorithms produce near-optimal morphologies in the three unique environments. Among the three algorithms, the HypE algorithm converges faster than the other two algorithms, which could be useful for generating optimal morphologies with lower computational time. Future work includes: identifying multiple optimal morphologies in a given map based on obstacle density, developing a robust controller for the $Smorphi$ robot and implementing a reinforcement learning-based approach for generating optimal morphologies.