A metaheuristic approach to optimal morphology in reconfigurable tiling robots

Reconfigurable robots are suitable for cleaning applications due to their high flexibility and ability to change shape according to environmental needs. However, continuous change in morphology is not an energy-efficient approach, with the limited battery capacity. This paper presents a metaheuristic-based framework to identify the optimal morphology of a reconfigurable robot, aiming to maximize the area coverage and minimize the energy consumption in the given map. The proposed approach exploits three different metaheuristic algorithms, namely, SMPSO, NSGA-II, and MACO, to generate the optimal morphology for every unique layout of a two-dimensional grid map by considering the path-length as the energy consumption. The novel feature of our approach is the implementation of the footprint-based Complete Coverage Path Planning (CCPP) adaptable for all possible configurations of reconfigurable robots. We demonstrate the proposed method in simulations and experiments using a Tetris-inspired robot with four blocks named Smorphi, which can reconfigure into an infinite number of configurations by varying its hinge angle. The optimum morphologies were identified for three settings, i.e., 2D indoor map with obstacles and free spaces. The optimum morphology is compared with the standard Tetris shapes in the simulation and the real-world experiment. The results show that the proposed framework efficiently produces non-dominated solutions for choosing the optimal energy-efficient morphologies.


Introduction
Cleaning is an essential function of daily routine, yet it is considered monotonous, dirty, and time-consuming. Moreover, humans prefer to prioritize other tasks over cleaning, which fosters the development of cleaning robots that can free up humans to do creative tasks. Several cleaning robots have been developed to tackle these problems, including Roomba, Xiaomi, Samsung, Dyson, etc. However, the existing robots' performance is limited by their morphology, primarily because they are not custom-made to work on the different layouts in the home. To address this limitation, reconfigurable robots are designed to change their morphology or shape to clean efficiently and access difficult-to-reach regions. When compared to fixed morphology robots, these reconfigurable robots provide better area coverage. However, at the same time, they consume more power to often change shapes to clean a given area, and they may not be able to function continuously for a long time since they draw power from finite source batteries. Hence, it will be useful to look for a method to find an optimal shape of a reconfigurable robot that can be used during, say, a cleaning or maintenance task according to the given map of the deployment area.
Reconfigurable systems have a wide range of advantages in terms of multi-ability (in various configurations, the system can execute multiple functions at different times), evolution (the system's configuration can be altered by removing, substituting, and adding new elements), and survivability (the system can still work despite a potential failure of one or more components) [1]. Reconfigurable systems have been widely adopted in various applications, including reconfigurable manufacturing systems [2], field-programmable gate arrays [3], various software concepts [4], aerospace [5], etc. One of the main applications of reconfigurable systems is demonstrated in self-reconfigurable robots, which can modify their morphology manually or automatically depending on the situation or environment. The development of reconfigurable robotic systems has gained attention since 1980 [6]. Self-reconfigurable robots are broadly divided into two main categories: intra-reconfigurable and inter-reconfigurable [7]. Inter reconfigurable robots change their function by adding a new module/system via assembly and disassembly. CONRO [8], Crystal [9], and M-Lattice [10] are a few examples. Intrareconfigurable robots transform their configuration without the requirement for external attachment to enable a new function or improve an existing function. Examples include Panthera [11], a pavement-sweeping robot that can change its footprint by expanding and collapsing to clean the pavements while overcoming different obstacles, a ship maintenance robot with different end-effector to remove barnacles from the hull [12], and sTetro [13,14], a staircase cleaning robot. Therefore, the potential of morphology change in reconfigurable robots are beneficial for several applications.
In recent years, there has been substantial progress in the mechanism design, control, perception, and autonomy in reconfigurable robots [15][16][17][18][19][20]. For example, the Tetrisinspired tiling robot reported in [21] can change into seven different configurations, namely, {I, J, L, O, S, T, Z} following the constraint of one-sided polyomino [22] to perform area coverage tasks [21,23,24]. Here the authors highlighted the advantage of reconfigurable robots over fixed morphology robots by demonstrating the higher area coverage task. Similarly, the concept proposed in [25] named pleomorphic h-Infi can change into an infinite number of configurations to perform cleaning tasks (as the constraint of a one-sided polyomino is not accounted). However, only a few authors have explored the energy optimization problem in reconfigurable robots. The energy consumption and time taken in self-reconfigurable robots are a crucial factor since these robots have a higher number of actuators than fixed morphology robots [26]. In [27], the author proposed a coverage path planning approach with the optimal energy consumption based on the Genetic Algorithm (GA) and the Ant Colony Optimization (ACO) technique, and the reported work in [28] introduced a Reinforcement Learning (RL) based path planning approach suitable for producing Pareto plans, which allow the robot to cover a specified region on the map with the least amount of energy consumed. The suggested solution in [29] assures complete area coverage of the robot while decreasing the number of reconfigurations carried out, reducing time and energy consumption. However, to the best of the author's knowledge, none of the aforementioned works address the identification of a single optimal morphology to cover a given area resulting in minimizing energy consumption and maximizing the area coverage.
Path planning is an integral part of the self-reconfigurable robotic system [30]. Establishing a path which traverses across all points of a defined region in the map is considered as coverage path planning [31]. Classical exact cellular decomposition is one of the prominent CPP methods [32]. This method breaks down the coverable region (excluding obstacles) into a non-overlapping region called cells. Trapezoidal and boustrophedon are the two popular offline decomposition methods [33]. Grid-based methods, on the other hand, utilize a representation of the map in the form of a set of uniform grid cells. Each cell holds a value in this gridbased arrangement, whether it has an obstacle or free space. A grid map is simple to generate since it is generally modeled as an array, with each element containing occupancy information corresponding to the cell [34]. As a result, coverage algorithms most commonly use grid-based representations. However, grid-based algorithms are only suitable for indoor environments (relatively small maps) since they require accurate localization and suffer from the exponential growth of memory usage [31].
This paper proposes a first-of-its-kind framework to identify an optimal morphology out of multiple morphologies taken by a Tetris-inspired reconfigurable robot for an area coverage of a given two-dimensional (2D) environmental map. The framework is devised to identify the best morphology accounting for maximizing task performance and minimizing energy consumption. The proposed framework uses metaheuristic algorithms to generate the optimal shape for the robot by treating it as a multi-objective optimization problem. The proposed framework is generic and can be applied to any class of tiling robot (polyominoes, heptiamonds, and hexiamonds [22]) that reconfigures in 2D space. However, this work demonstrates using a developed reconfigurable robot, named Smor phi, which can change its shape into infinite 1 configurations. On a larger scale, the proposed method allows end-users to customize the robot shape according to the environment where it has to be deployed. To the best of the author's knowledge, an optimal shape generation framework for the reconfigurable robot which compromises the area coverage and energy consumption has not been addressed.
The objectives of the proposed work are: The rest of the article is organized as follows; The proposed framework is introduced in Sect. "Opt-Morph frameworksec:2". Section "Design space" describes the mechanical design of the Smor phi robot along with the design principles and the kinematics of the robot. Section "Complete Area Coverage and Path Planning (CCPP) space" explains the implemented path planning and footprint-based area coverage. Section "inlinkOptimization spacesec:5" presents the optimization approach along with the objective functions. Results and discussion of the proposed framework, along with the experiments conducted, are presented in Sect. "Results and discussion". Finally, Sect. "Conclusion" concludes the paper by summarizing the key findings and highlighting future work.

Opt-Morph framework
The proposed Opt-Morph framework for determining the optimal shape of a reconfigurable robot is illustrated in Fig. 2. The proposed framework is divided into three layers as shown in Fig. 2: I. Design space of a reconfigurable robot (here, dealt in Sect. "Design space"), II. Complete Area Coverage and Path Planning (CCPP) space (in Sect. "Complete Area Coverage and Path Planning (CCPP) space"), and III. Optimization space schemes (in Sect. "Optimization space"). In design space, to generalize the framework, we used a set of three transformational principles, namely expand/collapse, expose/cover, and fuse/divide, proposed in [58]. These three principles cover the entire spectrum of reconfigurable products or systems based on their means of reconfiguration. The transformation design principles and the design of the robot named Smorphi used to demonstrate the Opt-Morph framework is described in Sect. "Design space".

Area coverage and Path length
The CCPP space focuses on the algorithms for coverage planning and computes the path-length and the percentage of area-covered by the robot based on the robot's footprint and map of the environment. The CCPP space follows the twostep approach to compute the percentage of area-covered and the path-length. First, the rectangular footprints ( Fig. 2, 'I') corresponding to the robot shape generated by the design space is estimated. Then, an occupancy matrix for the individual shape is computed such that the occupancy grid's size will equal the dimensions of the rectangular footprint ( Fig. 2, 'II'). The elements of the occupancy matrix are determined such that zero '0' represents the region unoccupied by the robot base and one '1' (Fig. 2, III) corresponds to the region occupied by the robot base. Then, with the footprint matrix and the 2D map as the input, the complete area coverage path planning is computed using the wavefront algorithm (Fig. 2, 'IV'). In the optimization space ( Fig. 2, 'V'), for the given objective function and constraints, the algorithm search for the best optimal shape in the infinite pool of configuration by optimizing the pathlength p min , and the percentage of area-covered a max (Fig. 2). This work uses metaheuristic-based search methods, namely, SMPSO, NSGA-II, and MACO, for the optimization. It is mainly because of the ability to produce an optimal solution with the least amount of information [59]. Finally, upon completing the optimization, the framework provides the non-dominated solutions for the given map. This work demonstrates the proposed framework using the reconfigurable tiling robot named Smor phi. A detailed discussion on the design of the Smor phi robot is discussed in the following section.

Design space
This section outlines the mechanical design process of the Smor phi robot along with the transformational design prin-ciples and facilitators used along with the kinematics of the robot.

Transformational design principles
In this work, we exploited transformational design principles as an enabler for designing the reconfigurable robot named Smor phi. [58] author empirically analyzed all the reconfigurable products, resulting in three transformational principles: expand/collapse, expose/cover, fuse/divide, and twenty transformational facilitators. The transformational principle is a guideline, when embodied singly, creates transformation. Below is a brief description of the transformation 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.
A transformation facilitator is a design element that facilitates or assists reconfiguration or transitions. A facilitator alone will not result in reconfiguration; the facilitator must be used in conjunction with at least one principle. Please refer to [58] for a detailed description of transformational facilitators.
Our previous work extended the transformational design principles by adding a generator layer to design a reconfigurable pavement sweeping robot [11,60]. The generator is the mechanism construct that helps realize the transformational facilitator in the form of a mechanism, such as a linkage,    gear, cam, belt, pulley, etc., that transforms the given input motion into the desired output motion by varying the joint configuration. Figure 3 illustrates the examples of three transformational principles in reconfigurable robots. For example, in Fig. 3a pavement sweeping robot expands its body to clean the pavement and collapses its body to allow the pedestrians to pass [61]. The floor-cleaning robot is reconfigured into a wall-cleaning robot by revealing a new surface in Fig. 3b. The modular robot in Fig. 3c can allow multiple units to attach/detach to perform various logistics tasks. Deriving inspiration from the transformational principle, the Smor phi robot reconfiguration behavior is conceived with the aid of Expose/Cover principle. As the name implies, the Smor phi robot is designed to reveal/conceal the side surface of each block in order to reconfigure from one shape to another. The comprehensive mechanical design of the Smor phi robot is discussed in the following section.

Mechanical design
The detailed mechanical design of the Smor phi robot is shown in Fig. 4. The Smor phi robot is designed as an individual modular block that shares the common mechanical structure that can be connected manually through a hinge. The more number of the blocks increases the total degree of freedom of the robot. We have used four identical square blocks with a length of 10 cm 2 and the footprint of 100 cm 2 in the LLR (Left, Left, Right) hinge configuration. Each block of the designed robot is built in a four-layer structure. The locomotion units are mounted on the bottom plate, and lowlevel controllers, batteries, and other accessories are on the middle two plates. Perception and localization sensors are installed on the top plate to avoid obstruction while reconfiguring. All the plates are fabricated with the 0.2 cm thickness acrylic sheet except the chassis plate to reduce the overall weight. The aluminum standoff is used to adjust the height between different plates to keep it modular. In addition, the 2D LiDAR is installed on top of the robot for mapping and localization, and the robot is controlled with the help of the Intel compute stick. The system specifications of the Smor phi robot is provided in the Table 1.
Smor phi has the ability to change into an infinite number of configurations by varying the three hinge angles. The Smor phi robot leverages on Mecanum wheel-based locomotion to navigate the environment, with each block equipped with four Meccanum wheels and motors. Since this locomotion type offers a holonomic drive where the total degree of freedom is equal to the controllable degree of freedom, the robot can move in all directions independently. Also, this locomotion type enables the robot to rotate with zero turning radius. The 7.4v DC motor with the encoder is used as a traction motor to drive the Mecanum wheel. As the Mecanum wheels drive at different speeds, the individual blocks rotate around the hinge axis to reconfigure. During this reconfiguration, one block should be kept stationary while the other three transform concerning that block. For Smorphi Block 2 (B2) is taken as the reference block and does not change its pose, while the rest of the block can reconfigure about its passive hinge.

Robot configurations, footprint, and kinematics
where C s, f p is the footprint corresponding to the configuration space, and l is each block's geometric length and width and are constant. The possible shape from the infinite number of configurations that can be formed by changing the three hinge angles is determined by Eq. 1. Each hinge angle can be varied between 0 and π , and the angles are 0 when all four blocks form a straight line or in I-configuration, as shown in Fig. 1. The kinematic of the individual block of Smorphi follows the holonomic locomotion supported by four Mecanum wheels as shown in Fig. 5a. The kinematics of a single block with four Mecanum wheels are given as: where φ m1 is the vector of the wheel velocities of module 1 (m1), #1, i.e., [φ 1φ2φ3φ4 ] T and ζ m1 ≡ [ẋẏθ ] T is the vector of robot velocity in the robot frame. To transfer this information to the world frame, the rotation matrix about Zaxis is included as R(θ )ζ m1 . J m 1 is the Jacobian matrix for a single holonomic robot with four Mecanum wheels with elements as where the constant k 1 = (l + w), i.e., basically the sum of length and breadth of each block dimension. Figure 5c shows the four block m = 1, 2, · · · , 4 and in total 16 Mecanum wheels. To establish the relation between the robot frame and the frame associated with individual blocks, the origin of the second block, i.e., #2 is selected as the robot frame with O R X R Y R . Relative to this frame, the location of other individual blocks are assigned with the vector d 1 , d 3 , d 4 and can be geometrically found given the angle between the two consecutive blocks are known. Then the kinematic relation mapping the wheel velocity of a given block #n (here n = 1, 2, 3) in the robot frame is given by: is the rotation about the Z −axes between the blocks, i.e., α, β, γ and T n is the translation matrix in 2D by the magnitudes of vector d expressed as: Moreover, in the inertial frame X I O I Y I the inverse kinematics equation of the four-block system is derived as: where,˙ m 1 ≡ [φ 1φ2φ3φ4 ] T , and the Jacobian matrix is a function of the angle between the two blocks represented in the robot frame, i.e., ψ n,n+1 , α n,n+1 J m , ∀m = 1, 2, 3, 4 is the Jacobian for four blocks. The connection constraints from the hinge joint between each block are basically the rotation and translation of each block w.r.t., #2. Matrix The robot configuration is mainly defined by the angle between each block, i.e., α, β, and γ . By considering the increment of each angle giving one configuration, the total number of possible configuration spaces with 0 ≤ (α, β, γ ) ≤ 180 is 180 3 = 5832000 (including self collision of block) is very large. Hence, the metaheuristic approach is used to find the optimal configuration of the reconfigurable robot Smorphi for the complete coverage path planning (CCPP) task, like, cleaning a given 2D space. In the next section CCPP approach is discussed.

Complete Area Coverage and Path Planning (CCPP) space
The tasks associated with the Complete Area Coverage and Path Planning (CCPP) are the footprint generation of a reconfigurable robot, coverage path planning, and area coverage estimation based on the footprint. The coverage planning algorithm generates a global trajectory for the robot by avoiding collision with obstacles in a given map. The path planning algorithm takes in two inputs for the trajectory generation; the robot footprint and the 2D environment map.

Footprint generation
This section details the implementation of the footprint generation for a given configuration of the Smor phi robot.
The Smor phi robot's footprint is represented by four squares hinged at LLR corners. This makes the footprint to have a 2D geometry. Assuming block 2 (B2) 's pose is fixed, the remaining square blocks can be rotated about their hinge axes passing as shown in Fig. 5. The projection of vector a onto the X − and Y − axes to determine the coordinates of the vertices, are given in the Eq. 6, and 7, a y = [a 1 · y a 2 · y . . . a n · y] T where a i are the position vector of the vertices of the polyomino-shaped Smorphi with total vertices of four blocks resulting in n = 16. Vector x and y are the unit vectors along X -and Y -axes, respectively. The dot product operator is used to calculate the magnitude of the projection of vector a 1 on x as a 1 · x.
Similarly, the dimension of the footprint bounding box, i.e. its length l and width w are determined by taking the difference between maximum and minimum values of the coordinate points along the X and Y axes, as shown in the Eqs. 8 and 9, where min and max are functions for finding the minimum and maximum of a given set of numbers and |.| is an absolute value operator.

Coverage path planning algorithm
For the coverage planning, we adapted the wavefront algorithm [35]. The implementation of the grid-based wavefront path planning algorithm for complete area coverage is explained in Algorithm 1. This Algorithm 1 works based on the breadth-first search method. As per the algorithm, the robot moves from the start point to the goal point while covering all the grids. Initially, all the grids are assigned with the 0 value except the grid corresponding to the obstacles, which are assigned with the -1 value. Then, the algorithm assigns the value from the goal point for all the grids with zero value in the order of increasing one until it fully covers the entire map. Each node assigns a cost to eight adjacent nodes in the existing wavefront algorithm while considering the robot's diagonal movement. However, since the Smor phi robot is assumed to move in only four directions, we modified the wavefront algorithm to assign costs to only four adjacent nodes (up, down, left, and right). Following, nodes have been assigned with a cost, the algorithm advances from the start node, choosing the neighbor with the highest cost until it reaches the goal node. Figure 6 shows the implemented coverage algorithm for the three different scenarios, the arrow mark on the Figure shows the direction of the path from the start node to the goal node.

Footprint-based path planning
After generating the footprint model, the input map is resized with respect to the width and length of the footprint size, as shown in the Eq. 10, where x and y are the dimensions of scaled map, and w f p and l f p are the dimensions of footprint respectively. The scaled map, including the obstacle data, is passed to the path planner, which computes the cost and generates waypoints to cover the whole map. Following that, the robot moves from the starting position to the goal point by covering all the cells based on the waypoints received. After completing the coverage path planning with respect to the robot's trials, the area with zeroes is regarded as unvisited, while an area greater than zero is considered visited. The resulting footprint, path planning, and area coverage are shown in Fig. 7 for the values (α = 0, β = 90, and γ = 0). Furthermore, the provided footprint generation approach avoids self-collision between individual blocks by determining the relative angle between each block. The pseudocode for the implemented footprint-based area coverage is shown in the Algorithm 2.

Optimization space
This section discusses the modeling of the problem, optimization objectives, and optimization algorithms' implementation in detail. The optimal shape-finding problem defined in this work is modeled as a Multi-Objective Optimization Problem (MOOP). For each iteration, the algorithm calculates the path-length and the percentage of area covered in a given map. The MOOP algorithms evaluate the possible shapes with respect to the objective function resulting in Pareto optimal solutions. In this MOOP, the path length is assumed to be proportional to the energy consumption. The overall environment is considered a m × n matrix map, with each cell representing the following information: free space, visited region, obstacle region, and overlap region.
The total cost function C t for the defined problem is stated in the Eq. 11. The functions f 1 and f 2 calculate the pathlength P l and the percentage of area-covered A c , respectively. This problem is modeled as a min-max optimization problem, as the algorithm should aim to minimize the path length and maximize the percentage of area-covered.

Metaheuristic algorithms for optimization
Metaheuristic is an iterative solution-finding process inspired by nature that guides a subordinate heuristic by incorporating various techniques for exploring the solution space. Metaheuristics algorithm works based on two principles; choosing the best solution and randomization. The best solution ensures optimal convergence, and randomization is used In this work, we exploited three different metaheuristic algorithms to optimize the objective function, namely, Speedconstrained Multi-objective Particle Swarm Optimisation (SMPSO), Non-Dominated Sorting Genetic Algorithm-II (NSGA-II), and Multi-objective Ant Colony Optimization (MACO). Using metaheuristic approaches a group of "noninferior" solutions can be identified that define a limit in the goal space where no objective can be improved without losing at least one of the others is known as a Pareto optimum solution. The working principle and the implementation of those algorithms are discussed in the following sections.

Speed-constrained Multi-objective Particle Swarm Optimisation (SMPSO)
Particle swarm optimization is the bio-inspired metaheuristic optimization algorithm introduced by Eberhart and Kennedy in 1995 [62]. It is categorized as a swarm-based optimization approach influenced by bird flocking. In the Particle Swarm Optimization method, the particle is referred to as each possible solution, and the swarm is the population of solutions in the objective space. In PSO, each particle is randomly distributed in the state space with the initial particle ( p 0 ) and an initial velocity ( v 0 ). For every iteration, all the individual particles record their best solution p best . As the particle starts to move, the new velocity v n+1 , and the particle's next solution p n+1 is updated as per the Equations 12, 13, and 14 until the maximum number of generation reaches [63].
The notations used in the above Equations are adapted from [63]. Speed Constrained Multi-Objective Optimization (SMPSO) is an improved version of the PSO algorithm [63], which facilitates the quicker exploration of the Pareto front and a more uniform examination of the search region.
As the name suggests, the SMPSO algorithm constrains the motion of the particles to avoid swarm explosion; swarm explosion makes the algorithm inefficient by skipping the non-visited region. This approach updates the new velocity of the particles from PSO by multiplying it with the constriction coefficient. The Equation for the constriction coefficient and dimensional speed range [63] is mentioned in Eqs. 15, and 16. Also, the pseudocode for the implemented SMPSO is shown in Algorithm 3.

Non-Dominated Sorting Genetic Algorithm II (NSGA-II)
Genetic algorithm (GA) is a search heuristic influenced by Charles Darwin's theory of biological evolution, and it is classified as an evolution-inspired metaheuristic optimization technique. NSGA-II is an improved version of the genetic algorithm proposed by Deb et al. [64]. It is the most prevalent form of GA, mainly used for multi-objective optimization. The NSGA-II algorithm is built on four operators: selection, crossover, mutation, and crowding distance. The selection operator is responsible for choosing solutions from a pool of options by eliminating the incompetent ones. The crossover operator creates a new copy by combing the good substrings from parent populations. By doing a local search, the mutation operator enhances the solution. Finally, the crowding distance operator helps maintain diversity among all the populations by supporting the distant solutions. The crowding distance [65] is defined as, The pseudocode for the implemented NSGA-II is shown in Algorithm 4. For the detailed scheme of NSGA-II, please refer to the article [64].

Multi-objective Ant Colony Optimization (MACO)
Ant Colony Optimization (ACO) is a swarm-based metaheuristic method developed by Marco Dorigo in 1996 [66,67], inspired by the behavior of real ants and their pheromone characteristics. It is mainly used for solving complex combinatorial optimization problems.
In ACO, all the pheromone values are initially assigned with a value of τ 0 , and the algorithm starts with constructing a set of possible solutions. The algorithm chooses a feasible solution during each construction and adds it to the partial solution. The feasible solution selection is based on the probability scheme described in Eq. 18. Further, the local demon search is performed to improve the solutions constructed by the ant. Finally, the pheromone values are increased with respect to the components found with the high-quality solution as described in the Eq. 19, The pseudocode for the implemented multi-objective ACO is detailed in the Algorithm 5. For the detailed scheme of MACO, please refer to the article [68,69].
In the next section, the results obtained from the metaheuristic algorithms discussed here are highlighted and discussed.

Results and discussion
The simulation results and validation of the defined problem using the three metaheuristic multi-objective optimization methods are discussed in this section. The simulation was performed on three different maps, each with a size of 500 × 500 cm. Each map is unique in terms of the obstacle shape, position, and orientation. In this approach, we chose a set of parameters to provide a fair comparison between the three methods. SMPSO method uses a particle size of 100. Similarly, NSGA-II and MACO use an internal population size of 100; and the number of generations is set to 100 for all approaches. We applied polynomial mutation and simulated binary crossover (SBX) as operators for mutation and crossover in the NSGA-II algorithm, using binary tournament selection for the parents. We discuss the optimal shapes obtained using the three algorithms next.   In the objective space, f 1 and f 2 denote the path length and the percentage of area covered, respectively. Similarly, Fig. 9 illustrates some of the Smorphi robot's optimal morphology From the outcome of the heuristic approaches, it was observed that the non-dominated solutions are mainly pop-ulated in the region-II with path-length ( f 1 ) values between 125 and 275 cm and the percentage of area-covered ( f 2 ) values between 50 and 78, respectively, as shown in the Fig. 8a, c, e. The NSGA-II algorithm generated fewer non-dominated solutions than the SMPSO and MACO in maps -1, -2, and -3, as shown in Fig. 8a, c, e. Region-I and region-II in the Pareto front mainly distinguishes the morphologies based on task efficiency. Over here, we differentiated region-1 and 2 in the Pareto front by considering cleaning as the main task the robot has to perform. The morphology in the region-1 Table 2 Comparison on the f 1 and f 2 based on the footprint size (C s, f p ) in C s and PF C s for Map-1 C s : is the configuration space defined by angles α, β, γ in degrees (refer Fig. 5) C s, f p : Footprint area by the configuration space, PF C s : Pareto front configuration space 1 SMPSO: 2 NSGA-II: 3 MACO Table 3 Dimensions, i.e., length l f p and width w f p of the configuration space footprint C s, f p (Fig. 5 of the tiling shapes I, J, O, S, T, Z is considered less suitable for cleaning applications as they can only cover less percentage of the area in a given map as shown in Fig. 8a, c, e. In contrast, the morphology produced in region-2 is more suitable for cleaning application, as they have a higher percentage of area-covered ( f 1 ) with the optimal path length ( f 1 ) compared to the morphology produced in region-I. Table 2 highlights some of the values from region-II obtained for Map-1 with the magnitude of the configuration space (α, β, γ ) and the objectives f 1 and f 2 . Besides, the finding indicates that the results produced by the MACO algorithm are superior to SMPSO and NSGA-II for generating optimal morphology, which compromises the path-length and the percentage of area covered. Overall, all three algorithms converged as per the objective function for generating optimal energy-efficient morphology. Similarly, in the configuration space highlighted in Fig. 8b,

Effects of morphologies
In addition, to validate the simulated findings, we compared the Pareto front to the widely adopted standard polyomino configurations in tiling robots as reported in [21,23]. Our simulated finding suggests that these standard polyomino configurations are considered to be sub-optimal compared to the shapes generated from our algorithm in terms of the path length and the percentage of area-covered as shown in Table 2, and 3. For example, in Table 3, row 3, the standard polyomino configuration 'O'with (α, β, γ ) = (0, 180, 0) covered 70.88% of the area in the map-1 with a path-length of 452 cm. Even though the shape has a higher percentage of area covered, the path length required to cover that given map is also higher, implying the energy consumption is also higher. On the other hand, the shape generated by the MACO algorithm with the PF C s (α, β, γ = 114.28, 67.96, 18.24) (Table 2, row 6); path-length ( f 1 = 180 cm) and percentage of areacovered ( f 2 = 69.22%) is produced in the optimal range as per the objective function. Hence this shape is considered optimal morphology for Map-1 compared to the standard polyomino configuration 'O'. Further, this reveals the usefulness of infinite configurations in area coverage problems.
Further, to validate our simulated findings, we compared the Pareto front solutions with the random configuration chosen from the Smorphi robot. Table 4 shows the path length and the area covered for the set of random shapes. The results indicate that the random shapes taken from the configuration space are also considered sub-optimal compared to the morphology generated in the Pareto front (   Table 2, row 5), which covers significantly more area than the random configuration ( f 2 = 68.33%) for nearly the same path-length ( f 1 -195 cm), making it a more optimal configuration for area coverage tasks with lesser energy consumption. These results substantiate the multi-objective optimisation algorithm's usefulness in identifying the optimal morphology.

Effects of the footprint size
From the simulation, we identified that the footprint size is inversely proportional to the path length; as the footprint size is larger, the path length is shorter. This makes the shape with a larger footprint a suitable candidate for the optimal morphology, and the morphology with a lesser footprint size (I, J, O, S, T, Z) is the least suitable candidate for the optimal morphology, as shown in the Tables 3, and 2. However, as per our objective function, the morphology with the shorter path length alone cannot be optimal. Since the problem is defined in such a way that the optimal shape generated is the trade-off between path length and percentage of area covered. On the other hand, the area coverage solely relies on the footprint occupancy matrix and the path generated rather than the footprint size. As shown in Table 2, shapes from the Pareto front are verified with the random values chosen from the configuration space (C s ). The results show that even the larger footprint size alone cannot be an optimal morphology. For example, in Table 2, row 2, the random value chosen from the C s (α, β, γ = 58, 120, 100) has a lesser path length ( f 1 = 126) with lower energy consumption; however, for that shape, the percentage of area covered ( f 2 = 60.16%) is also lesser, making it as a sub-optimal morphology. Sim-ilarly, in the case of the shape generated from the PF C s (α, β, γ = 74.28, 41.24, 82.81). Table 2, row 2) has the optimal path-length ( f 1 = 194) and the percentage of areacovered ( f 2 = 68.83%), making it an optimal morphology for map-1. Hence, the metaheuristic algorithms were useful in identifying the optimal value that compromises the percentage of area covered and path-length. Further, this example justifies the use case of these meta-heuristic algorithms for this combinatorial problem. Figure 10 summarizes the computational time taken 3 by the SMPSO, NSGA-II and MACO algorithms to complete the 100 generations in three different simulated environments. Among these three methods, the SMSPO algorithm has less computational cost than the other two algorithms; this is mainly because SMPSO uses mathematical operators instead of evolutionary operators for generating solutions. Similarly, in Map-2, the NSGA-II algorithm converges faster than the MACO and SMPSO algorithms. Further, from Fig. 10, we can observe that the MACO algorithm slightly took more computational cost than the other two algorithms in the three different maps. In addition, we observed that there are only slight changes in the computational cost between different environments even though the obstacle complexity in every map is different; this is mainly due to the size of the map being kept constant for the three simulations. However, on the other hand, heuristic-based optimization algorithms are stochastic in nature; we cannot conclude one type of optimization algorithm will always produce the best solution with less computational cost; it varies with respect to the environmental uncertainty, design parameters, nature of the algorithm, type of problem, computational power, etc.

Experiments
This section details the experiments conducted to validate the simulation study using the real Smor phi robot. The experiments were carried out in a 3 m × 3 m lab setup designed to resemble the simulated environment. Occupancy grids such as warning signs and cardboards are considered obstacles, and the remaining space in the bounding box is considered free space for the robot to navigate. The real experiments aim to show the optimal morphology generated through the simulation is energy efficient to cover the area while maximizing the area coverage and minimizing the path length. We conducted four set of experiments, including shapes from standard polyomino sets, SMPSO, NSGA-II, and MACO. For all the experiments, the robot is teleoperated with the Fig. 10 Computational time taken by the optimization algorithm in the three different environments ran on a given personal computing device same speed of 0.1 m/s. For every shape, the path length ( f 1 ) and the percentage of area covered( f 2 ) are tracked using the overhead camera mounted on the ceiling. Besides, during the experiment, the angle between the hinges is locked using an acrylic mechanical stopper to restrict the hinges during navigation. Figures 11, 12, 13, and 14 depict actual experiments performed using four different shapes. To fairly compare the three optimization algorithms, among the multiple morphologies generated from the Pareto front, we chose the morphology with the highest area coverage and lesser path length for the real-world experiment from the respective algorithm. The path-length ( f 1 ) and percentage of area-covered ( f 2 ) for each of the four shapes are shown in Table 5. The results indicate that the forms taken from the Pareto front outperformed the standard polyomino configuration. Furthermore, the shape generated from the MACO algorithm has a higher percentage of area covered with lesser path-length  Fig. 12 Real-world experiment using the optimal morphology (refer Table 5) generated from the SMPSO algorithm Fig. 13 Experimental results using the optimal morphology (refer Table 5) generated from the NSGA-II algorithm Fig. 14 Real-world experiment using the optimal morphology (refer Table 5) generated from the MACO algorithm compared to the other two algorithms. Besides, as mentioned in Table 5 the difference in the area coverage between the actual and simulation is mainly due to the wheel slippage and the manual error during the teleoperation. When the robot is commanded to move in a straight line, the wheel slippage causes the robot to drift from the global path causing it to leave some area uncovered. In conclusion, the experiment results satisfy our simulation finding for generating optimal energy-efficient morphology. Overall, the suggested framework can determine the optimal morphology by making trade-offs between the path length and the percentage of area covered using the metaheuristic algorithm. Moreover, this work revealed how each possible shape in a reconfigurable robot could be exploited for area coverage applications.

Conclusion
Given the current limitations of the existing approach for reconfigurable robots in performing efficient cleaning with optimal energy consumption, we propose a framework for generating an optimal morphology of a reconfigurable robot. The reconfigurable robot considered in this work consists of four blocks and is referred to as Smor phi. The robot can take several morphologies apart from the set of standard polyomino shapes, i.e., I, J, L, O, S, T, Z. The objective was to maximize the area coverage and minimizes energy consumption. The proposed framework is validated through simulation and real-world experiments using three different metaheuristics algorithms: SMPSO, NSGA-II, and MACO. Moreover, the proposed method can be extended for different classes of reconfigurable robots to select an optimal configuration depending on the given application such as cleaning, surveillance or reconnaissance task, among others.
Furthermore, insight into the optimal morphologies is gained through the simulation and experimental results. The results show that the each approach finds a near-optimal energy-efficient morphology for the given map. Besides, the shape generated from the MACO approach is superior to the other two algorithms, which can be considered a suitable algorithm for generating the optimal-energy efficient morphology. Overall, the proposed method opens a new avenue for further research on identifying Opt-Morph for different classes of reconfigurable robots. The future work includes a reinforcement learning-based approach to determine the optimal shape in reconfigurable robots and developing a platform-agnostic framework for optimal shape generation in a different class of reconfigurable robots, developing a suitable control strategy, and accounting for the rotational and diagonal motion of the reconfigurable robot in the global path planner.

Data availability
The data used during the current study are available from the first author and corresponding author upon reasonable request.
Code availability Not applicable.

Conflict of interest The authors declare that they have no conflict of interest
Ethics approval Not applicable.

Consent for publication: Yes
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