The presence of symmetric and periodic configurations introduces complexities while solving various types of tasks on a grid graph using autonomous robots. Though symmetry and periodicity can be easily broken in some scenarios, it can be difficult and sometimes impossible to break the same in certain scenarios. This paper studies all possible types of symmetric and periodic configurations that exist in a grid and presents techniques that successfully break symmetry and periodicity in a finite grid within finite time. It highlights the scenarios where breaking of the same is impossible. This paper considers oblivious and asynchronous robots on grids that are endowed with unlimited visibility and pebbles but do not have a sense of direction or any communication among themselves. Though the proposed techniques may be used as a preliminary step while solving various types of tasks on a grid, we specifically present the use of these techniques for exploration of a grid using an already existing exploration algorithm.

In this paper, the solutions to the target searching problem on a ring by a swarm of anonymous, oblivious, asynchronous robots that are initially deployed over the ring randomly are addressed. A particular node in the ring is specified as the target node and the robots from their arbitrary initial distribution are required to find out the target node and gather at that location. In the proposed solution, apart from finally gathering at the target node, the robots collectively explore the complete ring within a finite amount of time. The ring nodes here are considered to have persistent memory, known as whiteboards, which may be used to store an integer that would finally indicate the distance of that node from the target node. In this work, the robots are considered to have the strongest possible myopia; they can see only up to their neighbouring nodes. However, a robot cannot modify the whiteboard marking of its neighbours. Two different algorithms are proposed in the paper. In the first solution, it is assumed that in the initial distribution at least one robot must be present either at the target node or at any one of its neighbours. However, in the second solution, the initial distribution is assumed to be completely random.

This paper studies the terminating exploration of a finite grid where k autonomous robots traverse the grid such that each node is visited at least once and robots eventually stop when the task is completed. Practically, there may exist nodes that are either not reachable or considered to be insignificant for exploration, denoted as blocked nodes. We study the exploration of a grid that has one or more blocked nodes using oblivious, non-communicating and fully-synchronous robots that have unlimited visibility and follow the look-compute-move model. Each node has a limited amount of space called whiteboard where robots can read and write some information. The proposed algorithm enables robots to collectively explore all the non-blocked nodes of the grid in finite time. During exploration, though a node may be visited more than once, the algorithm ensures that all visits to a particular node are made by the same robot.

This paper inspects the performance of two existing assembling algorithms after calibrating some useful parameters associated to the environment and robot characteristics in a swarm robotic system by using simulation. The impact of parameters like synchrony, mobility, number of robots and number of obstacles are observed and analyzed for two models of axes agreement namely, full compass model and direction only model. The comparison is made on the basis of maximum number of computation cycles taken by each algorithm to assemble the robots in the boundary. Results of the simulation show that when non-rigidity is assumed in the semi-synchronous model, the performance of both the models deteriorate significantly compared to other combinations.

This paper addresses the partition problem under the continuous domain in swarm robotics. In this problem, a swarm of n robots, randomly deployed over a rectangular workspace, are required to form K size-balanced groups within a finite amount of time. We aim to look into the problem in the presence of horizontal line obstacles. The obstacles are scattered randomly over the bounded region, and their positions remain unaltered throughout the process. In the proposed solutions, the robots are assumed to be identical, autonomous, and do not have any direct communication among themselves. The robots are memoryless, except they retain only the information of the two parameters n and K throughout the process and their states among three possible ones. Two different partition algorithms are proposed assuming full-compass axis agreement and half-compass axis agreement among the local coordinate systems of the robots. In the first case, the proposed algorithm works for both synchronous and semi-synchronous models, whereas, in the second case, the robots are assumed to be synchronous. As a pre-processing step of the partition algorithms, an assembling algorithm for the half-compass axis agreement model has also been proposed for synchronous and semi-synchronous robots.

We address the terminating exploration problem where a team of k autonomous robots explores a grid such that each node of the grid is visited at least once and all robots stop moving when the task is completed. Robots considered for the task are anonymous, asynchronous, oblivious and non-communicating that have unlimited visibility and a pebble each. Visiting a node more than once by the robots during exploration is a waste of energy due to energy constraints. In this work we aim at minimizing the total number of wasteful repeated visits of the nodes that occur during exploration. Our proposed algorithm explores a grid with minimum repeated visits of nodes.

This article studies the problem of painting of a rectangular region cluttered with horizontal obstacles, by a swarm of mobile robots. Initially, the robots are deployed randomly within the target area. From the initial configuration, the robots are assembled on the left boundary of the region. Upon assembling on the left boundary, the robots first explore the entire area collectively to make a virtual partition of it and then collectively paint the target region in a subsequent phase. The proposed algorithm assumes that robots work in look-compute-move model. The robots follow a completely distributed algorithm to paint the region. The robots are either synchronous or semi-synchronous. The outcome of the proposed algorithm is a complete painting of an area without any repetition and collision. [GRAPHICS] .

This paper addresses the problem of assembling semi-synchronous oblivious robots in presence of horizontal line obstacles under direction only axes agreement model. In this article, a distributed algorithm is proposed for a swarm of autonomous mobile robots that are required to assemble over the boundaries of a rectangular bounded region within a finite amount of time. The robots are initially deployed randomly within the region and assumed to support the non-rigid motion. All the line-shaped obstacles are randomly scattered within the region and are assumed to be positioned horizontally, parallel to the top and bottom boundaries of the rectangular region. Though the robots are assumed to have unlimited visibility, these opaque obstacles restrict their vision. The existing solution to this problem assumes full axes agreement among the robots, whereas, the proposed solution assumes direction only axis agreement, a much weaker model. Our proposed algorithm is fault-tolerant and supports the collision-free movement of the robots. It successfully assembles robots on the left and right boundaries of the region.

There are three models of automated robots based on their activation schedules and time notions. They are FSYNC, SSYNC and ASYNC Model. The FSYNC-fully synchronous robots have the common notion of time and follow a global round of time. In every turn, all the robots are active and perform their job in synchronization. All the robots have the same notion of time in the SSYNC-semi-synchronous model, whereas in this model, all robots may not be active in each round. The activation time of ASYNC-asynchronous is independent and unpredictable. Research is ongoing to vanish the difference between asynchrony, semi-synchrony and synchrony by equipping the robots with various combination of lights and snapshots. Most of the research has been done on robots with unlimited visibility capabilities. In these works, a robot can view the colors of all other robots and based on that the decision is taken to change its own color thus achieves synchronization. Decision taking becomes challenging if the robots have limited view of the other robots. This paper proposed a technique of synchronizing the robots which are asynchronous by nature, using observable colored lights, where the robots have limited range of visibility.

Swarm of mobile robots is actually a large number of small robots imitating the behavior of social insects that perform certain tasks in a group. This chapter considers the problem of area coverage by a swarm of mobile robots. Initially, the robots occupy random positions in a target area. The objective is to physically cover/scan each accessible location of that area by at least one robot of the swarm. After discussing, in brief, different models and their challenges this chapter summarizes the research works carried out to solve this problem. The existing literature is classified into two categories, namely, team based approach and individual approach. The pros and cons of both the approaches are indicated and finally a comparative study of the addressed works in terms of computational model, synchrony, characteristics of robots, etc. is presented.

In a swarm of robots, each robot has certain capabilities to perform their computations to achieve a global objective. One such capability is the sensing capability, known as vision. This enables a robot to sense the positions of the other robots in the system. The sensing capability may be restricted by two factors: (i) the sensing range and (ii) the opacity of the robots. The sensing range may be limited or unlimited and the robots may be transparent or opaque. When robots have limited sensing range, a robot can sense all the robots within a fixed radius around it. If three opaque robots are collinear, the middle robot obstructs the vision of the two other robots. This chapter deals with these two constraints on the vision of the robots. A model with such a constraint is referred to as the restricted visibility model. This chapter presents different geometric formation problems for swarm robots under the restricted visibility model.

COopeRative Distributed Asynchronous model or CORDA model is a basic computational model in the field of robot swarm. The objective of this work is to justify popularity and suitability of the CORDA model as a basic model of computation vis-a-vis other computational models. The problem of covering a target area has been taken up as an area of focus. This chapter presents a critical review of the various solutions of the coverage problem under CORDA model.

This article studies the problem of painting an obstacle free rectangular region by a swarm of mobile robots. Initially the robots are deployed randomly within the target area subject to the condition that the distribution is d *-dense, where d * = 3 d 2 , and a robot can view up to a distance d . By d *-dense, it is meant that if all the robots are projected on a horizontal line, then the distance between two consecutive robots must be less than or equal to d *. Non-consideration of the popular CORDA (computational) model in the field of area coverage by swarm robots has been addressed here. The proposed algorithm assumes CORDA model. The robots follow a completely distributed algorithm to paint the region. The robots do not need to be synchronous, but they are assumed to have equal velocities. However, the proposed algorithm supports the robots with different speed. In that case, if r is the given upper bound on the ratios of the speeds of any two robots, then the initial distribution has to be D *-dense, where D * = d 4 ( 3 − r ) ( r + 5 ) 1 ≤ r < 3 .

Discrete domain swarm robotics is an emerging and challenging field of research. Unlike continuous domain where working place of the robots is a two-dimensional plane, in the discrete domain their working place is modelled by a graph. Robots are deployed on the nodes of a given graph, and they are allowed to move only along the edges of that graph. Consequently, the models used in the continuous domain are not always applicable in the discrete domain. Exemplified by the gathering problem, this article critically reviews models, assumptions, and approaches that have been proposed in solving the problems in the discrete domain.

Swarm robotic research in discrete domain assumes that robots in the swarm are randomly deployed over any graph and the robots can move only through the edges of the graph. In target searching, all the robots in the swarm are required to gather at the specially designated node, termed as target node. Moreover, during target search if the graph is guaranteed to be explored completely, the scope of the solution increases. An algorithm for tree searching by swarm of asynchronous robots of limited visibility has been proposed in this paper. An O(1) memory is assumed to be attached to each node of the tree. The target node is initially visible to at least one robot in the swarm. However, if it is executed on synchronous system, the algorithm takes O(n) computational cycles to gather all the robots at the target node after exploration of the tree completely, where n is the number of nodes in the graph.

This paper presents a distributed algorithm for assembling a swarm of autonomous mobile robots on a common boundary of a given polygonal region in presence of opaque horizontal line obstacles. Robots and obstacles are initially scattered in an unknown environment and they do not have direct communication among themselves. The algorithm guarantees successful assembling of all the robots on the left boundary of the given region within finite amount of time and without facing any collision during their movement. The intermediate distances among the assembled robots are not fixed. In this proposed algorithm, the robots follow the basic Wait-Observe-Compute-Move model together with the Full-Compass and Synchronous/Semi-synchronous timing models.

Objective: In the material world, objects like railway tracks, bridges, roofs etc., are constructed by collections of nonelastic rigid rods, beams, etc.. A structure is said to be rigid if there is no continuous motion of the structure that changes its shape without changing the shapes of its components like rods or beams. In this survey work, we accumulate the fundamental concepts on graph rigidity. Methods and Analysis: We give the analytical definition of rigid graphs using the idea of rigid motions. The Laman’s theorem and Hendrickson’s algorithm are presented as methods for testing graph rigidity in the plane. The construction of the rigid graphs is also analyzed using the Henneberg’s operations. We describe how in distributed environments the rigidity of graphs can be checked using the vertex ordering in the graph. For frameworks lying in the higher dimensional spaces rigidity testing method is presented in form of a theorem. Novelty and Improvements: The results of this review works may help the readers to better understand the graph rigidity theory from different perspectives. This study founds a positive association between the analytical and combinatorial concepts of graph rigidity investigated so far. Keywords: Graph Realization, Localizability Testing, Network Localization, Rigidity of Graphs

An emerging and challenging area of research in swarm robotics is to consider swarms deployed in discrete domains. In the continuous domain, it has already been established that different computational and behavioral models of the robot swarm play an important role in solvability of different fundamental problems. Due to some basic differences, not all existing models in the continuous domain are relevant or significant with respect to the discrete case. In this paper we draw an analogy between the models already existing in both the domains and propose a few relevant models for the discrete domain.

Localization is an important issue for Wireless Sensor Networks (WSN). A mobile sensor may change its position rapidly and thus require localization calls frequently. A localization may require network wide information and increase traffic over the network. It dissipates valuable energy for message communication. Thus localization is very costly. The control of the number of localization calls may save energy consumption, as it is rather expensive. To reduce the frequency of localization calls for a mobile sensor, we propose a technique that involves \textit{Mobility Aware Interpolation} (MAINT) for position estimation. It controls the number of localizations which gives much better result than the existing localization control schemes using mobility aware extrapolation. The proposed method involves very low arithmetic computation overheads. We find analytical expressions for the expected error in position estimation. A parameter, the time interval, has been introduced to externally control the energy dissipation. Simulation studies are carried out to compare the performances of the proposed method with some existing localization control schemes as well as the theoretical results. The simulation results shows that the expected error at any point of time may be computed from this expression. We have seen that constant error limit can be maintained increasing the time period of localization proportional to rate of change of direction of its motion. Increasing time period, the energy may be saved with a stable error limit.

This paper presents a distributed painting algorithm for painting a priori known rectangular region by swarm of autonomous mobile robots. We assume that the region is obstacle free and of rectangular in shape. The basic approach is to divide the region into some cells, and to let each robot to paint one of these cells. Assignment of different cells to the robots is done by ranking the robots according to their relative positions. In this algorithm, the robots follow the basic Wait-Observe-Compute-Move model together with the synchronous timing model. This paper also presents a simulation of the proposed algorithm. The simulation is performed using the Player/Stage Robotic Simulator on Ubuntu 10.04 (Lucid Lynx) platform.