Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
  • Marco Dorigo received the Laurea, Master of Technology, degree in industrial technologies engineering in 1986, and th... moreedit
When selecting a resource to exploit, an insect colony must take into account at least two constraints: the resource must be abundant enough to sustain the whole group, but not too large to limit exploitation costs, and risks of conflicts... more
When selecting a resource to exploit, an insect colony must take into account at least two constraints: the resource must be abundant enough to sustain the whole group, but not too large to limit exploitation costs, and risks of conflicts with other colonies. Following recent results on cockroaches and ants, we introduce here a behavioral mechanism that satisfies these two constraints. Individuals simply modulate their probability to switch to another resource as a function of the local density of conspecifics locally detected.
Abstract Swarm robotics draws inspiration from decentralized self-organizing biological systems in general and from the collective behavior of social insects in particular. In social insect colonies, many tasks are performed by higher... more
Abstract Swarm robotics draws inspiration from decentralized self-organizing biological systems in general and from the collective behavior of social insects in particular. In social insect colonies, many tasks are performed by higher order group or team entities, whose task-solving capacities transcend those of the individual participants. In this paper, we investigate the emergence of such higher order entities. We report on an experimental study in which a team of physical robots performs a foraging task.
Dario Florcano's review of Robot SIItI~ JIrI~ T: All E. uprrinuut ill l3rlraoior EtyL~ irmrrinET is very stimulating and deserves a long answer. Howevcr, wc do not want to abuse our privilege of having the last word, which the editor of... more
Dario Florcano's review of Robot SIItI~ JIrI~ T: All E. uprrinuut ill l3rlraoior EtyL~ irmrrinET is very stimulating and deserves a long answer. Howevcr, wc do not want to abuse our privilege of having the last word, which the editor of this special issue has accorded to us.
EvoSTIM Papers-Design of Iterated Local Search Algorithms. Matthijs den Besten, Thomas Stutzle, Marco Dorigo Lecture Notes in Computer Science 2037, 441-451, Berlin: Springer-Verlag, 1973-, 2001.
Abstract. Ant-based clustering and sorting is a nature-inspired heuristic for general clustering tasks. It has been applied variously, from problems arising in commerce, to circuit design, to text-mining, all with some promise. However,... more
Abstract. Ant-based clustering and sorting is a nature-inspired heuristic for general clustering tasks. It has been applied variously, from problems arising in commerce, to circuit design, to text-mining, all with some promise. However, although early results were broadly encouraging, there has been very limited analytical evaluation of antbased clustering.
In this paper we present an approach to the cooperative transport of multiple objects in swarm robotics. The approach is motivated by the observation that the performance of cooperative transport in insect colonies as well as in groups of... more
In this paper we present an approach to the cooperative transport of multiple objects in swarm robotics. The approach is motivated by the observation that the performance of cooperative transport in insect colonies as well as in groups of robots grows in a super linear way with the number of individuals participating in the transport. The transport relies on a cart in which multiple objects are collected and stored before being moved to destination.
For any robotic entity to complete a task efficiently, its morphology must be appropriate to the task. If the task is well-defined in advance, the morphology of a robotic entity can be pre-specified accordingly. If, however, some of the... more
For any robotic entity to complete a task efficiently, its morphology must be appropriate to the task. If the task is well-defined in advance, the morphology of a robotic entity can be pre-specified accordingly. If, however, some of the task parameters are not known in advance, or if the same robotic system is required to solve several different tasks, morphological flexibility may be required. It is easy to imagine, for example, that navigating on uneven terrain and hole-crossing are likely to require different morphologies (see Fig. 1).
Abstract: A combination of distributed computation, positive feedback and constructive greedy heuristic is proposed as a new approach to stochastic optimization and problem solving. Positive feedback accounts for rapid discovery of very... more
Abstract: A combination of distributed computation, positive feedback and constructive greedy heuristic is proposed as a new approach to stochastic optimization and problem solving. Positive feedback accounts for rapid discovery of very good solutions, distributed computation avoids premature convergence, and greedy heuristic helps the procedure to find acceptable solutions in the early stages of the search process.
Abstract Dynamic optimisation problems are problems where the search space does not remain constant over time. Evolutionary algorithms aimed at static optimisation problems often fail to effectively optimise dynamic problems. The main... more
Abstract Dynamic optimisation problems are problems where the search space does not remain constant over time. Evolutionary algorithms aimed at static optimisation problems often fail to effectively optimise dynamic problems. The main reason for this is that the algorithms converge to a single optimum in the search space, and then lack the necessary diversity to locate new optima once the environment changes.
In this chapter, we study a collective robotic system that is inspired by the behavior of social insects. The system consists of mobile robots that can autonomously perceive and modify their environment. We investigate the problem of... more
In this chapter, we study a collective robotic system that is inspired by the behavior of social insects. The system consists of mobile robots that can autonomously perceive and modify their environment. We investigate the problem of controlling a number of these robots in cooperation based tasks that are too difficult for the robots to solve when operating alone.
Abstract. Ant-based clustering and sorting is a nature-inspired heuristic for general clustering tasks. It has been applied variously, from problems arising in commerce, to circuit design, to text-mining, all with some promise. However,... more
Abstract. Ant-based clustering and sorting is a nature-inspired heuristic for general clustering tasks. It has been applied variously, from problems arising in commerce, to circuit design, to text-mining, all with some promise. However, although early results were broadly encouraging, there has been very limited analytical evaluation of the algorithm. Toward this end, we first propose a scheme that enables unbiased interpretation of the clustering solutions obtained, and then use this to conduct a full evaluation of the algorithm.
Abstract A number of methodological papers published during the last years, testify that a need for a thorough revision of the research methodology is felt by the operations research community—see, for example, Barr et al.(1995), Hooker... more
Abstract A number of methodological papers published during the last years, testify that a need for a thorough revision of the research methodology is felt by the operations research community—see, for example, Barr et al.(1995), Hooker (1995), and Rardin and Uzsoy (2001). In particular, the performance evaluation of nondeterministic methods, such as the methods known as metaheuristics, require the definition of new experimental protocols.
Colonies of social insects can exhibit an amazing variety of complex behaviors and have captured, since ever, the interest of biologists and entomologists. More recently, computer scientists have found in the study of social insects... more
Colonies of social insects can exhibit an amazing variety of complex behaviors and have captured, since ever, the interest of biologists and entomologists. More recently, computer scientists have found in the study of social insects behavior a source of inspiration for the design and implementation of novel distributed multi-agent algorithms. In particular, the study of ant colonies behavior turned out to be very fruitful, giving raise to a completely novel field of research, now known as ant algorithms.
Abstract We introduce a high-performing composite particle swarm optimization (PSO) algorithm. In an analogy to the popular character of Mary Shelley's famous novel, we call our algorithm Frankenstein's PSO, as it consists of different... more
Abstract We introduce a high-performing composite particle swarm optimization (PSO) algorithm. In an analogy to the popular character of Mary Shelley's famous novel, we call our algorithm Frankenstein's PSO, as it consists of different algorithmic components drawn from other PSO variants. Frankenstein's PSO constituents were selected after careful evaluation of their impact on speed and reliability.
We propose a modular architecture for autonomous robots which allows for the implementation of basic behavioral modules by both programming and training, and accommodates for an evolutionary development of the interconnections among... more
We propose a modular architecture for autonomous robots which allows for the implementation of basic behavioral modules by both programming and training, and accommodates for an evolutionary development of the interconnections among modules. This architecture can implement highly complex controllers and allows for incremental shaping of the robot behavior.
The quadratic assignment problem (QAP) is an important problem in theory and practice. It was, which was introduced by Koopmans and Beckmann in 1957 [28] and is a model for many practical problems like backboard wiring [53], campus and... more
The quadratic assignment problem (QAP) is an important problem in theory and practice. It was, which was introduced by Koopmans and Beckmann in 1957 [28] and is a model for many practical problems like backboard wiring [53], campus and hospital layout [15, 17], typewriter keyboard design [9], scheduling [23] and many others [16, 29] can be formulated as QAPs.
Abstract We introduce a new robotic system, called swarm-bot. The system consists of a swarm of mobile robots with the ability to connect to/disconnect from each other to self-assemble into different kinds of structures. First, we... more
Abstract We introduce a new robotic system, called swarm-bot. The system consists of a swarm of mobile robots with the ability to connect to/disconnect from each other to self-assemble into different kinds of structures. First, we describe our vision and the goals of the project. Then we present preliminary results on the formation of patterns obtained from a grid-world simulation of the system.
Abstract We have designed and built a new open hardware/software board that lets miniaturized robots communicate and at the same time obtain the range and bearing of the source of emission. The open E-puck Range & Bearing board improves... more
Abstract We have designed and built a new open hardware/software board that lets miniaturized robots communicate and at the same time obtain the range and bearing of the source of emission. The open E-puck Range & Bearing board improves an existing infrared relative localization/communication software library (libIrcom) developed for the e-puck robot and based on its on-board infrared sensors. The board allows the robots to have an embodied, decentralized and scalable communication system.
An insect-based algorithm inspired by the division of labor in insect colonies is proposed and applied to the solution of an online scheduling problem. A painting facility is considered for illustrating the problem: Trucks leave an... more
An insect-based algorithm inspired by the division of labor in insect colonies is proposed and applied to the solution of an online scheduling problem. A painting facility is considered for illustrating the problem: Trucks leave an assembly line to get painted in painting booths. The goal is to minimize the makespan, that is, the time needed for painting all given trucks. In this paper we address two issues. First, we propose and analyze four modifications of an insect-based algorithm previously introduced by Cicirello and Smith.
Abstract The probabilistic traveling salesman problem (PTSP), a paradigmatic example of a stochastic combinatorial optimization problem, is used to study routing problems under uncertainty. Recently, we introduced a new estimation-based... more
Abstract The probabilistic traveling salesman problem (PTSP), a paradigmatic example of a stochastic combinatorial optimization problem, is used to study routing problems under uncertainty. Recently, we introduced a new estimation-based iterative improvement algorithm for the PTSP and we showed that it outperforms for a number of instance classes the previous state-of-the-art algorithms.
In this paper, we study coordinated motion in a swarm robotic system, called a swarm-bot. A swarm-bot is a self-assembling and self-organising artifact, composed of a swarm of s-bots, mobile robots with the ability to connect to and... more
In this paper, we study coordinated motion in a swarm robotic system, called a swarm-bot. A swarm-bot is a self-assembling and self-organising artifact, composed of a swarm of s-bots, mobile robots with the ability to connect to and disconnect from each other. The swarm-bot concept is particularly suited for tasks that require all-terrain navigation abilities, such as space exploration or rescue in collapsed buildings.
Abstract. In this paper, we study the cooperative transport of a heavy object by a group of robots towards a goal. We investigate the case in which robots have partial and noisy knowledge of the goal direction and can not perceive the... more
Abstract. In this paper, we study the cooperative transport of a heavy object by a group of robots towards a goal. We investigate the case in which robots have partial and noisy knowledge of the goal direction and can not perceive the goal itself. The robots have to coordinate their motion to apply enough force on the object to move it.
In this paper we present the results of a research relative to the ascertainment of limits and potentialities of genetic algorithms ([Dorigo, 1989],[DeJong-Spears, 1989],[Goldberg, 1989]) in addressing highly constrained problems, that is... more
In this paper we present the results of a research relative to the ascertainment of limits and potentialities of genetic algorithms ([Dorigo, 1989],[DeJong-Spears, 1989],[Goldberg, 1989]) in addressing highly constrained problems, that is optimization problems, where a minimal change to a feasible solution is very likely to generate an unfeasible one.
Communication is often required for coordination of collective behaviours. Social insects like ants, termites or bees make use of different forms of communication, which can be roughly classified in three classes: indirect (stigmergic)... more
Communication is often required for coordination of collective behaviours. Social insects like ants, termites or bees make use of different forms of communication, which can be roughly classified in three classes: indirect (stigmergic) communication, direct interaction and direct communication. The use of stigmergic communication is predominant in social insects (eg, the pheromone trails in ants), but also direct interactions (eg, antennation in ants) and direct communication can be observed (eg, the waggle dance of honey bee workers).
Abstract This paper presents a new robotic concept, called SWARM-BOT, based on a swarm of autonomous mobile robots with self-assembling capabilities. This concept has been developed to ensure robust navigation, search and transportation... more
Abstract This paper presents a new robotic concept, called SWARM-BOT, based on a swarm of autonomous mobile robots with self-assembling capabilities. This concept has been developed to ensure robust navigation, search and transportation in rough terrain. The SWARM-BOT concept takes advantage from collective and distributed approaches to ensure robustness to failures and to hard environment conditions.
Page 1. Swarm Robotics Marco Dorigo FNRS Research Director IRIDIA Université Libre de Bruxelles Page 2. Swarm robotics What is swarm robotics? It is the application of swarm intelligence principles to collective robotics It is research in... more
Page 1. Swarm Robotics Marco Dorigo FNRS Research Director IRIDIA Université Libre de Bruxelles Page 2. Swarm robotics What is swarm robotics? It is the application of swarm intelligence principles to collective robotics It is research in collective robotics: – that is relevant for the control and coordination of large numbers of robots – in which robots are relatively simple and incapable, so that the tasks they tackle require cooperation – in which the robots have only local and limited sensing and communication abilities 3 Page 3.
This paper addresses the cooperative transport of a heavy object, called prey, towards a sporadically changing target location by a group of robots. The study is focused on the situation in which some robots are given the opportunity to... more
This paper addresses the cooperative transport of a heavy object, called prey, towards a sporadically changing target location by a group of robots. The study is focused on the situation in which some robots are given the opportunity to localize the target, while the others (called the blind ones) are not. We propose the use of relatively simple robots capable of self-assembling into structures which pull or push the prey.
Abstract Group transport is performed in many natural systems and has become a canonical task for studying cooperation in robotics. We simulate a system of simple, insect-like robots that can move autonomously and grasp objects as well as... more
Abstract Group transport is performed in many natural systems and has become a canonical task for studying cooperation in robotics. We simulate a system of simple, insect-like robots that can move autonomously and grasp objects as well as each other. We use artificial evolution to produce solitary transport and group transport behaviors. We show that robots, even though not aware of each other, can be effective in group transport. Group transport can even be performed by robots that behave as in solitary transport.
Abstract In this paper, we review half a century of research on the design of systems displaying (physical) self-assembly of macroscopic components. We report on the experience gained in the design of 21 such systems, exhibiting... more
Abstract In this paper, we review half a century of research on the design of systems displaying (physical) self-assembly of macroscopic components. We report on the experience gained in the design of 21 such systems, exhibiting components ranging from passive mechanical parts to mobile robots. We present a taxonomy of the systems and discuss design principles and functions. Finally, we summarize the main achievements and indicate potential directions for future research.
The Ant Colony Optimization metaheuristic (ACO) is a recent metaheuristic [2] that has been successfully applied to a number of combinatorial optimization problems (see [3] for an overview). In this paper we present and discuss an... more
The Ant Colony Optimization metaheuristic (ACO) is a recent metaheuristic [2] that has been successfully applied to a number of combinatorial optimization problems (see [3] for an overview). In this paper we present and discuss an application of ACO to Constraint Satisfaction Problems (CSPs).
Abstract In this article, we propose a distributed control mechanism for a self-propelled, self-assembling robotic system that allows robots to form specific, connected morphologies. Global morphologies are grown using only local visual... more
Abstract In this article, we propose a distributed control mechanism for a self-propelled, self-assembling robotic system that allows robots to form specific, connected morphologies. Global morphologies are grown using only local visual perception. Robots that are part of the connected entity indicate where new robots should attach to grow the local structure appropriately. We demonstrate the efficacy of the mechanism by letting groups of seven real robots self-assemble into four different morphologies: line, star, arrow, and rectangle.
In this paper, we aim to design decision-making mechanisms for a simulated Khepera robot equipped with simple sensors, which integrates over time its perceptual experience in order to initiate a simple signalling response. Contrary to... more
In this paper, we aim to design decision-making mechanisms for a simulated Khepera robot equipped with simple sensors, which integrates over time its perceptual experience in order to initiate a simple signalling response. Contrary to other previous similar studies, in this work the decision-making is uniquely controlled by the time-dependent structures of the agent controller, which in turn are tightly linked to the mechanisms for sensorimotor coordination.
Abstract Boolean networks, in spite of their structural simplicity, seem to be able to simulate the dynamics of complex biological and nonbiological systems. Learning algorithms in neural networks have shown to be a very promising... more
Abstract Boolean networks, in spite of their structural simplicity, seem to be able to simulate the dynamics of complex biological and nonbiological systems. Learning algorithms in neural networks have shown to be a very promising approach to some problems connected to artificial intelligence. Positive feedback has been successfully used by the genetic algorithm and the ant system. In this paper we propose an adaptive Boolean network that takes advantage of all these properties
This paper considers some issues related to the apportionment of credit problem in Genetic Based Machine Learning systems (GBML). A GBML system is composed of three major subsystems. The first one, the performance subsystem, is a parallel... more
This paper considers some issues related to the apportionment of credit problem in Genetic Based Machine Learning systems (GBML). A GBML system is composed of three major subsystems. The first one, the performance subsystem, is a parallel adaptive rule-based system where the knowledge base is a set of rules expressed in a low-level syntax.
Consider a group of autonomous, mobile robots with the ability to physically connect to one another (self-assemble). The group is said to exhibit functional self-assembly if the robots can choose to self-assemble in response to the... more
Consider a group of autonomous, mobile robots with the ability to physically connect to one another (self-assemble). The group is said to exhibit functional self-assembly if the robots can choose to self-assemble in response to the demands of their task and environment [15]. We present the first robotic controller capable of functional self-assembly implemented on a real robotic platform. The task we consider requires a group of robots to navigate over an area of unknown terrain towards a target light source.
Abstract—In this paper, we present a comprehensive study on autonomous self-assembly. In particular, we discuss the selfassembling capabilities of the swarm-bot, a distributed robotics concept that lies at the intersection between... more
Abstract—In this paper, we present a comprehensive study on autonomous self-assembly. In particular, we discuss the selfassembling capabilities of the swarm-bot, a distributed robotics concept that lies at the intersection between collective and selfreconfigurable robotics. A swarm-bot comprises autonomous mobile robots called s-bots. S-bots can either act independently or self-assemble into a swarm-bot by using their grippers.
Prey retrieval, also known as foraging, is a widely used test application in collective robotics. The task consists in searching for objects spread in the environment and in bringing them to a specific place called nest. Scientific issues... more
Prey retrieval, also known as foraging, is a widely used test application in collective robotics. The task consists in searching for objects spread in the environment and in bringing them to a specific place called nest. Scientific issues usually concern efficient exploration, mapping, communication among agents, task coordination and allocation, and conflict resolution. In particular, interferences among robots reduce the efficiency of the group in performing the task.
Abstract. Up to now, only a few collective or modular robot systems have proven capable of letting separate and autonomous units, or groups of units, self-assemble. In each case, ad hoc control algorithms have been developed. The aim of... more
Abstract. Up to now, only a few collective or modular robot systems have proven capable of letting separate and autonomous units, or groups of units, self-assemble. In each case, ad hoc control algorithms have been developed. The aim of this paper is to show that a control algorithm for autonomous self-assembly can be ported from a source multi-robot platform (ie, the swarm-bot system) to a different target multirobot platform (ie, a super-mechano colony system).
Abstract A number of methodological papers published during the last years testify that a need for a thorough revision of the research methodology is felt by the operations research community–see, for example,[Barr et al., J. Heuristics 1... more
Abstract A number of methodological papers published during the last years testify that a need for a thorough revision of the research methodology is felt by the operations research community–see, for example,[Barr et al., J. Heuristics 1 (1995) 9–32; Eiben and Jelasity, Proceedings of the 2002 Congress on Evolutionary Computation (CEC'2002) 582–587; Hooker, J. Heuristics 1 (1995) 33–42; Rardin and Uzsoy, J. Heuristics 7 (2001) 261–304].
Abstract This article is on the subject of evolving neural network controllers for cooperative, mobile robots. We evolve controllers for combined hole-avoidance and phototaxis in a group of physically connected, autonomous robots called... more
Abstract This article is on the subject of evolving neural network controllers for cooperative, mobile robots. We evolve controllers for combined hole-avoidance and phototaxis in a group of physically connected, autonomous robots called s-bots, each with limited sensing capabilities.
Abstract Incremental social learning (ISL) was proposed as a way to improve the scalability of systems composed of multiple learning agents. In this paper, we show that ISL can be very useful to improve the performance of population-based... more
Abstract Incremental social learning (ISL) was proposed as a way to improve the scalability of systems composed of multiple learning agents. In this paper, we show that ISL can be very useful to improve the performance of population-based optimization algorithms.
Abstract Communication is of central importance in collective robotics, as it is integral to the switch from solitary to social behavior. In this article, we study emergent communication behaviors that are not predetermined by the... more
Abstract Communication is of central importance in collective robotics, as it is integral to the switch from solitary to social behavior. In this article, we study emergent communication behaviors that are not predetermined by the experimenter, but are shaped by artificial evolution, together with the rest of the behavioral repertoire of the robots.
Abstract We consider a heterogeneous swarm consisting of aerial and wheeled robots. We present a system that enables spatially targeted communication. Our system enables aerial robots to establish dedicated communication links with... more
Abstract We consider a heterogeneous swarm consisting of aerial and wheeled robots. We present a system that enables spatially targeted communication. Our system enables aerial robots to establish dedicated communication links with individual wheeled robots or with selected groups of wheeled robots based on their position in the environment. The system does not rely on any form of global information.
Abstract In this paper, we present a heterogeneous recruitment system which allows aerial robots to recruit groups of wheeled robots. The system is novel because it is self-organized, is based only on simple probabilistic rules and relies... more
Abstract In this paper, we present a heterogeneous recruitment system which allows aerial robots to recruit groups of wheeled robots. The system is novel because it is self-organized, is based only on simple probabilistic rules and relies only on local communication. Our approach is inspired by the aggregation behavior of cockroaches. The system allows aerial robots to recruit wheeled robot groups of different sizes in parallel.
Abstract In this paper, we present our first steps towards applying swarm intelligence methods for solving exploration and navigation tasks performed by a swarm of robots in unknown environments. Our approach consists in using chains of... more
Abstract In this paper, we present our first steps towards applying swarm intelligence methods for solving exploration and navigation tasks performed by a swarm of robots in unknown environments. Our approach consists in using chains of visually connected robots that collectively explore their environment. We adopt the idea of robotic chains from Goss et al.[5], and realize our system stressing the swarm intelligence approach.
ABSTRACT In this paper we present an application of ALECSYS, a distributed learning classifier system, to the control of a robot arm. ALECSYS is initialised with a set of randomly generated rules and is trained to control a robot arm... more
ABSTRACT In this paper we present an application of ALECSYS, a distributed learning classifier system, to the control of a robot arm. ALECSYS is initialised with a set of randomly generated rules and is trained to control a robot arm whose task is to reach a non moving light source. At this point of our research our results are relative to the simulation of a real robot arm (IBM 7547 with a SCARA geometry), which will be the target of the final implementation of our learning system.
In this paper, we study how an opinion dynamics model can be the core of a collective decision-making mechanism for swarm robotics. Our main result is that when opinions represent action choices, the opinion associated with the action... more
In this paper, we study how an opinion dynamics model can be the core of a collective decision-making mechanism for swarm robotics. Our main result is that when opinions represent action choices, the opinion associated with the action that is the fastest to execute spreads in the population. Moreover, the spread of the best choice happens even when only a minority is initially advocating for it. The key elements involved in this process are consensus building and positive feedback.
Abstract. In Biology/Psychology, the capability of natural organisms to learn from the observation of conspecifics is referred to as social learning. Roboticists have recently developed an interest on social learning, since it might... more
Abstract. In Biology/Psychology, the capability of natural organisms to learn from the observation of conspecifics is referred to as social learning. Roboticists have recently developed an interest on social learning, since it might represent an e ective strategy to enhance the adaptivity of a team of autonomous robots.

And 248 more

Building structures is a remarkable collective process but its automation remains an open challenge. Robot swarms provide a promising solution to this challenge. However, collective construction involves a number of difficulties regarding... more
Building structures is a remarkable collective process but its automation remains an open challenge. Robot swarms provide a promising solution to this challenge. However, collective construction involves a number of difficulties regarding efficient robots allocation to the different activities, particularly if the goal is to reach an optimal construction rate. In this paper, we study an abstract construction scenario, where a swarm of robots is engaged in a collective perception process to estimate the density of building blocks around a construction site. The goal of this perception process is to maintain a minimum density of blocks available to the robots for construction. To maintain this density, the allocation of robots to the foraging task needs to be adjusted such that enough blocks are retrieved. Our results show a robust collective perception that enables the swarm to maintain a minimum block density under different rates of construction and foraging. Our approach leads the system to stabilize around a state in which the robots allocation allows the swarm to maintain a tile density that is close to or above the target minimum.