Swarm robotics: a review from the swarm engineering perspective <s> Macroscopic models <s> Abstract This paper presents an experiment of clustering implemented at three different levels: in a hardware implementation, in a sensor-based simulation and in a probabilistic model. The experiment consists of small reactive autonomous robots gathering and clustering randomly distributed objects. It is shown that, while the behaviour of the real robots can be faithfully reproduced in a sensor-based simulation, the evolution of the cluster sizes is perfectly described, both qualitatively and quantitatively, by a simple probabilistic model. Rather than simulating robots moving within an environment, the probabilistic model represents the clustering activity as a sequence of probabilistic events during which cluster sizes can be modified depending on simple geometrical considerations. <s> BIB001 </s> Swarm robotics: a review from the swarm engineering perspective <s> Macroscopic models <s> In this article, we present a macroscopic analytical model of collaboration in a group of reactive robots. The model consists of a series of coupled differential equations that describe the dynamics of group behavior. After presenting the general model, we analyze in detail a case study of collaboration, the stick-pulling experiment, studied experimentally and in simulation by Ijspeert et al. (Autonomous Robots, 11, 149-171). The robots' task is to pull sticks out of their holes, and it can be successfully achieved only through the collaboration of two robots. There is no explicit communication or coordination between the robots. Unlike microscopic simulations (sensor-based or using a probabilistic numerical model), in which computational time scales with the robot group size, the macroscopic model is computationally efficient, because its solutions are independent of robot group size. Analysis reproduces several qualitative conclusions of Ijspeert et al.: namely, the different dynamical regimes for different values of the ratio of robots to sticks, the existence of optimal control parameters that maximize system performance as a function of group size, and the transition from superlinear to sublinear performance as the number of robots is increased. <s> BIB002 </s> Swarm robotics: a review from the swarm engineering perspective <s> Macroscopic models <s> In this article we consider a discrete time one-dimensional asynchronous swarm. First, we describe the mathematical model for motions of the swarm members. Then, we analyze the stability properties of that model. The stability concept that we consider, which matches exactly with stability of equilibria in control theory, characterizes stability of a particular position (relative arrangement) of the swarm members, that we call the comfortable position (with comfortable intermember distance). Our stability analysis employs some results on contractive mappings from the parallel and distributed computation literature. <s> BIB003 </s> Swarm robotics: a review from the swarm engineering perspective <s> Macroscopic models <s> Coordinated dynamical swarm behavior occurs when certain types of animals forage for food or try to avoid predators. Analogous behaviors can occur in engineering systems (e.g. in groups of autonomous mobile robots or UAVs). In this paper, we study a model of an M-dimensional (M /spl ges/ 2) asynchronous swarm with a communication topology, where each member only communicate with fixed neighbors, to provide conditions under which collision-free convergence can be achieved with finite-size swarm members that have proximity sensors and neighbor position sensors that only provide delayed position information. Moreover, we give conditions under which an M-dimensional asynchronous mobile swarm following an "edge-leader" can maintain cohesion during movements even in the presence of sensing delays and asynchronism, and analyze the swarm movement flexibility. Such stability analysis is of fundamental importance if one wants to understand the coordination mechanisms for groups of autonomous vehicles or robots where inter-member communication channels are less than perfect and collisions must be avoided. <s> BIB004 </s> Swarm robotics: a review from the swarm engineering perspective <s> Macroscopic models <s> In this paper, we present a time-discrete, incremental methodology for modeling, at the microscopic and macroscopic level, the dynamics of distributed manipulation experiments using swarms of autonomous robots endowed with reactive controllers. The methodology is well-suited for nonspatial metrics since it does not take into account robots’ trajectories or the spatial distribution of objects in the environment. The strength of the methodology lies in the fact that it has been generated by considering incremental abstraction steps, from real robots to macroscopic models, each with well-defined mappings between successive implementation levels. Precise heuristic criteria based on geometrical considerations and systematic tests with one or two real robots prevent the introduction of free parameters in the calibration procedure of models. As a consequence, we are able to generate highly abstracted macroscopic models that can capture the dynamics of a swarm of robots at the behavioral level while still being closely anchored to the characteristics of the physical set-up. Although this methodology has been and can be applied to other experiments in distributed manipulation (e.g., object aggregation and segregation, foraging), in this paper we focus on a strictly collaborative case study concerned with pulling sticks out of the ground, an action that requires the collaboration of two robots to be successful. Experiments were carried out with teams consisting of two to 600 individuals at different levels of implementation (real robots, embodied simulations, microscopic and macroscopic models). Results show that models can deliver both qualitatively and quantitatively correct predictions in time lapses that are at least four orders of magnitude smaller than those required by embodied simulations and that they represent a useful tool for generalizing the dynamics of these highly stochastic, asynchronous, nonlinear systems, often outperforming intuitive reasoning. Finally, in addition to discussing subtle numerical effects, small prediction discrepancies, and difficulties in generating the mapping between different abstractions levels, we conclude the paper by reviewing the intrinsic limitations of the current modeling methodology and by proposing a few suggestions for future work. <s> BIB005 </s> Swarm robotics: a review from the swarm engineering perspective <s> Macroscopic models <s> Bacteria, bees, and birds often work together in groups to find food. A group of robots can be designed to coordinate their activities to search for and collect objects. Networked cooperative uninhabited autonomous vehicles are being developed for commercial and military applications. Suppose that we refer to all such groups of entities as "social foraging swarms". In order for such multiagent systems to succeed it is often critical that they can both maintain cohesive behaviors and appropriately respond to environmental stimuli (e.g., by optimizing the acquisition of nutrients in foraging for food). In this paper, we characterize swarm cohesiveness as a stability property and use a Lyapunov approach to develop conditions under which local agent actions will lead to cohesive foraging even in the presence of "noise" characterized by uncertainty on sensing other agent's position and velocity, and in sensing nutrients that each agent is foraging for. The results quantify earlier claims that social foraging is in a certain sense superior to individual foraging when noise is present, and provide clear connections between local agent-agent interactions and emergent group behavior. Moreover, the simulations show that very complicated but orderly group behaviors, reminiscent of those seen in biology, emerge in the presence of noise. <s> BIB006 </s> Swarm robotics: a review from the swarm engineering perspective <s> Macroscopic models <s> In this article we specify an M-member "individual-based" continuous time swarm model with individuals that move in an n-dimensional space according to an attractant/repellent or a nutrient profile. The motion of each individual is determined by three factors: i) attraction to the other individuals on long distances; ii) repulsion from the other individuals on short distances; and iii) attraction to the more favorable regions (or repulsion from the unfavorable regions) of the attractant/repellent profile. The emergent behavior of the swarm motion is the result of a balance between inter-individual interactions and the simultaneous interactions of the swarm members with their environment. We study the stability properties of the collective behavior of the swarm for different profiles and provide conditions for collective convergence to more favorable regions of the profile. <s> BIB007 </s> Swarm robotics: a review from the swarm engineering perspective <s> Macroscopic models <s> It is a characteristic of swarm robotics that specifying overall emergent swarm behaviours in terms of the low-level behaviours of individual robots is very difficult. Yet if swarm robotics is to make the transition from the laboratory to real-world engineering realisation we need such specifications. This paper explores the use of temporal logic to formally specify, and possibly also prove, the emergent behaviours of a robotic swarm. The paper makes use of a simplified wireless connected swarm as a case study with which to illustrate the approach. Such a formal approach could be an important step toward a disciplined design methodology for swarm robotics. <s> BIB008 </s> Swarm robotics: a review from the swarm engineering perspective <s> Macroscopic models <s> The biologically-inspired swarm paradigm is being used to design self-organizing systems of locally interacting artificial agents. A major difficulty in designing swarms with desired characteristics is understanding the causal relation between individual agent and collective behaviors. Mathematical analysis of swarm dynamics can address this difficulty to gain insight into system design. This paper proposes a framework for mathematical modeling of swarms of microscopic robots that may one day be useful in medical applications. While such devices do not yet exist, the modeling approach can be helpful in identifying various design trade-offs for the robots and be a useful guide for their eventual fabrication. Specifically, we examine microscopic robots that reside in a fluid, for example, a bloodstream, and are able to detect and respond to different chemicals. We present the general mathematical model of a scenario in which robots locate a chemical source. We solve the scenario in one-dimension and show how results can be used to evaluate certain design decisions. <s> BIB009 </s> Swarm robotics: a review from the swarm engineering perspective <s> Macroscopic models <s> It is a characteristic of swarm robotics that modelling the overall swarm behaviour in terms of the low-level behaviours of individual robots is very difficult. Yet if swarm robotics is to make the transition from the laboratory to real-world engineering realisation such models would be critical for both overall validation of algorithm correctness and detailed parameter optimisation. We seek models with predictive power: models that allow us to determine the effect of modifying parameters in individual robots on the overall swarm behaviour. This paper presents results from a study to apply the probabilistic modelling approach to a class of wireless connected swarms operating in unbounded environments. The paper proposes a probabilistic finite state machine (PFSM) that describes the network connectivity and overall macroscopic behaviour of the swarm, then develops a novel robot-centric approach to the estimation of the state transition probabilities within the PFSM. Using measured data from simulation the paper then carefully validates the PFSM model step by step, allowing us to assess the accuracy and hence the utility of the model. <s> BIB010 </s> Swarm robotics: a review from the swarm engineering perspective <s> Macroscopic models <s> Designing and analyzing self-organizing systems such as robotic swarms is a challenging task even though we have complete knowledge about the robot’s interior. It is difficult to determine the individual robot’s behavior based on the swarm behavior and vice versa due to the high number of agent–agent interactions. A step towards a solution of this problem is the development of appropriate models which accurately predict the swarm behavior based on a specified control algorithm. Such models would reduce the necessary number of time-consuming simulations and experiments during the design process of an algorithm. In this paper we propose a model with focus on an explicit representation of space because the effectiveness of many swarm robotic scenarios depends on spatial inhomogeneity. We use methods of statistical physics to address spatiality. Starting from a description of a single robot we derive an abstract model of swarm motion. The model is then extended to a generic model framework of communicating robots. In two examples we validate models against simulation results. Our experience shows that qualitative correctness is easily achieved, while quantitative correctness is disproportionately more difficult but still possible. <s> BIB011 </s> Swarm robotics: a review from the swarm engineering perspective <s> Macroscopic models <s> We present a biologically inspired approach to the dynamic assignment and reassignment of a homogeneous swarm of robots to multiple locations, which is relevant to applications like search and rescue, environmental monitoring, and task allocation. Our work is inspired by experimental studies of ant house hunting and empirical models that predict the behavior of the colony that is faced with a choice between multiple candidate nests. We design quorum based stochastic control policies that enable the team of agents to distribute themselves among multiple candidate sites in a specified ratio, and compare our results to the linear stochastic policies described in (Halasz et al., in Proceedings of the International Conference on Intelligent Robots and Systems (IROS’07), pp. 2320–2325, 2007). We show how our quorum model consistently performs better than the linear models while minimizing computational requirements and now it can be implemented without the use of inter-agent wireless communication. <s> BIB012 </s> Swarm robotics: a review from the swarm engineering perspective <s> Macroscopic models <s> We implement a self-organized flocking behavior in a group of mobile robots and analyze its transition from an aligned state to an unaligned state. We briefly describe the robot and the simulator platform together with the observed flocking dynamics. By experimenting with robotic and numerical systems, we find that an aligned-to-unaligned phase transition can be observed in both physical and simulated robots as the noise level is increased, and that this transition depends on the characteristics of the heading sensors. We extend the Vectorial Network Model to approximate the robot dynamics and show that it displays an equivalent phase transition. By computing analytically the critical noise value and numerically the steady state solutions of this model, we show that the model matches well the results obtained using detailed physics-based simulations. <s> BIB013 </s> Swarm robotics: a review from the swarm engineering perspective <s> Macroscopic models <s> We present a scalable approach to dynamically allocating a swarm of homogeneous robots to multiple tasks, which are to be performed in parallel, following a desired distribution. We employ a decentralized strategy that requires no communication among robots. It is based on the development of a continuous abstraction of the swarm obtained by modeling population fractions and defining the task allocation problem as the selection of rates of robot ingress and egress to and from each task. These rates are used to determine probabilities that define stochastic control policies for individual robots, which, in turn, produce the desired collective behavior. We address the problem of computing rates to achieve fast redistribution of the swarm subject to constraint(s) on switching between tasks at equilibrium. We present several formulations of this optimization problem that vary in the precedence constraints between tasks and in their dependence on the initial robot distribution. We use each formulation to optimize the rates for a scenario with four tasks and compare the resulting control policies using a simulation in which 250 robots redistribute themselves among four buildings to survey the perimeters. <s> BIB014 </s> Swarm robotics: a review from the swarm engineering perspective <s> Macroscopic models <s> Purpose – The purpose of this paper is to examine and illustrate the development of a methodology for generating swarms using lossless flocking.Design/methodology/approach – A general methodology for swarm design is described. Examples of this approach in the literature are examined. A general requirement for lossless flocking is developed. The requirement is used in developing two swarm behaviors.Findings – It is possible to apply the approach to the lossless flocking and to use the swarm condition to develop two swarm behaviors which satisfy this condition in many situations.Research limitations/implications – This paper illustrates the general swarm engineering method and demonstrates how it can be properly applied.Originality/value – The swarm engineering method is used to develop the “quark” model, a new physicomimetic model. <s> BIB015 </s> Swarm robotics: a review from the swarm engineering perspective <s> Macroscopic models <s> Understanding the effect of individual parameters on the collective performance of swarm robotic systems in order to design and optimize individual robot behaviors is a significant challenge. In this paper we present a macroscopic probabilistic model of adaptive collective foraging in a swarm of robots, where each robot in the swarm is capable of adjusting its time threshold parameters following the rules described by Liu et al. 2007. The swarm adapts the ratio of foragers to resters (division of labor) in order to maximize the net swarm energy for a given food density. A probabilistic finite state machine (PFSM) and a number of difference equations are developed to describe collective foraging at a macroscopic level. To model adaptation we introduce the new concepts of the sub-PFSM and private/public time thresholds. The model has been validated extensively with simulation trials, and results show that the model achieves very good accuracy in predicting the group performance of the swarm. Finally, a real-coded genetic algorithm is used to explore the parameter spaces and optimize the parameters of the adaptation algorithm. Although this paper presents a macroscopic probabilistic model for adaptive foraging, we argue that the approach could be applied to any adaptive swarm system in which the heterogeneity of the system is coupled with its time parameters. <s> BIB016 </s> Swarm robotics: a review from the swarm engineering perspective <s> Macroscopic models <s> 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. We show how a spatially targeted one-to-one communication link can be established using a simple LED and camera based communication modality. We provide a probabilistic model of our approach to derive an upper bound on the average time required for establishing communication. In simulation, we show that our approach scales well. Furthermore, we show how our approach can be extended to establish a spatially targeted one-to-many communication link between an aerial robot and a specific number of co-located wheeled robots. The heterogeneous swarm robotic hardware is currently under development. We therefore demonstrate the proposed approach on an existing multirobot system consisting of only wheeled robots by letting one of the wheeled robots assume the role of an aerial robot. <s> BIB017 </s> Swarm robotics: a review from the swarm engineering perspective <s> Macroscopic models <s> We present a scalable approach to optimizing robot control policies for a target collective behavior in a spatially inhomogeneous robotic swarm. The approach can incorporate robot feedback to maintain system performance in an unknown environmental flow field. We consider systems in which the robots follow both deterministic and random motion and transition stochastically between tasks. Our methodology is based on an abstraction of the swarm to a macroscopic continuous model, whose dimensionality is independent of the population size, that describes the expected time evolution of swarm subpopulations over a discretization of the environment. We incorporate this model into a stochastic optimization method and map the optimized model parameters onto the robot motion and task transition control policies to achieve a desired global objective. We illustrate our methodology with a scenario in which the behaviors of a swarm of robotic bees are optimized for both uniform and nonuniform pollination of a blueberry field, including in the presence of an unknown wind. <s> BIB018 </s> Swarm robotics: a review from the swarm engineering perspective <s> Macroscopic models <s> This paper examines the dynamic interplay between decentralized controllers and mesh networking protocols for controlling groups of robots. A proportional controller is used to maintain robots in a formation based on estimates of the robots' states observed through the network. The state information is propagated through the network using a flooding algorithm, which introduces topology-dependent time delays. The coupled interaction of information flow over the network with the dynamics of the robots is modeled as a linear dynamical system. With this model it is shown that systems made up of robots with stable first order dynamics are stable for all network update times, positive feedback gains, and connected communication graphs. With higher order robot dynamics it is found that stability is a complex and counter intuitive function of feedback gain and network update time. A performance metric is proposed for analyzing the convergence rate of the multi-robot system. Experiments with flying quadrotor robots verify the predictions of the model and the performance metric. <s> BIB019 </s> Swarm robotics: a review from the swarm engineering perspective <s> Macroscopic models <s> A robot swarm is a collection of simple robots designed to work together to carry out some task. Such swarms rely on: the simplicity of the individual robots; the fault tolerance inherent in having a large population of often identical robots; and the self-organised behaviour of the swarm as a whole. Although robot swarms are being deployed in increasingly sophisticated areas, designing individual control algorithms that can guarantee the required global behaviour is difficult. In this paper we apply and assess the use of formal verification techniques, in particular that of model checking, for analysing the emergent behaviours of robotic swarms. These techniques, based on the automated analysis of systems using temporal logics, allow us to analyse all possible behaviours and so identify potential problems with the robot swarm conforming to some required global behaviour. To show this approach we target a particular swarm control algorithm, and show how automated temporal analysis can help to refine and analyse such an algorithm. <s> BIB020 </s> Swarm robotics: a review from the swarm engineering perspective <s> Macroscopic models <s> We compare our previously developed deterministic [7] and stochastic [3], [4] strategies for allocating tasks in robotic swarms1 consisting of very large populations of highly resource-constrained robots. We study our two task allocation approaches in a simulated scenario in which a collective of insect-inspired micro-aerial vehicles (MAVs) must produce a specified spatial distribution of pollination activity over a crop field. We investigate the approaches' requirements, advantages, and disadvantages under realistic conditions of error in robot localization, navigation, and sensing in simulation. Our results show that the deterministic approach, which requires region-based robot navigation, yields higher task progress in all cases. For robots without such navigation capabilities, the stochastic approach is a feasible alternative, and its resulting task progress is less sensitive to error in localization, error in navigation, and a combination of high error in localization, navigation, and sensing. <s> BIB021 </s> Swarm robotics: a review from the swarm engineering perspective <s> Macroscopic models <s> An alternative to deploying a single robot of high complexity can be to utilise robot swarms comprising large numbers of identical, and much simpler, robots. Such swarms have been shown to be adaptable, fault-tolerant and widely applicable. However, designing individual robot algorithms to ensure effective and correct overall swarm behaviour is actually very difficult. While mechanisms for assessing the effectiveness of any swarm algorithm before deployment are essential, such mechanisms have traditionally involved either computational simulations of swarm behaviour, or experiments with robot swarms themselves. However, such simulations or experiments cannot, by their nature, analyse all possible swarm behaviours. In this paper, we will develop and apply the use of automated probabilistic formal verification techniques to robot swarms, involving an exhaustive mathematical analysis, in order to assess whether swarms will indeed behave as required. In particular we consider a foraging robot scenario to which we apply probabilistic model checking. <s> BIB022 </s> Swarm robotics: a review from the swarm engineering perspective <s> Macroscopic models <s> In this paper, we propose a novel top-down design method for the development of collective behaviors of swarm robotics systems called property-driven design. Swarm robotics systems are usually designed and developed using a code-and-fix approach, that is, the developer devises, tests and modifies the individual robot behaviors until a desired collective behavior is obtained. The code-and-fix approach can be very time consuming and relies completely on the ingenuity and expertise of the designer. The idea of property-driven design is that a swarm robotics system can be described by specifying formally a set of desired properties. In an iterative process similar to test-driven development, the developer produces a model of the system that satisfies the desired properties. Subsequently, the system is implemented in simulation and using real robots. Property-driven design helps to minimize the risk of developing a system that does not satisfy the required properties, and to promote the reuse of hardware independent models. In this paper, we start by giving a general description of the method. We then present a possible way to apply it by using Discrete Time Markov Chains (DTMC) and Probabilistic Computation Tree Logic* (PCTL*). Finally, we conclude by presenting the application of the proposed method to the design and development of a swarm robotics system performing aggregation. <s> BIB023 </s> Swarm robotics: a review from the swarm engineering perspective <s> Macroscopic models <s> The search for generally applicable methods in swarm intelligence aims to gain new insights about natural swarms and to develop design methodologies for artificial swarms. The ideal would be a 'swarm calculus' that allows to calculate key features such as expected swarm performance and robustness on the basis of a few parameters. A path towards this ideal is to find methods and models that have maximal generality. We report two models that might be examples of exceptional generality. First, we present an abstract model that describes the performance of a swarm depending on the swarm density based on the dichotomy between cooperation and interference. Second, we give an abstract model for decision making that is inspired by urn models. A parameter, that controls the feedback based on the current consensus, allows to understand the effects of an increasing probability for positive feedback over time in a decision making system. <s> BIB024
Macroscopic models consider swarm robotics systems as a whole. The individual elements of the systems are not taken into account in favor of a description of the system at a higher level. In this section we provide a broad overview of the main contributions in this area. We classify works in macroscopic modeling into three categories. In the first category, we consider works resorting to rate or differential equations. In the second category, we consider works where classical control and stability theory are used to prove properties of the swarm. In the third category, we consider other approaches. Rate and differential equations One of the first works that uses rate equations for modeling swarm robotics systems is by BIB001 . In this and in follow-up works, the term rate equations was used to denote such models. Rate equations describe the time evolution of the proportion of robots in a particular state over the total number of robots. Rate equations can be used to derive a macroscopic model of a collective behavior, starting from an individual-level PFSM. The procedure is the following: (i) First, a set of variables is defined. Usually, one variable is defined for each state of the individual-level PFSM. These variables are used to track the proportion of the robots in the corresponding states. (ii) Second, for each variable, an equation is defined  ). This equation is called rate equation because it is used to describe the time evolution of that variable, that is, the time evolution of the proportion of the robots in the corresponding state. The rate equation contains a set of parameters, one for each input and output transition of the corresponding state. Numerically, these parameters can be derived either from the description of the system or empirically. The rate equations method was used to model many swarm robotics systems. In their seminal work, BIB001 used rate equations to model a clustering task where robots gather objects. BIB002 and BIB005 used rate equations to model the stick pulling experiment, a task where two robots need to cooperate in order to pull sticks out of their holes.  modeled the foraging task under the effect of interference. In this case, the authors were correctly able to model the individual performance in foraging to be a decreasing function of group size.  used rate equations to model a chain formation behavior and an aggregation behavior, implemented using probabilistic finite state machines.  modeled the collective behavior of robots performing multi-foraging (i.e., foraging with more than one food source). BIB010 used rate equations to model a swarm of robots whose goal is to stay together while avoiding collisions. BIB016 used the rate equations to model another foraging task involving the collection of energy units. Finally,  used rate equations to model an aggregation collective behavior. In this work, a flying robot can actively control the number of robots aggregating beneath it. The model included probabilities to join and to leave an aggregate, which were then directly used in the behavior design. The main advantage of the rate equation approach is that it is a systematic method to translate microscopic models into macroscopic models. Its main limit is that, in general, it is difficult to model space and time: robots' positions in space are not explicitly modeled and discrete time is usually assumed; furthermore, the two mentioned limitations imply that each robot can change its position to any other location in the environment at each timestep. BIB009 extended the rate equation model to include spatiality in a task where robots search for a chemical substance. However, the authors did not perform any experiments, simulated or with real robots, to validate their approach. A recent advancement in macroscopic modeling based on differential equations is due to BIB011 . Their models include noise, stochasticity, and spatiality. The basic building block of this modeling method are the Langevin equation and the FokkerPlanck equation, both borrowed from the statistical physics literature. The Langevin equation is a family of stochastic differential equations that describes the motion of a particle in a fluid. The Langevin equation is a mesoscopic model (intermediate level between micro and macro). In fact, the motion of the particle is modeled using two components: a deterministic component, which represents the microscopic laws of motion of that particle, and a stochastic component, which represents the interaction of the particle with the environment (in this case the ensemble of particles composing the fluid). In the case of a robot, the deterministic component of the Langevin equation models the deterministic motion of the robot influenced by its individual behavior, whereas the stochastic part models the interaction of the robot with the other robots (considered as a flow) and with the environment. The Fokker-Plank equation can be used to describe the dynamics of the entire swarm. It models the time-evolution of the probability density function that describes the state (for example the position or the velocity) of all particles, or robots, in the environment. The derivation of the Fokker-Planck equation starting from the Langevin equation is possible using tools of statistical mechanics plus some problem-dependent intuition. BIB011 applied this modeling method to analyze coordinated motion (which they call collective taxis), aggregation (which they call collective perception) and foraging. Recently, the authors modeled aggregation in presence of a temperature gradient in the environment, and provided a comparison with another model called Stock & Flow (Schmickl et al. 2009 ). The Fokker-Planck equation approach has the advantage that it can be used, in principle, to model any swarm robotics collective behavior. The two main disadvantages, however, are the following: the Fokker-Plank equation is difficult to be solved analytically and sometimes requires computationally demanding numerical algorithms; communication aspects, at present, are very difficult to model. A similar approach was adopted also by BIB014 , who used a set of advection-diffusion-reaction partial differential equations to derive the individual behaviors of a swarm performing task allocation. In BIB018 , this approach is applied to an area coverage task. BIB021 compared the results obtained in this work with those obtained from a simulation of the same behavior. The goal of the authors was to understand the effects of noise and errors on the collective behavior. Another interesting study on the use of the Fokker-Planck equation was done by  . In their work, the authors compared four different models of an area coverage behavior. They did it by measuring the area covered by the robots. Each model is characterized by being microscopic or macroscopic, and spatial or non-spatial. The predictions obtained from the models are compared with the results of both simulated and real-robot experiments. The authors showed that predictions of the spatial and non-spatial models differ for short time spans, for which the results of spatial models are more accurate, but are very similar for long time spans. Classical control and stability theory The second set of works uses classical control and stability theory to prove properties of the swarm. BIB004 and BIB003 modeled a swarm of agents in a one-dimensional space using discrete-time discreteevent dynamical systems. BIB006 and BIB007 used Lyapunov stability theory to prove that the behavior studied was able to let a swarm achieve coherent social foraging in presence of noise. Similarly, Passino (2003, 2004a) proved that, in specific conditions, a swarm of agents aggregates in one point of the environment. BIB019 modeled a swarm of communicating robots as a linear, discrete-time dynamical system. The authors then used their model to study how different communication topologies affect the Lyapunov stability of the system. Finally, BIB012 used delay differential equations to model task allocation (agents allocating and reallocating to different physical sites), proving the stability of the reached configuration. In the same work, the authors also proposed a method to compute the optimal transition matrix in order to obtain a swarm that reaches the desired configuration. All these modeling methods have the advantage to be based on strong mathematical formulations. However, the main problem with these methods is that they rely on many assumptions that, in swarm robotics, are often violated because of noise, asynchronicity, stochasticity, and absence of global information. Extensions of classical methods that tackle one of these elements individually do exist, but we are not aware of any work that tackles all these elements together. Other modeling approaches In the third and final category we consider works in modeling that resort to other mathematical frameworks. BIB008 and BIB020 used linear time temporal logic to define properties of individual robots and of the swarm. The authors defined and proved two properties of the system: safety and liveness. The safety property is verified when the robots do not exhibit undesirable behaviors. The liveness property is verified when the swarm dynamics actually do evolve over time. A similar approach was used by BIB022 and BIB023 . These authors analyzed swarm robotic systems by using probabilistic model checking, a technique that allows one to formally verify whether a given property is satisfied by a given model.  used Bio-PEPA to analyze consensus achievement in a robot swarm. Bio-PEPA is a high level modeling language. From a description of a system formulated in Bio-PEPA, one can automatically derive different models apt to perform stochastic simulation, fluid flow (ODE) analysis, and model checking. These models are guaranteed to be logically consistent. BIB015 expressed the properties of a swarm using a mathematical language and proved their validity. The author proposed a way to define properties of a swarm robotics problem which he calls "model independent", that is, they do not depend on the actual implementation of the agent/robot. He proposed model-independent properties for two collective behaviors: shape formation and flocking. Soysal andŞahin (2007) modeled aggregation using Markov chains and validated the prediction using simulation. The work of BIB013 represents one of the first modeling attempts to bridge studies of flocking within physics with studies of flocking within robotics. In their study, the authors modeled alignment in flocking. The model shows that there is a phase transition from aligned to non-aligned state corresponding to a critical value of noise. The results were validated using simulation. Correll (2008) used a population dynamics model to find the parameters used in two task-allocation behaviors. Using the model and an optimization algorithm, the authors could estimate the parameters that lead to the optimal distribution of robots. BIB017 modeled a problem in which a flying robot selects one mobile robot within a group to establish a communication channel with it. To model the system, the authors used the theory of branching processes  , which is used in social sciences to model phenomena such as population growth and virus spread. Finally, BIB024 developed two simple models for swarm robotics systems. In the first, the performance of a generic swarm robotics systems is explained using the interaction between cooperation and interference. In the second, the consensus achievement behavior is studied using a simple probabilistic model based on the urn problem.