## A Space- and Time-Continuous Model of Self-organizing Robot Swarms

### BibTeX

@MISC{_aspace-,

author = {},

title = {A Space- and Time-Continuous Model of Self-organizing Robot Swarms},

year = {}

}

### OpenURL

### Abstract

for Design Support Abstract — Designing and implementing artificial self-organizing systems is a challenging task since they typically behave nonintuitive and no theoretical foundations exist. Predicting a system of many components with a huge amount of interactions is beyond human skills. The currently common use of simulations for design support is not satisfying, as it is time-consuming and the results are most likely suboptimal. In this work, we present the derivation of an analytical, time-, and space-continuous model for a swarm of autonomous robots based on the Fokker-Planck equation. While the motion model is in most parts physically motivated, the communication model is based on a heuristic approach. A showcase application to a recently proposed scenario of collective perception in a huge swarm of robots with very limited abilities is given and the simulation results are compared to the model. Despite the high level of abstraction, the prediction discrepancies are small and the parameters can be mapped one-to-one from the model to the control algorithm. Finally, we give an outlook on the capabilities of the proposed model, discuss its limitations and suggest an improvement that could reduce the number of empirically determined parameters. I.

### Citations

1138 | A survey on sensor networks
- Akyildiz, Su, et al.
- 2002
(Show Context)
Citation Context ...the number of empirically determined parameters. I. INTRODUCTION The ongoing advances in electronics and robotics have made it possible to build small sensor nodes as well as small robots at low cost =-=[2]-=-, [4], [17]. By this evolution it became feasible to implement large sensor networks and groups of 50 or more robots, so-called artificial robot swarms [1], [5]. While the hardware is available, the d... |

547 |
Stochastic Processes
- Doob
- 1953
(Show Context)
Citation Context ...al context, here, it is clearly defined by the underlying control algorithm that the value of Ri before the jump has to be used. We omit the derivation of the Fokker-Planck equation here and refer to =-=[7]-=-, [19] for further details. The related equation for the robot density ρ(r,t) is: � ∂ ρ(r,t) = − ∇ m(r,t) ∂t ∇P(r,t) �∇P(r,t)� vρ(r,t) � + 1 2 ∇2 �� � � 2vρ(r,t) 1 − m(r,t) . (7) To simplify the forma... |

438 |
The Fokker-Planck Equation
- Risken
- 1989
(Show Context)
Citation Context ...(r,t)� � v + (1 − m(Ri,t))ξ(t)v. Ri (6) v is the constant nominal velocity. ξ(t) is a normalized noise term �ξ(t)� = 1, with mean 〈ξ(t)〉 = 0, and uncorrelated in time 〈ξ(t)ξ(t ′ )〉 = δ(t − t ′ ) (see =-=[13]-=- for details). This equation is interpreted that the robots move with velocity v either towards the maximum of P with a probability of m(Ri,t) or otherwise to a random direction. From this microscopic... |

239 |
Stochastic processes in physics and chemistry
- Kampen
- 1998
(Show Context)
Citation Context ...ntext, here, it is clearly defined by the underlying control algorithm that the value of Ri before the jump has to be used. We omit the derivation of the Fokker-Planck equation here and refer to [7], =-=[19]-=- for further details. The related equation for the robot density ρ(r,t) is: � ∂ ρ(r,t) = − ∇ m(r,t) ∂t ∇P(r,t) �∇P(r,t)� vρ(r,t) � + 1 2 ∇2 �� � � 2vρ(r,t) 1 − m(r,t) . (7) To simplify the formalism w... |

174 |
An Introduction
- Synergetics
- 1983
(Show Context)
Citation Context ...applications. See for example the framework of so-called Brownian agents by Schweitzer [16] or the science of synergetics founded by Haken that describes self-organization of patterns (chapter six in =-=[8]-=-). A. PDE of the Robot Density The behavior of a single robot basically consists of two distinct parts: navigation and communication. First we derive the model describing the navigation part. Since th... |

83 | Modeling of swarm robotic systems: A case study in collaborative distributed manipulation
- Martinoli, Easton, et al.
(Show Context)
Citation Context ...thods available that support the engineer in designing and optimizing the control algorithm. First steps to establish such a theory of self-organizing behavior in robotics have been proposed recently =-=[12]-=-, [9], [11]. The fundamental substance of these Heiko Hamann and Heinz Wörn Institute for Process Control and Robotics Universität Karlsruhe 76128 Karlsruhe, Germany Email: {hamann, woern}@ira.uka.de ... |

46 | Analysis of dynamic task allocation in multi-robot systems
- Lerman, Jones, et al.
- 2006
(Show Context)
Citation Context ...able that support the engineer in designing and optimizing the control algorithm. First steps to establish such a theory of self-organizing behavior in robotics have been proposed recently [12], [9], =-=[11]-=-. The fundamental substance of these Heiko Hamann and Heinz Wörn Institute for Process Control and Robotics Universität Karlsruhe 76128 Karlsruhe, Germany Email: {hamann, woern}@ira.uka.de approaches ... |

31 |
Poincaré and the Three Body Problem
- Barrow-Green
- 1996
(Show Context)
Citation Context ...ts unpredictability [6]. Alternatively to the new field of emergence consider the three body problem, as it is already hard to to find the exact motions of three defined bodies by Newtonian mechanics =-=[3]-=-. Naturally predicting emergent behavior is a tough task but whether it is feasible or not is unanswered presently. A similar question can be asked for the task of approximately predicting emergence. ... |

24 | Emergent phenomena and complexity
- Darley
- 1994
(Show Context)
Citation Context ...tely predicts the behavior of the swarm. At least the exact prediction of emergence is considered a contradiction in terms by some researchers as they define emergent behavior by its unpredictability =-=[6]-=-. Alternatively to the new field of emergence consider the three body problem, as it is already hard to to find the exact motions of three defined bodies by Newtonian mechanics [3]. Naturally predicti... |

16 | Critical density thresholds in distributed wireless networks
- Krishnamachari, Wicker, et al.
- 2002
(Show Context)
Citation Context ...age densities as the robots abandon certain areas and aggregate in this scenario. The change from total connectivity to several clusters with decreasing density is not a smooth but a phase transition =-=[10]-=-. Therefore, the proposed model is only inaccurate concerning connectivity for the small interval below the critical density. Additionally, just the mean of the potential field is modeled. This mean i... |

16 |
H.: The I-SWARM project: Intelligent small world autonomous robots for micro-manipulation. Swarm Robotics 3342
- Seyfried, Szymanski, et al.
- 2005
(Show Context)
Citation Context ... of empirically determined parameters. I. INTRODUCTION The ongoing advances in electronics and robotics have made it possible to build small sensor nodes as well as small robots at low cost [2], [4], =-=[17]-=-. By this evolution it became feasible to implement large sensor networks and groups of 50 or more robots, so-called artificial robot swarms [1], [5]. While the hardware is available, the development ... |

9 |
Trophallaxis among swarm-robots: A biologically inspired strategy for swarm robotics
- Schmickl, Crailsheim
- 2006
(Show Context)
Citation Context ...e the bigger one, and to aggregate there, for example, to do some work. See Fig. 1 for a screen-shot of a typical situation in the simulation (using the simulation platform LaRoSim by Schmickl et al. =-=[14]-=-). The strategy that solves this problem is inspired by trophallaxis, a frequently observed behavior in social insects, as reported [15]. It is assumed that the robots can communicate by infra-red dev... |

9 | K.: Collective perception in a robot swarm
- Schmickl, Möslinger, et al.
(Show Context)
Citation Context .... II. A SCENARIO OF COLLECTIVE PERCEPTION AND THE TROPHALLAXIS STRATEGY As an example we choose the scenario of collective perception in a swarm of robots as described and analyzed by Schmickl et al. =-=[15]-=-. In this scenario a big group of up to 1000 robots moves within a rectangular area bounded by walls. Initially, the robots are uniformly distributed. Their task is to discriminate collectively betwee... |

8 | Brownian Agents and Active Particles. On the Emergence of - Schweitzer - 2003 |

5 |
Swarm embodiment - a new way for deriving emergent behavior in artificial swarms”, Autonome Mobile Systeme
- Kornienko, Kornienko, et al.
- 2005
(Show Context)
Citation Context ...available that support the engineer in designing and optimizing the control algorithm. First steps to establish such a theory of self-organizing behavior in robotics have been proposed recently [12], =-=[9]-=-, [11]. The fundamental substance of these Heiko Hamann and Heinz Wörn Institute for Process Control and Robotics Universität Karlsruhe 76128 Karlsruhe, Germany Email: {hamann, woern}@ira.uka.de appro... |

4 |
The autonomous microbot ‘alice’: a platform for scientific and commercial applications
- Caprari, Balmer, et al.
- 1998
(Show Context)
Citation Context ...umber of empirically determined parameters. I. INTRODUCTION The ongoing advances in electronics and robotics have made it possible to build small sensor nodes as well as small robots at low cost [2], =-=[4]-=-, [17]. By this evolution it became feasible to implement large sensor networks and groups of 50 or more robots, so-called artificial robot swarms [1], [5]. While the hardware is available, the develo... |

4 | SelfOrganized Embedded Sensor/Actuator Networks for “Smart” Turbines
- Correll, Cianci, et al.
- 2006
(Show Context)
Citation Context ...des as well as small robots at low cost [2], [4], [17]. By this evolution it became feasible to implement large sensor networks and groups of 50 or more robots, so-called artificial robot swarms [1], =-=[5]-=-. While the hardware is available, the development of the control software is still a challenge. To minimize the complexity of the entire system and due to the limited capabilities of an individual ro... |

3 |
robot - project website
- Jasmine
- 2007
(Show Context)
Citation Context ...or nodes as well as small robots at low cost [2], [4], [17]. By this evolution it became feasible to implement large sensor networks and groups of 50 or more robots, so-called artificial robot swarms =-=[1]-=-, [5]. While the hardware is available, the development of the control software is still a challenge. To minimize the complexity of the entire system and due to the limited capabilities of an individu... |

1 |
Distributed shortest-path finding by a micro-robot swarm
- Szymanski, Breitling, et al.
- 2006
(Show Context)
Citation Context ...gation threshold δaggr 0 performed in a negotiation between neighboring robots via the infra-red communication devices that additionally allow the robots to determine the direction of their neighbors =-=[18]-=-. C is the consumption amount C(Ri,t + ∆t) = P(Ri,t)rconsmpt, (4) with consumption rate rconsmpt. It is intuitive and substantiated by simulation runs that in the average the maximum of the potential ... |