# Levy Flights in the Stochastic Dynamics of Robot Swarm Gathering

### Yechiel J. Crispin

#### Abstract

We consider the problem of gathering a swarm of robots which is initially randomly dispersed over a domain in the plane. A stochastic method for the cooperative control of a swarm of mobile robots is presented. The network of mobile robots is modeled by a swarm performing a directed random walk. The swarm dynamics are governed by a system of stochastic difference equations. The motion is controlled by a robot leader, which transmits the coordinates of the gathering point to the swarm as the network cooperative control signal. We study the case where the control signal is corrupted by noise and find that the gathering process is robust to noise and efficient. The swarm dynamics display anomalous diffusion and Levy flights, where the robots move along straight lines over many time steps, followed by short random walks in the vicinity of the gathering point.

#### References

- Bonabeau, E., Dorigo, M. and Theraulaz, G.: Swarm Intelligence, From Natural to Artificial Systems. Santa Fe Institute in the Sciences of Complexity, Oxford University Press, New York, 1999.
- De Sousa, J.B. and Pereira, F.L.: On Coordinated Control Strategies for Networked Dynamic Control Systems: an Application to AUVs. Proceedings of the 15th International Symposium of Mathematical Theory of Networks and Systems, 2002.
- Dorigo, M., Di Caro, G. and Gambardella, L.M.: Ant Algorithms for Discrete Optimization. Artificial Life, Vol.5, pp.137-172, 1999.
- Eberhart, R.C. and Kennedy, J.: A New Optimizer Using Particle Swarm Theory. Proceedings of the 6th Symposium on Micro Machine and Human Science, IEEE Service Center, Piscataway, NJ, 1995.
- Kennedy, J. and Eberhart, R.C., Shi, Y.: Swarm Intelligence. Morgan Kaufman Publishers, Academic Press, San Francisco, 2001.
- Mandelbrot, B. B.: The Fractal Geometry of Nature. W.H. Freeman and Company, San Francisco, CA, Revised Edition, 1983.
- Parsopoulos, K.E. and Vrahatis, M.N.: Recent Approaches to Global Optimization Problems Through Particle Swarm Optimization. Natural Computing, Vol.1, pp.235-306, 2002.
- Silva, J., Speranzon, A., de Sousa, J.B. and Johansson, K.H.: Hierarchical Search Strategy for a Team of Autonomous Vehicles. IFAC 2004.
- Solomon, T.H., Weeks, E.R. and Swinney, H.L.: Observation of Anomalous Diffusion and Levy Flights in a Two-Dimensional Rotating Flow. Physical Review Letters, Vol.71, p.3975, 1993.
- Speranzon, A.: On Control Under Communication Constraints in Autonomous MultiRobot Systems. Licentiate Thesis, KTH Signals, Sensors and Systems, Kungliga Tekniska Hogskolan, Stockholm, Sweden, 2004.

#### Paper Citation

#### in Harvard Style

J. Crispin Y. (2006). **Levy Flights in the Stochastic Dynamics of Robot Swarm Gathering** . In *Proceedings of the 2nd International Workshop on Multi-Agent Robotic Systems - Volume 1: MARS, (ICINCO 2006)* ISBN 978-972-8865-66-5, pages 3-12. DOI: 10.5220/0001225200030012

#### in Bibtex Style

@conference{mars06,

author={Yechiel J. Crispin},

title={Levy Flights in the Stochastic Dynamics of Robot Swarm Gathering},

booktitle={Proceedings of the 2nd International Workshop on Multi-Agent Robotic Systems - Volume 1: MARS, (ICINCO 2006)},

year={2006},

pages={3-12},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0001225200030012},

isbn={978-972-8865-66-5},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 2nd International Workshop on Multi-Agent Robotic Systems - Volume 1: MARS, (ICINCO 2006)

TI - Levy Flights in the Stochastic Dynamics of Robot Swarm Gathering

SN - 978-972-8865-66-5

AU - J. Crispin Y.

PY - 2006

SP - 3

EP - 12

DO - 10.5220/0001225200030012