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A Correlated Random Walk Model to Rapidly Approximate Hitting Time Distributions in Multi-robot Systems

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Intelligent Autonomous Systems 17 (IAS 2022)

Part of the book series: Lecture Notes in Networks and Systems ((LNNS,volume 577))

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Abstract

Multi-robot systems are frequently used for tasks involving searching, so it is important to be able to estimate the searching time. Yet, simulation approaches and real-world experiments to determine searching time can be cumbersome and even impractical. In this work, we propose a correlated-random-walk based model to efficiently approximate hitting time distributions of multi-robot systems in large arenas. We verified the computational results by using ARGoS, a physics-based simulator. We found that the Gamma distribution can provide a good fit to the hitting time distributions of random walkers.

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References

  1. Alon, N., Avin, C., Kouckỳ, M., Kozma, G., Lotker, Z., Tuttle, M.R.: Many random walks are faster than one. Comb. Probab. Comput. 20(4), 481–502 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barlow, M.T., Peres, Y., Sousi, P.: Collisions of random walks. Ann. l’IHP Probab. Stat. 48(4), 922–946 (2012)

    MathSciNet  MATH  Google Scholar 

  3. Besenczi, R., Bátfai, N., Jeszenszky, P., Major, R., Monori, F., Ispány, M.: Large-scale simulation of traffic flow using Markov model. PLOS One 16(2) (2021)

    Google Scholar 

  4. Boley, D.: On fast computation of directed graph Laplacian pseudo-inverse. Linear Algebra Appl. 623, 128–148 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  5. Boley, D., Ranjan, G., Zhang, Z.L.: Commute times for a directed graph using an asymmetric Laplacian. Linear Algebra Appl. 435(2), 224–242 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brightwell, G., Winkler, P.: Maximum hitting time for random walks on graphs. Random Struct. Algorithms 1(3), 263–276 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  7. Codling, E.A., Bearon, R.N., Thorn, G.J.: Diffusion about the mean drift location in a biased random walk. Ecology 91(10), 3106–3113 (2010)

    Article  Google Scholar 

  8. Codling, E.A., Plank, M.J., Benhamou, S.: Random walk models in biology. J. R. Soc. Interface 5(25), 813–834 (2008)

    Article  Google Scholar 

  9. Dayar, T., Akar, N.: Computing moments of first passage times to a subset of states in Markov chains. SIAM J. Matrix Anal. 27(2), 396–412 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dimidov, C., Oriolo, G., Trianni, V.: Random walks in swarm robotics: an experiment with kilobots. In: International Conference on Swarm Intelligence, pp. 185–196. Springer, Berlin (2016)

    Google Scholar 

  11. Efremenko, K., Reingold, O.: How well do random walks parallelize? In: Dinur, I., Jansen, K., Naor, S., Rolim, J. (eds.) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, vol. 5687. Springer, Berlin (2009)

    Google Scholar 

  12. Gaudillière, A.: Collision probability for random trajectories in two dimensions. Stoch. Process. Appl. 119(3), 775–810 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Grinstead, C.M., Snell, J.L.: Introduction to Probability. American Mathematical Society (1997)

    Google Scholar 

  14. Hamann, H.: Modeling swarm systems and formal design methods. In: Swarm Robotics: A Formal Approach, pp. 95–127. Springer, Berlin (2018)

    Google Scholar 

  15. Hill, N., Häder, D.P.: A biased random walk model for the trajectories of swimming micro-organisms. J. Theor. Biol. 186(4), 503–526 (1997)

    Article  Google Scholar 

  16. Jeong, M., Harwell, J., Gini, M.: Analysis of exploration in swarm robotic systems. IAS-16 (2021)

    Google Scholar 

  17. Kemeny, J.G., Snell, J.L.: Finite Markov Chains. Springer, Berlin (1976)

    Google Scholar 

  18. Khalastchi, E., Kalech, M.: Fault detection and diagnosis in multi-robot systems: a survey. Sensors 19(18) (2019)

    Google Scholar 

  19. Lancaster, J.P., Gustafson, D.A.: Predicting the behavior of robotic swarms in search and tag tasks. Procedia Comput. Sci. 20, 77–82 (2013)

    Article  Google Scholar 

  20. Lau, H.W., Szeto, K.Y.: Asymptotic analysis of first passage time in complex networks. EPL (Europhys. Lett.) 90(4), 40005 (2010)

    Google Scholar 

  21. Lovász, L.: Random walks on graphs. Combinatorics 2(1–46), 4 (1993)

    Google Scholar 

  22. Masuda, N., Porter, M.A., Lambiotte, R.: Random walks and diffusion on networks. Phys. Rep. 716, 1–58 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Morgan, R.B.: GMRES with deflated restarting. SIAM J. Sci. Comput. 24(1), 20–37 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Nain, R., Sen, K.: Transition probability matrices for correlated random walks. J. Appl. Probab. 17(1), 253–258 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  25. Patel, R., Carron, A., Bullo, F.: The hitting time of multiple random walks. SIAM J. Matrix Anal. 37(3), 933–954 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. Pinciroli, C., Trianni, V., O’Grady, R., Pini, G., Brutschy, A., Brambilla, M., Mathews, N., Ferrante, E., Di Caro, G., Ducatelle, F., et al.: ARGoS: a modular, multi-engine simulator for heterogeneous swarm robotics. In: IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 5027–5034. IEEE (2011)

    Google Scholar 

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Acknowledgements

This research was partially supported by the University of Minnesota Robotics Institute.

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Authors and Affiliations

  1. Computer Science and Engineering, University of Minnesota, Minneapolis, MN, 55455, USA

    Yi Zhang, Daniel Boley, John Harwell & Maria Gini

Authors
  1. Yi Zhang
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  2. Daniel Boley
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  3. John Harwell
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  4. Maria Gini
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Corresponding author

Correspondence to Daniel Boley .

Editor information

Editors and Affiliations

  1. Faculty of Electrical Engineering, University of Zagreb, Zagreb, Croatia

    Ivan Petrovic

  2. Department of Information Engineering, University of Padua, Padua, Italy

    Emanuele Menegatti

  3. Faculty of Electrical Engineering, University of Zagreb, Zagreb, Croatia

    Ivan Marković

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Zhang, Y., Boley, D., Harwell, J., Gini, M. (2023). A Correlated Random Walk Model to Rapidly Approximate Hitting Time Distributions in Multi-robot Systems. In: Petrovic, I., Menegatti, E., Marković, I. (eds) Intelligent Autonomous Systems 17. IAS 2022. Lecture Notes in Networks and Systems, vol 577. Springer, Cham. https://doi.org/10.1007/978-3-031-22216-0_48

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