Continuous vs. Discrete Asynchronous Moves: A Certified Approach for Mobile Robots

  • Thibaut Balabonski
  • Pierre Courtieu
  • Robin Pelle
  • Lionel Rieg
  • Sébastien Tixeuil
  • Xavier UrbainEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11704)


Oblivious Mobile Robots have been studied both in continuous Euclidean spaces, and discrete spaces (that is, graphs). However the obtained literature forms distinct sets of results for the two settings. In our view, the continuous model reflects well the physicality of robots operating in some real environment, while the discrete model reflects well the digital nature of autonomous robots, whose sensors and computing capabilities are inherently finite.

We explore the possibility of bridging results between the two models. Our approach is certified using the Coq proof assistant and the Pactole framework, which we extend to the most general asynchronous model without compromising its genericity. Our extended framework is then used to formally prove the equivalence between atomic moves in a discrete space (the classical “robots on graphs” model) and non-atomic moves in a continuous unidimensional space when robot vision sensors are discrete (robots move in straigth lines between positions, but their observations are at source and destination positions only), irrespective of the problem being solved. Our effort consolidates the integration between the model, the problem specification, and its proof that is advocated by the Pactole framework.


Formal proof Proof assistant Coq Mobile autonomous robots Distributed algorithms 


  1. 1.
    Auger, C., Bouzid, Z., Courtieu, P., Tixeuil, S., Urbain, X.: Certified impossibility results for byzantine-tolerant mobile robots. In: Higashino, T., Katayama, Y., Masuzawa, T., Potop-Butucaru, M., Yamashita, M. (eds.) SSS 2013. LNCS, vol. 8255, pp. 178–190. Springer, Cham (2013). Scholar
  2. 2.
    Balabonski, T., Courtieu, P., Pelle, R., Rieg, L., Tixeuil, S., Urbain, X.: Brief announcement continuous vs. discrete asynchronous moves: a certified approach for mobile robots. In: Izumi, T., Kuznetsov, P. (eds.) SSS 2018. LNCS, vol. 11201, pp. 404–408. Springer, Cham (2018). Scholar
  3. 3.
    Balabonski, T., Delga, A., Rieg, L., Tixeuil, S., Urbain, X.: Synchronous gathering without multiplicity detection: a certified algorithm. In: Bonakdarpour, B., Petit, F. (eds.) SSS 2016. LNCS, vol. 10083, pp. 7–19. Springer, Cham (2016). Scholar
  4. 4.
    Balabonski, T., Pelle, R., Rieg, L., Tixeuil, S.: A foundational framework for certified impossibility results with mobile robots on graphs. In: Bellavista, P., Garg, V.K., (eds.) Proceedings of the 19th International Conference on Distributed Computing and Networking, ICDCN 2018, Varanasi, India, 4–7 January 2018, pp. 5:1–5:10. ACM (2018)Google Scholar
  5. 5.
    Baldoni, R., Bonnet, F., Milani, A., Raynal, M.: Anonymous graph exploration without collision by mobile robots. Inf. Process. Lett. 109(2), 98–103 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bérard, B., et al.: Formal methods for mobile robots: current results and open problems. Int. J. Inform. Soc. 7(3), 101–114 (2015). Invited PaperGoogle Scholar
  7. 7.
    Bérard, B., Lafourcade, P., Millet, L., Potop-Butucaru, M., Thierry-Mieg, Y., Tixeuil, S.: Formal verification of mobile robot protocols. Distrib. Comput. 29(6), 459–487 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Blin, L., Burman, J., Nisse, N.: Exclusive graph searching. Algorithmica 77(3), 942–969 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Blin, L., Milani, A., Potop-Butucaru, M., Tixeuil, S.: Exclusive perpetual ring exploration without chirality. In: Lynch, N.A., Shvartsman, A.A. (eds.) DISC 2010. LNCS, vol. 6343, pp. 312–327. Springer, Heidelberg (2010). Scholar
  10. 10.
    Bonnet, F., Défago, X., Petit, F., Potop-Butucaru, M., Tixeuil, S.: Discovering and assessing fine-grained metrics in robot networks protocols. In 33rd IEEE International Symposium on Reliable Distributed Systems Workshops, SRDS Workshops 2014, Nara, Japan, 6–9 October 2014, pp. 50–59. IEEE (2014)Google Scholar
  11. 11.
    Bonnet, F., Milani, A., Potop-Butucaru, M., Tixeuil, S.: Asynchronous exclusive perpetual grid exploration without sense of direction. In: Fernàndez Anta, A., Lipari, G., Roy, M. (eds.) OPODIS 2011. LNCS, vol. 7109, pp. 251–265. Springer, Heidelberg (2011). Scholar
  12. 12.
    Bonnet, F., Potop-Butucaru, M., Tixeuil, S.: Asynchronous gathering in rings with 4 robots. In: Mitton, N., Loscri, V., Mouradian, A. (eds.) ADHOC-NOW 2016. LNCS, vol. 9724, pp. 311–324. Springer, Cham (2016). Scholar
  13. 13.
    Bouzid, Z., Dolev, S., Potop-Butucaru, M., Tixeuil, S.: RoboCast: asynchronous communication in robot networks. In: Lu, C., Masuzawa, T., Mosbah, M. (eds.) OPODIS 2010. LNCS, vol. 6490, pp. 16–31. Springer, Heidelberg (2010). Scholar
  14. 14.
    Caron, G., Mouaddib, E.M., Marchand, É.: 3D model based tracking for omnidirectional vision: a new spherical approach. Robot. Auton. Syst. 60(8), 1056–1068 (2012)CrossRefGoogle Scholar
  15. 15.
    Chalopin, J., Flocchini, P., Mans, B., Santoro, N.: Network exploration by silent and oblivious robots. In: Thilikos, D.M. (ed.) WG 2010. LNCS, vol. 6410, pp. 208–219. Springer, Heidelberg (2010). Scholar
  16. 16.
    Cieliebak, M., Flocchini, P., Prencipe, G., Santoro, N.: Distributed computing by mobile robots: gathering. SIAM J. Comput. 41(4), 829–879 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Courtieu, P., Rieg, L., Tixeuil, S., Urbain, X.: Impossibility of gathering, a certification. Inf. Process. Lett. 115, 447–452 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Courtieu, P., Rieg, L., Tixeuil, S., Urbain, X.: Certified universal gathering in \(\mathbb{R} ^2\) for oblivious mobile robots. In: Gavoille, C., Ilcinkas, D. (eds.) DISC 2016. LNCS, vol. 9888, pp. 187–200. Springer, Heidelberg (2016). Scholar
  19. 19.
    D’Angelo, G., Navarra, A., Nisse, N.: A unified approach for gathering and exclusive searching on rings under weak assumptions. Distrib. Comput. 30(1), 17–48 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    D’Angelo, G., Stefano, G.D., Navarra, A., Nisse, N., Suchan, K.: Computing on rings by oblivious robots: a unified approach for different tasks. Algorithmica 72(4), 1055–1096 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Devismes, S., Lamani, A., Petit, F., Raymond, P., Tixeuil, S.: Optimal grid exploration by asynchronous oblivious robots. In: Richa, A.W., Scheideler, C. (eds.) SSS 2012. LNCS, vol. 7596, pp. 64–76. Springer, Heidelberg (2012). Scholar
  22. 22.
    Devismes, S., Lamani, A., Petit, F., Tixeuil, S.: Optimal torus exploration by oblivious robots. In: Bouajjani, A., Fauconnier, H. (eds.) NETYS 2015. LNCS, vol. 9466, pp. 183–199. Springer, Cham (2015). Scholar
  23. 23.
    Devismes, S., Petit, F., Tixeuil, S.: Optimal probabilistic ring exploration by semi-synchronous oblivious robots. Theoret. Comput. Sci. 498, 10–27 (2013)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Doan, H.T.T., Bonnet, F., Ogata, K.: Model checking of a mobile robots perpetual exploration algorithm. In: Liu, S., Duan, Z., Tian, C., Nagoya, F. (eds.) SOFL+MSVL 2016. LNCS, vol. 10189, pp. 201–219. Springer, Cham (2017). Scholar
  25. 25.
    Doan, H.T.T., Bonnet, F., Ogata, K.: Model checking of robot gathering. In: Aspnes, J., Felber, P. (edS.) Principles of Distributed Systems - 21th International Conference (OPODIS 2017), Leibniz International Proceedings in Informatics (LIPIcs), Lisbon, Portugal, December 2017. Schloss Dagstuhl-Leibniz-Zentrum fuer InformatikGoogle Scholar
  26. 26.
    Flocchini, P., Ilcinkas, D., Pelc, A., Santoro, N.: Remembering without memory: tree exploration by asynchronous oblivious robots. Theoret. Comput. Sci. 411(14–15), 1583–1598 (2010)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Flocchini, P., Ilcinkas, D., Pelc, A., Santoro, N.: Computing without communicating: ring exploration by asynchronous oblivious robots. Algorithmica 65(3), 562–583 (2013)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Flocchini, P., Prencipe, G., Santoro, N.: Distributed Computing by Oblivious Mobile Robots. Synthesis Lectures on Distributed Computing Theory. Morgan & Claypool Publishers, California (2012)CrossRefGoogle Scholar
  29. 29.
    Flocchini, P., Prencipe, G., Santoro, N., Widmayer, P.: Arbitrary pattern formation by asynchronous, anonymous, oblivious robots. Theoret. Comput. Sci. 407(1–3), 412–447 (2008)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Fujinaga, N., Yamauchi, Y., Kijima, S., Yamashita, M.: Asynchronous pattern formation by anonymous oblivious mobile robots. In: Aguilera, M.K. (ed.) DISC 2012. LNCS, vol. 7611, pp. 312–325. Springer, Heidelberg (2012). Scholar
  31. 31.
    Izumi, T., Bouzid, Z., Tixeuil, S., Wada, K.: Brief announcement: the BG-simulation for Byzantine mobile robots. In: Peleg, D. (ed.) DISC 2011. LNCS, vol. 6950, pp. 330–331. Springer, Heidelberg (2011). Scholar
  32. 32.
    Izumi, T., Izumi, T., Kamei, S., Ooshita, F.: Mobile robots gathering algorithm with local weak multiplicity in rings. In: Patt-Shamir, B., Ekim, T. (eds.) SIROCCO 2010. LNCS, vol. 6058, pp. 101–113. Springer, Heidelberg (2010). Scholar
  33. 33.
    Kamei, S., Lamani, A., Ooshita, F., Tixeuil, S.: Asynchronous mobile robot gathering from symmetric configurations without global multiplicity detection. In: Kosowski, A., Yamashita, M. (eds.) SIROCCO 2011. LNCS, vol. 6796, pp. 150–161. Springer, Heidelberg (2011). Scholar
  34. 34.
    Kamei, S., Lamani, A., Ooshita, F., Tixeuil, S.: Gathering an even number of robots in an odd ring without global multiplicity detection. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 542–553. Springer, Heidelberg (2012). Scholar
  35. 35.
    Lamani, A., Potop-Butucaru, M.G., Tixeuil, S.: Optimal deterministic ring exploration with oblivious asynchronous robots. In: Patt-Shamir, B., Ekim, T. (eds.) SIROCCO 2010. LNCS, vol. 6058, pp. 183–196. Springer, Heidelberg (2010). Scholar
  36. 36.
    Millet, L., Potop-Butucaru, M., Sznajder, N., Tixeuil, S.: On the synthesis of mobile robots algorithms: the case of ring gathering. In: Felber, P., Garg, V. (eds.) SSS 2014. LNCS, vol. 8756, pp. 237–251. Springer, Cham (2014). Scholar
  37. 37.
    Prencipe, G.: Impossibility of gathering by a set of autonomous mobile robots. Theoret. Comput. Sci. 384(2–3), 222–231 (2007)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Aminof, B., Murano, A., Rubin, S., Zuleger, F.: Verification of asynchronous mobile-robots in partially-known environments. In: Chen, Q., Torroni, P., Villata, S., Hsu, J., Omicini, A. (eds.) PRIMA 2015. LNCS (LNAI), vol. 9387, pp. 185–200. Springer, Cham (2015). Scholar
  39. 39.
    Sangnier, A., Sznajder, N., Potop-Butucaru, M., Tixeuil, S.: Parameterized verification of algorithms for oblivious robots on a ring. In: Formal Methods in Computer Aided Design, Vienna, Austria, October 2017Google Scholar
  40. 40.
    Suzuki, I., Yamashita, M.: Distributed anonymous mobile robots: formation of geometric patterns. SIAM J. Comput. 28(4), 1347–1363 (1999)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Tomita, Y., Yamauchi, Y., Kijima, S., Yamashita, M.: Plane formation by synchronous mobile robots without chirality. In: Aspnes, J., Bessani, A., Felber, P., Leitão, J. (eds.) 21st International Conference on Principles of Distributed Systems, OPODIS 2017. LIPIcs, vol. 95, Lisbon, Portugal, 18–20 December 2017, pp. 13:1–13:17. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2017)Google Scholar
  42. 42.
    Yamashita, M., Suzuki, I.: Characterizing geometric patterns formable by oblivious anonymous mobile robots. Theoret. Comput. Sci. 411(26–28), 2433–2453 (2010)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Yamauchi, Y., Uehara, T., Kijima, S., Yamashita, M.: Plane formation by synchronous mobile robots in the three-dimensional Euclidean space. J. ACM 64(3), 16:1–16:43 (2017)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Yamauchi, Y., Uehara, T., Yamashita, T.: Brief announcement: pattern formation problem for synchronous mobile robots in the three dimensional euclidean space. In: Giakkoupis, G. (ed.) Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing, PODC 2016, Chicago, IL, USA, 25–28 July 2016, pp. 447–449. ACM (2016)Google Scholar
  45. 45.
    Yamauchi, Y., Yamashita, M.: Pattern formation by mobile robots with limited visibility. In: Moscibroda, T., Rescigno, A.A. (eds.) SIROCCO 2013. LNCS, vol. 8179, pp. 201–212. Springer, Cham (2013). Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Thibaut Balabonski
    • 1
  • Pierre Courtieu
    • 2
  • Robin Pelle
    • 1
  • Lionel Rieg
    • 3
  • Sébastien Tixeuil
    • 4
  • Xavier Urbain
    • 5
    Email author
  1. 1.LRI, CNRS UMR 8623, Université Paris-Sud, Université Paris-SaclayOrsayFrance
  2. 2.CÉDRIC – Conservatoire national des arts et métiersParisFrance
  3. 3.Université Grenoble Alpes, Grenoble INP, VERIMAGSaint Martin d’HèresFrance
  4. 4.Sorbonne Université, CNRS, LIP6ParisFrance
  5. 5.Université Claude Bernard Lyon-1, LIRIS CNRS UMR 5205, Université de LyonLyonFrance

Personalised recommendations