Advertisement

Geometric Aspects of Robot Navigation: From Individual Robots to Massive Particle Swarms

  • Sándor P. FeketeEmail author
Chapter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11340)

Abstract

We describe a spectrum of challenges and results related to geometric aspects of robot navigation, advancing from centralized methods for difficult offline problems (such as the Art Gallery Problem), to online tasks for many robots (as in online exploration by a swarm of robots), locally managing the connectivity and shape of a large swarm (i.e., cohesive control), all the way to controlling massive swarms of particles by global forces.

References

  1. 1.
    Abrahamsen, M., Adamaszek, A., Miltzow, T.: The art gallery problem is \(\exists \)\({\rm I\!R}\)-complete. In: Proceedings of the 50th Annual ACM Symposium on Theory of Computing (STOC), pp. 65–73 (2018)Google Scholar
  2. 2.
    Akash, A.K., Fekete, S.P., Lee, S.K., López-Ortiz, A., Maftuleac, D., McLurkin, J.: Lower bounds for graph exploration using local policies. J. Graph Algorithms Appl. 21(3), 371–387 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Balch, T., Hybinette, M.: Social potentials for scalable multi-robot formations. In: Proceedings of the 17th IEEE/RSJ International Conference on Intelligent Robots and Systems (ICRA), pp. 73–80 (2000)Google Scholar
  4. 4.
    Becker, A., Demaine, E.D., Fekete, S.P., Habibi, G., McLurkin, J.: Reconfiguring massive particle swarms with limited, global control. In: Flocchini, P., Gao, J., Kranakis, E., Meyer auf der Heide, F. (eds.) ALGOSENSORS 2013. LNCS, vol. 8243, pp. 51–66. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-642-45346-5_5CrossRefGoogle Scholar
  5. 5.
    Becker, A.T., Demaine, E.D., Fekete, S.P., Lonsford, J., Morris-Wright, R.: Particle computation: complexity, algorithms, and logic. Natural Computing (to appear)Google Scholar
  6. 6.
    Becker, A.T., Demaine, E.D., Fekete, S.P., Shad, S.H.M., Morris-Wright, R.: Tilt: the video. Designing worlds to control robot swarms with only global signals. In: Proceedings of the 31st International Symposium on Computational Geometry (SoCG), pp. 16–18 (2015). https://youtu.be/H6o9DTIfkn0
  7. 7.
    Becker, A.T., et al.: Tilt assembly: algorithms for micro-factories that build objects with uniform external forces. In: Proceedings of the 28th International Symposium on Algorithms and Computation (ISAAC 2017), pp. 11:1–11:13 (2017). Full version to appear in: AlgorithmicaGoogle Scholar
  8. 8.
    Becker, A.T., Ou, Y., Kim, P., Kim, M.J., Julius, A.: Feedback control of many magnetized: tetrahymena pyriformis cells by exploiting phase inhomogeneity. In: Proceedings of the 26th IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 3317–3323 (2013)Google Scholar
  9. 9.
    Bonabeau, E., Meyer, C.: Swarm intelligence: a whole new way to think about business. Harvard Bus. Rev. 79, 106–114 (2001)Google Scholar
  10. 10.
    Borrmann, D., et al.: Point guards and point clouds: solving general art gallery problems. In: Proceedings of the 29th Symposium on Computational Geometry (SoCG), pp. 347–348 (2013)Google Scholar
  11. 11.
    Borrmann, D., Elseberg, J., Lingemann, K., Nüchter, A., Hertzberg, J.: Globally consistent 3D mapping with scan matching. Robot. Auton. Syst. 56(2), 130–142 (2008)CrossRefGoogle Scholar
  12. 12.
    Chanu, A., Felfoul, O., Beaudoin, G., Martel, S.: Adapting the clinical MRI software environment for real-time navigation of an endovascular untethered ferromagnetic bead for future endovascular interventions. Magn. Reson. Med. 59(6), 1287–1297 (2008)CrossRefGoogle Scholar
  13. 13.
    Chazelle, B.: The convergence of bird flocking. J. ACM 61(4), 21 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chuang, Y.-L., Huang, Y.R., D’Orsogna, M.R., Bertozzi, A.L.: Multi-vehicle flocking: scalability of cooperative control algorithms using pairwise potentials. In: Proceedings of the 24th IEEE/RSJ International Conference on Intelligent Robots and Systems (ICRA), pp. 2292–2299 (2007)Google Scholar
  15. 15.
    Chvátal, V.: A combinatorial theorem in plane geometry. J. Comb. Theory Ser. B 18, 39–41 (1974)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Cortes, J., Martinez, S., Karatas, T., Bullo, F.: Coverage control for mobile sensing networks. IEEE Trans. Robot. Autom. 20(2), 243–255 (2004)CrossRefGoogle Scholar
  17. 17.
    Couto, M.C., de Souza, C.C., de Rezende, P.J.: An exact and efficient algorithm for the orthogonal art gallery problem. In: Proceedings of the XX Brazilian Symposium on Computer Graphics and Image Processing (SIBGRAPI), pp. 87–94 (2007)Google Scholar
  18. 18.
    Demaine, E.D., Demaine, M.L., O’Rourke, J.: PushPush and Push-1 are NP-hard in 2D. In: Proceedings of the 12th Canadian Conference on Computational Geometry (CCCG), pp. 211–219 (2000)Google Scholar
  19. 19.
    Efrat, A., Fekete, S.P., Mitchell, J.S., Polishchuk, V., Suomela, J.: Improved approximation algorithms for relay placement. ACM Trans. Algorithms 12, 20:1–20:28 (2016)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Eidenbenz, S., Stamm, C., Widmayer, P.: Inapproximability results for guarding polygons and terrains. Algorithmica 31(1), 79–113 (2001)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Engels, B., Kamphans, T.: Randolphs robot game is NP-hard. Electron. Notes Discrete Math. 25, 49–53 (2006)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ernestus, M., Fekete, S.P., Hemmer, M., Krupke, D.: Continuous geometric algorithms for robot swarms with multiple leaders. In: Proceedings of the 31st European Workshop on Computational Geometry (EuroCG), pp. 69–72 (2015)Google Scholar
  23. 23.
    Ernestus, M., Krupke, D.: Distributed, scalable algorithmic methods for swarms with multiple leader robots. Bachelor thesis, TU Braunschweig (2014)Google Scholar
  24. 24.
    Fekete, S.P., Friedrichs, S., Kröller, A., Schmidt, C.: Facets for art gallery problems. Algorithmica 73, 411–440 (2015)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Fekete, S.P., Kamphans, T., Kröller, A., Mitchell, J.S.B., Schmidt, C.: Exploring and triangulating a region by a swarm of robots. In: Goldberg, L.A., Jansen, K., Ravi, R., Rolim, J.D.P. (eds.) APPROX/RANDOM -2011. LNCS, vol. 6845, pp. 206–217. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-22935-0_18CrossRefzbMATHGoogle Scholar
  26. 26.
    Fekete, S.P., Kamphans, T., Kröller, A., Schmidt, C.: Robot swarms for exploration and triangulation of unknown environments. In: Proceedings of the 25th European Workshop on Computational Geometry, pp. 153–156 (2010)Google Scholar
  27. 27.
    Fekete, S.P., Kröller, A.: Geometry-based reasoning for a large sensor network. In: Proceedings of the 22nd Symposium on Computational Geometry (SoCG), pp. 475–476 (2006)Google Scholar
  28. 28.
    Fekete, S.P., Kröller, A., Lee, S., McLurkin, J., Schmidt, C.: Triangulating unknown environments using robot swarms. In: Proceedings of the 29th Symposium on Computational Geometry (SoCG), pp. 345–346 (2013)Google Scholar
  29. 29.
    Fekete, S.P., Mitchell, J.S.B., Schmidt, C.: Minimum covering with travel cost. J. Comb. Optim. 24, 32–51 (2012)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Fekete, S.P., Rex, S., Schmidt, C.: Online exploration and triangulation in orthogonal polygonal regions. In: Ghosh, S.K., Tokuyama, T. (eds.) WALCOM 2013. LNCS, vol. 7748, pp. 29–40. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-36065-7_5CrossRefzbMATHGoogle Scholar
  31. 31.
    Fekete, S.P., Schmidt, C.: Polygon exploration with time-discrete vision. Comput. Geom.: Theory Appl. 43(2), 148–168 (2010)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Fisk, S.: A short proof of Chvátal’s watchman theorem. J. Comb. Theory Ser. B 24, 374 (1978)CrossRefGoogle Scholar
  33. 33.
    Garey, M.R., Graham, R.L., Johnson, D.S.: The complexity of computing Steiner minimal trees. SIAM J. Appl. Math. 32(4), 835–859 (1977)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Gazi, V.: Swarm aggregations using artificial potentials and sliding-mode control. IEEE Trans. Robot. 21(6), 1208–1214 (2005)CrossRefGoogle Scholar
  35. 35.
    Hamann, H., Wörn, H.: Aggregating robots compute: an adaptive heuristic for the euclidean steiner tree problem. In: Asada, M., Hallam, J.C.T., Meyer, J.-A., Tani, J. (eds.) SAB 2008. LNCS (LNAI), vol. 5040, pp. 447–456. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-69134-1_44CrossRefGoogle Scholar
  36. 36.
    Hayes, A.T., Dormiani-Tabatabaei, P.: Self-organized flocking with agent failure: off-line optimization and demonstration with real robots. In: Proceedings of the 19th IEEE/RSJ International Conference on Intelligent Robots and Systems (ICRA), pp. 3900–3905 (2002)Google Scholar
  37. 37.
    Hearn, R.A., Demaine, E.D.: PSPACE-completeness of sliding-block puzzles and other problems through the nondeterministic constraint logic model of computation. Theor. Comput. Sci. 343(1–2), 72–96 (2005)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Hoffmann, M.: Motion planning amidst movable square blocks: Push-* is NP-hard. In: Proceedings of the 12th Canadian Conference on Computational Geometry (CCCG), pp. 205–210 (2000)Google Scholar
  39. 39.
    Holzer, M., Schwoon, S.: Assembling molecules in ATOMIX is hard. Theor. Comput. Sci. 313(3), 447–462 (2004)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Howard, A., Matarić, M.J., Sukhatme, G.S.: Mobile sensor network deployment using potential fields: a distributed, scalable solution to the area coverage problem. In: Asama, H., Arai, T., Fukuda, T., Hasegawa, T. (eds.) Proceedings 6th International Symposium on Distributed Autonomous Robotics Systems (DARS), pp. 299–308. Springer, Tokyo (2002).  https://doi.org/10.1007/978-4-431-65941-9_30CrossRefGoogle Scholar
  41. 41.
    Hwang, F.K., Richards, D.S., Winter, P.: The Steiner Tree Problem. Annals of Discrete Mathematics, vol. 53. Elsevier, Amsterdam (1992)zbMATHGoogle Scholar
  42. 42.
    Jerrum, M.R.: The complexity of finding minimum-length generator sequences. Theor. Comput. Sci. 36, 265–289 (1985)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Kamimura, A., Murata, S., Yoshida, E., Kurokawa, H., Tomita, K., Kokaji, S.: Self-reconfigurable modular robot-experiments on reconfiguration and locomotion. In: Proceedings of the 14th IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 606–612 (2001)Google Scholar
  44. 44.
    Keldenich, P., et al.: On designing 2D discrete workspaces to sort or classify 2D polyominoes. In: Proceedings of the 35th IEEE/RSJ International Conference on Intelligent Robots and Systems (ICRA) (2018, to appear)Google Scholar
  45. 45.
    Khalil, I.S.M., Pichel, M.P., Reefman, B.A., Sukas, O.S., Abelmann, L., Misra, S.: Control of magnetotactic bacterium in a micro-fabricated maze. In: Proceedings of the 30th IEEE/RSJ International Conference on Intelligent Robots and Systems (ICRA), pp. 5488–5493 (2013)Google Scholar
  46. 46.
    Kröller, A., Baumgartner, T., Fekete, S.P., Schmidt, C.: Exact solutions and bounds for general art gallery problems. J. Exp. Algorithms 17, 2–3 (2012)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Kröller, A., Fekete, S.P., Pfisterer, D., Fischer, S.: Deterministic boundary recognition and topology extraction for large sensor networks. In: Proceedings of the ACM/SIAM Symposium on Discrete Algorithms (SODA), pp. 1000–1009 (2006)Google Scholar
  48. 48.
    Krupke, D.M., Ernestus, M., Hemmer, M., Fekete, S.P.: Distributed cohesive control for robot swarms: maintaining good connectivity in the presence of exterior forces. In: Proceedings of the 28th IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 413–420 (2015)Google Scholar
  49. 49.
    La, H.M., Sheng, W.: Adaptive flocking control for dynamic target tracking in mobile sensor networks. In: Proceedings of the 22nd IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 4843–4848 (2009)Google Scholar
  50. 50.
    La, H.M., Sheng, W.: Flocking control of a mobile sensor network to track and observe a moving target. In: Proceedings of the 22nd IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 3129–3134 (2009)Google Scholar
  51. 51.
    Lee, D.-T., Lin, A.K.: Computational complexity of art gallery problems. IEEE Trans. Inf. Theory 32(2), 276–282 (1986)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Lee, S.K., Becker, A.T., Fekete, S.P., Kröller, A., McLurkin, J.: Exploration via structured triangulation by a multi-robot system with bearing-only low-resolution sensors. In: Proceedings of the 31st IEEE/RSJ International Conference on Intelligent Robots and Systems (ICRA), pp. 2150–2157 (2014)Google Scholar
  53. 53.
    Lee, S.K., Fekete, S.P., McLurkin, J.: Virtual-agent coverage control in triangulated environments using geodesic Voronoi tessellation. In: Proceedings of the 27th IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 3858–3865 (2014)Google Scholar
  54. 54.
    Lee, S.K., Fekete, S.P., McLurkin, J.: Structured triangulation in multi-robot systems: coverage, patrolling, Voronoi partitions, and geodesic centers. Int. J. Robot. Res. 9(35), 1234–1260 (2016)CrossRefGoogle Scholar
  55. 55.
    Lee, S.K., McLurkin, J.: Distributed cohesive configuration control for swarm robots with boundary information and network sensing. In: Proceedings of the 27th IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 1161–1167. IEEE (2014)Google Scholar
  56. 56.
    Lindhé, M., Ögren, P., Johansson, K.H.: Flocking with obstacle avoidance: a new distributed coordination algorithm a new distributed coordination algorithm based on Voronoi partitions. In: Proceedings of the 22nd IEEE/RSJ International Conference on Intelligent Robots and Systems (ICRA), pp. 1785–1790 (2005)Google Scholar
  57. 57.
    Maftulec, D., Lee, S.K., Fekete, S.P., Akash, A.K., López-Ortiz, A., McLurkin, J.: Local policies for efficiently patrolling a triangulated region be a robot swarm. In: Proceedings of the 32nd IEEE/RSJ International Conference on Intelligent Robots and Systems (ICRA), pp. 1809–1815 (2015)Google Scholar
  58. 58.
    Mahadev, A.V., Krupke, D., Fekete, S.P., Becker, A.T.: Mapping, foraging, and coverage with a particle swarm controlled by uniform inputs. In: Proceedings of the 30th IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 1097–1104 (2017)Google Scholar
  59. 59.
    Mahadev, A.V., Krupke, D., Reinhardt, J.-M., Fekete, S.P., Becker, A.T.: Collecting a swarm in a 2D environment using shared, global inputs. In: Proceedings of the 13th Conference on Automation Science and Engineering (CASE 2016), pp. 1231–1236 (2016)Google Scholar
  60. 60.
    McLurkin, J., Demaine, E.D.: A distributed boundary detection algorithm for multi-robot systems. In: Proceedings of the 22th IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 4791–4798 (2009)Google Scholar
  61. 61.
    McLurkin, J., et al.: A low-cost multi-robot system for research, teaching, and outreach. In: Martinoli, A., et al. (eds.) Proceedings 10th International Symposium on Distributed Autonomous Robotics Systems (DARS), vol. 83, pp. 597–609. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-32723-0_43CrossRefGoogle Scholar
  62. 62.
    McLurkin, J., et al.: A robot system design for low-cost multi-robot manipulation. In: Proceedings of the 27th IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 912–918 (2014)Google Scholar
  63. 63.
    Olfati-Saber, R.: Flocking for multi-agent dynamic systems: algorithms and theory. IEEE Trans. Autom. Control 51, 401–420 (2006)MathSciNetCrossRefGoogle Scholar
  64. 64.
    O’Rourke, J.: Art Gallery Theorems and Algorithms. International Series of Monographs on Computer Science. Oxford University Press, New York (1987)zbMATHGoogle Scholar
  65. 65.
    Prömel, H.J., Steger, A.: The Steiner Tree Problem: A Tour Through Graphs, Algorithms, and Complexity. Vieweg, Braunschweig (2002)CrossRefGoogle Scholar
  66. 66.
    Reynolds, C.W.: Flocking, herds, and schools: a distributed behavioral model. Comput. Graph. 21(4), 25–34 (1987)CrossRefGoogle Scholar
  67. 67.
    Schmidt, A., Manzoor, S., Huang, L., Becker, A.T., Fekete, S.P.: Efficient parallel self-assembly under uniform control inputs. IEEE Robot. Autom. Lett. 3, 3521–3528 (2018)CrossRefGoogle Scholar
  68. 68.
    Schmidt, C.: Algorithms for mobile agents with limited capabilities. Ph.D. thesis, TU Braunschweig (2011)Google Scholar
  69. 69.
    Shad, H.M., Moris-Wright, R., Demaine, E.D., Fekete, S.P., Becker, A.T.: Particle computation: device fan-out and binary memory. In: Proceedings of the 32nd IEEE/RSJ International Conference on Intelligent Robots and Systems (ICRA), pp. 5384–5389 (2015)Google Scholar
  70. 70.
    Shi, H., Wang, L., Chu, T., Xu, M.: Flocking coordination of multiple mobile autonomous agents with asymmetric interactions and switching topology. In: Proceedings of the 18th IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 935–940 (2005)Google Scholar
  71. 71.
    Spears, W.M., Spears, D.F., Hammann, J.C., Heil, R.: Distributed, physics-based control of swarms of vehicles. Auton. Robots 17(2–3), 137–162 (2004)CrossRefGoogle Scholar
  72. 72.
    ThinkFun: Tilt: gravity fed logic maze. http://www.thinkfun.com/tilt
  73. 73.
    Tozoni, D.C., de Rezende, P.J., de Souza, C.C.: The quest for optimal solutions for the art gallery problem: a practical iterative algorithm. In: Bonifaci, V., Demetrescu, C., Marchetti-Spaccamela, A. (eds.) SEA 2013. LNCS, vol. 7933, pp. 320–336. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-38527-8_29CrossRefGoogle Scholar
  74. 74.
    Vartholomeos, P., Akhavan-Sharif, M., Dupont, P.E.: Motion planning for multiple millimeter-scale magnetic capsules in a fluid environment. In: Proceedings of the 29th IEEE/RSJ International Conference on Intelligent Robots and Systems (ICRA), pp. 1927–1932 (2012)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Computer ScienceTU BraunschweigBraunschweigGermany

Personalised recommendations