Distributed Multi-robot Circumnavigation with Dynamic Spacing and Time Delay

  • Huimin LuEmail author
  • Weijia Yao
  • Liangming Chen


Circumnavigation is the process whereby a single agent or multiple agents rotate around a target while preserving a circular formation, which has promising potential in real-world applications such as entrapping a malicious target or escorting an important member. For the multi-robot circumnavigation problem, spacing (i.e., the angle differences) among robots plays an important role in forming a desirable circular formation. The spacing is usually assumed to be a unified constant in most of the studies. However, when robots have different or even time-varying kinematic capabilities, a fixed and equal spacing is probably not effective for accomplishing such task as preventing an enclosed target from fleeing, and thus dynamic spacing is naturally proposed and preferred. The variations of spacing are caused by the “weights” (termed utilities) of robots. This paper relaxes the condition of piecewise constant utilities and provides the ultimate bound and the input-to-state (ISS) stability conditions for the spacing error and its dynamics respectively. In addition, since time delay is ubiquitous in practical engineering systems while seldom considered in the current studies on circumnavigation, the maximum allowable time delay within which the circumnavigation remains stable is derived using both the frequency domain method and the Lambert-W function. Finally, the theoretical results are validated by a practical simulation system.


Circumnavigation Dynamic spacing Utility Time-delay 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



This work is supported by National Science Foundation of China (No. 61773393, No. U1813205).

Supplementary material

10846_2019_1111_MOESM1_ESM.mp4 (6.4 mb)
(MP4 6.39 MB)

(WMV 3.81 MB)


  1. 1.
    Cao, Y.: Uav circumnavigating an unknown target under a gps-denied environment with range-only measurements. Automatica 55, 150–158 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Clarke, F.H.: Optimization and nonsmooth analysis, vol. 5. Siam (1990)Google Scholar
  3. 3.
    Corless, R.M., Gonnet, G.H., Hare, D.E., Jeffrey, D.J., Knuth, D.E.: On the lambertw function. Adv. Comput. Math. 5(1), 329–359 (1996)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dai, W., Yao, W., Luo, S., Ma, J., Wang, R., Hong, S., Zhou, Z., Li, X., Han, B., Xiao, J., et al.: Nubot team description paper (2018)Google Scholar
  5. 5.
    Deghat, M., Shames, I., Anderson, B.D., Yu, C.: Localization and circumnavigation of a slowly moving target using bearing measurements. IEEE Trans. Autom. Control 59(8), 2182–2188 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dong, Y., Hu, X.: Distributed control of periodic formations for multiple under-actuated autonomous vehicles. IET Control Theory Appl. 11(1), 66–72 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dugard, L., Verriest, E.I.: Stability and control of time-delay systems, vol. 228. Springer (1998)Google Scholar
  8. 8.
    Fax, J.A., Murray, R.M.: Information flow and cooperative control of vehicle formations. IEEE Trans. Autom. Control 35(1), 115–120 (2002)Google Scholar
  9. 9.
    Franchi, A., Stegagno, P., Oriolo, G.: Decentralized multi-robot encirclement of a 3D target with guaranteed collision avoidance. Auton. Robot. 40(2), 245–265 (2016). CrossRefGoogle Scholar
  10. 10.
    Goodwin, G.C., Graebe, S.F., Salgado, M.E.: Control System Design, 1st edn. Prentice Hall PTR, Upper Saddle River (2000)Google Scholar
  11. 11.
    Gu, K., Chen, J., Kharitonov, V.L.: Stability of time-delay systems. Springer Science & Business Media (2003)Google Scholar
  12. 12.
    Horn, R.A., Johnson, C.R.: Matrix analysis. Cambridge University Press (2012)Google Scholar
  13. 13.
    Khalil, H.K.: Noninear systems. Prentice-Hall New Jersey 2(5), 5–1 (1996)Google Scholar
  14. 14.
    Krasovskii, N.N.: Stability of motion. Stanford University Press (1963)Google Scholar
  15. 15.
    Li, R., Shi, Y., Song, Y.: Localization and circumnavigation of multiple agents along an unknown target based on bearing-only measurement: a three dimensional solution. Automatica 94, 18–25 (2018)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Luo, S., Yao, W., Yu, Q., Xiao, J., Lu, H., Zhou, Z.: Object Detection Based on Gpu Parallel Computing for Robocup Middle Size League. In: 2017 IEEE International Conference on Robotics and Biomimetics (ROBIO), pp. 86–91. IEEE (2017)Google Scholar
  17. 17.
    Ma, J., Yao, W., Dai, W., Lu, H., Xiao, J., Zheng, Z.: Cooperative Encirclement Control for a Group of Targets by Decentralized Robots with Collision Avoidance. In: 2018 37Th Chinese Control Conference (CCC), pp. 6848–6853. IEEE (2018)Google Scholar
  18. 18.
    Marshall, B.J.A., Broucke, M.E., Francis, B.A.: Formation of vehicles in cyclic pursuit. IEEE Trans. Autom. Control 49(11), 1963–1974 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Matveev, A.S., Semakova, A.A., Savkin, A.V.: Range-only based circumnavigation of a group of moving targets by a non-holonomic mobile robot. Automatica 65, 76–89 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mesbahi, M., Egerstedt, M.: Graph theoretic methods in multiagent networks. Princeton University Press (2010)Google Scholar
  21. 21.
    Miao, Z., Wang, Y., Fierro, R.: Cooperative circumnavigation of a moving target with multiple nonholonomic robots using backstepping design. Syst. Control Lett. 103, 58–65 (2017)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Oh, K.K., Park, M.C., Ahn, H.S.: A survey of multi-agent formation control. Automatica 53, 424–440 (2015)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Olfati-Saber, R., Murray, R.M.: Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans. Autom. Control 49(9), 1520–1533 (2004)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Olfati-Saber, R., Murray, R.M.: Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans. Autom. Control 49(9), 1520–1533 (2004). MathSciNetCrossRefGoogle Scholar
  25. 25.
    Pavone, M., Frazzoli, E.: Decentralized Policies for Geometric Pattern Formation and Path Coverage. J. Dyn. Syst. Measur. Control 129(5), 633–643 (2007)CrossRefGoogle Scholar
  26. 26.
    Razumikhin, B.S.: On the stability of systems with a delay. Prikl. Mat. Mekh. 20(4), 500–512 (1956)Google Scholar
  27. 27.
    Richard, J.P.: Time-delay systems: an overview of some recent advances and open problems. Automatica 39 (10), 1667–1694 (2003)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Sastry, S.: Nonlinear systems: analysis, stability, and control, vol. 10. Springer Science & Business Media (2013)Google Scholar
  29. 29.
    Tang, S., Shinzaki, D., Lowe, G.C., Clark, C.M.: Multi-robot control for circumnavigation of particle distributions. Springer Tracts Adv. Robot. 104, 149–152 (2014)CrossRefGoogle Scholar
  30. 30.
    Wang, C., Xie, G., Cao, M.: Forming Circle Formations of Anonymous Mobile Agents With Order Preservation. IEEE Trans. Autom. Control 58(12), 3248–3254 (2013)CrossRefGoogle Scholar
  31. 31.
    Wang, C., Xie, G., Cao, M.: Controlling anonymous mobile agents with unidirectional locomotion to form formations on a circle. Automatica 50(4), 1100–1108 (2014)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Wang, X., Zeng, Z., Cong, Y.: Multi-agent distributed coordination control: Developments and directions via graph viewpoint. Neurocomputing 199, 204–218 (2016). CrossRefGoogle Scholar
  33. 33.
    Xiao, J., Lu, H., Zeng, Z., Xiong, D., Yu, Q., Huang, K., Cheng, S., Yang, X., Dai, W., Ren, J., et al.: Nubot team description paper 2015. Proceedings of RoboCup (2015)Google Scholar
  34. 34.
    Xiao, J., Xiong, D., Yao, W., Yu, Q., Lu, H., Zheng, Z.: Building Software System and Simulation Environment for Robocup Msl Soccer Robots Based on Ros and Gazebo. In: Robot Operating System (ROS), pp. 597–631. Springer (2017)Google Scholar
  35. 35.
    Xiong, D., Xiao, J., Lu, H., Zeng, Z., Yu, Q., Huang, K., Yi, X., Zheng, Z.: The design of an intelligent soccer-playing robot. Ind. Robot. 43, 91–102 (2016). CrossRefGoogle Scholar
  36. 36.
    Yao, W., Dai, W., Xiao, J., Lu, H., Zheng, Z.: A Simulation System Based on Ros and Gazebo for Robocup Middle Size League. In: 2015 IEEE International Conference On Robotics and Biomimetics (ROBIO), pp. 54–59. IEEE (2015)Google Scholar
  37. 37.
    Yao, W., Kapitanyuk, Y.A., Cao, M.: Robotic Path Following in 3D Using a Guiding Vector Field. In: 2018 IEEE Conference on Decision and Control (CDC), pp. 4475–4480. IEEE (2018)Google Scholar
  38. 38.
    Yao, W., Lu, H., Zeng, Z., Xiao, J., Zheng, Z.: Distributed static and dynamic circumnavigation control with arbitrary spacings for a heterogeneous multi-robot system. Journal of Intelligent & Robotic Systems 94 (3-4), 883–905 (2019)CrossRefGoogle Scholar
  39. 39.
    Yao, W., Luo, S., Lu, H., Xiao, J.: Distributed circumnavigation control with dynamic spacings for a heterogeneous multi-robot system 2018. RoboCup Symposium (2018)Google Scholar
  40. 40.
    Yao, W., Zeng, Z., Wang, X., Lu, H., Zheng, Z.: Distributed Encirclement Control with Arbitrary Spacing for Multiple Anonymous Mobile Robots. In: 2017 Chinese Control Conference (CCC). Dalian, China (2017)Google Scholar
  41. 41.
    Yi, S., Nelson, P.W., Ulsoy, A.G.: Time-delay systems: analysis and control using the Lambert W function. World Scientific (2010)Google Scholar
  42. 42.
    Yu, X., Liu, L.: Distributed circular formation control of ring-networked nonholonomic vehicles. Automatica 68, 92–99 (2016). MathSciNetCrossRefGoogle Scholar
  43. 43.
    Zheng, R., Lin, Z., Fu, M., Sun, D.: Distributed Circumnavigation by Unicycles with Cyclic Repelling Strategies. In: 2013 9Th Asian Control Conference (ASCC), pp. 1–6. IEEE (2013)Google Scholar
  44. 44.
    Zheng, R., Lin, Z., Fu, M., Sun, D.: Distributed control for uniform circumnavigation of ring-coupled unicycles. Automatica 53, 23–29 (2015). MathSciNetCrossRefGoogle Scholar
  45. 45.
    Zheng, R., Liu, Y., Sun, D.: Enclosing a target by nonholonomic mobile robots with bearing-only measurements. Automatica 53, 400–407 (2015)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Zhou, Z., Yao, W., Ma, J., Lu, H., Xiao, J., Zheng, Z.: Simatch: a Simulation System for Highly Dynamic Confrontations between Multi-Robot Systems. In: 2018 Chinese Automation Congress (CAC), pp. 3934–3939. IEEE (2018)Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Robotics Research Center, College of Intelligence Science and TechnologyNational University of Defense TechnologyChangshaChina
  2. 2.Department of Control Science and EngineeringHarbin Institute of TechnologyHarbinPeople’s Republic of China

Personalised recommendations