Herding by caging: a formation-based motion planning framework for guiding mobile agents


We propose a solution to the problem of herding by caging: given a set of mobile robots (called herders) and a group of moving agents (called sheep), we guide the sheep to a target location without letting them escape from the herders along the way. We model the interaction between the herders and the sheep by defining virtual “repulsive forces” pushing the sheep away from the herders. This enables the herders to partially control the motion of the sheep. We formalize this behavior topologically by applying the notion of caging, a concept used in robotic manipulation. We demonstrate that our approach is provably correct in the sense that the sheep cannot escape from the robots under our assumed motion model. We propose an RRT-based path planning algorithm for herding by caging, demonstrate its probabilistic completeness, and evaluate it in simulations as well as on a group of real mobile robots.

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  1. 1.

    The number of robots n does not depend on m, but, in our work, should not be smaller than 3 in the case of a 2D workspace. Robots are all the same and indistinguishable to the sheep.

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    By interior of a closed curve in \({\mathbb {R}}^2\) we mean the union of bounded connected components in its complement. Note that the term “connected components” is used in the topological sense, and a closed curve can potentially have self-intersections.

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    We assume that the robots do not move slower than at their maximum speed until they reach their final positions, so we assume that it is fixed and equal to \({\mathrm {v}}_{{\mathrm {r}}}\).

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    We compute homology with coefficients in \({\mathbb {Z}}_2\).

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  1. Bacon, M., & Olgac, N. (2012). Swarm herding using a region holding sliding mode controller. Journal of Vibration and Control, 18(7), 1056–1066.

    MathSciNet  Article  Google Scholar 

  2. Balch, T., & Arkin, R. C. (1998). Behavior-based formation control for multirobot teams. IEEE Transactions on Robotics and Automation, 14(6), 926–939.

    Article  Google Scholar 

  3. Beard, R. W., Lawton, J., & Hadaegh, F. Y. (2001). A coordination architecture for spacecraft formation control. IEEE Transactions on Control Systems Technology, 9(6), 777–790.

    Article  Google Scholar 

  4. Bhattacharya, S., Kim, S., Heidarsson, H., Sukhatme, G. S., & Kumar, V. (2015). A topological approach to using cables to separate and manipulate sets of objects. The International Journal of Robotics Research, 34(6), 799–815.

    Article  Google Scholar 

  5. Butler, Z., Corke, P., Peterson, R., & Rus, D. (2004). Virtual fences for controlling cows. In International conference on robotics and automation (Vol. 5, pp. 4429–4436). New York: IEEE.

  6. Cortes, J., Martinez, S., Karatas, T., & Bullo, F. (2004). Coverage control for mobile sensing networks. IEEE Transactions on Robotics and Automation, 20(2), 243–255.

    Article  Google Scholar 

  7. Dantam, N. T., Kingston, Z. K., Chaudhuri, S., & Kavraki, L. E. (2018). An incremental constraint-based framework for task and motion planning. The International Journal of Robotics Research, 37(10), 1134–1151.

    Article  Google Scholar 

  8. Durrant-Whyte, H., Roy, N., & Abbeel, P. (2012). Controlling wild bodies using linear temporal logic. Robotics: Science and Systems, 7, 17–24.

    Google Scholar 

  9. Edelsbrunner, H., & Harer, J. (2010). Computational topology: An introduction. New York: American Mathematical Society.

    MATH  Google Scholar 

  10. Egerstedt, M., & Hu, X. (2001a). Formation constrained multi-agent control. IEEE Transactions on Robotics and Automation, 17(6), 947–951.

    Article  Google Scholar 

  11. Egerstedt, M., & Hu, X. (2001b). Formation constrained multi-agent control. IEEE transactions on Robotics and Automation, 17(6), 947–951.

    Article  Google Scholar 

  12. Elamvazhuthi, K., Kakish, Z., Shirsat, A., & Berman, S. (2020). Controllability and stabilization for herding a robotic swarm using a leader: A mean-field approach. IEEE Transactions on Robotics, 2020, 1–15.

    Article  Google Scholar 

  13. Ferrari-Trecate, G., Egerstedt, M., Buffa, A., & Ji, M. (2006). Laplacian sheep: A hybrid, stop-go policy for leader-based containment control. In International workshop on hybrid systems: Computation and control (pp. 212–226). London: Springer.

  14. Fine, B. T., & Shell, D. A. (2013). Eliciting collective behaviors through automatically generated environments. In 2013 IEEE/RSJ international conference on intelligent robots and systems (pp. 3303–3308).

  15. Fink, J., Hsieh, M. A., & Kumar, V. (2008). Multi-robot manipulation via caging in environments with obstacles. In International conference on robotics and automation (pp. 1471–1476). New York: IEEE.

  16. Garrell, A., Sanfeliu, A., & Moreno-Noguer, F. (2009). Discrete time motion model for guiding people in urban areas using multiple robots. In International conference on intelligent robots and systems. New York: IEEE.

  17. Garrido, S., Moreno, L., & Lima, P. U. (2011). Robot formation motion planning using fast marching. Robotics and Autonomous Systems, 59(9), 675–683.

    Article  Google Scholar 

  18. Gazi, V., & Passino, K. M. (2004). A class of attractions/repulsion functions for stable swarm aggregations. International Journal of Control, 77(18), 1567–1579.

    MathSciNet  Article  Google Scholar 

  19. LaValle, S. M., & Kuffner, J. J., Jr. (2001). Randomized kinodynamic planning. The International Journal of Robotics Research, 20(5), 378–400.

    Article  Google Scholar 

  20. Lee, W., & Kim, D. (2017). Autonomous shepherding behaviors of multiple target steering robots. Sensors (Basel, Switzerland), 17, 2729.

    Article  Google Scholar 

  21. Lien, J. M., Rodriguez, S., Malric, J. P., & Amato, N. M. (2005). Shepherding behaviors with multiple shepherds. In International conference on robotics and automation (pp. 3402–3407).

  22. Lu, Z. (2010). Cooperative optimal path planning for herding problems. Ph.D. thesis, Texas A&M University.

  23. Mahler, J., Pokorny, F. T., McCarthy, Z., van der Stappen, A. F., & Goldberg, K. (2016a). Energy-bounded caging: Formal definition and 2-D energy lower bound algorithm based on weighted alpha shapes. IEEE Robotics and Automation Letters, 1(1), 508–515.

    Article  Google Scholar 

  24. Mahler, J., Pokorny, F. T., Niyaz, S., & Goldberg, K. (2016b). Synthesis of energy-bounded planar caging grasps using persistent homology. In Workshop on the algorithmic foundations of robotics.

  25. Makita, S., & Wan, W. (2017). A survey of robotic caging and its applications. Advanced Robotics, 31(19–20), 1071–1085.

    Article  Google Scholar 

  26. Pereira, G. A. S., Das, A. K., Kumar, R. V., & Campos, M. F. M. (2003). Decentralized motion planning for multiple robots subject to sensing and communication constraints. Departmental Papers (MEAM) University of Pennsylvania.

  27. Pereira, G. A. S., Campos, M. F. M., & Kumar, V. (2004). Decentralized algorithms for multi-robot manipulation via caging. The International Journal of Robotics Research, 23(7–8), 783–795.

    Article  Google Scholar 

  28. Pierson, A., & Schwager, M. (2018). Controlling noncooperative herds with robotic herders. IEEE Transactions on Robotics, 34(2), 517–525.

    Article  Google Scholar 

  29. Rodriguez, A., Mason, M. T., & Ferry, S. (2012). From caging to grasping. The International Journal of Robotics Research, 31(7), 886–900.

    Article  Google Scholar 

  30. Schultz, A., Grefenstette, J. J., & Adams, W. (1996). Robo-shepherd: Learning complex robotic behaviors. In Robotics and manufacturing: Recent trends in research and applications (Vol. 6, pp. 763–768). New York: ASME Press.

  31. Shedied, S. A. (2002). Optimal control for a two player dynamic pursuit evasion game: The herding problem. Ph.D. thesis, Virginia Tech.

  32. Strömbom, D., Mann, R. P., Wilson, A. M., Hailes, S., Morton, A. J., Sumpter, D., & King, A. J. (2014). Solving the shepherding problem: Heuristics for herding autonomous, interacting agents. Journal of The Royal Society Interface, 11(100), 20140719.

    Article  Google Scholar 

  33. Tanner, H. G. (2004). ISS properties of nonholonomic vehicles. Systems and Control Letters, 53(3–4), 229–235.

    MathSciNet  Article  Google Scholar 

  34. Tanner, H. G., Jadbabaie, A., & Pappas, G. J. (2007). Flocking in fixed and switching networks. IEEE Transactions on Automatic control, 52(5), 863–868.

    MathSciNet  Article  Google Scholar 

  35. Varava, A., Hang, K., Kragic, D., & Pokorny, F. T. (2017). Herding by caging: A topological approach towards guiding moving agents via mobile robots. In Proceedings of robotics: science and systems.

  36. Vaughan, R., Sumpter, N., Henderson, J., Frost, A., & Cameron, s. (2000). Experiments in automatic flock control. Robotics and Autonomous Systems, 31(1–2), 109–117.

    Article  Google Scholar 

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This research work was supported by the Innovation and Technology Fund of the Government of the Hong Kong Special Administrative Region (Project No. ITS/104/19FP), Knut and Alice Wallenberg Foundation, Swedish Research Council, and European Research Council (Project 884807 BIRD).

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Correspondence to Haoran Song.

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This is one of the several papers published in Autonomous Robots comprising the Special Issue on Topological Methods in Robotics.

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Song, H., Varava, A., Kravchenko, O. et al. Herding by caging: a formation-based motion planning framework for guiding mobile agents. Auton Robot (2021). https://doi.org/10.1007/s10514-021-09975-8

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  • Topological representation and abstraction of configuration spaces
  • Computational geometry
  • Path planning for multiple mobile robots or agents
  • Motion and path planning