Autonomous Robots

, Volume 42, Issue 7, pp 1389–1404 | Cite as

On the relationship between dynamics and complexity in multi-agent collision avoidance

  • Jeffrey Kane Johnson
Part of the following topical collections:
  1. Special Issue on Robotics: Science and Systems 2016


This work examines how dynamics and complexity are related in multi-agent collision avoidance. Motivated particularly by work in the field of automated driving, this work considers a variant of the reciprocal n-body collision avoidance problem. In this problem, agents must avoid collision while moving according to individual reward functions in a crowded environment. The main contribution of this work is the result that there is a quantifiable relationship between system dynamics and the requirement for agent coordination, and that this requirement can change the complexity class of the problem dramatically: from P to NEXP or even \(\hbox {NEXP}^{\text {NP}}\). A constructive proof is provided that demonstrates the relationship, and potential practical applications are discussed.


Complexity Dynamics Collision avoidance 



The author would like to acknowledge the reviewers for providing many insightful comments and corrections, as well as Philipp Robbel and Elmar Mair for comments and feedback, and Valerie Aquila for assistance with figures and editing.


  1. Abrams, S., & Allen, P. K. (2000). Computing swept volumes. Journal of Visualization and Computer Animation, 11(2), 69–82.;2-7.CrossRefGoogle Scholar
  2. Allen, R. E., Clark, A. A., Starek, J. A., & Pavone, M. (2014). A machine learning approach for real-time reachability analysis. In 2014 IEEE/RSJ international conference on intelligent robots and systems (pp. 2202–2208), Chicago, IL, USA, 14–18 September 2014. IEEE.
  3. Alonso-Mora, J., Breitenmoser, A., Rufli, M., Beardsley, P. A., & Siegwart, R. (2010). Optimal reciprocal collision avoidance for multiple non-holonomic robots. In A. Martinoli, F. Mondada, N. Correll, G. Mermoud, M. Egerstedt, M. A. Hsieh, L. E. Parker & K. Støy (Eds.), The 10th international symposium distributed autonomous robotic systems, DARS 2010, Lausanne, Switzerland, 1–3 November 2010, Springer tracts in advanced robotics (Vol. 83, pp. 203–216). Springer.
  4. Alterovitz, R., Siméon, T., & Goldberg, K. Y. (2007). The stochastic motion roadmap: A sampling framework for planning with markov motion uncertainty. In W. Burgard , O. Brock & C. Stachniss (Eds.), Robotics: Science and systems III, 27–30 June 2007, Georgia Institute of Technology, Atlanta, Georgia, USA. The MIT Press.
  5. Bacha, A., Bauman, C., Faruque, R., Fleming, M., Terwelp, C., Reinholtz, C. F., et al. (2008). Odin: Team victortango’s entry in the DARPA urban challenge. Journal of field Robotics, 25(8), 467–492. Scholar
  6. Baek, N., Shin, S. Y., & Chwa, K. (1999). On computing translational swept volumes. International Journal of Computational Geometry & Applications, 9(3), 293–317. Scholar
  7. Bekris, K. E. (2010). Avoiding inevitable collision states: Safety and computational efficiency in replanning with sampling-based algorithms. In International conference on robotics and automation (ICRA-10). Accessed 15 Feb 2017.
  8. Bekris, K. E., Grady, D. K., Moll, M., & Kavraki, L. E. (2012). Safe distributed motion coordination for second-order systems with different planning cycles. The International Journal of Robotics Research, 31(2), 129–150. Scholar
  9. Bernstein, D. S., Givan, R., Immerman, N., & Zilberstein, S. (2002). The complexity of decentralized control of Markov decision processes. Mathematics of Operations Research, 27(4), 819–840. Scholar
  10. Blass, A., & Gurevich, Y. (1982). On the unique satisfiability problem. Information and Control, 55(1–3), 80–88. Scholar
  11. Boutilier, C. (1996). Planning, learning and coordination in multiagent decision processes. In Y. Shoham (Ed.), Proceedings of the sixth conference on theoretical aspects of rationality and knowledge (pp. 195–210). De Zeeuwse Stromen, The Netherlands, 17–20 March 1996, Morgan Kaufmann.Google Scholar
  12. Chazelle, B., & Dobkin, D. P. (1980). Detection is easier than computation (extended abstract). In R. E. Miller, S. Ginsburg, W. A. Burkhard & R. J. Lipton (Eds.), Proceedings of the 12th annual ACM symposium on theory of computing (pp. 146–153), 28–30 April 1980. Los Angeles, California, USA, ACM.
  13. Daskalakis, K., & Papadimitriou, C. H. (2005). The complexity of games on highly regular graphs. In G. S. Brodal & S. Leonardi (Eds.), 13th annual European symposium algorithms—ESA 2005 (Vol. 3669, pp. 71–82), Palma de Mallorca, Spain, 3–6 October 2005, Proceedings, Springer, Lecture Notes in Computer Science.
  14. Eberly, D. (2008). Intersection of convex objects: The method of separating axes. Accessed 2 Jan 2017.
  15. Erickson, L. H., & LaValle, S. M. (2013). A simple, but NP-hard, motion planning problem. In M. desJardins & M. L. Littman (Eds.), Proceedings of the twenty-seventh AAAI conference on artificial intelligence, 14–18 July 2013, Bellevue, Washington, USA. AAAI Press. Accessed 15 Feb 2017.
  16. Ericson, C. (2005). Real-time collision detection. In Morgan Kaufmann series in interactive 3D technology. Amsterdam: Elsevier.
  17. Fiorini, P., & Shiller, Z. (1998). Motion planning in dynamic environments using velocity obstacles. The International Journal of Robotics Research, 17(7), 760–772. Scholar
  18. Fraichard, T., & Asama, H. (2004). Inevitable collision states—A step towards safer robots? Advanced Robotics, 18(10), 1001–1024. Scholar
  19. Gilbert, E. G., Johnson, D. W., & Keerthi, S. S. (1988). A fast procedure for computing the distance between complex objects in three-dimensional space. IEEE Journal on Robotics and Automation, 4(2), 193–203. Scholar
  20. Goldsmith, J., & Mundhenk, M. (2007). Competition adds complexity. In J. C. Platt, D. Koller, Y. Singer & S. T. Roweis (Eds.), Advances in neural information processing systems 20, Proceedings of the twenty-first annual conference on neural information processing systems (pp. 561–568), Vancouver, British Columbia, Canada, 3–6 December 2007. Curran Associates, Inc.
  21. Halperin, D., & Sharir, M. (1996). A near-quadratic algorithm for planning the motion of a polygon in a polygonal environment. Discrete & Computational Geometry, 16(2), 121–134. Scholar
  22. Harding, J., Powell, G., Yoon, R., Fikentscher, J., Doyle, C., Sade, D., Lukuc, M., Simons, J., & Wang, J. (2014). Vehicle-to-vehicle communications: Readiness of V2V technology for application. Technical Report DOT HS 812 014, U.S. Department of Transportation, National Highway Transportation Safety Administration, National Highway Traffic Safety Administration, 1200 New Jersey Avenue SE. Washington, DC 20590. Accessed 15 Feb 2017.
  23. Hauser, K. K. (2012). The minimum constraint removal problem with three robotics applications. In E. Frazzoli, T. Lozano-Pérez, N. Roy & D. Rus (Eds.), Algorithmic foundations of robotics X-proceedings of the tenth workshop on the algorithmic foundations of robotics, WAFR 2012 (Vol. 86, pp. 1–17). MIT, Cambridge, Massachusetts, USA, 13–15 June 2012. Springer Tracts in Advanced Robotics. Springer.
  24. Hopcroft, J., Schwartz, J., & Sharir, M. (1984). On the complexity of motion planning for multiple independent objects; PSPACE-hardness of the “Warehouseman’s Problem”. The International Journal of Robotics Research, 3(4), 76–88.
  25. Hornung, A., Wurm, K. M., Bennewitz, M., Stachniss, C., & Burgard, W. (2013). OctoMap: An efficient probabilistic 3D mapping framework based on octrees. Autonomous Robots.,, software available at
  26. Jansson, J. (2005). Collision avoidance theory: with application to automotive collision mitigation. Linköping studies in science and technology: Dissertations, Department of Electrical Enginering, University.
  27. Jiménez, P., Thomas, F., & Torras, C. (2001). 3D collision detection: A survey. Computers & Graphics, 25(2), 269–285. Scholar
  28. Johnson, J. K. (2016). A novel relationship between dynamics and complexity in multi-agent collision avoidance. In Proceedings of robotics: Science and systems. Ann Arbor, Michigan.
  29. Johnson, J., & Hauser, K. K. (2012). Optimal acceleration-bounded trajectory planning in dynamic environments along a specified path. In IEEE international conference on robotics and automation, ICRA 2012 (pp. 2035–2041), 14–18 May 2012, St. Paul, Minnesota, USA. IEEE.
  30. Kaelbling, L. P., & Lozano-Pérez, T. (2013). Integrated task and motion planning in belief space. The International Journal of Robotics Research, 32(9–10), 1194–1227. Scholar
  31. Kamat, V. V. (1993). A survey of techniques for simulation of dynamic collision detection and response. Computers & Graphics, 17(4), 379–385. Scholar
  32. Kambhampati, S., Cutkosky, M. R., Tenenbaum, M., & Lee, S. H. (1991). Combining specialized reasoners and general purpose planners: A case study. In T. L. Dean & K. McKeown (Eds.), Proceedings of the 9th national conference on artificial intelligence, Anaheim, CA, USA (Vol. 1, pp. 199–205), 14–19 July 1991. AAAI Press/The MIT Press.
  33. Kim, Y. J., Varadhan, G., Lin, M. C., & Manocha, D. (2004). Fast swept volume approximation of complex polyhedral models. Computer-Aided Design, 36(11), 1013–1027. Scholar
  34. LaValle, S. M. (2006). Planning algorithms. Cambridge: Cambridge University Press.CrossRefzbMATHGoogle Scholar
  35. LaValle, S. M., & Kuffner, J. J. (2001). Randomized kinodynamic planning. The International Journal of Robotics Research, 20(5), 378–400. Scholar
  36. Lin, M. C., & Canny, J. F. (1991). A Fast Algorithm for incremental distance calculation. In IEEE international conference on robotics and automation (pp. 1008–1014). Accessed 15 Feb 2017.
  37. Littman, M. L., Dean, T. L., & Kaelbling, L. P. (1995) On the complexity of solving Markov decision problems. In UAI ’95: Proceedings of the eleventh annual conference on uncertainty in artificial intelligence (pp. 394–402), Montreal, Quebec, Canada, 18–20 August 1995. Accessed 15 Feb 2017.
  38. Mazer, E., Ahuactzin, J. M., & Bessière, P. (1998). The ariadne’s clew algorithm. Journal of Artificial Intelligence Research (JAIR), 9, 295–316. Scholar
  39. Mitter, S., & Sahai, A. (1999). Information and control: Witsenhausen revisited (pp. 281–293). London: Springer. Scholar
  40. Montemerlo, M., Becker, J., Bhat, S., Dahlkamp, H., Dolgov, D., Ettinger, S., et al. (2008). Junior: The stanford entry in the urban challenge. Journal of field Robotics, 25(9), 569–597. Scholar
  41. Ó’Dúnlaing, C. (1987). Motion planning with inertial constraints. Algorithmica, 2, 431–475. Scholar
  42. Oliehoek, F. A., Witwicki, S. J., & Kaelbling, L. P. (2012). Influence-based abstraction for multiagent systems. In Proceedings of the twenty-sixth AAAI conference on artificial intelligence, 22–26 July 2012, Toronto, Ontario, Canada.
  43. Paden, B., Cáp, M., Yong, S. Z., Yershov, D. S., & Frazzoli, E. (2016). A survey of motion planning and control techniques for self-driving urban vehicles. CoRR
  44. Papadimitriou, C. H., & Tsitsiklis, J. N. (1987). The complexity of markov decision processes. Mathematics of Operations Research, 12(3), 441–450.
  45. Petti, S., & Fraichard, T. (2005). Safe motion planning in dynamic environments. In 2005 IEEE/RSJ international conference on intelligent robots and systems (pp. 2210–2215), Edmonton, Alberta, Canada, 2–6 August 2005. IEEE.
  46. Reif, J. H. (1979). Complexity of the mover’s problem and generalizations (extended abstract). In 20th annual symposium on foundations of computer science (pp. 421–427), San Juan, Puerto Rico, 29–31 October 1979. IEEE Computer Society.
  47. Reif, J. H., & Sharir, M.(1985). Motion planning in the presence of moving obstacles. In 26th annual symposium on foundations of computer science (pp. 144–154), Portland, Oregon, USA, 21–23 October 1985. IEEE Computer Society.
  48. Schoettle, B., & Sivak, M. (2015). A preliminary analysis of real-world crashes involving self-driving vehicles. Technical report, The University of Michigan Transportation Research Institute.Google Scholar
  49. Shamos, M. (1978). Computational geometry. Ph.D. thesis, Yale University.Google Scholar
  50. Shiller, Z., Gal, O., & Fraichard, T. (2010). The nonlinear velocity obstacle revisited: The optimal time horizon. In Guaranteeing safe navigation in dynamic environments workshop, Anchorage, United States. Accessed 15 Feb 2017.
  51. Shoham, Y., & Tennenholtz, M. (1995). On social laws for artificial agent societies: Off-line design. Artificial Intelligence, 73(1–2), 231–252. Scholar
  52. Solovey, K., & Halperin, D. (2015). On the hardness of unlabeled multi-robot motion planning. In Robotics: Science and systems XI, Sapienza University of Rome, Rome, Italy, 13–17 July 2015.
  53. Täubig, H., Bäuml, B., & Frese, U. (2011). Real-time swept volume and distance computation for self collision detection. In 2011 IEEE/RSJ international conference on intelligent robots and systems, IROS 2011 (pp. 1585–1592), San Francisco, CA, USA, 25–30 September 2011. IEEE.
  54. Urmson, C., Anhalt, J., Bagnell, J. A. D., Baker, C. R., Bittner, R. E., Dolan, J. M., Duggins, D., Ferguson, D., Galatali, T., Geyer, H., Gittleman, M., Harbaugh, S., Hebert, M., Howard, T., Kelly, A., Kohanbash, D., Likhachev, M., Miller, N., Peterson, K., Rajkumar, R., Rybski, P., Salesky, B., Scherer, S., Seo, Y. W., Simmons, R,, Singh, S., Snider, J. M., Stentz, A. T., Whittaker, W. R. L., & Ziglar, J. (2007). Tartan racing: A multi-modal approach to the DARPA urban challenge. Technical Report CMU-RI-TR-, Robotics Institute, Pittsburgh, PA. Accessed 15 Feb 2017.
  55. van den Berg, J., Guy, S. J., Lin, M. C., & Manocha, D. (2009). Reciprocal n-body collision avoidance. In C. Pradalier, R. Siegwart & G. Hirzinger (Eds.), The 14th international symposium robotics research, ISRR 2009, August 31–September 3, 2009, Lucerne, Switzerland, Springer tracts in advanced robotics (Vol. 70, pp. 3–19). Springer.
  56. van den Berg J. P., Snape, J., Guy, S. J., & Manocha, D. (2011). Reciprocal collision avoidance with acceleration-velocity obstacles. In IEEE international conference on robotics and automation, ICRA 2011, Shanghai, China, 9–13 May 2011 (pp. 3475–3482). IEEE.
  57. Valtazanos, A., & Ramamoorthy, S. (2011). Online motion planning for multi-robot interaction using composable reachable sets. In T. Röfer, N. M. Mayer, J. Savage, U. Saranli (Eds.), RoboCup 2011: Robot Soccer World Cup XV [papers from the 15th annual RoboCup international symposium, Istanbul, Turkey, July 2011]. Lecture Notes in Computer Science (Vol. 7416, pp. 186–197). Springer.
  58. von Dziegielewski, A., Erbes, R., & Schömer, E. (2010). Conservative swept volume boundary approximation. In G. Elber, A. Fischer, J. Keyser & M. Kim (Eds.), ACM symposium on solid and physical modeling, Proceedings of the 14th ACM symposium on solid and physical modeling, SPM 2010 (pp. 171–176), Haifa, Israel, 1–3 September 2010. ACM.
  59. Weller, R. (2013). New geometric data structures for collision detection and haptics. Springer series on touch and haptic systems. Springer International Publishing. Accessed 15 Feb 2017.
  60. Wilkerson, J. L., Bobinchak, J., Culp, M., Clark, J., Halpin-Chan, T., Estabridis, K., & Hewer, G. (2014). Two-dimensional distributed velocity collision avoidance. Technical Report NAWCWD TP 8786, Physics Division, Research and Intelligence Department, Naval Air Warfare Center Weapons Division, China Lake, CA 93555-6100. Accessed 15 Feb 2017.
  61. Wilkie, D., van den Berg, J. P., & Manocha, D. (2009). Generalized velocity obstacles. In 2009 IEEE/RSJ international conference on intelligent robots and systems (pp. 5573–5578), 11–15 October 2009, St. Louis, MO, USA. IEEE.
  62. Witsenhausen, H. S. (1968). A counterexample in stochastic optimum control. SIAM Journal on Control, 6(1), 131–147. Scholar
  63. Ziegler, J., Bender, P., Dang, T., & Stiller, C. (2014). Trajectory planning for bertha—A local, continuous method. In 2014 IEEE intelligent vehicles symposium proceedings (pp. 450–457), Dearborn, MI, USA, 8–11 June 2014. IEEE.

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Authors and Affiliations

  1. 1.School of Informatics, Computing, and EngineeringIndiana UniversityBloomingtonUSA

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