The Reach-Avoid Problem for Constant-Rate Multi-mode Systems

  • Shankara Narayanan Krishna
  • Aviral Kumar
  • Fabio Somenzi
  • Behrouz Touri
  • Ashutosh TrivediEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10482)


A constant-rate multi-mode system is a hybrid system that can switch freely among a finite set of modes, and whose dynamics is specified by a finite number of real-valued variables with mode-dependent constant rates. Alur, Wojtczak, and Trivedi have shown that reachability problems for constant-rate multi-mode systems for open and convex safety sets can be solved in polynomial time. In this paper we study the reachability problem for non-convex state spaces, and show that this problem is in general undecidable. We recover decidability by making certain assumptions about the safety set. We present a new algorithm to solve this problem and compare its performance with the popular sampling based algorithm rapidly-exploring random tree (RRT) as implemented in the Open Motion Planning Library (OMPL).


  1. 1.
    Alur, R., Courcoubetis, C., Henzinger, T.A., Ho, P.-H.: Hybrid automata: an algorithmic approach to the specification and verification of hybrid systems. In: Grossman, R.L., Nerode, A., Ravn, A.P., Rischel, H. (eds.) HS 1991-1992. LNCS, vol. 736, pp. 209–229. Springer, Heidelberg (1993). doi: 10.1007/3-540-57318-6_30 CrossRefGoogle Scholar
  2. 2.
    Alur, R., Dill, D.: A theory of timed automata. Theoret. Comput. Sci. 126, 183–235 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alur, R., Forejt, V., Moarref, S., Trivedi, A.: Safe schedulability of bounded-rate multi-mode systems. In: HSCC, pp. 243–252 (2013)Google Scholar
  4. 4.
    Alur, R., Trivedi, A., Wojtczak, D.: Optimal scheduling for constant-rate multi-mode systems. In: HSCC, pp. 75–84 (2012)Google Scholar
  5. 5.
    Asarin, E., Oded, M., Pnueli, A.: Reachability analysis of dynamical systems having piecewise-constant derivatives. TCS 138, 35–66 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Branicky, M.S., Borkar, V.S., Mitter, S.K.: A unified framework for hybrid control: model and optimal control theory. Autom. Control 43(1), 31–45 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Clarke, E., Biere, A., Raimi, R., Zhu, Y.: Bounded model checking using satisfiability solving. Formal Methods Syst. Des. 19(1), 7–34 (2001)CrossRefzbMATHGoogle Scholar
  8. 8.
    de Moura, L., Bjørner, N.: Z3: an efficient SMT solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-78800-3_24 CrossRefGoogle Scholar
  9. 9.
    Firby, R.J.: Adaptive execution in complex dynamic worlds. Ph.D. thesis, Yale University, New Haven, CT, USA (1989). AAI9010653Google Scholar
  10. 10.
    Frazzoli, E., Dahleh, M.A., Feron, E.: Robust hybrid control for autonomous vehicle motion planning. In: Proceedings of the 39th IEEE Conference on Decision and Control, vol. 1, pp. 821–826. IEEE (2000)Google Scholar
  11. 11.
    Gat, E.: Three-layer architectures. In: Kortenkamp, D., Bonasso, R.P., Murphy, R. (eds.) Artificial Intelligence and Mobile Robots, pp. 195–210. MIT Press, Cambridge (1998)Google Scholar
  12. 12.
    Henzinger, T.A.: The theory of hybrid automata. In: LICS 1996, Washington, DC, USA, p. 278. IEEE Computer Society (1996)Google Scholar
  13. 13.
    Henzinger, T.A., Kopke, P.W.: Discrete-time control for rectangular hybrid automata. TCS 221(1–2), 369–392 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Henzinger, T.A., Kopke, P.W., Puri, A., Varaiya, P.: What’s decidable about hybrid automata? J. Comput. Syst. Sci. 57, 94–124 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kato, S., Takeuchi, E., Ishiguro, Y., Ninomiya, Y., Takeda, K., Hamada, T.: An open approach to autonomous vehicles. IEEE Micro 35(6), 60–68 (2015)CrossRefGoogle Scholar
  16. 16.
    Krishna, S.N., Kumar, A., Somenzi, F., Touri, B., Trivedi, A.: The reach-avoid problem for constant-rate multi-mode systems. CoRR, abs/1707.04151 (2017)Google Scholar
  17. 17.
    Latombe, J.: Robot Motion Planning, vol. 124. Springer, Heidelberg (2012)Google Scholar
  18. 18.
    LaValle, S.M.: Planning Algorithms. Cambridge University Press, Cambridge (2006).
  19. 19.
    Minsky, M.L.: Computation: Finite and Infinite Machines. Prentice-Hall, Inc., Upper Saddle River (1967)zbMATHGoogle Scholar
  20. 20.
    O’Kelly, M., Abbas, H., Gao, S., Shiraishi, S., Kato, S., Mangharam, R.: Apex: a tool for autonomous vehicle plan verification and execution. In: Society of Automotive Engineers (SAE) World Congress and Exhibition (2016)Google Scholar
  21. 21.
    Saha, I., Ramaithitima, R., Kumar, V., Pappas, G.J., Seshia, S.A.: Implan: scalable incremental motion planning for multi-robot systems. In: ICCPS 2016, pp. 43:1–43:10 (2016)Google Scholar
  22. 22.
    Schwartz, J.T., Sharir, M.: On the “piano movers” problem. II. general techniques for computing topological properties of real algebraic manifolds. Adv. Appl. Math. 4(3), 298–351 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Şucan, I.A., Moll, M., Kavraki, L.E.: The open motion planning library. IEEE Robot. Autom. Mag. 19(4), 72–82 (2012). CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Shankara Narayanan Krishna
    • 1
  • Aviral Kumar
    • 1
  • Fabio Somenzi
    • 2
  • Behrouz Touri
    • 2
  • Ashutosh Trivedi
    • 2
    Email author
  1. 1.Indian Institute of Technology BombayMumbaiIndia
  2. 2.University of Colorado BoulderBoulderUSA

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