Autonomous Underwater Vehicles: Singular Extremals and Chattering

  • M. Chyba
  • T. Haberkorn
Part of the IFIP International Federation for Information Processing book series (IFIPAICT, volume 202)


In this paper, we consider the time minimal problem for an Autonomous Underwater Vehicle. We investigate, on a simplified model, the existence of singular extremals and discuss their optimality status. Moreover, we prove that singular extremals corresponding to the angular acceleration are of order 2. We produce in this case a semi-canonical form of our Hamiltonian system and we can conclude the existence of chattering extremals..


Underwater Vehicles Time optimal Singular Extremals Chattering 


  1. [1]
    M. Chyba, N.E. Leonard, E.D. Sontag. Singular trajectories in the multi-input time-optimal problem: Application to controlled mechanical systems. Journal on Dynamical and Control Systems 9(l):73–88, 2003.MathSciNetGoogle Scholar
  2. [2]
    M. Chyba, N.E. Leonard, E.D. Sontag). Optimality for underwater vehicles. In Proceedings of the 40th IEEE Conf. on Decision and Control, Orlando, 2001.Google Scholar
  3. [3]
    M. Chyba. Underwater vehicles: a surprising non time-optimal path. In 42th IEEE Conf. on Decision and Control, Maui 2003.Google Scholar
  4. [4]
    M. Chyba, H. Maurer, H.J. Sussmann, G. Vossen. Underwater Vehicles: The Minimum Time Problem. In Proceedings of the 43th IEEE Conf. on Decision and Control, Bahamas, 2004.Google Scholar
  5. [5]
    J.P. Mcdanell and W.F. Powers. Necessary conditions for joining optimal singular and nonsingular subarcs. SIAM J. Control, 4(2): 161–173, 1971.MathSciNetCrossRefGoogle Scholar
  6. [6]
    R. Fourer, D.M. Gay, B.W. Kernighan. AMPL: A Modeling Language for Mathematical Programming. Duxbury Press, Brooks-Cole Publishing Company, 1993.Google Scholar
  7. [7]
    T.I. Fossen. Guidance and control of ocean vehicles. Wiley, New York, 1994Google Scholar
  8. [8]
    A.T. Fuller. Study of an optimum nonlinear control system. J. Electronics Control, 15:63–71, 1963.MathSciNetGoogle Scholar
  9. [9]
    L.S. Pontryagin, B. Boltyanski, R. Gamkrelidze, E. Michtchenko. The Mathematical Theory of Optimal Processes. Interscience, New-York, 1962.MATHGoogle Scholar
  10. [10]
    H.M. Robbins. A generalized Legendre-Clebsh condition for the singular cases of optimal control. 1BMJ. Res. Develop 11:361–372, 1967.MATHCrossRefGoogle Scholar
  11. [11]
    E.D. Sontag, H.J. Sussmann. Time-Optimal Control of Manipulators. In IEEE Int. Conf on Robotics and Automation., San Francisco: 1962–1697, 1986.Google Scholar
  12. [12]
    A. Waechter, L. T. Biegler. On the Implementation of an Interior-Point Filter-Line Search Algorithm for Large-Scale Nonlinear Programming. Research Report RC 23149, IBM T.J. Watson Research Center, Yorktown, New-York.Google Scholar
  13. [13]
    M.I. Zelikin, V.F. Borisov. Theory of Chattering Control. Birkhäuser, Boston, 1994.Google Scholar

Copyright information

© International Federation for Information Processing 2006

Authors and Affiliations

  • M. Chyba
    • 1
  • T. Haberkorn
    • 1
  1. 1.Department of MathematicsUniversity of HawaiiHonoluluHawaii

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