Journal of Mathematical Sciences

, Volume 220, Issue 3, pp 301–317 | Cite as

Two-Point Problem for Systems Satisfying the Controllability Condition with Lie Brackets of the Second Order

  • V. V. Grushkovskaya
  • A. L. Zuev

We study a two-point control problem for systems linear in control. The class of problems under consideration satisfies a controllability condition with Lie brackets up to the second order, inclusively. To solve the problem, we use trigonometric polynomials whose coefficients are computed by expanding the solutions into the Volterra series. The proposed method allows one to reduce the two-point control problem to a system of algebraic equations. It is shown that this algebraic system has (locally) at least one real solution. The proposed method for the construction of control functions is illustrated by several examples.


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  1. 1.
    P. E. Crouch, “Spacecraft attitude control and stabilization: applications of geometric control theory to rigid body models,” IEEE Trans. Automat. Contr., 29, 321–331 (1984).CrossRefzbMATHGoogle Scholar
  2. 2.
    R. W. Brockett, “Control theory and singular Riemannian geometry,” in: P. J. Hilton and G. S. Young (editors), New Directions of Applied Mathematics, Springer, New York (1981), pp. 11–27.Google Scholar
  3. 3.
    L. G. Bushnell, D. M. Tilbury, and S. S. Sastry, “Steering three-input chained form nonholonomic systems using sinusoids: the fire truck example,” Int. J. Robot. Res., 14, 366–381 (1995).CrossRefGoogle Scholar
  4. 4.
    J. F. Carinena, J. Clemente-Gallardo, and A. Ramos, “Motion on Lie groups and its applications in control theory,” Repts. Math. Phys., 51, 159–170 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Y. Chitour, F. Jean, and R. Long, “A global steering method for nonholonomic systems,” J. Different. Equat., 254, 1903–1956 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    T. Chumachenko and A. Zuyev, “Application of the return method to the steering of nonlinear systems,” in: K. Kozłowski (editor), Robot Motion and Control, Springer, Berlin (2009), pp. 83–91.Google Scholar
  7. 7.
    I. Duleba, W. Khefifi, and I. Karcz-Duleba, “Layer, Lie algebraic method of motion planning for nonholonomic systems,” J. Franklin Inst., 349, 201–215 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    V. V. Grushkovskaya and A. L. Zuev, “Approximate solution of the boundary-value problem for controlled systems satisfying the rank condition,” Zb. Prats. Inst. Matem. NAN Ukr., 11, No. 4, 86–101 (2014).zbMATHGoogle Scholar
  9. 9.
    L. Gurvits and Z. X. Li, “Smooth time-periodic feedback solutions for nonholonomic motion planning,” in: Z. Li and J. F. Canny (Eds.), Nonholonomic Motion Planning, Springer, New York (1993), pp. 53–108.CrossRefGoogle Scholar
  10. 10.
    F. Jean, Control of Nonholonomic Systems: from sub-Riemannian Geometry to Motion Planning, Springer, Cham (2014).Google Scholar
  11. 11.
    M. A. Krasnosel’skii and P. P. Zabreiko, Geometric Methods of Nonlinear Analysis, Springer, Berlin (1984).CrossRefGoogle Scholar
  12. 12.
    G. Lafferriere and H. J. Sussmann, “Motion planning for controllable systems without drift,” in: Proc. IEEE Internat. Conf. on Robotics and Automation, Sacramento (1991), IEEE (1991), pp. 1148–1153.Google Scholar
  13. 13.
    Z. Li and J. Canny, “Motion of two rigid bodies with rolling constraint,” IEEE Trans. Robot. Automat., 6, 62–71 (1990).CrossRefGoogle Scholar
  14. 14.
    R. M. Murray and S. S. Sastry, “Steering nonholonomic systems using sinusoids,” in: Proc. 29th IEEE CDC, Honolulu, HI (1990), pp. 2097–2101.Google Scholar
  15. 15.
    H. Nijmeijer and A. J. van der Schaft, Nonlinear Dynamical Control Systems, Springer, New York (1990).CrossRefzbMATHGoogle Scholar
  16. 16.
    W. Liu, “An approximation algorithm for nonholonomic systems,” SIAM J. Contr. Optim., 35, 1328–1365 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    R. Yang, P. S. Krishnaprasad, and W. Dayawansa, “Optimal control of a rigid body with two oscillators,” in: Mech. Day, Fields Inst. Comm., Vol. 7, Amer. Math. Soc., Providence, RI (1996), pp. 233–260.Google Scholar
  18. 18.
    V. I. Zubov, Lectures on the Control Theory [in Russian], Nauka, Moscow (1975).Google Scholar

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  • V. V. Grushkovskaya
    • 1
  • A. L. Zuev
    • 2
  1. 1.Institute of MathematicsUkrainian Natonal Academy of SciencesKievUkraine
  2. 2.Max Planck Institute for Dynamics of Complex Technical SystemsMagdeburgGermany

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