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Nonlinear Dynamics

, Volume 76, Issue 2, pp 1411–1422 | Cite as

Modeling and analysis of nonholonomic dynamic systems with a class of rheonomous affine constraints

  • Tatsuya Kai
Original Paper

Abstract

This paper is devoted to modeling and theoretical analysis of dynamic control systems subject to a class of rheonomous affine constraints, which are called \(A\)-rheonomous affine constraints. We first define \(A\)-rheonomous affine constraints and explain their geometric representation. Next, a necessary and sufficient condition for complete nonholonomicity of \(A\)-rheonomous affine constraints is shown. Then, we derive nonholonomic dynamic systems with \(A\)-rheonomous affine constraints (NDSARAC), which are included in the class of nonlinear control systems. We also analyze linear approximated systems and accessibility for the NDSARAC. Finally, the results are applied to some physical examples in order to check the application potentiality.

Keywords

Rheonomous affine constraints Rheonomous bracket Nonlinear control systems Nonholonomic dynamic systems Accessibility 

References

  1. 1.
    Neimark, J.I., Fafaev, N.A.: Dynamics of Nonholonomic Systems. American Mathematical Society, Providence (1972)MATHGoogle Scholar
  2. 2.
    Jurdjevic, V.: Geometric Control Theory. Cambridge University Press, Cambridge (1996)CrossRefGoogle Scholar
  3. 3.
    Sastry, S.S.: Nonlinear Systems. Springer-Verlag, New York (1999)CrossRefMATHGoogle Scholar
  4. 4.
    Cortés, J.: Geometric, Control and Numerical Aspects of Nonholonomic Systems. Springer-Verlag, New York (2002)CrossRefMATHGoogle Scholar
  5. 5.
    Bloch, A.M.: Nonholonomic Mechanics and Control. Springer-Verlag, New York (2005)Google Scholar
  6. 6.
    Bullo, F., Rewis, A.D.: Geometric Control of Mechanical Systems. Springer-Verlag, New York (2004)Google Scholar
  7. 7.
    Bloch, A.M., Reyhanoglu, M., McClamroch, H.: Control and stabilization of nonholonomic dynamic systems. IEEE Trans. Autom. Control 37(11), 1746–1757 (1992)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Murray, R.M., Sastry, S.S.: Nonholonomic motion planning: steering using sinusoids. IEEE Trans. Autom. Control 38(5), 700–716 (1993)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Rand, R.H., Ramani, D.V.: Nonlinear normal modes in a system with nonholonomic constraints. Nonlinear Dyn. 25, 49–64 (2001)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Wang, P.: Perturbation to symmetry and adiabatic invariants of discrete nonholonomic nonconservative mechanical system. Nonlinear Dyn. 68, 53–62 (2012)CrossRefMATHGoogle Scholar
  11. 11.
    Chang, C.M., Ge, Z.M.: Complete identification of chaos of nonlinear nonholonomic systems. Nonlinear Dyn. 73, 2103–2109 (2013)CrossRefGoogle Scholar
  12. 12.
    Brockett, R.W.: Asymptotic Stability and Feedback Stabilization. Differential Geometric Control Theory. Birkhäuser, Boston (1983)Google Scholar
  13. 13.
    Kai, T., Kimura, H.: Theoretical analysis of affine constraints on a configuration manifold—part I: integrability and nonintegrability conditions for affine constraints and foliation structures of a configuration manifold. Trans. Soc. Instrum. Control Eng. 42(3), 212–221 (2006)Google Scholar
  14. 14.
    Kai, T., Kimura, H.: Theoretical analysis of affine constraints on a configuration manifold—part II: accessibility of kinematic asymmetric affine control systems with affine constraints. Trans. Soc. Instrum. Control Eng. 42(3), 222–231 (2006)Google Scholar
  15. 15.
    Kai, T.: Integrating algorithms for integrable affine constraints, IEICE Transactions on Fundamentals of Electronics. Commun. Comput. Sci. E94–A(1), 464–467 (2011)Google Scholar
  16. 16.
    Kai, T., Kimura, H., Hara, S.: Nonlinear control analysis on kinematically asymmetrically affine control systems with nonholonomic affine constraints, In: Proceedings of 16th IFAC World Congress, Prague, 4–8 July, Paper No. Mo-M08-TO/5 (2005)Google Scholar
  17. 17.
    Kai, T., Kimura, H., Hara, S.: Nonlinear control analysis on nonholonomic dynamic systems with affine constraints, In: Proceedings of 44th IEEE Conference on Decision and Control and European Control Conference 2005, Seville, pp. 1459–1464 (2005)Google Scholar
  18. 18.
    Kai, T.: Affine constraints in nonlinear control theory, In: Proceedings of 3rd Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Nagoya, pp. 251–256 (2005)Google Scholar
  19. 19.
    Kai, T.: Extended chained forms and their applications to nonholonomic kinematic systems with affine constraints, In: Proceedings of 45th IEEE Conference on Decision and Control, San Diego, pp. 6104–6109 (2006)Google Scholar
  20. 20.
    Kai, T.: Derivation and analysis of nonholonomic Hamiltonian systems with affine constraints, In: Proceedings of European Control Conference 2007, Kos, pp. 4805–4810 (2007)Google Scholar
  21. 21.
    Kai, T.: Generalized canonical transformations and passivity-based control for nonholonomic Hamiltonian systems with affine constraints, In: Proceedings of 46th IEEE Conference on Decision and Control, New Orleans, pp. 3369–3374 (2007)Google Scholar
  22. 22.
    Kai, T.: Mathematical modelling and theoretical analysis of nonholonomic kinematic systems with a class of rheonomous affine constraints. Appl. Math. Modell. 36, 3189–3200 (2012)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Kai, T.: Theoretical analysis for a class of rheonomous affine constraints on configuration manifolds—part I: fundamental properties and integrability/nonintegrability conditions. Math. Prob. Eng. 2012, 543098 (2012)MathSciNetGoogle Scholar
  24. 24.
    Kai, T.: Theoretical analysis for a class of rheonomous affine constraints on configuration manifolds—part II: foliation structures and integrating algorithms. Math. Prob. Eng. 2012, 345942 (2012)MathSciNetGoogle Scholar
  25. 25.
    Nijmeijer, H., Schaft, A.J.van der: Nonlinear Dynamical Control Systems. Springer-Verlag, New York (1990)CrossRefMATHGoogle Scholar
  26. 26.
    Isidori, A.: Nonlinear Control Systems, 3rd edn. Springer-Verlag, New York (1995) Google Scholar
  27. 27.
    Haddad, W.M., Chellaboina, V.: Nonlinear Dynamical Systems and Control. Princeton University Press, Princeton (2008)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Applied Electronics, Faculty of Industrial Science and TechnologyTokyo University of ScienceKatsushika-kuJapan

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