\(\mathcal{H}_{\infty }\) Generation of Periodic Motion of Mechanical Systems of One Degree of Underactuation

  • Yury V. Orlov
  • Luis T. Aguilar
Part of the Systems & Control: Foundations & Applications book series (SCFA)


In contrast to fully actuated systems, in underactuated systems, the presence of unactuated links, whose dynamics depend on those of actuated links, does not allow one to independently control each link of such a system. A powerful analytical tool of generating a periodic motion in underactuated mechanical systems relies on the virtual constraint approach to be presented in this chapter in terms of the joint generalized coordinates. Although the configuration is generally given in Cartesian coordinates and orientation, however, using inverse kinematics, they can be transformed into joint coordinates. Coupled with the virtual constraint approach, the \(\mathcal{H}_{\infty }\) generation of periodic motion is further developed and applied to a three-degree-of-freedom (3-DOF) helicopter prototype of one degree of underactuation.


Underactuated system Underactuation degree Virtual constraint method \(\mathcal{H}_{\infty }\)-periodic-motion generation Helicopter prototype 


  1. 6.
    Andronov, A., Vitt, A.: On Lyapunov stability. Exp. Theor. Phys. 3, 373–374 (1933)Google Scholar
  2. 7.
    Arimoto, S.: Control Theory of Non-linear Mechanical Systems: A Passivity-Based and Circuit-Theoretic Approach. Oxford University Press, New York (1996)zbMATHGoogle Scholar
  3. 11.
    Avila-Vilchis, J., Brogliato, B., Dzul, A., Lozano, R.: Nonlinear modelling and control of helicopters. Automatica 39, 1526–1530 (2003)CrossRefMathSciNetGoogle Scholar
  4. 26.
    Canudas-de-Wit, C.: On the concept of virtual constraints as a tool for walking robot control and balancing. Annu. Rev. Control 28, 157–166 (2004)CrossRefGoogle Scholar
  5. 28.
    Chaturvedi, N., McClamroch, N., Bernstein, D.: Stabilization of a 3D axially symmetric pendulum. Automatica 44(9), 2258–2265 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 41.
    Dzul, A., Lozano, R., Castillo, P.: Adaptive control for a radio-controlled helicopter in a vertical flying stand. Int. J. Adapt. Control Signal Process. 18, 473–485 (2004)CrossRefzbMATHGoogle Scholar
  7. 46.
    Freidovich, L., Mettin, U., Shiriaev, A., Spong, M.: A passive 2-DOF walker: hunting for gaits using virtual holonomic constraints. IEEE Trans. Robot. 25(5), 1202–1208 (2009)CrossRefGoogle Scholar
  8. 47.
    Freidovich, L., Robersson, A., Shiriaev, A., Johansson, R.: Periodic motions of the pendubot via virtual holonomic constraints: theory and experiments. Automatica 44, 785–791 (2008)CrossRefGoogle Scholar
  9. 69.
    Isidori, A., Marconi, L., Serrani, A.: Robust nonlinear motion control of a helicopter. IEEE Trans. Automat. Contr. 48(3), 413–426 (2003)CrossRefMathSciNetGoogle Scholar
  10. 77.
    Leonov, G.: Generalization of Andronov–Vitt theorem. Regul. Chaotic Dyn. 11(2), 282–289 (2006)CrossRefGoogle Scholar
  11. 82.
    Meza, I., Aguilar, L., Shiriaev, A., Freidovich, L., Orlov, Y.: Periodic motion planning and nonlinear \(\mathcal{H}_{\infty }\) tracking control of a 3-DOF underactuated helicopter. Int. J. Syst. Sci. 42(5), 829–838 (2011)CrossRefzbMATHGoogle Scholar
  12. 99.
    Quanser: 3D helicopter system with active disturbance. Tech. rep. (2004)Google Scholar
  13. 110.
    Shiriaev, A., Freidovich, L., Gusev, S.: Transverse linearization for controlled mechanical systems with severe passive degrees of freedom. IEEE Trans. Automat. Contr. 55(4), 893–906 (2010)CrossRefMathSciNetGoogle Scholar
  14. 111.
    Shiriaev, A., Freidovich, L., Manchester, I.: Can we make a robot ballerina perform a pirouette? Orbital stabilization of periodic motions of underactuated mechanical systems. Annu. Rev. Control 32(2), 200–211 (2008)Google Scholar
  15. 112.
    Shiriaev, A., Perram, J., Canudas-de-Wit, C.: Contructive tool for an orbital stabilization of underactuated nonlinear systems: virtual constraint approach. IEEE Trans. Automat. Contr. 50(8), 1164–1176 (2005)CrossRefMathSciNetGoogle Scholar
  16. 113.
    Shiriaev, A., Robertsson, A., Perram, J., Sandberg, A.: Periodic motion planning for virtually constrained Euler Lagrange systems. Syst. Control Lett. 55(11), 900–907 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 115.
    Spong, M., Hutchinson, S., Vidyasagar, M.: Robot Modeling and Control. Wiley, New York (2006)Google Scholar
  18. 130.
    Westerberg, S., Mettin, U., Shiriaev, A., Freidovich, L., Orlov, Y.: Motion planning and control of a simplified helicopter model based on virtual holonomic constraints. In: Proceedings of the 14th International Conference on Advanced Robotics, Munich, pp. 1–6 (2009)Google Scholar

Copyright information

© Springer Science+Business Media, New York 2014

Authors and Affiliations

  • Yury V. Orlov
    • 1
  • Luis T. Aguilar
    • 2
  1. 1.Electronics and TelecommunicationCICESE Research CenterEnsenadaMexico
  2. 2.Centro de Investigación y Desarrollo de Tecnología DigitalInstituto Politécnico NacionalTijuanaMexico

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