Multibody System Dynamics

, 21:71 | Cite as

Ball on a beam: stabilization under saturated input control with large basin of attraction

  • Yannick AoustinEmail author
  • Alexander Formal’skii


This article is devoted to the stabilization of two underactuated planar systems, the well-known straight beam-and-ball system and an original circular beam-and-ball system. The feedback control for each system is designed, using the Jordan form of its model, linearized near the unstable equilibrium. The limits on the voltage, fed to the motor, are taken into account explicitly. The straight beam-and-ball system has one unstable mode in the motion near the equilibrium point. The proposed control law ensures that the basin of attraction coincides with the controllability domain. The circular beam-and-ball system has two unstable modes near the equilibrium point. Therefore, this device, never considered in the past, is much more difficult to control than the straight beam-and-ball system. The main contribution is to propose a simple new control law, which ensures by adjusting its gain parameters that the basin of attraction arbitrarily can approach the controllability domain for the linear case. For both nonlinear systems, simulation results are presented to illustrate the efficiency of the designed nonlinear control laws and to determine the basin of attraction.


Saturated control Controllability domain Jordan form Stabilization Basin of attraction 


  1. 1.
    Hauser, J., Sastry, S., Meyer, G.: Nonlinear control design for slightly nonminimum phase systems. Application to V/STOL aircraft. Automatica 28(4), 665–679 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alimir, M., Boyer, F.: Fast generation of attractive trajectories for an under-actuated satellite. Application to feedback control design. J. Optim. Eng. 4, 225–244 (2003) Google Scholar
  3. 3.
    Book, W.J., Majette, M.: Controller design for flexible, distributed parameter mechanical arms via combined state space and frequency domain techniques. Trans. ASME, J. Dyn. Syst. Meas. Control 105, 245–254 (1983) zbMATHCrossRefGoogle Scholar
  4. 4.
    De Luca, A., Siciliano, B.: Closed-form dynamic model of planar multi-link lightweight robots. IEEE Trans. Syst. Man Cybern. 21(4), 826–839 (1991) CrossRefGoogle Scholar
  5. 5.
    Astrom, K.J., Furuta, K.: Swinging up a pendulum by energy control. Automatica 36(2), 287–295 (2000) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Grishin, A.A., Lenskii, A.V., Okhotsimsky, D.E., Panin, D.A., Formal’sky, A.M.: A control synthesis for an unstable object. An inverted pendulum. J. Comput. Syst. Sci, Int. 41(5), 685–694 (2002) Google Scholar
  7. 7.
    Spong, M.W.: The swing up control problem for the acrobot. IEEE Control Syst. Mag. 14(1), 49–55 (1995) CrossRefGoogle Scholar
  8. 8.
    Grizzle, J.W., Moog, C.M., Chevallereau, C.: Nonlinear control of mechanical systems with an unactuated cyclic variable. IEEE Trans. Autom. Control 50(5), 559–576 (2005) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Aoustin, Y., Formal’sky, A.M.: Design of reference trajectory to stabilize desired nominal cyclic gait of a biped. In: Proceedings International Workshop on Robot Motion and Control, ROMOCO’99, pp. 159–165 (1999) Google Scholar
  10. 10.
    Aoustin, Y., Formal’sky, A.M.: Control design for a biped reference trajectory based on driven angles as functions of the undriven angle. J. Comput. Syst. Sci. Int. 42(4), 159–176 (2003) MathSciNetGoogle Scholar
  11. 11.
    Plestan, F., Grizzle, J.W., Westervelt, W., Abba, G.: Stable walking of a 7-dof biped robot. IEEE Trans. Robot. Autom. 19, 653–668 (2003) CrossRefGoogle Scholar
  12. 12.
    Acosta, J., Ortega, R., Astolfo, A.: Position feedback stabilization of mechanical systems with underactuation degree one. In: Proceedings of the 6th IFAC Symposium Nonlinear Control Systems, NOCOLS’04 (2004) Google Scholar
  13. 13.
    Fantoni, I., Lozano, R.: Non Linear Control for Underactuated Mechanical Systems. Communications and Control Engineering Series. Springer, London (2002) Google Scholar
  14. 14.
    Olfati-Saber, R.: Nonlinear control of underactuated mechanical systems with application to robotics and aerospace vehicles. PhD thesis, Massachusetts Institute of Technology (2001) Google Scholar
  15. 15.
    Hauser, J., Sastry, S., Kokotović, P.: Nonlinear control via approximate input–output linearization. IEEE Trans. Autom. Control 37, 392–398 (1992) CrossRefGoogle Scholar
  16. 16.
    Teel, A.R., Praly, L.: Tools for semiglobal stabilization by partial state and output feedback. SIAM J. Control Optim. 33, 1443–1488 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Teel, A.R.: Using saturation to stabilize a class of single-input partially linear composite systems. In: IFAC NOLCOS’92 Symposium, May 1992, pp. 369–374 (1992) Google Scholar
  18. 18.
    Sepulchre, R., Janković, M., Kokotović, P.: Constructive Nonlinear Control. Springer, Berlin (1997) zbMATHGoogle Scholar
  19. 19.
    Aoustin, Y., Formal’sky, A.M.: On the stabilization of biped vertical posture in single support using internal torques. Robotica 23(1), 65–74 (2005) CrossRefGoogle Scholar
  20. 20.
    Aoustin, Y., Formal’sky, A.M., Martynenko, Y.: Stabilisation of unstable equilibrium postures of a two-link pendulum using a flywheel. J. Comput. Syst. Sci. Int. 2, 16–23 (2006) Google Scholar
  21. 21.
    Formal’skii, A.M.: Controllability and Stability of Systems with Limited Resources. Nauka, Moscow (1974) (in Russian) Google Scholar
  22. 22.
    Hu, T., Lin, Z., Qiu, L.: Stabilization of exponentially unstable linear systems with saturating actuators. IEEE Trans. Autom. Control 46(6), 973–979 (2001) zbMATHMathSciNetGoogle Scholar
  23. 23.
    Gorinevsky, D.M., Formal’sky, A.M., Schneider, A.Yu.: Force Control of Robotics Systems. CRC Press, New York (1997) zbMATHGoogle Scholar
  24. 24.
    Kalman, R.E., Falb, P.L., Arbib, M.A.: Topics in Mathematical System Theory. McGraw-Hill, New York (1969) zbMATHGoogle Scholar
  25. 25.
    Korn, G.A., Korn, T.M.: Mathematical Handbook for Engineers and Scientists. McGraw-Hill, New York (1968) Google Scholar
  26. 26.
    Khalil, H.K.: Nonlinear System. Prentice Hall, New Jersey (2002) Google Scholar
  27. 27.
    Boltyansky, V.G.: Mathematical Methods of Optimal Control. Nauka, Moscow (1966) (in Russian) Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Institut de Recherche en Communications et Cybernétique de Nantes U.M.R. 6597Nantes Cedex 3France
  2. 2.Institute of MechanicsMoscow Lomonosov State UniversityMoscowRussia

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