Advertisement

Regular and Chaotic Dynamics

, Volume 19, Issue 5, pp 556–575 | Cite as

On the method of interconnection and damping assignment passivity-based control for the stabilization of mechanical systems

  • Dong Eui Chang
Article

Abstract

Interconnection and damping assignment passivity-based control (IDA-PBC) is an excellent method to stabilize mechanical systems in the Hamiltonian formalism. In this paper, several improvements are made on the IDA-PBC method. The skew-symmetric interconnection submatrix in the conventional form of IDA-PBC is shown to have some redundancy for systems with the number of degrees of freedom greater than two, containing unnecessary components that do not contribute to the dynamics. To completely remove this redundancy, the use of quadratic gyroscopic forces is proposed in place of the skew-symmetric interconnection submatrix. Reduction of the number of matching partial differential equations in IDA-PBC and simplification of the structure of the matching partial differential equations are achieved by eliminating the gyroscopic force from the matching partial differential equations. In addition, easily verifiable criteria are provided for Lyapunov/exponential stabilizability by IDA-PBC for all linear controlled Hamiltonian systems with arbitrary degrees of underactuation and for all nonlinear controlled Hamiltonian systems with one degree of underactuation. A general design procedure for IDA-PBC is given and illustrated with examples. The duality of the new IDA-PBC method to the method of controlled Lagrangians is discussed. This paper renders the IDA-PBC method as powerful as the controlled Lagrangian method.

Keywords

feedback control stabilization energy shaping mechanical system 

MSC2010 numbers

70Q05 93C10 93D15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Seipel, J. and Holmes, P., A Simple Model for Clock-Actuated Legged Locomotion, Regul. Chaotic Dyn., 2007, vol. 12, no. 5, pp. 502–520.MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Gregg, R.D., Bretl, T., and Spong, M.W., A Control Theoretic Approach to Robot-Assisted Locomotor Therapy, in Proc. of the Conf. on Decision and Control, 2010, pp. 1679–1686.Google Scholar
  3. 3.
    Holm, J.K., Lee, D., and Spong, M.W., Time-Scaling Trajectories of Passive-Dynamic Bipedal Robots, in Proc. of the IEEE Internat. Conf. on Robotics and Automation, 2007, pp. 3603–3608.Google Scholar
  4. 4.
    Arimoto, S. and Miyazaki, F., Stability and Robustness of PID Feedback Control for Robot Manipulators of Sensory Capability, in Robotics Research: 1st Internat. Symp., M. Brady, R.P. Paul (Eds.), Cambridge: MIT Press, 1983, pp. 783–799.Google Scholar
  5. 5.
    Bloch, A. M., Krishnaprasad, P. S., Marsden, J.E., and Sánchez de Alvarez, G., Stabilization of Rigid Body Dynamics by Internal and External Torques, Automatica, 1992, vol. 28, no. 4, pp. 745–756.MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bloch, A. M., Leonard, N.E., and Marsden, J.E., Stabilization of Mechanical Systems Using Controlled Lagrangians, in Proc. of the 36th IEEE Conf. on Decision and Control, 1997, pp. 2356–2361.Google Scholar
  7. 7.
    Bloch, A. M., Leonard, N.E., and Marsden, J.E., Controlled Lagrangians and the Stabilization of Mechanical Systems: 1. The First Matching Theorem, IEEE Trans. Automat. Contr., 2000, vol. 45, no. 12, pp. 2253–2270.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bloch, A.M., Chang, D.E., Leonard, N. E., and Marsden, J. E., Controlled Lagrangians and the Stabilization of Mechanical Systems: 2. Potential Shaping, IEEE Trans. Automat. Contr., 2001, vol. 46, no. 10, pp. 1556–1571.MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bloch, A. M., Leonard, N.E., and Marsden, J.E., Controlled Lagrangians and the Stabilization of Euler-Poincaré Mechanical Systems, Internat. J. Robust Nonlinear Control, 2001, vol. 11, pp. 191–214.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Chang, D.E., Some Results on Stabilizability of Controlled Lagrangian Systems by Energy Shaping, in Proc. of the 17th IFAC World Congr. (Seoul, Korea, 2008), pp. 3161–3165.Google Scholar
  11. 11.
    Chang, D. E., The Method of Controlled Lagrangians: Energy Plus Force Shaping, SIAM J. Control and Optimization, 2010, vol. 48, no. 8, pp. 4821–4845.MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Chang, D.E., Stabilizability of Controlled Lagrangian Systems of Two Degrees of Freedom and One Degree of Under-Actuation, IEEE Trans. Automat. Contr., 2010, vol. 55, no. 8, pp. 1888–1893.CrossRefGoogle Scholar
  13. 13.
    Chang, D.E., Pseudo-Energy Shaping for the Stabilization of a Class of Second-Order Systems, Internat. J. Robust Nonlinear Control, 2012, vol. 22, no. 18, pp. 1999–2013.MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Chang, D. E., Bloch, A.M., Leonard, N. E., Marsden, J.E., and Woolsey, C., The Equivalence of Controlled Lagrangian and Controlled Hamiltonian Systems, ESAIM Control Optim. Calc. Var., 2002, vol. 8, pp. 393–422.MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Gharesifard, B., Lewis, A.D., and Mansouri, A.-R., A Geometric Framework for Stabilization by Energy Shaping: Sufficient Conditions for Existence of Solutions, Commun. Inf. Syst., 2008, vol. 8, no. 4, pp. 353–398.MathSciNetMATHGoogle Scholar
  16. 16.
    Ng, W., Chang, D.E., and Labahn, G., Energy Shaping for Systems with Two Degrees of Underactuation and More than Three Degrees of Freedom, SIAM J. Control and Optimization, 2013, vol. 51, no. 2, pp. 881–905.MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Zenkov, D. V., Matching and Stabilization of Linear Mechanical Systems, in Proc. of the 15th Internat. Symp. on Mathematical Theory of Networks and System, 2002, 8 pp.Google Scholar
  18. 18.
    Chang, D.E., Generalization of the IDA-PBC Method for Stabilization of Mechanical Systems, in Proc. of the 18th Mediterranean Conf. on Control & Automation, 2010, pp. 226–230.Google Scholar
  19. 19.
    Acosta, J.A., Ortega, R., Astolfi, A., and Mahindrakar, A., Interconnection and Damping Assignment Passivity-Based Control of Mechanical Systems with Underactuation Degree One, IEEE Trans. Automat. Contr., 2005, vol. 50, no. 12, pp. 1936–1955.MathSciNetCrossRefGoogle Scholar
  20. 20.
    Ortega, R. and García-Canseco, E., Interconnection and Damping Assignment Passivity-Based Control: A Survey, Eur. J. Control, 2004, vol. 10, pp. 432–450.MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Romero, J.G., Donaire, A., and Ortega, R., Robust Energy Shaping Control of Mechanical Systems, Syst. Control Lett., 2013, vol. 62, no. 9, pp. 770–780.MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Pommaret, J. F., Partial Differential Equations and Group Theory: New Perspectives for Applications, Dortrecht: Kluwer, 1994.CrossRefMATHGoogle Scholar
  23. 23.
    Lee, J.M., Introduction to Smooth Manifolds, New York: Springer, 2002.MATHGoogle Scholar
  24. 24.
    Ng, W., Chang, D.E., and Song, S.H., Four Representative Applications of the Energy Shaping Method for Controlled Lagrangian Systems, J. Electrical Engineering and Technology, 2013, vol. 8, no. 6, pp. 1579–1589.CrossRefGoogle Scholar
  25. 25.
    Liu, Y. and Yu, H., A Survey of Underactuated Mechanical Systems, IET Control Theory and Applications, 2013, vol. 7, no. 7, pp. 921–935.MathSciNetCrossRefGoogle Scholar
  26. 26.
    Respondek, W. and Ricardo, S., Equivariants of Mechanical Control Systems, SIAM J. Control and Optimization, 2013, vol. 51, no. 4, pp. 3027–3055.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of WaterlooWaterlooCanada

Personalised recommendations