Total Energy Shaping Control of Mechanical Systems: Simplifying the Matching Equations Via Coordinate Changes

  • Giuseppe Viola
  • Romeo Ortega
  • Ravi Banavar
  • José Ángel Acosta
  • Alessandro Astolfi
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 366)


Total Energy Shaping is a controller design methodology that achieves (asymptotic) stabilization of mechanical systems endowing the closed-loop system with a Lagrangian or Hamiltonian structure with a desired energy function—that qualifies as Lyapunov function for the desired equilibrium. The success of the method relies on the possibility of solving two PDEs which identify the kinetic and potential energy functions that can be assigned to the closed-loop. Particularly troublesome is the PDE associated to the kinetic energy which is nonlinear and non-homogeneous and the solution, that defines the desired inertia matrix, must be positive definite. In this paper we prove that we can eliminate or simplify the forcing term in this PDE modifying the target dynamics and introducing a change of coordinates in the original system. Furthermore, it is shown that, in the particular case of transformation to the Lagrangian coordinates, the possibility of simplifying the PDEs is determined by the interaction between the Coriolis and centrifugal forces and the actuation structure. The example of a pendulum on a cart is used to illustrate the results.


Mechanical System Centrifugal Force Force Term Potential Energy Function Inertia Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    J. A. Acosta, R. Ortega, A. Astolfi, and A. M. Mahindrakar. Interconnection and damping assignment passivity-based control of mechanical systems with underactuation degree one. IEEE Trans. Automat. Contr., 50, 2005.Google Scholar
  2. 2.
    A. Ailon and R. Ortega. An observer-based controller for robot manipulators with flexible joints. Syst. & Cont. Letters, 21:329–335, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    D. Auckly and L. Kapitanski. On the λ-equations for matching controllaws. SIAM J. Control and Optimization, 41:1372–1388, 2002.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    D. Auckly, L. Kapitanski, and W. White. Control of nonlinear underactuated systems. Comm. Pure Appl. Math., 3:354–369, 2000.CrossRefMathSciNetGoogle Scholar
  5. 5.
    G. Blankenstein, R. Ortega, and A.J. van der Schaft. The matching conditions of controlled lagrangians and interconnection assigment passivity based control. Int J of Control, 75:645–665, 2002.zbMATHCrossRefGoogle Scholar
  6. 6.
    A. Bloch, N. Leonard, and J. Marsden. Controlled lagrangians and the stabilization of mechanical systems. IEEE Trans. Automat. Contr., 45, 2000.Google Scholar
  7. 7.
    D.E. Chang, A.M. Block, N.E. Leonard, J.E. Marsden, and C.A. Woolsey. The equivalence of controlled lagrangian and controlled hamiltonian systems for simple mechanical systems. ESAIM: Control, Optimisation, and Calculus of Variations, 45:393–422, 2002.CrossRefGoogle Scholar
  8. 8.
    K. Fujimoto and T. Sugie. Canonical transformations and stabilization of generalized hamiltonian systems. Systems and Control Letters, 42:217–227, 2001.CrossRefMathSciNetGoogle Scholar
  9. 9.
    R. Kelly, V. Santibanez, and A. Loria. Control of robot manipulators in joint space. Springer-Verlag, London, 2005.Google Scholar
  10. 10.
    A. Lewis. Notes on energy shaping. 43rd IEEE Conf Decision and Control, Dec 14–17, 2004, Paradise Island, Bahamas., 2004.Google Scholar
  11. 11.
    A. D. Mahindrakar, A. Astolfi, R. Ortega, and G. Viola. Further constructive results on interconnection and damping assignments control of mechanical systems: The acrobot example. American Control Conference, Minneapolis, USA, 14–16 June 2006, 2006.Google Scholar
  12. 12.
    R. Ortega and E. Garcia-Canseco. Interconnection and damping assignment passivity-based control: A survey. European J of Control, 10:432–450, 2004.CrossRefMathSciNetGoogle Scholar
  13. 13.
    R. Ortega, M. Spong, F. Gomez, and G. Blankenstein. Stabilization of underactuated mechanical systems via interconnection and damping assignment. IEEE Trans. Automat. Contr., AC.-47:1218–1233, 2002.CrossRefGoogle Scholar
  14. 14.
    E.T. Whittaker. A treatise on the analytical dynamics of particles and rigid bodies. Cambridge University Press, 1988.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Giuseppe Viola
    • 1
  • Romeo Ortega
    • 2
  • Ravi Banavar
    • 3
  • José Ángel Acosta
    • 4
  • Alessandro Astolfi
    • 1
    • 5
  1. 1.Dipartimento di Informatica, Sistemie ProduzioneUniversitä di Roma “Tor Vergata”RomaItaly
  2. 2.Laboratoire des Signaux et Systèmes, Supélec, Plateau du MoulonGif-sur-YvetteFrance
  3. 3.Systems and Control EngineeringI.I.T.—BombayMumbaiIndia
  4. 4.Depto. de Ingeniería de Sistemas y AutomäticaEscuela Superior de IngenierosSevillaSpain
  5. 5.Electrical Engineering DepartmentImperial CollegeLondonUK

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