Authors

J.A. Acosta, E. Panteley, R. Ortega

Abstract

In this brief note a new strict Lyapunov function for mechanical systems controlled by the well-known Passivity-based Control technique of Interconnection and Damping Assignment is proposed. The general, total energy-shaping, formulation of the control technique is considered, which yields a port-Hamiltonian closed-loop system with non-fixed symplectic structure. To construct the proposed Lyapunov function a new systematic mathematical machinery is introduced. The resulting Lyapunov function contains, as particular cases, previous functions obtained for robot manipulators controlled by potential energy-shaping (plus damping injection) schemes. An additional contribution of our work is that, in contrast with most of the existing literature on this topic that is restricted to robot manipulators with only revolute joints, our analysis is applicable to robots with both revolute and prismatic joints. As an illustration example, practical bounds for a two-link direct drive robot manipulator are computed.

Citation

  • Journal: 2009 IEEE International Conference on Control Applications
  • Year: 2009
  • Volume:
  • Issue:
  • Pages: 519–524
  • Publisher: IEEE
  • DOI: 10.1109/cca.2009.5280704

BibTeX

@inproceedings{Acosta_2009,
  title={{A new strict Lyapunov function for fully-actuated mechanical systems controlled by IDA-PBC}},
  DOI={10.1109/cca.2009.5280704},
  booktitle={{2009 IEEE International Conference on Control Applications}},
  publisher={IEEE},
  author={Acosta, J.A. and Panteley, E. and Ortega, R.},
  year={2009},
  pages={519--524}
}

Download the bib file

References

  • Nijmeijer, H. & van der Schaft, A. Nonlinear Dynamical Control Systems. (Springer New York, 1990). doi:10.1007/978-1-4757-2101-0 – 10.1007/978-1-4757-2101-0
  • Ghorbel, F., Srinivasan, B. & Spong, M. W. On the uniform boundedness of the inertia matrix of serial robot manipulators. Journal of Robotic Systems vol. 15 17–28 (1998) – 10.1002/(sici)1097-4563(199812)15:1<17::aid-rob2>3.0.co;2-v
  • teel, Matrosov s theorem using a family of auxiliary functions An analysis tool to aid time-varying nonlinear control design (2003)
  • Ortega, R., Spong, M. W., Gomez-Estern, F. & Blankenstein, G. Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment. IEEE Transactions on Automatic Control vol. 47 1218–1233 (2002) – 10.1109/tac.2002.800770
  • Acosta, J. A., Ortega, R., Astolfi, A. & Mahindrakar, A. D. Interconnection and damping assignment passivity-based control of mechanical systems with underactuation degree one. IEEE Transactions on Automatic Control vol. 50 1936–1955 (2005) – 10.1109/tac.2005.860292
  • Bloch, A. M., Leonard, N. E. & Marsden, J. E. Controlled Lagrangians and the stabilization of mechanical systems. I. The first matching theorem. IEEE Transactions on Automatic Control vol. 45 2253–2270 (2000) – 10.1109/9.895562
  • Ailon, A. & Ortega, R. An observer-based set-point controller for robot manipulators with flexible joints. Systems & Control Letters vol. 21 329–335 (1993) – 10.1016/0167-6911(93)90076-i
  • Auckly, D., Kapitanski, L. & White, W. Control of nonlinear underactuated systems. Communications on Pure and Applied Mathematics vol. 53 354–369 (2000) – 10.1002/(sici)1097-0312(200003)53:3<354::aid-cpa3>3.3.co;2-l
  • Arteaga, M. A. & Kelly, R. Robot Control Without Velocity Measurements: New Theory and Experimental Results. IEEE Transactions on Robotics and Automation vol. 20 297–308 (2004) – 10.1109/tra.2003.820872
  • Santibáñez, V. & Kelly, R. Strict Lyapunov functions for control of robot manipulators. Automatica vol. 33 675–682 (1997) – 10.1016/s0005-1098(96)00194-x
  • koditschek, Strict Globlal Lyapunov Function for Mechanical Systems. Proc American Control Conference (1988)
  • Takegaki, M. & Arimoto, S. A New Feedback Method for Dynamic Control of Manipulators. Journal of Dynamic Systems, Measurement, and Control vol. 103 119–125 (1981) – 10.1115/1.3139651
  • Astolfi, A., Ortega, R. & Sepulchre, R. Stabilization and Disturbance Attenuation of Nonlinear Systems Using Dissipativity Theory. European Journal of Control vol. 8 408–431 (2002) – 10.3166/ejc.8.408-431
  • Mazenc, F. & Malisoff, M. Further Constructions of Control-Lyapunov Functions and Stabilizing Feedbacks for Systems Satisfying the Jurdjevic–Quinn Conditions. IEEE Transactions on Automatic Control vol. 51 360–365 (2006) – 10.1109/tac.2005.863500
  • santiba?n?ez, A new Saturated Nonlinear PID Global Regulator for Robot Manipulators. IFAC World Congress (2008)