Port-Based Modeling of a Flexible Link

Authors

A. Macchelli, C. Melchiorri, S. Stramigioli

Abstract

In this paper, a simple way to model flexible robotic links is presented. This is different from classical approaches and from the Euler-Bernoulli or Timoshenko theory, in that the proposed model is able to describe large deflections in 3D space and does not rely on any finite-dimensional approximation (e.g., modal approximation). The model has been formulated within the port Hamiltonian formalism because intuitive considerations on the geometric behavior of the elastic link naturally define a Stokes-Dirac structure, the kernel of a port Hamiltonian system. Moreover, port Hamiltonian systems can be easily interconnected, thus allowing the description of complex systems as a composition of parts in an object-oriented way. By combining rigid bodies, springs, dampers, joints and, finally, flexible links, it is virtually possible to model and mathematically describe whatever complex mechanical structure formed by beams. In order to demonstrate the dynamical properties of the model and how complex mechanisms can be obtained by port interconnection, simulations of 1-DoF and 2-DoF serial manipulators and of a 2-DoF flexible closed kinematic chain are presented.

Citation

• Journal: IEEE Transactions on Robotics
• Year: 2007
• Volume: 23
• Issue: 4
• Pages: 650–660
• Publisher: Institute of Electrical and Electronics Engineers (IEEE)
• DOI: 10.1109/tro.2007.898990

BibTeX

@article{Macchelli_2007,
  title={{Port-Based Modeling of a Flexible Link}},
  volume={23},
  ISSN={1941-0468},
  DOI={10.1109/tro.2007.898990},
  number={4},
  journal={IEEE Transactions on Robotics},
  publisher={Institute of Electrical and Electronics Engineers (IEEE)},
  author={Macchelli, A. and Melchiorri, C. and Stramigioli, S.},
  year={2007},
  pages={650--660}
}

Download the bib file

References

• Stanway, J., Sharf, I. & Damaren, C. Comparison and Validation of Dynamics Simulation Models for a Structurally Flexible Manipulator. Journal of Dynamic Systems, Measurement, and Control vol. 120 404–409 (1998) – 10.1115/1.2805417
• fasse, some applications of screw theory to lumped-parameter modeling of visco-elastically coupled rigid bodies. Proc Symp Commemorating the Legacy Works and Life of Sir Robert Stawell Ball upon the 100th Anniversary of a Treatise on the Theory of Screws (2000)
• bassi, an algorithm to discretize one-dimensional distributed port hamiltonian systems. Proc 3rd IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control (2006)
• Golo, G., Talasila, V., van der Schaft, A. & Maschke, B. Hamiltonian discretization of boundary control systems. Automatica vol. 40 757–771 (2004) – 10.1016/j.automatica.2003.12.017
• Putting energy back in control. IEEE Control Systems vol. 21 18–33 (2001) – 10.1109/37.915398
• Macchelli, A., Stramigioli, S. & Melchiorri, C. Port-based modelling of manipulators with flexible links. Proceedings 2006 IEEE International Conference on Robotics and Automation, 2006. ICRA 2006. 1886–1891 doi:10.1109/robot.2006.1641981 – 10.1109/robot.2006.1641981
• Frankel, T. The Geometry of Physics. (2003) doi:10.1017/cbo9780511817977 – 10.1017/cbo9780511817977
• selig, Geometric Fundamentals of Robotics (2005)
• 20-Sim (2006)
• paynter, Analysis and Design of Engineering Systems (1961)
• Wang, D. & Vidyasagar, M. Modeling a class of multilink manipulators with the last link flexible. IEEE Transactions on Robotics and Automation vol. 8 33–41 (1992) – 10.1109/70.127237
• Macchelli, A. & Melchiorri, C. Control by interconnection of mixed port Hamiltonian systems. IEEE Transactions on Automatic Control vol. 50 1839–1844 (2005) – 10.1109/tac.2005.858656
• Nicosia, S., Valigi, P. & Zaccarian, L. Dynamic modelling of a two link flexible robot and experimental validation. Proceedings of IEEE International Conference on Robotics and Automation vol. 3 1953–1958 – 10.1109/robot.1996.506158
• Book, W. J. Recursive Lagrangian Dynamics of Flexible Manipulator Arms. The International Journal of Robotics Research vol. 3 87–101 (1984) – 10.1177/027836498400300305
• Ueno, H., Yangsheng Xu & Yoshida, T. Modeling and control strategy of a 3-D flexible space robot. Proceedings IROS ’91:IEEE/RSJ International Workshop on Intelligent Robots and Systems ’91 978–983 doi:10.1109/iros.1991.174616 – 10.1109/iros.1991.174616
• Li, C.-J. & Sankar, T. S. Systematic methods for efficient modeling and dynamics computation of flexible robot manipulators. IEEE Transactions on Systems, Man, and Cybernetics vol. 23 77–95 (1993) – 10.1109/21.214769
• Simo, J. C. A finite strain beam formulation. The three-dimensional dynamic problem. Part I. Computer Methods in Applied Mechanics and Engineering vol. 49 55–70 (1985) – 10.1016/0045-7825(85)90050-7
• Selig, J. M. & Ding, X. A screw theory of static beams. Proceedings 2001 IEEE/RSJ International Conference on Intelligent Robots and Systems. Expanding the Societal Role of Robotics in the the Next Millennium (Cat. No.01CH37180) vol. 1 312–317 – 10.1109/iros.2001.973376
• golo, hamiltonian formulation of planar beams. Proc 2nd IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control (2003)
• van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics vol. 42 166–194 (2002) – 10.1016/s0393-0440(01)00083-3
• Macchelli, A., van der Schaft, A. J. & Melchiorri, C. Port Hamiltonian formulation of infinite dimensional systems I. Modeling. 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601) 3762-3767 Vol.4 (2004) doi:10.1109/cdc.2004.1429324 – 10.1109/cdc.2004.1429324
• Maschke, B. M. & van der Schaft, A. J. Interconnection of systems: the network paradigm. Proceedings of 35th IEEE Conference on Decision and Control vol. 1 207–212 – 10.1109/cdc.1996.574297
• golo, a hamiltonian formulation of the timoshenko beam model. Proc Mechatronics 2002 (2002)
• maschke, Modelling and Control of Mechanism and Robots (1996)
• Kim, J. U. & Renardy, Y. Boundary Control of the Timoshenko Beam. SIAM Journal on Control and Optimization vol. 25 1417–1429 (1987) – 10.1137/0325078
• Luo, Z.-H., Guo, B.-Z. & Morgul, O. Stability and Stabilization of Infinite Dimensional Systems with Applications. Communications and Control Engineering (Springer London, 1999). doi:10.1007/978-1-4471-0419-3 – 10.1007/978-1-4471-0419-3
• stramigioli, Modeling and IPC Control of Interactive Mechanical Systems A Coordinate Free Approach (2001)
• Macchelli, A. & Melchiorri, C. Modeling and Control of the Timoshenko Beam. The Distributed Port Hamiltonian Approach. SIAM Journal on Control and Optimization vol. 43 743–767 (2004) – 10.1137/s0363012903429530
• Chait, Y., Miklavcic, M., Maccluer, C. R. & Radcliffe, C. J. A natural modal expansion for the flexible robot arm problem via a self-adjoint formulation. IEEE Transactions on Robotics and Automation vol. 6 601–603 (1990) – 10.1109/70.62050
• Tomei, P. & Tornambe, A. Approximate modeling of robots having elastic links. IEEE Transactions on Systems, Man, and Cybernetics vol. 18 831–840 (1988) – 10.1109/21.21610
• Khadem, S. E. & Pirmohammadi, A. A. Analytical development of dynamic equations of motion for a three-dimensional flexible link manipulator with revolute and prismatic joints. IEEE Transactions on Systems, Man and Cybernetics, Part B (Cybernetics) vol. 33 237–249 (2003) – 10.1109/tsmcb.2003.810439
• De Luca, A. & Siciliano, B. Closed-form dynamic model of planar multilink lightweight robots. IEEE Transactions on Systems, Man, and Cybernetics vol. 21 826–839 (1991) – 10.1109/21.108300
• de luca, Theory of Robot Control (1996)
• macchelli, distributed port-hamiltonian formulation of infinite dimensional systems. Proc 16th International Symposium on Mathematical Theory of Networks and Systems (MTNS2004) (2004)
• van der schaft, -Gain and Passivity Techniques in Nonlinear Control ser Communication and Control Engineering (2000)
• Dalsmo, M. & van der Schaft, A. On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems. SIAM Journal on Control and Optimization vol. 37 54–91 (1998) – 10.1137/s0363012996312039
• maschke, port controlled hamiltonian systems: modeling origins and system theoretic properties. Proc Third Conf on Nonlinear Control Systems (NOLCOS) (1992)
• maschke, port controlled hamiltonian representation of distributed parameter sytems. Workshop on Modeling and Control of Lagrangian and Hamiltonian Systems (2000)
• Macchelli, A., Melchiorri, C. & Bassi, L. Port-based Modelling and Control of the Mindlin Plate. Proceedings of the 44th IEEE Conference on Decision and Control 5989–5994 doi:10.1109/cdc.2005.1583120 – 10.1109/cdc.2005.1583120
• Macchelli, A., van der Schaft, A. J. & Melchiorri, C. Multi-variable port Hamiltonian model of piezoelectric material. 2004 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) (IEEE Cat. No.04CH37566) vol. 1 897–902 – 10.1109/iros.2004.1389466