Port-Based Modeling and Simulation of Mechanical Systems With Rigid and Flexible Links

Authors

A. Macchelli, C. Melchiorri, S. Stramigioli

Abstract

In this paper, a systematic procedure for the definition of the dynamical model in port-Hamiltonian form of mechanical systems is presented as the result of the power-conserving interconnection of a set of basic components (rigid bodies, flexible links, and kinematic pairs). Since rigid bodies and flexible links are described within the port-Hamiltonian formalism, their interconnection is possible once a proper relation between the power-conjugated port variables is deduced. These relations are the analogous of the Kirchhoff laws of circuit theory. From the analysis of a set of oriented graphs that describe the topology of the mechanism, an automatic procedure for deriving the dynamical model of a mechanical system is illustrated. The final model is a mixed port-Hamiltonian system, because of the presence of a finite-dimensional subsystem (modeling the rigid bodies) and an infinite-dimensional one (describing the flexible links). Besides facilitating the deduction of the dynamical equations, it is shown how the intrinsic modularity of this approach also simplifies the simulation phase.

Citation

• Journal: IEEE Transactions on Robotics
• Year: 2009
• Volume: 25
• Issue: 5
• Pages: 1016–1029
• Publisher: Institute of Electrical and Electronics Engineers (IEEE)
• DOI: 10.1109/tro.2009.2026504

BibTeX

@article{Macchelli_2009,
  title={{Port-Based Modeling and Simulation of Mechanical Systems With Rigid and Flexible Links}},
  volume={25},
  ISSN={1941-0468},
  DOI={10.1109/tro.2009.2026504},
  number={5},
  journal={IEEE Transactions on Robotics},
  publisher={Institute of Electrical and Electronics Engineers (IEEE)},
  author={Macchelli, A. and Melchiorri, C. and Stramigioli, S.},
  year={2009},
  pages={1016--1029}
}

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References

• (2008)
• b, (2008)
• Nakamura, Y. & Yamane, K. Dynamics computation of structure-varying kinematic chains and its application to human figures. IEEE Transactions on Robotics and Automation vol. 16 124–134 (2000) – 10.1109/70.843167
• maschke, Modeling and Control of Mechanisms and Robots (1996)
• Lang, S. Y. T. & Kesavan, H. K. Graph theoretic modeling and analysis of multibody planar mechanical systems. IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans vol. 31 97–111 (2001) – 10.1109/3468.911367
• Baciu, G. & Kesavan, H. K. From particle-mass to multibody systems: graph-theoretic modeling. IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans vol. 27 244–250 (1997) – 10.1109/3468.554686
• fritzson, Principles of Object-Oriented Modeling and Simulation with Modelica 2 1 (2004)
• Mattsson, S. E., Elmqvist, H. & Otter, M. Physical system modeling with Modelica. Control Engineering Practice vol. 6 501–510 (1998) – 10.1016/s0967-0661(98)00047-1
• 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
• stramigioli, Modeling and IPC Control of Interactive Mechanical Systems A Coordinate Free Approach (2001)
• Rodriguez, H., van der Schaft, A. J. & Ortega, R. On stabilization of nonlinear distributed parameter port-controlled Hamiltonian systems via energy shaping. Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228) vol. 1 131–136 – 10.1109/cdc.2001.980086
• Macchelli, A., Melchiorri, C. & Stramigioli, S. Port-Based Modeling of a Flexible Link. IEEE Transactions on Robotics vol. 23 650–660 (2007) – 10.1109/tro.2007.898990
• 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
• Book, W. J. Recursive Lagrangian Dynamics of Flexible Manipulator Arms. The International Journal of Robotics Research vol. 3 87–101 (1984) – 10.1177/027836498400300305
• Sunada, W. H. & Dubowsky, S. On the Dynamic Analysis and Behavior of Industrial Robotic Manipulators With Elastic Members. Journal of Mechanisms, Transmissions, and Automation in Design vol. 105 42–51 (1983) – 10.1115/1.3267343
• duindam, Modeling and Control of Complex Physical Systems The Port-Hamiltonian Approach (0)
• 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
• 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
• 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
• 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
• 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
• golo, hamiltonian formulation of planar beams. Proc IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control (2004)
• koopman, port-hamiltonian formulation and analysis of the lugre friction model. Proc 45th IEEE Conf Decision 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
• slocum, Precision Machine Design (1992)
• karnopp, System Dynamics Modeling and Simulation of Mechatronic Systems (2006)
• Huang, T. C. The Effect of Rotatory Inertia and of Shear Deformation on the Frequency and Normal Mode Equations of Uniform Beams With Simple End Conditions. Journal of Applied Mechanics vol. 28 579–584 (1961) – 10.1115/1.3641787
• golo, a hamiltonian formulation of the timoshenko beam model. 8th Mechatronics Forum (2002)
• bassi, an algorithm to discretize one-dimensional distributed port hamiltonian systems. presented at the 3rd IFAC Workshop Lagrangian Hamiltonian Methods Nonlinear Control (2006)
• Shabana, A. A. Multibody System Dynamics vol. 1 189–222 (1997) – 10.1023/a:1009773505418
• Antman, S. S. Nonlinear Problems of Elasticity. Applied Mathematical Sciences (Springer New York, 1995). doi:10.1007/978-1-4757-4147-6 – 10.1007/978-1-4757-4147-6
• (2006)
• (2008)
• (2008)
• (2008)
• (2008)
• DynaFlexPro for Maple. IEEE Control Systems vol. 26 127–138 (2006) – 10.1109/mcs.2006.252839
• (2008)
• maschke, port controlled hamiltonian systems: modeling origins and system theoretic properties. Proc 3rd IFAC Symp Nonlinear Contr Syst Design (1992)
• van der Schaft, A. L2 - Gain and Passivity Techniques in Nonlinear Control. Communications and Control Engineering (Springer London, 2000). doi:10.1007/978-1-4471-0507-7 – 10.1007/978-1-4471-0507-7
• Nakamura, Y. & Ghodoussi, M. Dynamics computation of closed-link robot mechanisms with nonredundant and redundant actuators. IEEE Transactions on Robotics and Automation vol. 5 294–302 (1989) – 10.1109/70.34765
• Bayo, E. A finite‐element approach to control the end‐point motion of a single‐link flexible robot. Journal of Robotic Systems vol. 4 63–75 (1987) – 10.1002/rob.4620040106
• Asada, H., Ma, Z.-D. & Tokumaru, H. Inverse Dynamics of Flexible Robot Arms: Modeling and Computation for Trajectory Control. Journal of Dynamic Systems, Measurement, and Control vol. 112 177–185 (1990) – 10.1115/1.2896124
• Shabana, A. A. Dynamics of Flexible Bodies Using Generalized Newton-Euler Equations. Journal of Dynamic Systems, Measurement, and Control vol. 112 496–503 (1990) – 10.1115/1.2896170
• Theodore, R. J. & Ghosal, A. Comparison of the Assumed Modes and Finite Element Models for Flexible Multilink Manipulators. The International Journal of Robotics Research vol. 14 91–111 (1995) – 10.1177/027836499501400201
• 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
• macchelli, port-based finite element model of a flexible link. Proc 5th IFAC Symp Nonlinear Control Syst (2007)
• luca, Theory of Robot Control (1996)
• macchelli, distributed port-hamiltonian formulation of infinite dimensional systems. presented at the 16th Int Symp Math Theory Netw Syst (2004)
• 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
• Jonghoon Park & Wan-Kyun Chung. Geometric integration on Euclidean group with application to articulated multibody systems. IEEE Transactions on Robotics vol. 21 850–863 (2005) – 10.1109/tro.2005.852253
• Olver, P. J. Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics (Springer New York, 1993). doi:10.1007/978-1-4612-4350-2 – 10.1007/978-1-4612-4350-2
• selig, Geometric Fundamentals of Robotics (2005)
• Ferretti, G., Schiavo, F. & Vigano, L. Modular Modelling of Flexible Thin Beams in Multibody Systems. Proceedings of the 44th IEEE Conference on Decision and Control 3363–3368 doi:10.1109/cdc.2005.1582681 – 10.1109/cdc.2005.1582681
• fasse, some applications of screw theory to lumped-parameter modeling of visco-elastically coupled rigid bodies. Symp Commemorating Legacy Works Life Sir Robert Stawell Ball Upon 100th Anniversary A Treatise on the Theory of Screws ‘ London (2000)
• 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