Research Papers

An Algorithm to Study the Elastodynamics of Parallel Kinematic Machines With Lower Kinematic Pairs

[+] Author and Article Information
Alessandro Cammarata

e-mail: acamma@diim.unict.it

Rosario Sinatra

Department of Industrial Engineering,
University of Catania,
Viale A. Doria 6,
95125 Catania, Italy

Hereafter, the letter i will be referred to bodies, while the letter j to joints.

Notice that stiffness units change according to the corresponding degrees of freedom, as instance: position coordinates, slope coordinates and so on.

If the base platform is considered fixed, the expressions of case (a) are simplified deleting all terms containing the nodal-displacement array of the base.

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the Journal of Mechanisms and Robotics. Manuscript received February 10, 2011; final manuscript received September 11, 2012; published online October 19, 2012. Assoc. Editor: Vijay Kumar.

J. Mechanisms Robotics 5(1), 011004 (Jan 19, 2012) (9 pages) Paper No: JMR-11-1016; doi: 10.1115/1.4007705 History: Received February 10, 2011; Revised September 11, 2012

In this paper, an algorithm to help designers to integrate the elastodynamics analysis along with the inverse positioning and orienting problems of a parallel kinematic machine (PKM) into a single package is conceived. The proposed algorithm applies concepts from the matrix structural analysis (MSA) and finite element analysis (FEA) to determine the generalized stiffness matrix and the linearized elastodynamics equations of a PKM with only lower kinematic pairs. A PKM is modeled as a system of flexible links and rigid bodies connected by means of joints. Three cases are analyzed to consider the combinations between flexible and rigid bodies in order to find the local stiffness matrices. The latter are combined to obtain the limb matrices and, then, the global stiffness matrix of the whole robotic system. The same nodes coming from the links discretization are considered to include joint masses/inertias into the model. Finally, a case study is proposed to show some feasible applications and to compare results to commercial software for validation.

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Zienkiewicz, O. C., and Taylor, R. L., 2000, Solid Mechanics—Volume 2, Butterworth Heinemann, London.
Corradini, C., Fauroux, J. C., Krut, S., and Company, O., 2004, “Evaluation of a 4 Degree of Freedom Parallel Manipulator Stiffness,” Proceedings of the 11th World Congress in Mechanism and Machine Science, IFTOMM2004, Tianjin, China, April 1–4.
Bouzgarrou, B. C., Fauroux, J. C., Gogu, G., and Heerah, Y., 2004, “Rigidity Analysis of T3R1 Parallel Robot With Uncoupled Kinematics,” Proceedings of the 35th International Symposium on Robotics, Paris, France, March.
Zhao, Y., 2011, “Kineto-Elastodynamic Characteristics of the Six-Degree-of-Freedom Parallel Structure Seismic Simulator,” J. Rob., 2011, pp. 1–17. [CrossRef]
Martin, H. C., 1966, Introduction to Matrix Methods of Structural Analysis, McGraw-Hill Book Company, New York, 29(9), p. 4.
Wang, C. K., 1966, Matrix Methods of Structural Analysis, International Textbook Company, Scranton, PA.
Przemieniecki, J. S., 1985, Theory of Matrix Structural Analysis, Dover Publications, Inc, New York.
Huang, T., Zhao, X., and Withehouse, D. J., 2001, “Stiffness Estimation of a Tripod-Based Parallel Kinematic Machine,” Proceedings 2001 ICRA. IEEE International Conference on Robotics and Automation (Cat. No.01CH37164), 4, p. 3280.
Li, Y. W., Wang, J. S., and Wang, L. P., 2002, “Stiffness Analysis of a Stewart Platform-Based Parallel Kinematic Machine,” Proceedings of IEEE ICRA: International Conference on Robotics and Automation, Washington, USA, May 11–15.
Clinton, C. M., Zhang, G., and Wavering, A. J., 1997, “Stiffness Modeling of a Stewart-Platform-Based Milling Machine,” Transactions of the North America Manufacturing Research Institution of SME, Lincoln, NB, USA, May 20–23, Vol. XXV, pp. 335–340.
Al Bassit, L., Angeles, J., Al-Wydyan, K., and Morozov, A., 2002, “The Elastodynamics of a Schönflies -Motion Generator,” Centre for Intelligent Machines, McGill University, Montreal, Canada, Technical Report No. TR-CIM-02-06.
Deblaise, D., Hernot, X., and Maurine, P., 2006, “A Systematic Analytical Method for PKM Stiffness Matrix Calculation,” Proceedings of the 2006 IEEE International Conference on Robotics and Automation, Orlando, FL, May.
Gonalves, R. S., and Carvalho, J. C. M., 2008, “Stiffness Analysis of Parallel Manipulator Using Matrix Structural Analysis,” Second European Conference on Mechanism Science, EUCOMES 2008, Cassino, Italy.
Wittbrodt, E., Adamiec-Wójcik, I., and Wojciech, S., 2006, Dynamics of Flexible Multibody Systems, Springer, New York.
Gosselin, C. M., 1990, “Stiffness Mapping for Parallel Manipulator,” IEEE Trans. Rob. Autom., 6, pp. 377–382. [CrossRef]
Briot, S., Pashkevich, A., and Chablat, D., 2009, “On the Optimal Design of Parallel Robots Taking Into Account Their Deformations and Natural Frequencies,” Proceedings of the ASME 2009 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE, San Diego, CA, USA, Aug. 30–Sept. 2.
Yoon, W. K., Suehiro, T., Tsumaki, Y., and Uchiyama, M., 2004, “Stiffness Analysis and Design of a Compact Modified Delta Parallel Mechanism,” Robotica, 22(5), pp. 463–475. [CrossRef]
Gosselin, C. M., and Zhang, D., 2002, “Stiffness Analysis of Parallel Mechanisms Using a Lumped Model,” Int. J. Rob. Autom., 17, pp. 17–27.
Pashkevich, A., Chablat, D., and Wenger, P., 2009, “Stiffness Analysis of Overconstrained Parallel Manipulators,” Mech. Mach. Theory, 44, pp. 966–982. [CrossRef]
Pashkevich, A., Klimchik, A., and Chablat, D., 2011, “Enhanced Stiffness Modeling of Manipulators With Passive Joints,” Mech. Mach. Theory, 46(5), pp. 662–679. [CrossRef]
Angeles, J., 2007, Fundamentals of Robotic Mechanical Systems, Springer, New York.
Pashkevich, A., Chablat, D., and Wenger, P., 2009, “Stiffness Analysis of Multi-Chain Parallel Robotic Systems,” J. Autom. Mobile Rob. Intell. Syst., 3(3), pp. 75–82.
Shabana, A. A., 2005, Dynamics of Multibody Systems, 3rd ed., Cambridge University Press, Cambridge, MA.
Cammarata, A., 2012, “On the Stiffness Analysis and Elastodynamics of Parallel Kinematic Machines,” Serial and Parallel Robot Manipulators: Kinematic Dynamics and Control, Serdar Küçük, ed., In-Tech, Rijeka, Croatia.
Zhou, Z., Mechefske, C. K., and Xi, F., 2007, “Nonstationary Vibration of a Fully Flexible Parallel Kinematic Machine,” ASME J. Vibr. Acoust., 129, pp. 623–630. [CrossRef]
Gosselin, C. M., Tale-Masouleh, M., Duchaine, V., Richard, P.-L., Foucault, S., and Kong, A., 2007, “Tripteron, Quadrupteron, and Pentapteron: Kinematic Analysis and Benchmarking of a Family of Parallel Mechanisms,” Proceeding of the IEEE International Conference on Robotics and Automation (ICRA07), April 10–14, Rome, Italy.


Grahic Jump Location
Fig. 1

Notation of the algorithm

Grahic Jump Location
Fig. 2

Possible combinations of bodies and nodes inside a limb: (a) rigid body–flexible body; (b) flexible–flexible; and (c) flexible body–rigid body

Grahic Jump Location
Fig. 3

Application to a kinematic chain

Grahic Jump Location
Fig. 5

The first nine modes and the associated level of strain energy V

Grahic Jump Location
Fig. 6

Relative error versus mode number of the algorithm compared to FEA commercial software results

Grahic Jump Location
Fig. 4

CAD model of tripteron: undeformed pose



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