Research Papers

Natural Frequency Computation of Parallel Robots

[+] Author and Article Information
Coralie Germain

Institut de Recherche en Communications
et Cybernétique de Nantes IRCCyN,
UMR CNRS 6597,
1, rue de la Noë,
Nantes 44321, France
e-mail: coralie.germain@ens-rennes.fr

Sébastien Briot

Institut de Recherche en Communications
et Cybernétique de Nantes IRCCyN,
UMR CNRS 6597,
1, rue de la Noë,
Nantes 44321, France
e-mail: Sebastien.Briot@irccyn.ec-nantes.fr

Stéphane Caro

Institut de Recherche en Communications
et Cybernétique de Nantes IRCCyN,
UMR CNRS 6597,
1, rue de la Noë,
Nantes 44321, France
e-mail: Stephane.Caro@irccyn.ec-nantes.fr

Philippe Wenger

Institut de Recherche en Communications
et Cybernétique de Nantes IRCCyN,
UMR CNRS 6597,
1, rue de la Noë,
Nantes 44321, France
e-mail: Philippe.Wenger@irccyn.ec-nantes.fr

Note that each robot link can be composed of one element or several elements.

It is assumed that the generalized velocities are equal to d/dt(δqt)=q·t.

Note that index ij is written i, j in this section for a better understanding of the equations.

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received July 30, 2013; final manuscript received September 11, 2014; published online January 12, 2015. Assoc. Editor: Arend L. Schwab.

J. Comput. Nonlinear Dynam 10(2), 021004 (Mar 01, 2015) (11 pages) Paper No: CND-13-1194; doi: 10.1115/1.4028573 History: Received July 30, 2013; Revised September 11, 2014; Online January 12, 2015

The characterization of the elastodynamic behavior and natural frequencies of parallel robots is a crucial point. Accurate elastodynamic models of parallel robots are useful at both their design and control stages in order to define their optimal dimensions and shapes while improving their vibratory behavior. Several methods exist to write the elastodynamic model of manipulators. However, those methods do not provide a straightforward way to write the Jacobian matrices related to the kinematic constraints of parallel manipulators. Therefore, the subject of this paper is about a systematic method for the determination of the mass and stiffness matrices of any parallel robot in stationary configurations. The proposed method is used to express the mass and stiffness matrices of the Nantes Variable Actuation Robot (NaVARo), a three-degree-of-freedom (3DOF) planar parallel robot with variable actuation schemes, developed at IRCCyN. Then, its natural frequencies are evaluated and compared with those obtained from both Cast3m software and experimentally.

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Brogardh, T., 2007, “Present and Future Robot Control Development—An Industrial Perspective,” Annu. Rev. Control, 31(1), pp. 69–79. [CrossRef]
Chanal, H.,Duc, E., and Ray, P., 2006, “A Study of the Impact of Machine Tool Structure on Machining Processes,” Int. J. Mach. Tools Manuf., 46(2), pp. 98–106, [CrossRef]
Voglewede, P. A., and Ebert-Uphoff, I., 2005, “Overatching Framework for Measuring the Closeness to Singularities of Parallel Manipulators,” IEEE Trans. Rob., 21(6), pp. 1037–1045. [CrossRef]
Bouzgarrou, B. C., Ray, P., and Gogu, G., 2005, “New Approach for Dynamic Modelling of Flexible Manipulators,” Part K: J. Multi-Body Dyn., 219(3), pp. 285–298. [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 and Computers and Information in Engineering Conference IDETC/CIE, San Diego, CA, Aug. 30–Sept. 2, pp. 367–376.
Singer, N. C., and Seering, W. P., 1990, “Preshaping Command Inputs to Reduce System Vibration,” ASME J. Dyn. Sys., Meas., Control, 112(1), pp. 76–82. [CrossRef]
Singh, T., and Singhose, W. E., 2002, “Tutorial on Input Shaping/Time Delay Control of Maneuvering Flexible Structures,” American Control Conference, Anchorage, AK, May 8–10, pp. 1717–1731. [CrossRef]
Pelaez, G., Pelaez, Gu., Perez, J. M., Vizan, A., and Bautista, E., 2005, “Input Shaping Reference Commands for Trajectory Following Cartesian Machines,” Control Eng. Pract., 13(8), pp. 941–958. [CrossRef]
Bouzgarrou, B. C., Fauroux, J. C., Gogu, G., and Heerah, Y., 2004, “Rigidity Analysis of t3r1 Parallel Robot Uncoupled Kinematics,” Proceedings of the 35th International Symposium on Robotics, Paris, France.
Bettaieb, F.,Cosson, P., and Hascoët, J.-Y., 2007, “Modeling of a High-Speed Machining Center With a Multibody Approach: The Dynamic Modeling of Flexible Manipulators,” Proceedings of the 6th International Conference on High Speed Machining, San Sebastian, Spain.
Shabana, A., 2005, Dynamics of Multibody Systems, Cambridge University Press.
Rognant, M., Courteille, E., and Maurine, P., 2010, “A Systematic Procedure for the Elastodynamic Modeling and Identification of Robot Manipulators,” IEEE Trans. Rob., 26(6), pp. 1085–1093. [CrossRef]
Bauchau, O. A., 2011, Flexible Multibody Dynamics, Springer, Dordrecht, Heidelberg, London, New York.
Cammarata, A.,Condorelli, D., and Sinatra, R., 2013, “An Algorithm to Study the Elastodynamics of Parallel Kinematic Machines With Lower Kinematic Pairs,” ASME J. Mech. Rob., 5(1), p. 011004. [CrossRef]
Wittbrodt, E., Adamiec-Wójcik, I., and Wojciech, S., 2006, Dynamics of Flexible Multibody Systems, Springer-Verlag, Berlin, Heidelberg, New York.
Ibrahim, O., and Khalil, W., 2010, “Inverse and Direct Dynamic Models of Hybrid Robots,” Mech. Mach. Theory, 45(4), pp. 627–640. [CrossRef]
Rakotomanga, N.,Chablat, D., and Caro, S., “Kinetostatic Performance of a Planar Parallel Mechanism With Variable Actuation,” Advances in Robot Kinematics, Springer, The Netherlands, pp. 311–320.
Caro, S.,Chablat, D., Wenger, P., and Kong, X., 2014, “Kinematic and Dynamic Modeling of a Parallel Manipulator With Eight Actuation Modes,” New Trends in Medical and Service Robots, Springer, pp. 315–329. [CrossRef]
Castem3000. The Castem Software, http:\\www-cast3m.cea.fr. Webpage accessed November 2012.
Boyer, F., and Khalil, W., 1996, “Dynamics of a 3DOF Spatial Parallel Manipulator With Flexible Links,” IEEE International Conference on Robotics and Automation, 1996. Proceedings of IEEE International Conference on, 1995, Nagoya, May 21–27, pp. 627–633. [CrossRef]
Blevins, R. D., 2001, “Formulas for Natural Frequency and Mode Shape,” ASME J. Appl. Mech., 47(2), pp. 461–462. [CrossRef]
Pashkevich, A., Chablat, D., and Wenger, P., 2009, “Stiffness Analysis of Overconstrained Parallel Manipulators,” Mech. Mach. Theory, 44(5), pp. 966–982. [CrossRef]
Khalil, W., and Dombre, E., 2002, Modeling, Identification, and Control of Robots, Hermes Penton London, London, UK.
Arakelian, V., Briot, S., and Glazunov, V., 2008, “Increase of Singularity-Free Zones in the Workspace of Parallel Manipulators Using Mechanisms of Variable Structure,” Mech. Mach. Theory, 43(9), pp. 1129–1140. [CrossRef]


Grahic Jump Location
Fig. 1

Schematic of a parallel robot for its dynamic modeling: (a) kinematic chain and (b) virtual tree structure

Grahic Jump Location
Fig. 2

Schematics of the flexible elements into consideration: (a) parameters of one flexible body j and (b) assembly of two flexible bodies

Grahic Jump Location
Fig. 3

The NaVARo (a) prototype of the NaVARo located at IRCCyN, Nantes, France and (b) Shematics of the NaVARo

Grahic Jump Location
Fig. 4

The eight configurations used for the experiments: (a) pose 1 x = 0 m, y = 0 m, θ = 0 rad; (b) pose 2 x = 0 m, y = 0 m, θ = −π/3 rad; (c) pose 3 x = 0.117 m, y = 0.068 m, θ = −π/3 rad; (d) pose 4 x = 0.182 m, y = 0.105 m, θ = −π/3 rad;(e) pose 5 x = −0.117 m, y = 0.068 m, θ = −π/3 rad; (f) Pose 6 x = −0.182 m, y = 0.105 m, θ = −π/3 rad; (g) pose 7 x = 0 m, y = −0.135 m, θ = −π/3 rad; and (h) pose 8 x = 0 m, y = −0.21 m, θ = −π/3 rad

Grahic Jump Location
Fig. 5

Experimental setup: DataBox




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