Abstract
Natural frequencies of a Symmetric Gough-Stewart Platform (SGSP) mechanically limit its bandwidth and precision e.g. in CNCs or optical collimation systems. Hence, the required vibrational behavior at the neutral configuration of an SGSP can be regarded as an essential property to be optimized. However, due to the complexity of its geometry, the analysis of the vibrational behavior, using analytical methods, is quite challenging and in the literature a complete joint-space formulation of SGSP vibrations has not yet been addressed. In this paper, we present an analytical and parametric formulation of this problem in the joint space. We parametrically formulate the Jacobian matrix, the linearized equations of motion and calculate the eigenvectors and eigenfrequencies in terms of the design variables of the system. The parametric model presented in this study can be directly employed for design, optimization and control of SGSPs. It is concluded that for SGSPs, the joint-space formulation gives additional insights to the modal properties complementing the Cartesian-space analysis.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
J.J. Zierer, J.H. Beno, D.A. Weeks, I.M. Soukup, J.M. Good, J.A. Booth, G.J. Hill, M.D. Rafal, Design, testing, and installation of a high-precision hexapod for the Hobby-Eberly Telescope Dark Energy Experiment (HETDEX), Ground-Based and Airborne Telescopes Iv, 8444 (2012).
Y.D. Patel, Parallel Manipulators Applications—A Survey, Modern Mechanical Engineering, 02 (2012) 57-64.
Y. Ting, Y.S. Chen, H.C. Jar, Modeling and control for a Gough-Stewart platform CNC machine, Journal of Robotic Systems, 21 (2004) 609-623.
V.T. Portman, V.S. Chapsky, Y. Shneor, Workspace of parallel kinematics machines with minimum stiffness limits: Collinear stiffness value based approach, Mechanism and Machine Theory, 49 (2012) 67-86.
J. Chen, F. Lan, Instantaneous stiffness analysis and simulation for hexapod machines, Simulation Modelling Practice and Theory, 16 (2008) 419-428.
B.S. El-Khasawneh, P.M. Ferreira, Computation of stiffness and stiffness bounds for parallel link manipulators, International Journal of Machine Tools & Manufacture, 39 (1999) 321-342.
C. Gosselin, Stiffness Mapping for Parallel Manipulators, Ieee Transactions on Robotics and Automation, 6 (1990) 377-382.
S. Pedrammehr, M. Mahboubkhah, N. Khani, A study on vibration of Stewart platform-based machine tool table, International Journal of Advanced Manufacturing Technology, 65 (2013) 991-1007.
M. Mahboubkhah, M.J. Nategh, S.E. Khadem, A comprehensive study on the free vibration of machine tools' hexapod table, International Journal of Advanced Manufacturing Technology, 40 (2009) 1239-1251.
M. Mahboubkhah, M.J. Nategh, S.E. Khadem, Vibration analysis of machine tool's hexapod table, International Journal of Advanced Manufacturing Technology, 38 (2008) 1236-1243.
P. Mukherjee, B. Dasgupta, A.K. Mallik, Dynamic stability index and vibration analysis of a flexible Stewart platform, Journal of Sound and Vibration, 307 (2007) 495-512.
B. Afzali-Far, P. Lidström, K. Nilsson, Parametric damped vibrations of Gough–Stewart platforms for symmetric configurations, Mechanism and Machine Theory, 80 (2014) 52-69.
T.X. Tian, H.Z. Jiang, J.F. He, Z.Z. Tong, Influence of passive joint damping on modal space decoupling for a class of symmetric spatial parallel mechanisms, Applied Mechanics and Materials, 2013, pp. 1152-1157.
Z.Z. Tong, J.F. He, H.Z. Jiang, G.R. Duan, Optimal design of a class of generalized symmetric Gough-Stewart parallel manipulators with dynamic isotropy and singularity-free workspace, Robotica, 30 (2012) 305-314.
B. Afzali-Far, P. Lidström, K. Nilsson, Analytical Stiffness Optimization of High-Precision Hexapods for Large Optical Telescope Applications, Proceedings of the 25th Nordic Seminar on Computational Mechanics, 2012.
H.Z. Jiang, J.F. He, Z.Z. Tong, Characteristics analysis of joint space inverse inertia matrix for the optimal design of a 6-DOF parallel manipulator, Mechanism and Machine Theory, 45 (2010) 722-739.
H.M. Lin, J.E. McInroy, Disturbance attenuation in precise hexapod pointing using positive force feedback, Control Engineering Practice, 14 (2006) 1377-1386.
Y.X. Chen, J.E. McInroy, Decoupled control of flexure-jointed hexapods using estimated joint-space mass-inertia matrix, Ieee Transactions on Control Systems Technology, 12 (2004) 413-421.
J.E. McInroy, J.C. Hamann, Design and control of flexure jointed hexapods, Ieee Transactions on Robotics and Automation, 16 (2000) 372-381.
A. Cantoni, P. Butler, Eigenvalues and Eigenvectors of Symmetric Centrosymmetric Matrices, Linear Algebra and Its Applications, 13 (1976) 275-288.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Afzali-Far, B., Lidström, P. (2015). A Joint-Space Parametric Formulation for the Vibrations of Symmetric Gough-Stewart Platforms. In: Selvaraj, H., Zydek, D., Chmaj, G. (eds) Progress in Systems Engineering. Advances in Intelligent Systems and Computing, vol 366. Springer, Cham. https://doi.org/10.1007/978-3-319-08422-0_48
Download citation
DOI: https://doi.org/10.1007/978-3-319-08422-0_48
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-08421-3
Online ISBN: 978-3-319-08422-0
eBook Packages: EngineeringEngineering (R0)