Parametric damped vibrations of Gough–Stewart platforms for symmetric configurations

Abstract Modal behavior of a Gough–Stewart Platform (GSP) is sensitive to several variables related to its inertia, damping and stiffness as well as its complex 3-D geometry. To optimize its dynamical performance, due to the complications of this system, it is crucial to have the equations parametrically at the neutral configuration. However, in the literature, no complete parametric solution to this problem is presented. In this paper, we establish a fully-parametric and closed-form model for the damped vibrations of GSPs. In particular, this analytical model can be used in order to design, optimize and control GSPs in high-precision/bandwidth applications. Parametric expressions of the damped eigenfrequencies and the corresponding eigenvectors as well as the Jacobian, stiffness and damping matrices are developed. Interestingly, despite the complexity of the system, it is shown how well-structured algebraic expressions are obtained using the Cartesian-space approach. Having analytically studied the eigenvectors, the conditions for decoupled vibrations are also analytically formulated. Finally, using a reference GSP, the sensitivity of the damped eigenfrequencies to stiffness and damping variations are investigated accompanied by a cross-check with an ABAQUS® simulation.

[1]  Jing-feng He,et al.  Characteristics analysis of joint space inverse mass matrix for the optimal design of a 6-DOF parallel manipulator , 2010 .

[2]  A. K. Mallik,et al.  Dynamic stability index and vibration analysis of a flexible Stewart platform , 2007 .

[3]  Clément Gosselin,et al.  Kinematic Synthesis of Hexapods with Specified Orientation Capability and Well-Conditioned Dexterity , 2000 .

[4]  Mehran Mahboubkhah,et al.  A study on vibration of Stewart platform-based machine tool table , 2013 .

[5]  M. Mahboubkhah,et al.  A comprehensive study on the free vibration of machine tools’ hexapod table , 2009 .

[6]  Ben-Zion Sandler,et al.  Rigid 6-DOF parallel platform for precision 3-D micromanipulation , 2001 .

[7]  Wei Wang,et al.  Dynamic isotropic design for modified Gough–Stewart platforms lying on a pair of circular hyperboloids , 2011 .

[8]  P. M. George,et al.  Parallel Manipulators Applications—A Survey , 2012 .

[9]  P. Xu,et al.  Vibration damping modeling of Stewart platform through Newton-Euler approach , 2005, SPIE Smart Structures and Materials + Nondestructive Evaluation and Health Monitoring.

[10]  D. Chetwynd,et al.  A general and novel approach for parameter identification of 6-DOF parallel kinematic machines , 2005 .

[11]  James D. Lee,et al.  A dynamic model of a flexible stewart platform , 1993 .

[12]  Frank L. Lewis,et al.  The singularities and dynamics of a Stewart platform manipulator , 1993, J. Intell. Robotic Syst..

[13]  T. Chondros,et al.  Analytical Methods in Rotor Dynamics , 1983 .

[14]  Placid Mathew Ferreira,et al.  Computation of stiffness and stiffness bounds for parallel link manipulators 1 This research was sup , 1999 .

[15]  V. De Sapio.  Some approaches for modeling and analysis of a parallel mechanism with stewart platform architecture , 1998 .

[16]  Bhaskar Dasgupta,et al.  A canonical formulation of the direct position kinematics problem for a general 6-6 stewart platform , 1994 .

[17]  Per Lidström,et al.  Analytical Stiffness Optimization of High-Precision Hexapods for Large Optical Telescope Applications , 2012 .

[18]  Leonard S. Haynes,et al.  On the dynamic model and kinematic analysis of a class of Stewart platforms , 1992, Robotics Auton. Syst..

[19]  Bhaskar Dasgupta,et al.  A Newton-Euler Formulation for the Inverse Dynamics of the Stewart Platform Manipulator , 1998 .

[20]  Jean-Pierre Merlet,et al.  Parallel Robots , 2000 .

[21]  Y. Shneor,et al.  Workspace of parallel kinematics machines with minimum stiffness limits: Collinear stiffness value based approach , 2012 .

[22]  M. Mahboubkhah,et al.  Vibration analysis of machine tool’s hexapod table , 2008 .

[23]  Bhaskar Dasgupta,et al.  Closed-Form Dynamic Equations of the General Stewart Platform through the Newton–Euler Approach , 1998 .

[24]  Nicholas G. Dagalakis,et al.  Kinematic modeling and calibration of a flexure based hexapod nanopositioner , 2013 .

[25]  D. Stewart.  A Platform with Six Degrees of Freedom , 1965 .

[26]  Clément Gosselin,et al.  Stiffness mapping for parallel manipulators , 1990, IEEE Trans. Robotics Autom..

[27]  Bodo Heimann,et al.  Computational efficient inverse dynamics of 6-DOF fully parallel manipulators by using the Lagrangian formalism , 2009 .

[28]  Jean-Yves Hascoët,et al.  Qualification of parallel kinematics machines in high-speed milling on free form surfaces , 2004 .

[29]  Bokman Lim,et al.  Minimum Vibration Mechanism Design Via Convex Programming , 2009 .

[30]  M. Mahboubkhah,et al.  Improved dynamic equations for the generally configured Stewart platform manipulator , 2012 .

[31]  T. Chondros,et al.  Identification of cracks in welded joints of complex structures , 1980 .

[32]  André Preumont,et al.  A six-axis single-stage active vibration isolator based on Stewart platform , 2005 .

[33]  Hong Zhou Jiang,et al.  Influence of Passive Joint Damping on Modal Space Decoupling for a Class of Symmetric Spatial Parallel Mechanisms , 2013 .

[34]  Guangren Duan,et al.  Optimal design of a class of generalized symmetric Gough-Stewart parallel manipulators with dynamic isotropy and singularity-free workspace , 2012, Robotica.

[35]  Tian Huang,et al.  A general approach for error modeling of machine tools , 2014 .

[36]  L. K. Liu,et al.  Octo-strut vibration isolation platform and its application to whole spacecraft vibration isolation , 2006 .

[37]  Chunhong Zheng,et al.  Genetic design of kinematically optimal fine tuning Stewart platform for large spherical radio telescope , 2001 .

[38]  Gexue Ren,et al.  The multi-body system modelling of the Gough–Stewart platform for vibration control , 2004 .

[39]  Tian Huang,et al.  The Local Dexterity, Optimal Architecture and Design Criteria of Parallel Machine Tools , 1998 .

[40]  Thomas G. Chondros,et al.  Archimedes life works and machines , 2010 .

[41]  Carlo Menon,et al.  Geometrical Optimization of Parallel Mechanisms Based on Natural Frequency Evaluation: Application to a Spherical Mechanism for Future Space Applications , 2009, IEEE Transactions on Robotics.

[42]  J. Chen,et al.  Instantaneous stiffness analysis and simulation for hexapod machines , 2008, Simul. Model. Pract. Theory.