A general method for the analytical elastostatic stiffness modeling of overconstrained parallel manipulators (PMs) using geometric algebra and strain energy is proposed. First, an analytical solution of the constraint and actuation wrenches exerted on the moving platform is obtained using the outer product and dual operation of geometric algebra, which avoids solving complex symbolic linear equations. Second, considering the compliances of the limbs, an analytical elastostatic model is established using the strain energy to obtain the stiffness matrices of the limbs. Finally, the deformation compatibility equations are added into equilibrium equations to obtain the overall stiffness matrix of the PM, which has a concise expression and a clear physical meaning. The proposed method is applied to the Tex3 overconstrained PM and the Tex4 overconstrained PM with redundant actuation to prove its validity. Comparable results between the theoretical analysis and the finite-element analysis (FEA) show that the former could be used as an effective alternative to the FEA method in the predesign stage. This new approach is universally applicable to the elastostatic stiffness analysis of overconstrained PMs.
Issue Section:
Research Papers
References
1.
Jin
, Y.
, Kong
, X.
, and Higgins
, C.
, 2012
, “Kinematic Design of a New Parallel Kinematic Machine for Aircraft Wing Assembly
,” Proceedings of the 10th IEEE International Conference on Industrial Informatics
, Beijing, China
, July 25–27
, pp. 669
–674
. 2.
Kiper
, C.
, and Bağdadioğlu
, B.
, 2015
, “Function Generation Synthesis with a 2-DoF Overconstrained Double-Spherical 7R Mechanism Using the Method of Decomposition and Least Squares Approximation
,” P.
Flores
, and F. Viadero eds., New Trends in Mechanism and Machine Science: From Fundamentals to Industrial Applications
, Springer International Publishing
, Switzerland
, pp. 175
–183
.3.
Siciliano
, B.
, and Khatib
, O.
, 2008
, Handbook of Robotics
, Springer
, Berlin, Heidelberg
.4.
Klimchik
, A.
, Chablat
, D.
, and Pashkevich
, A.
, 2014
, “Stiffness Modeling for Perfect and Non-Perfect Parallel Manipulators Under Internal and External Loadings
,” Mech. Mach. Theory
, 79
, pp. 1
–28
. 5.
Khasawneh
, B. E.
, and Ferreira
, P. M.
, 1999
, “Computation of Stiffness and Stiffness Bounds for Parallel Link Manipulators
,” Int. J. Mach, Tool. Manuf.
, 39
, pp. 321
–342
. 6.
Akin
, J. E.
, 2005
, Finite Element Analysis With Error Estimators: An Introduction to the FEM and Adaptive Error Analysis for Engineering Students
, Butterworth-Heinemann
, Burlington
.7.
Klimchik
, A.
, Pashkevich
, A.
, and Chablat
, D.
, 2013
, “CAD-Based Approach for Identification of Elasto-Static Parameters of Robotic Manipulators
,” Finite. Elem. Anal. Des.
, 75
, pp. 19
–30
. 8.
Huang
, T.
, Zhao
, X. Y.
, and Whitehouse
, D. J.
, 2002
, “Stiffness Estimation of a Tripod-Based Parallel Kinematic Machine
,” IEEE Trans. Robot. Autom.
, 18
(1
), pp. 50
–58
. 9.
Martin
, H. C.
, 1966
, Introduction to Matrix Methods of Structural Analysis
, McGraw-Hill
, New York
.10.
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
, FL
, May 15–19
, pp. 4213
–4219
. 11.
Pashkevich
, A.
, Chablat
, D.
, and Wenger
, P.
, 2009
, “Stiffness Analysis of Overconstrained Parallel Manipulators
,” Mech. Mach. Theory
, 44
, pp. 966
–982
. 12.
Soares Júnior
, G. D. L.
, Carvalho
, J. C. M.
, and Goncalves
, R. S.
, 2016
, “Stiffness Analysis of Multibody Systems Using Matrix Structural Analysis—MSA
,” Robotica
, 34
, pp. 2368
–2385
. 13.
Yu
, G.
, Wang
, L. P.
, Wu
, J.
, Wang
, D.
, and Hu
, C. J.
, 2018
, “Stiffness Modeling Approach for a 3-DOF Parallel Manipulator With Consideration of Nonlinear Joint Stiffness
,” Mech. Mach. Theory
, 123
, pp. 137
–152
. 14.
Gosselin
, C.
, 1990
, “Stiffness Mapping for Parallel Manipulators
,” IEEE Trans. Robot. Autom.
, 6
(3
), pp. 377
–382
. 15.
Gosselin
, C.
, and Zhang
, D.
, 2002
, “Stiffness Analysis of Parallel Mechanisms Using a Lumped Model
,” Int. J. Robot. Autom.
, 17
(1
), pp. 17
–27
.16.
Majou
, F.
, Cosselin
, C.
, Wenger
, P.
, and Chablat
, D.
, 2007
, “Parametric Stiffness Analysis of the Orthoglide
,” Mech. Mach. Theory
, 42
, pp. 296
–311
. 17.
Hoevenaars
, A. G.
, Lambert
, P.
, and Herder
, J. L.
, 2016
, “Jacobian-Based Stiffness Analysis Method for Parallel Manipulators With Non-Redundant Legs
,” Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci.
, 230
(3
), pp. 341
–352
. 18.
Sun
, T.
, Lian
, B. B.
, and Song
, Y. M.
, 2016
, “Stiffness Analysis of a 2-DoF Over-Constrained RPM With an Articulated Traveling Platform
,” Mech. Mach. Theory
, 96
, pp. 165
–178
. 19.
Xu
, P.
, Li
, B.
, Cheung
, C.-F.
, and Zhang
, J.-F.
, 2017
, “Stiffness Modeling and Optimization of a 3-DOF Parallel Robot in a Serial-Parallel Polishing Machine
,” Int. J. Precis. Enf. Man.
, 18
(4
), pp. 497
–507
. 20.
Lian
, B. B.
, Sun
, T.
, Song
, Y. M.
, Jin
, Y.
, and Price
, M.
, 2015
, “Stiffness Analysis and Experiment of a Novel 5-DoF Parallel Kinematic Machine Considering Gravitational Effects
,” Int. J. Mach. Tool. Manuf.
, 92
, pp. 82
–96
. 21.
Yan
, S. J.
, Ong
, S. K.
, and Nee
, A. Y. C.
, 2016
, “Stiffness Analysis of Parallelogram-Type Parallel Manipulators Using a Strain Energy Method
,” Robot. Com-Int. Manuf.
, 37
, pp. 13
–22
. 22.
Wu
, J.
, Wang
, J. S.
, Wang
, L. P.
, Li
, T. M.
, and You
, Z.
, 2009
, “Study on the Stiffness of a 5-DOF Hybrid Machine Tool With Actuation Redundancy
,” Mech. Mach. Theory
, 44
, pp. 289
–305
. 23.
Wu
, J.
, Li
, T. M.
, Wang
, J. S.
, and Wang
, L. P.
, 2013
, “Stiffness and Natural Frequency of a 3-DOF Parallel Manipulator With Consideration of Additional leg Candidates
,” Robot. Auton. Syst.
, 61
(8
), pp. 868
–875
. 24.
Xu
, Y. D.
, Liu
, W. L.
, Yao
, J. T.
, and Zhao
, Y. S.
, 2015
, “A Method for Force Analysis of the Overconstrained Lower Mobility Parallel Mechanism
,” Mech. Mach. Theory
, 88
, pp. 31
–48
. 25.
Rezaei
, A.
, Akbarzadeh
, A.
, and Akbarzadeh
, M. R.
, 2012
, “An Investigation on Stiffness of a 3-PSP Spatial Parallel Mechanism With Flexible Moving Platform Using Invariant Form
,” Mech. Mach. Theory
, 51
, pp. 195
–216
. 26.
Rezaei
, A.
, and Akbarzadeh
, A.
, 2013
, “Position and Stiffness Analysis of a New Asymmetric 2PRR-PRR Parallel CNC Machine
,” Adv. Robot.
, 27
(2
), pp. 133
–145
. 27.
Li
, X.
, Zlatonov
, D.
, Zoppi
, M.
, and Molfino
, R.
, 2012
, “Stiffness Estimation and Experiments for the Exechon Parallel Self-Reconfiguring Fixture Mechanism
,” Proceedings of the ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
, Chicago, IL
, Aug. 28–31
, pp. 637
–645
. 28.
Bi
, Z. M.
, 2014
, “Kinetostatic Modeling of Exechon Parallel Kinematic Machine for Stiffness Analysis
,” Int. J. Adv. Manuf. Technol.
, 71
(1–4
), pp. 325
–335
. 29.
Zhang
, J.
, Zhao
, Y. Q.
, and Jin
, Y.
, 2016
, “Kinetostatic-Model-Based Stiffness Analysis of Exechon PKM
,” Robot. Comput. Integr. Manuf.
, 30
, pp. 208
–220
. 30.
Cao
, W. A.
, Ding
, H. F.
, and Yang
, D. H.
, 2017
, “A Method for Compliance Modeling of Five Degree-of-Freedom Overconstrained Parallel Robotic Mechanisms With 3T2R Output Motion
,” ASME J. Mech. Robot.
, 9
, p. 011011. 31.
Briot
, S.
, and Khalil
, W.
, 2015
, Dynamics of Parallel Robots——From Rigid Bodies to Flexible Elements
, Springer International Publishing
, Switzerland
.32.
Briot
, S.
, and Khalil
, W.
, 2014
, “Recursive and Symbolic Calculation of the Elastodynamic Model of Flexible Parallel Robots
,” Int. J. Rob. Res.
, 33
(3
), pp. 469
–483
. 33.
Briot
, S.
, and Khalil
, W.
, 2014
, “Recursive and Symbolic Calculation of the Stiffness and Mass Matrices of Parallel Robots
,” Proceedings of the 20th CISM-IFToMM Symposium on Theory and Practice of Robots and Manipulators
, Moscow, Russia
, pp. 123
–132
. 34.
Shabana
, A. A.
, 2013
, Dynamics of Multibody Systems
, Cambridge University Press
, New York
.35.
Bauchau
, O. A.
, 2010
, Flexible Multibody Dynamics
, Springer Science & Business Media
, The Netherlands
.36.
Huang
, Z.
, and Li
, Q. C.
, 2003
, “Type Synthesis of Symmetrical Lower Mobility Parallel Mechanisms Using the Constraint-Synthesis Method
,” Int. J. Rob. Res.
, 22
(1
), pp. 59
–82
. 37.
Xu
, L. M.
, Li
, Q. C.
, Zhang
, N. B.
, and Chen
, Q. H.
, 2017
, “Mobility, Kinematic Analysis, and Dimensional Optimization of New Three-Degrees-of-Freedom Parallel Manipulator With Actuation Redundancy
,” ASEM J. Mech. Robot.
, 9
(4
), p. 041008
. 38.
Xu
, L. M.
, Li
, Q. C.
, Tong
, J. H.
, and Chen
, Q. H.
, 2018
, “Tex3: An 2R1T Parallel Manipulator With Minimum DOF of Joints and Fixed Linear Actuators
,” Int. J. Precis. Eng. Man.
, 19
(2
), pp. 227
–238
. 39.
Li
, Q. C.
, Xu
, L. M.
, Chen
, Q. H.
, and Ye
, W.
, 2017
, “New Family of RPR-Equivalent Parallel Mechanisms: Design and Application
,” Chin. J. Mech. Eng.
, 30
(2
), pp. 217
–221
. 40.
Li
, Q. C.
, Chai
, X. X.
, and Xiang
, J. N.
, 2016
, “Mobility Analysis of Limited-Degrees-of-Freedom Parallel Mechanisms in the Framework of Geometric Algebra
,” ASME J. Mech. Robot.
, 8
(4
), p. 041005
. 41.
Chai
, X. X.
, and Li
, Q. C.
, 2017
, “Analytical Mobility Analysis of Bennett Linkage Using Geometric Algebra
,” Adv. Appl. Clifford. Al.
, 27
(3
), pp. 2083
–2095
. 42.
Yao
, H. J.
, Chen
, Q. H.
, Chai
, X. X.
, and Li
, Q. C.
, 2017
, “Singularity Analysis of 3-RPR Parallel Manipulators Using Geometric Algebra
,” Adv. Appl. Clifford. Al.
, 27
(3
), pp. 2097
–2113
. 43.
Aristidou
, A.
, and Lasenby
, J.
, 2011
, “FABRIK: A Fast, Iterative Solver for the Inverse Kinematics Problem
,” Graph. Models
, 73
(5
), pp. 243
–260
. 44.
Bayro-Corrochano
, E.
, 2010
, “Geometric Algebra for Modeling in Robot Physics,” Geometric Computing: For Wavelet Transforms, Robot Vision, Learning, Control and Action
, Springer
, London
, pp. 45
–59
.45.
Campos-Macías
, L.
, Carbajal-Espinosa
, O.
, Loukianov
, A.
, and Bayro-Corrochano
, E.
, 2016
, “Inverse Kinematics for a 6-DOF Walking Humanoid Robot Leg
,” Adv. Appl. Clifford. Al.
, 27
(1
), pp. 581
–597
. 46.
Hrdina
, J.
, Návrat
, A.
, and Vašík
, P.
, 2015
, “Control of 3-Link Robotic Snake Based on Conformal Geometric Algebra
,” Adv. Appl. Clifford. Al.
, 26
(3
), pp. 1069
–1080
. 47.
Hestenes
, D.
, 1999
, New Foundations for Classical Mechanics
, Kluwer Academic Publishers
, Dordrecht
.48.
Beer
, F. P.
, Johnston
, J. E. R.
, and DeWolf
, J. T.
, 2009
, Mechanics of Materials
, McGraw-Hill Press
, New York
.49.
Joshi
, S. A.
, and Tsai
, L. W.
, 2002
, “Jacobian Analysis of Limited-DOF Parallel Manipulators
,” ASME J. Mech. Des.
, 124
(2
), pp. 254
–258
. Copyright © 2019 by ASME
You do not currently have access to this content.
