In this article, the closed-form dynamic equations of planar flexible link manipulators (FLMs), with revolute joints and constant cross sections, are derived combining Lagrange’s equations and the assumed mode shape method. To overcome the lengthy and complicated derivative calculation of the Lagrangian function of a FLM, these computations are done only once for a single flexible link manipulator with a moving base (SFLMB). Employing the Lagrange multipliers and the dynamic equations of the SFLMB, the equations of motion of the FLM are derived in terms of the dependent generalized coordinates. To obtain the closed-form dynamic equations of the FLM in terms of the independent generalized coordinates, the natural orthogonal complement of the Jacobian constraint matrix, which is associated with the velocity constraints in the linear homogeneous form, is used. To verify the proposed closed-form dynamic model, the simulation results obtained from the model were compared with the results of the full nonlinear finite element analysis. These comparisons showed sound agreement. One of the main advantages of this approach is that the derived dynamic model can be used for the model based end-effector control and the vibration suppression of planar FLMs.
Issue Section:
Research Papers
1.
Wang
, F. Y.
, and Gao
, Y.
, 2003, Advanced Studies of Flexible Robotic Manipulators: Modeling, Design, Control and Applications: Series in Intelligent Control and Intelligent Automation
, World Scientific
, River Edge, NJ.
2.
Low
, K. H.
, and Vidyasagar
, M.
, 1988, “A Lagrangian Formulation of the Dynamic Model for Flexible Manipulator Systems
,” ASME J. Dyn. Syst., Meas., Control
0022-0434, 110
(2
), pp. 175
–181
.3.
Zhang
, X.
, Xu
, W.
, Nair
, S. S.
, and Cellabonia
, V.
, 2005, “PDE Modeling and Control of a Flexible Two-Link Manipulator
,” IEEE Trans. Control Syst. Technol.
1063-6536, 13
(2
), pp. 301
–312
.4.
Bathe
, K. J.
, 1996, Finite Element Procedures
, Prentice-Hall
, Upper Saddle River, NJ.
5.
Hoa
, S. V.
, 1979, “Vibration of a Rotating Beam With Tip Mass
,” J. Sound Vib.
0022-460X, 67
(3
), pp. 369
–381
.6.
Naganthan
, G.
, and Soni
, A. H.
, 1987, “Coupling Effects of Kinematics and Flexibility in Manipulators
,” Int. J. Robot. Res.
0278-3649, 6
(1
), pp. 75
–84
.7.
Khulief
, Y. A.
, 1992, “On the Finite Element Dynamic Response of Flexible Mechanism
,” Comput. Methods Appl. Mech. Eng.
0045-7825, 97
(1
), pp. 23
–32
.8.
Khulief
, Y. A.
, 2001, “Vibration Suppression in Rotating Beams Using Active Modal Control
,” J. Sound Vib.
0022-460X, 242
(4
), pp. 681
–699
.9.
Yang
, J. B.
, Jiang
, L. J.
, and Chen
, D. Ch.
, 2004, “Dynamic Modeling and Control of a Rotating Euler-Bernoulli Beam
,” J. Sound Vib.
0022-460X, 274
(3
), pp. 863
–875
.10.
Mayo
, J.
, and Dominguez
, J.
, 1997, “Finite Element Geometrically Nonlinear Dynamic Formulation of Flexible Multibody System Using a New Displacement Representation
,” ASME J. Vibr. Acoust.
0739-3717, 119
(4
), pp. 573
–581
.11.
Zhang
, X.
, and Erdman
, A. G.
, 2006, “Optimal Placement of Piezoelectric Sensors and Actuators for Controlled Flexible Linkage Mechanisms
,” ASME J. Vibr. Acoust.
0739-3717, 128
(2
), pp. 256
–260
.12.
Prezemienecki
, J. S.
, 1967, Theory of Matrix Structural Analysis
, McGraw-Hill
, New York
.13.
Meirovitch
, L.
, 1986, Elements of Vibration Analysis
, McGraw-Hill
, New York
.14.
Book
, W. J.
, 1984, “Recursive Lagrangian Dynamics of Flexible Manipulator Arms
,” Int. J. Robot. Res.
0278-3649, 3
(3
), pp. 87
–101
.15.
De Luca
, A.
, and Siciliano
, B.
, 1991, “Closed-Form Dynamic Model of Planar Multilink Lightweight Robots
,” IEEE Trans. Syst. Man Cybern.
0018-9472, 21
(4
), pp. 826
–839
.16.
Yuan
, B. S.
, Book
, W. J.
, and Huggins
, J. D.
, 1993, “Dynamics of Flexible Manipulator Arms: Alternative Derivation, Verification, and Characteristics for Control
,” ASME J. Dyn. Syst., Meas., Control
0022-0434, 115
(3
), pp. 394
–404
.17.
Chen
, W.
, 2001, “Dynamic Modeling of Multi-Link Flexible Robotic Manipulators
,” Comput. Struct.
0045-7949, 79
(2
), pp. 183
–195
.18.
Gosavi
, S. V.
, and Kelkar
, A. G.
, 2004, “Modelling, Identification and Passivity-Based Robust Control of Piezo-Actuated Flexible Beam
,” ASME J. Vibr. Acoust.
0739-3717, 126
(2
), pp. 260
–271
.19.
Theodore
, R. J.
, and Ghosal
, A.
, 1995, “Comparison of the Assumed Modes and Finite Element Models for Flexible Multilink Manipulators
,” Int. J. Robot. Res.
0278-3649, 14
(2
), pp. 91
–111
.20.
Hale
, J. K.
, and Verduyn Lunel
, S. M.
, 2003, “Stability and control of feedback systems with time delays
,” Int. J. Syst. Sci.
0020-7721, 34
(8-9
), pp. 497
–504
.21.
Benosman
, M.
, Le Vey
, G.
, Lanari
, L.
, and De Luca
, A.
, 2004, “Rest-to-Rest Motion for Planar Multi-Link Flexible Manipulator through Backward Recursion
,” ASME J. Dyn. Syst., Meas., Control
0022-0434, 126
(1
), pp. 115
–123
.22.
Cheong
, J.
, Chung
, W. K.
, and Youm
, Y.
, 2004, “Inverse Kinematic of Multilink Flexible Robotics for High-Speed Application
,” IEEE Trans. Rob. Autom.
1042-296X, 20
(2
), pp. 269
–282
.23.
Moallem
, M.
, Patel
, R. V.
, and Khorasani
, K.
, 2001, “Nonlinear Tip-Position Control of a Flexible-Link Manipulator: Theory and Experiments
,” Automatica
0005-1098, 37
(11
), pp. 1825
–1834
.24.
Shan
, J.
, Hong
, T. L.
, and Sun
, D.
, 2005, “Slewing and Vibration Control of a Single-Link Flexible Manipulator by Positive Position Feedback (PPF)
,” Mechatronics
0957-4158, 15
(4
), pp. 487
–503
.25.
Morita
, Y.
, Ukia
, H.
, and Kando
, H.
, 1997, “Robust Trajectory Tracking Control of Elastic Robot Manipulators
,” ASME J. Dyn. Syst., Meas., Control
0022-0434, 119
(4
), pp. 727
–735
.26.
Moallem
, M.
, Khorasani
, K.
, and Patel
, R. V.
, 1998, “Inversion—Based Sliding Control of a Flexible-Link Manipulator
,” Int. J. Control
0020-7179, 71
(3
), pp. 477
–490
.27.
Ginsberg
, J. H.
, 1995, Advanced Engineering Dynamics
, Cambridge University Press
, New York
.28.
Greenwood
, D. T.
, 1965, Principles of Dynamics
, Prentice-Hall
, New York
.29.
Centinkunt
, S.
, and Ittoop
, B.
, “Computer-Automated Symbolic Modeling of Dynamics of Robotic Manipulators With Flexible Link
,” IEEE Trans. Rob. Autom.
1042-296X 8
(1
), pp. 94
–105
.30.
Li
, C. J.
, and Sankar
, T. S.
, 1993, “Systematic Methods for Efficient Modeling and Dynamics of Flexible Robot Manipulators
,” IEEE Trans. Syst. Man Cybern.
0018-9472, 23
(1
), pp. 77
–95
.31.
Angeles
, J.
, and Lee
, S.
, 1988, “The Formulation of Dynamical Equations of Holonomic Mechanical Systems Using a Natural Orthogonal Complement
,” ASME J. Appl. Mech.
0021-8936, 55
(1
), pp. 243
–244
.32.
Saha
, S. K.
, and Angeles
, J.
, 1991, “Dynamics of Nonholonomic Mechanical Systems Using a Natural Orthogonal Complement
,” ASME J. Appl. Mech.
0021-8936, 58
(1
), pp. 238
–243
.33.
Fotouhi
, R.
, 2007, “Dynamic Analysis of a Very Flexible Beam
,” J. Sound Vib.
0022-460X, 305
(3
), pp. 521
–533
.34.
Vakil
, M.
, Fotouhi
, R.
, and Nikiforuk
, P. N.
, 2007, “Application of the Integral Manifold Concept for the End-Effector Trajectory Tracking of a Flexible Link Manipulator
,” American Control Conference
, New York
, Jul. 11–13, pp. 741
–747
.35.
Vakil
, M.
, Fotouhi
, R.
, and Nikiforuk
, P. N.
, 2007, “End-Effector Trajectory Tracking of a Flexible Link Manipulator Using Integral Manifold Concept
,” J. Sound Vib.
0022-460X, to be published.36.
Oguamanam
, D. C. D.
, Heppler
, G. R.
, and Hansen
, J. S.
, 1998, “Modelling of a Flexible Slewing Link
,” ASME J. Vibr. Acoust.
0739-3717, 120
(4
), pp. 994
–996
.37.
Alberts
, T. A.
, Xia
, H.
, and Chen
, Y.
, 1992, “Dynamic Analysis to Evaluate Viscoelastic Passive Damping Augmentation for the Space Shuttle Remote Manipulator System
,” ASME J. Dyn. Syst., Meas., Control
0022-0434, 114
(3
), pp. 468
–474
.38.
Fraser
, R.
, and Daniel
, R. W.
, 1991, Perturbation Techniques for Flexible Manipulators
, Kluwer Academic
, Boston
.39.
Trindade
, M. A.
, and Sampaio
, R.
, 2002, “Dynamics of Beams Undergoing Large Rotations Accounting for Arbitrary Axial Deformation
,” J. Guid. Control Dyn.
0731-5090, 25
(4
), pp. 634
–643
.40.
Shabana
, A. A.
, 2001, Computational Dynamics
, Wiley
, New York
.41.
Saha
, S. K.
, 1999, “Dynamics of Serial Multibody Systems Using the Decoupled Natural Orthogonal Complement Matrices
,” ASME J. Appl. Mech.
0021-8936, 66
(4
), pp. 986
–996
.42.
Khan
, A. W.
, Krovi
, V. N.
, Saha
, S. K.
, and Angeles
, J.
, 2005, “Recursive Kinematics and Inverse Dynamics for a Planar 3R Parallel Manipulator
,” ASME J. Dyn. Syst., Meas., Control
0022-0434, 127
(4
), pp. 529
–536
.43.
Bellezza
, F.
, Lanari
, L.
, and Ulivi
, G.
, 1990, “Exact Modeling of Flexible Slewing Link
,” Proceedings of IEEE International Conference on Robotics and Automation
, Los Angeles, CA
, pp. 734
–739
.44.
ANSYS, Release 10, “
Release 10 Documentation for ANSYS
,” Element Reference, www.ansys.comwww.ansys.com.45.
Fotouhi
, R.
, 1996, “Time Optimal Control of 2-Link Manipulators
,” Ph.D. thesis, University of Saskatchewan, Saskatoon, Canada.46.
Fotouhi
, R.
, Szyszkowski
, W.
, Nikiforuk
, P. N.
, and Gupta
, M. M.
, 1999, “Parameter Identification and Trajectory Following of a Two-Link Rigid Manipulator
,” Journal of Systems and Control Engineering, Proceedings Institution of Mechanical Engineering, Part I
, 213
(6
), pp. 455
–466
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