In consideration of the second-order coupling quantity of the axial displacement caused by the transverse displacement of flexible beam, the first-order approximation coupling model of planar 3-RRR flexible parallel robots is presented, in which the rigid body motion constraints, elastic deformation motion constraints, and dynamic constraints of the moving platform are considered. Based on the different speed of the moving platform, numerical simulation results using the conventional zero-order approximation coupling model and the proposed firstorder approximation coupling model show that the effect of “dynamic stiffening” term on dynamic characteristics of the system is insignificant and can be neglected, and the zero-order approximation coupling model is enough precisely for catching essentially dynamic characteristics of the system. Then, the commercial software ANSYS 13.0 is used to confirm the validity of the zero-order approximation coupling model. 1. Introduction In recent several decades, many researchers have paid more attention to the light flexible robots with high-speed, high-acceleration, and high-precision which are widely used in the assembly industry, the aerospace industry and the precision machining, and the measurement field. Essentially, the flexible parallel robots mechanism belongs to flexible multibody system. Dynamic modeling of flexible multibody system is a challenging task, in which not only rigid-flexible coupling effect must be studied but also elastic deformation coupling must be analyzed carefully. At present, dynamic modeling and control of the flexible multibody system have received considerable attention as seen in survey papers [1–4]. Unfortunately, most of published works in this area addressed the manipulators with one flexible link. Comparing with single-link flexible manipulator, two-link flexible manipulator, or four-bar linkage flexible mechanism [5, 6], the research works on the flexible parallel robots are rather few. Recently, few works have been done on dynamic modeling and control of complex mechanisms. Lee and Geng [7] developed a dynamic model of a flexible Stewart platform using Lagrange equations. Wang et al. [8–10] studied dynamic modeling and control of planar 3-Prismatic-joint-Revolute-joint-and-Revolute-joint (3-PRR) parallel robots. Zhang et al. [11, 12] studied dynamic modeling method and dynamic characteristics of planar 3-RRR flexible parallel robots. Q. H. Zhang and X. M. Zhang [13] also studied dynamic performance of planar 3-RRR flexible parallel robots under uniform temperature change. For a
S. K. Dwivedy and P. Eberhard, “Dynamic analysis of flexible manipulators, a literature review,” Mechanism and Machine Theory, vol. 41, no. 7, pp. 749–777, 2006.
A. G. Erdman and G. N. Sandor, “Kineto-elastodynamics—a review of the state of the art and trends,” Mechanism and Machine Theory, vol. 7, no. 1, pp. 19–33, 1972.
X. M. Zhang, C. J. Shao, and A. G. Erdman, “Active vibration controller design and comparison study of flexible linkage mechanism systems,” Mechanism and Machine Theory, vol. 37, no. 9, pp. 985–997, 2002.
X. M. Zhang, C. J. Shao, Y. W. Shen, and A. G. Erdman, “Complex mode dynamic analysis of flexible mechanism systems with piezoelectric sensors and actuators,” Multibody System Dynamics, vol. 8, no. 1, pp. 51–70, 2002.
X. Y. Wang and J. K. Mills, “FEM dynamic model for active vibration control of flexible linkages and its application to a planar parallel manipulator,” Applied Acoustics, vol. 66, no. 10, pp. 1151–1161, 2005.
X. Y. Wang and J. K. Mills, “Dynamic modeling of a flexible-link planar parallel platform using a substructuring approach,” Mechanism and Machine Theory, vol. 41, no. 6, pp. 671–687, 2006.
X. P. Zhang, J. K. Mills, and W. L. Cleghorn, “Experimental implementation on vibration mode control of a moving 3-PRR flexible parallel manipulator with multiple PZT transducers,” Journal of Vibration and Control, vol. 16, no. 13, pp. 2035–2054, 2010.
Q. H. Zhang, X. M. Zhang, and J. L. Liang, “Dynamic analysis of planar 3- RR flexible parallel robot,” in Proceedings of the IEEE International Conference on Robotics and Biomimetics (ROBIO '12), pp. 154–159, Guangzhou, China, December 2012.
Q. H. Zhang and X. M. Zhang, “Dynamic modeling and analysis of planar 3- RR flexible parallel robots,” Journal of Vibration Engineering, vol. 26, no. 2, pp. 239–245, 2013.
Q. H. Zhang and X. M. Zhang, “Dynamic analysis of planar 3- RR flexible parallel robots under uniform temperature change,” Journal of Vibration and Control, 2013.
G. P. Cai, J. Z. Hong, and S. X. Yang, “Model study and active control of a rotating flexible cantilever beam,” International Journal of Mechanical Sciences, vol. 46, no. 6, pp. 871–889, 2004.
T. R. Kane, R. R. Ryan, and A. K. Banerjee, “Dynamics of a cantilever beam attached to a moving base,” Journal of Guidance, Control, and Dynamics, vol. 10, no. 2, pp. 139–151, 1987.
J. M. Mayo, D. García-Vallejo, and J. Domínguez, “Study of the geometric stiffening effect: comparison of different formulations,” Multibody System Dynamics, vol. 11, no. 4, pp. 321–341, 2004.
U. Lugrís, M. A. Naya, J. A. Pérez, and J. Cuadrado, “Implementation and efficiency of two geometric stiffening approaches,” Multibody System Dynamics, vol. 20, no. 2, pp. 147–161, 2008.
D. J. Zhang, Y. W. Liu, and R. L. Huston, “On dynamic stiffening of flexible bodies having high angular velocity,” Mechanics of Structures and Machines, vol. 24, no. 3, pp. 313–329, 1996.
S. C. Wu and E. J. Haug, “Geometric non-linear substructure for dynamics of flexible mechanical systems,” International Journal for Numerical Methods in Engineering, vol. 26, no. 10, pp. 2211–2226, 1988.
H. Yang, Study of dynamic modeling theory and experiments for rigid-flexible coupling systems [Ph.D. thesis], Shanghai Jiaotong University, Shanghai, China, 2002. (Chinese).
J. Y. Liu and H. Lu, “Rigid-flexible coupling dynamics of three-dimensional hub-beams system,” Multibody System Dynamics, vol. 18, no. 4, pp. 487–510, 2007.
