Abstract
To gain a thorough understanding of the load state of parallel kinematic machines (PKMs), a methodology of elastodynamic modeling and joint reaction prediction is proposed. For this purpose, a Sprint Z3 model is used as a case study to illustrate the process of joint reaction analysis. The substructure synthesis method is applied to deriving an analytical elastodynamic model for the 3-PRS PKM device, in which the compliances of limbs and joints are considered. Each limb assembly is modeled as a spatial beam with non-uniform cross-section supported by lumped virtual springs at the centers of revolute and spherical joints. By introducing the deformation compatibility conditions between the limbs and the platform, the governing equations of motion of the system are obtained. After degenerating the governing equations into quasi-static equations, the effects of the gravity on system deflections and joint reactions are investigated with the purpose of providing useful information for the kinematic calibration and component strength calculations as well as structural optimizations of the 3-PRS PKM module. The simulation results indicate that the elastic deformation of the moving platform in the direction of gravity caused by gravity is quite large and cannot be ignored. Meanwhile, the distributions of joint reactions are axisymmetric and position-dependent. It is worthy to note that the proposed elastodynamic modeling method combines the benefits of accuracy of finite element method and concision of analytical method so that it can be used to predict the stiffness characteristics and joint reactions of a PKM throughout its entire workspace in a quick and accurate manner. Moreover, the present model can also be easily applied to evaluating the overall rigidity performance as well as statics of other PKMs with high efficiency after minor modifications.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
HENNES N, STAIMER D. Application of PKM in aerospace manufacturing-high performance machining centers ECOSPEED, ECOSPEED-F and ECOLINER [C]// Proceedings of the 4th Chemnitz Parallel Kinematics Seminar. Chemnitz: Verlag Wissenschaftliche Scripten, 2004: 557–577.
NEUMANN K E. Tricept application [C]// Proceedings-3rd Chemnitz Parallel Kinematics Seminar. Zwickau: Verlag Wissenschaftliche Scripten, 2002: 547–551.
CACCAVALE F, SICILIANO B. The tricept robot-dynamics and impedance control [J]. IEEE, 2003, 2: 263–268.
CARRETERO J A, PODHORODESKI R P, NAHON K A, GOSSELIN C M. Kinematic analysis and optimization of a new three degree-of-freedom spatial manipulator [J]. Journal of Mechanical Design, 2000, 122: 17–24.
LI Yan-min, XU Qing-song. Kinematic analysis of a 3-PRS parallel manipulator [J]. Robotics and Computer-Integrated Manufacturing, 2007, 23: 395–408.
POND G T, CARRETERO J A. Kinematic analysis and workspace determination of the inclined PRS parallel manipulator [C]// Proceedings of the 15th CISM - IFToMM Symposium on Robot Design, Dynamics and Control. Saint-Hubert: Springer Verlag, 2004: 1–6.
CHENG X, HUANG Y M, FAN Z M, SU J H. Workspace generation of the 3-PRS parallel robot based on the NN[C]// Proceedings of the First International Conference on Machine Learning and Cybernetics. Beijing: IEEE, 2002: 4–5.
GEOFFREY P, CARRETERO G A. Formulating Jacobian matrices for dexterity analysis of parallel mechanism [J]. Mechanism and Machine Theory, 2006, 41: 1505–1519.
LI Qin-chuan, CHENG Zhi, CHEN Qiao-hong, WU Chuan-yu, HU Xu-dong. Parasitic motion comparison of 3-PRS parallel mechanism with different limb arrangements [J]. Robotics and Computer-Integrated Manufacturing, 2011, 27: 389–396.
NIGUS H. Semi-analytical approach for stiffness estimation of 3-DOF PKM [J]. Modern Mechanical Engineering, 2014, 4: 108–118.
WANG You-yu, LIU Hai-tao, HUANG Tian, CHRTWYND D G. Stiffness modeling of the tricept robot using the overall Jacobian matrix [J]. ASME Journal of Mechanisms and Robotics, 2009, 1: 021002.
LI Yong-gang, SONG Yi-min, FENG Zhi-you, ZHANG Ce. Inverse dynamics of 3-RPS parallel mechanism by Newton-Euler formulation [J]. Acta Aeronautica et Astronautica Sinica, 2007, 28: 1210–1215. (in Chinese)
XI Feng-feng, ANGELICO O, SINATRA R. Tripod dynamics and its inertia effect [J]. ASME Journal of Mechanical Design, 2005, 127: 144–149.
WIENS G J, SHAMBLIN S A, OH Y H. Characterization of PKM dynamics in terms of system identification [J]. Journal of Multi-body Dynamics, 2002, 216(1): 59–72.
WU Pei-dong, XIONG He-gen, KONG Jian-yi. Dynamic analysis of 6-SPS parallel mechanism [J]. International Journal of Mechanics and Materials in Design, 2012, 8(2): 121–128.
JI Z M. Study of the effect of Leg Inertia in Stewart Platforms [C]// Proceedings of the IEEE Conference on Robotics and Automation. Altanta: IEEE, 1993: 121–126.
LEE J D, GENG Z. A dynamic model of a flexible Steward platform [J]. Computer and Structures, 1993, 48(3): 367–374.
LIU Shan-zeng, YU Yue-qing, ZHU Zhen-zai, SU Li-ying, LIU Qing-bo. Dynamic modeling and analysis of 3-RPS parallel manipulator with flexible links [J]. Journal of Central South University, 2010, 17(2): 323–331.
WANG Xiao-yu, MILLS J K. Dynamic modeling of a flexible-link planar parallel platform using a substructuring approach [J]. Mechanism and Machine Theory, 2006, 41(6): 671–687.
ZHANG Xu-ping, MILLS J K, CLEGHORN W L. Dynamic modeling and experimental validation of a 3-PRR parallel manipulator with flexible intermediate links [J]. Journal of Intelligent and Robotic Systems, 2007, 50(4): 323–340.
SHIAU T N, TSAI Y J, TSAI M S. Nonlinear dynamic analysis of a parallel mechanism with consideration of joint effects [J]. Mechanism and Machine Theory, 2008, 43(4): 491–505.
Author information
Authors and Affiliations
Corresponding author
Additional information
Foundation item: Project(Kfkt2013-12) supported by Open Research Fund of Key Laboratory of High Performance Complex Manufacturing of Central South University, China; Project(2014002) supported by the Open Fund of Shanghai Key Laboratory of Digital Manufacture for Thin-walled Structures, China; Project(51375013) supported by the National Natural Science Foundation of China
Rights and permissions
About this article
Cite this article
Zhang, J., Zhao, Yq. Elastodynamic modeling and joint reaction prediction for 3-PRS PKM. J. Cent. South Univ. 22, 2971–2979 (2015). https://doi.org/10.1007/s11771-015-2833-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11771-015-2833-y