Nonlinear Effects in Stiffness Modeling of Robotic Manipulators

Authors: A. Pashkevich, A. Klimchik, D. Chablat

Abstract:

The paper focuses on the enhanced stiffness modeling of robotic manipulators by taking into account influence of the external force/torque acting upon the end point. It implements the virtual joint technique that describes the compliance of manipulator elements by a set of localized six-dimensional springs separated by rigid links and perfect joints. In contrast to the conventional formulation, which is valid for the unloaded mode and small displacements, the proposed approach implicitly assumes that the loading leads to the non-negligible changes of the manipulator posture and corresponding amendment of the Jacobian. The developed numerical technique allows computing the static equilibrium and relevant force/torque reaction of the manipulator for any given displacement of the end-effector. This enables designer detecting essentially nonlinear effects in elastic behavior of manipulator, similar to the buckling of beam elements. It is also proposed the linearization procedure that is based on the inversion of the dedicated matrix composed of the stiffness parameters of the virtual springs and the Jacobians/Hessians of the active and passive joints. The developed technique is illustrated by an application example that deals with the stiffness analysis of a parallel manipulator of the Orthoglide family

Keywords: Robotic manipulators, Stiffness model, Loaded mode, Nonlinear effects, Buckling, Orthoglide manipulator

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1058353

References:

[1] S. Timoshenko and J. N. Goodier, Theory of elasticity, 3d ed., New York: McGraw-Hill, 1970.
[2] G. Piras, W.L. Cleghorn and J.K. Mills. "Dynamic finite-element analysis of a planar high-speed, high-precision parallel manipulator with flexible links," Mechanism and Machine Theory, vol. 40, No. 7, pp. 849-862, 2005.
[3] F. Majou, C. Gosselin, P. Wenger, D. Chablat, "Parametric stiffness analysis of the Orthoglide," Mechanism and Machine Theory, vol. 42, pp. 296-311. 2007.
[4] C.M. Gosselin, "Stiffness mapping for parallel manipulators," IEEE Transactions on Robotics and Automation, vol. 6, pp. 377-382, 1990.
[5] A. Pashkevich, D. Chablat, and P. Wenger, "Stiffness analysis of 3- d.o.f. overconstrained translational parallel manipulators," IEEE International Conference on Robotics and Automation, pp. 1562-1567,2008.
[6] A. Pashkevich, D. Chablat, P. Wenger, "Stiffness analysis of overconstrained parallel manipulators," Mechanism and Machine Theory, vol. 44, pp. 966-982, 2008
[7] Alici, G., Shirinzadeh, B., "Enhanced stiffness modeling, identification and characterization for robot manipulators," IEEE Transactions on Robotics vol. 21/4, pp. 554-564, 2005.
[8] C. Quennouelle and C. M. Gosselin, "Stiffness Matrix of Compliant Parallel Mechanisms," Springer Advances in Robot Kinematics: Analysis and Design, pp. 331-341, 2008
[9] A. Pashkevich, A. Klimchik, D. Chablat, "Stiffness analysis of multichain parallel robotic systems with loading," Journal of Automation, Mobile Robotics and Intelligent Systems, Vol. 3, No. 3, pp. 75-82,2009
[10] Y. Li and Q. Xu, "Stiffness analysis for a 3-PUU parallel kinematic machine,"Mechanism and Machine Theory, vol. 43, no. 2, pp. 186-200,2008.
[11] R. Clavel, "DELTA, a fast robot with parallel geometry," Proceedings, of the 18th International Symposium of Robotic Manipulators, IFR Publication, pp. 91-100, 1988.
[12] D. Chablat and P. Wenger, "Architecture Optimization of a 3-DOF Parallel Mechanism for Machining Applications, the Orthoglide," IEEE Transactions on Robotics and Automation, vol. 19, no. 3, pp. 403-410,2003.
[13] R. Vertechy, V. Parenti-Castelli. "Static and Stiffness Analyses of a Class of Over-Constrained Parallel Manipulators with Legs of Type US and UPS," In: Proceedings of IEEE International Conference on Robotics and Automation (ICRA), pp. 561 - 567, 2007.
[14] O. Company, S. Krut, F. Pierrot, "Modelling and Preliminary Design Issues of a 4-Axis Parallel Machine for Heavy Parts Handling," Journal of Multibody Dynamics, 216 (2002), pp. 1-11.
[15] S. M. LaValle, Planning Algorithms, Cambridge University Press 2006
[16] A. Pashkevich, A. Klimchik, D. Chablat, Ph. Wenger, "Accuracy Improvement for Stiffness Modeling of Parallel Manipulators," In: Proceedings of 42nd CIRP Conference on Manufacturing Systems, 2009