Abstract
Kinematics position problems in planar parallel continuum mechanisms, whose elements are elastic rods undergoing nonlinear large deformations, are ruled by a system of nonlinear differential equations. Under some conditions, those rods can be modelled as Kirchhoff rods whose equations can be solved using elliptic integrals. The resolution has to be numerical, and two approaches for that goal are shown in this paper. On the one hand, a method based in residuals evaluation finds multiple solutions at good computational rates but with no formal guarantee on the solving of all solutions. On the other hand, a procedure based on Interval Analysis constitutes a certified solution at a higher computational cost that can be improved with a Newton scheme.
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References
Wenger, P., Chablat, D.: Definition sets for the direct kinematics of parallel manipulators. ICAR97. Proceedings., 8th International Conference on Advanced Robotics, pp. 859864. IEEE, (1997) DOI:https://doi.org/10.1109/ICAR.1997.620282
Bryson, C.E., Rucker, D.C.: Toward Parallel Continuum Manipulators. Proc. IEEE Int. Conf. Robot. Autom., May 2014, pp. 778-785 (2014). doi:https://doi.org/10.1109/ICRA.2014.6906943
Till, J.. Bryson, C.E., Chung, S., Orekhov, A., Rucker, D.C.: Efficient Computation of Multiple Coupled Cosserat Rod Models for Real-Time Simulation and Control of Parallel Continuum Manipulators. Proc. IEEE Int. Conf. Robot. Autom., May 2015, pp. 5067-5074 (2015). doi:https://doi.org/10.1109/ICRA.2015.7139904
Altuzarra, O., Caballero, D., Campa, F.J., Pinto, Ch.: Position analysis in planar parallel continuum mechanisms. Mechanism and Machine Theory, 132, pp. 13-29, (2019). https://doi.org/10.1016/j.mechmachtheory.2018.10.014
Howell, L.L.: Compliant Mechanisms. Wiley, New York. (2001)
Holst, G. L. et al.: Modelling and Experiments of Buckling Modes and Deflection of Fixed-Guided Beams in Compliant Mechanisms. Journal of Mechanical Design, vol. 133, 051002-1–10 (2011)
Zhang, A., Chen, G.: A Comprehensive Elliptic Integral Solution to the Large Deflection Problems of Thin Beams in Compliant Mechanisms. Journal of Mechanisms. 5, pp. 021006-1-10. (2013). doi:https://doi.org/10.1115/1.4023558
Hansen, E.: Global optimization using interval analysis. Marcel Dekker. (2004)
Jaulin, L. and Kieffer, M. and Didrit, O. and Walter, E.: Applied Interval Analysis. Springer-Verlag. (2001)
Antman S.S.: Nonlinear Problems of Elasticity. Springer (2005) ISBN: 0-387-20880-1.
Merlet, J-P.: Solving the Forward Kinematics of a Gough-Type Parallel Manipulator with Interval Analysis. Int. J. of Robotics Research, vol. 23, 3, pp. 221-236, (2004). https://doi.org/10.1177/0278364904039806
Tapia, R.A.: The Kantorovitch theorem for Newton’s method. American Mathematic Monthly. vol. 78, 1.ea, pp. 389-392, (1971)DOI:https://doi.org/10.2307/2316909
Acknowledgments
The authors received financial support from the Spanish Government (DPI2015-64450-R) and the Regional Government of the Basque Country (Project IT949-16).
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Altuzarra, O., Merlet, J.P. (2019). Certified Kinematics Solution of 2-DOF Planar Parallel Continuum Mechanisms. In: Uhl, T. (eds) Advances in Mechanism and Machine Science. IFToMM WC 2019. Mechanisms and Machine Science, vol 73. Springer, Cham. https://doi.org/10.1007/978-3-030-20131-9_20
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DOI: https://doi.org/10.1007/978-3-030-20131-9_20
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