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Self-motions of pentapods with linear platform

Published online by Cambridge University Press:  02 December 2015

Georg Nawratil  and
Josef Schicho
Georg Nawratil*
Affiliation:
Institute of Discrete Mathematics and Geometry, Vienna University of Technology, Wiedner Hauptstrasse 8-10/104, 1040 Vienna, Austria
Josef Schicho
Affiliation:
Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenberger Strasse 69, 4040 Linz, Austria
*
*Corresponding author. E-mail: [email protected]
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Summary

We give a full classification of all pentapods with linear platform possessing a self-motion beside the trivial rotation about the platform. Recent research necessitates a contemporary and accurate re-examination of old results on this topic given by Darboux, Mannheim, Duporcq and Bricard, which also takes the coincidence of platform anchor points into account. For our study we use bond theory with respect to a novel kinematic mapping for pentapods with linear platform, beside the method of singular-invariant leg-rearrangements. Based on our results we design pentapods with linear platform, which have a simplified direct kinematics concerning their number of (real) solutions.

Keywords

PentapodBorel Bricard problemBond theorySelf-MotionDirect kinematicKinematic mapping
Type
Articles
Information
Robotica , Volume 35 , Issue 4 , April 2017 , pp. 832 - 860
Copyright
Copyright © Cambridge University Press 2015 

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References

1. Borras, J., Thomas, F. and Torras, C., “A Family of Quadratically-Solvable 5-SPU Parallel Robots,” Proceedings of IEEE International Conference on Robotics and Automation, Anchorage, Alaska (May 3–7, 2010) pp. 4703–4708.CrossRefGoogle Scholar
2. Borras, J. and Thomas, F., “Singularity-Invariant Leg Substitutions in Pentapods,” Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems, Taipei, Taiwan (October 18–22, 2010) pp. 2766–2771.CrossRefGoogle Scholar
3. Weiß, G. and Bär, G., “Singularity Investigation of a 5-leg Milling Robot,” Proceedings of EuCoMeS, the 1st European Conference on Mechanism Science, Obergurgl, Austria (February 21–26, 2006).Google Scholar
4. Pottmann, H. and Wallner, J., Computational Line Geometry (Springer, Berlin Heidelberg, 2001).Google Scholar
5. Grünwald, A., “Die kubische Kreisbewegung eines starren Körpers,” Z. Math. Phys. 55, 264–296 (1907).Google Scholar
6. Rulf, F., “Der kubische Kreis,” Monatshefte Math. Phys. 21 (1), 329–335 (1919).Google Scholar
7. Borel, E., “Mémoire sur les déplacements à trajectoires sphériques,” Mém.o présenteés par divers savants à l'Académie des Sci. de l'Institut Nat. de France 33 (1), 1–128 (1908).Google Scholar
8. Bricard, R., “Mémoire sur les déplacements à trajectoires sphériques,” J. École Polytech. (2) 11, 1–96 (1906).Google Scholar
9. Husty, M., E. Borel's and R. Bricard's Papers on Displacements with Spherical Paths and their Relevance to Self-Motions of Parallel Manipulators. International Symposium on History of Machines and Mechanisms (Ceccarelli, M. ed.) (Kluwer, 2000) pp. 163–172.Google Scholar
10. Krames, J., “Zur Bricardschen Bewegung, deren sämtliche Bahnkurven auf Kugeln liegen (Über symmetrische Schrotungen II),” Monatsheft Math. Phys. 45, 407–417 (1937).Google Scholar
11. Koenigs, G., Leçons de Cinématique (avec notes par G. Darboux) (Paris, Librairie Scientifique A. Hermann, 1897).Google Scholar
12. Mannheim, A., Principes et Développements de Géométrie Cinématique (Paris, Gauthier-Villars, 1894).Google Scholar
13. Duporcq, E., “Sur le déplacement le plus général d'une droite dont tous les points décrivent des trajectoires sphériques,” C. R. Paris 125, 762–763 (1897) and Journal de mathématiques pures et appliquées (5) 4, 121–136 (1898).Google Scholar
14. Nawratil, G. and Schicho, J., “Pentapods with Mobility 2,” ASME J. Mech. Robot. 7 (3), 031016 (2015).Google Scholar
15. Karger, A., “Architecturally singular non-planar parallel manipulators,” Mech. Mach. Theory 43 (3), 335–346 (2008).Google Scholar
16. Nawratil, G., Comments on “Architectural singularities of a class of pentapods,” Mech. Mach. Theory 57, 139 (2012).Google Scholar
17. Borras, J., Thomas, F. and Torras, C., Architectural singularities of a class of pentapods. Mech. Mach. Theory 46 (8), 1107–1120 (2011).Google Scholar
18. Röschel, O. and Mick, S., Characterisation of Architecturally Shaky Platforms, Advances in Robot Kinematics – Analysis and Control (Lenarcic, J. and Husty, M. L. eds.) (Kluwer, Dordrecht Bosten London, 1998), pp. 465–474.Google Scholar
19. Borras, J. and Thomas, F., “Kinematics of Line-Plane Subassemblies in Stewart Platforms,” Proceedings of IEEE International Conference on Robotics and Automation, Kobe, Japan (May 12–17, 2009) pp. 4094–4099.CrossRefGoogle Scholar
20. Nawratil, G., “Self-Motions of Planar Projective Stewart Gough Platforms,” Latest Advances in Robot Kinematics (Lenarcic, J. and Husty, M. eds.) (Springer, Dordrecht Heidelberg NewYork London, 2012) pp. 27–34.Google Scholar
21. Nawratil, G., “Introducing the theory of bonds for Stewart Gough platforms with self-motions,” ASME J. Mech. Robot. 6 (1), 011004 (2014).CrossRefGoogle Scholar
22. Nawratil, G., “Correcting Duporcq's theorem,” Mech. Mach. Theory 73, 282–295 (2014).Google Scholar
23. Schönflies, A., Geometrie der Bewegung in synthetischer Darstellung (Teubner, Leipzig, 1886).Google Scholar
24. Husty, M., Pfurner, M., Schröcker, H.-P. and Brunnthaler, K., “Algebraic methods in mechanism analysis and synthesis,” Robotica 25 (6), 661–675 (2007).Google Scholar
25. Husty, M. and Schröcker, H.-P., “Kinematics and algebraic geometry,” 21st Century Kinematics - The 2012 NSF Workshop (McCarthy, J. M. ed.), (Springer, London, 2012) pp. 85–123.Google Scholar
26. Husty, M.L., “An algorithm for solving the direct kinematics of general Stewart-Gough platforms,” Mech. Mach. Theory 31 (4), 365–380 (1996).Google Scholar
27. Zhang, C.-de and Song, S.-M., “Forward Kinematics of a Class of Parallel (Stewart) Platforms with Closed-Form Solution,” Proceedings of IEEE International Conference on Robotics and Automation, Sacramento, California (April 9–11, 1991) pp. 2676–2681.Google Scholar
28. Gallet, M., Nawratil, G. and Schicho, J., “Bond Theory for Pentapods and Hexapods,” J. Geom. 106 (2), 211–228 (2015).Google Scholar
29. Lee, C.-C. and Hervé, J.M., “Bricard one-DoF motion and its mechanical generation,” Mech. Mach. Theory 77, 35–49 (2014).Google Scholar
30. Hartmann, D., Singular Stewart-Gough Platforms, Master Thesis (Department of Mechanical Engineering, McGill University, Montreal, Canada, 1995).Google Scholar
31. Krames, J., “Die Borel-Bricard-Bewegung mit punktweise gekoppelten orthogonalen Hyperboloiden (Über symmetrische Schrotungen VI),” Monatsheft Math. Phys. 46, 172–195 (1937).Google Scholar
32. Borras, J., Thomas, F. and Torras, C., “New geometric approaches to the analysis and design of Stewart–Gough platforms,” IEEE/ASME Trans. Mechatronics 19 (2), 445–455 (2014).Google Scholar
33. Nawratil, G., “On the line-symmetry of self-motions of linear pentapods,” arXiv:1510.03567 (2015).Google Scholar
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