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Research Papers

Duporcq Pentapods

[+] Author and Article Information
Georg Nawratil

Institute of Discrete Mathematics and Geometry,
Vienna University of Technology,
Wiedner Hauptstrasse 8-10/104,
Vienna 1040, Austria
e-mail: nawratil@geometrie.tuwien.ac.at

Josef Schicho

Johann Radon Institute for Computational
and Applied Mathematics,
Austrian Academy of Sciences,
Altenberger Strasse 69,
Linz 4040, Austria
e-mail: josef.schicho@ricam.oeaw.ac.at

Manuscript received May 13, 2016; final manuscript received October 4, 2016; published online November 23, 2016. Assoc. Editor: Andreas Mueller.

J. Mechanisms Robotics 9(1), 011001 (Nov 23, 2016) (7 pages) Paper No: JMR-16-1141; doi: 10.1115/1.4035085 History: Received May 13, 2016; Revised October 04, 2016

In a foregoing publication, the authors studied pentapods with mobility 2, where neither all the platform anchor points nor all the base anchor points are located on a line. It turned out that the given classification is incomplete. This article is devoted to the discussion of the missing cases resulting in additional solutions already known to Duporcq.

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References

Nawratil, G. , and Schicho, J. , 2015, “ Pentapods With Mobility 2,” ASME J. Mech. Rob., 7(3), p. 031016. [CrossRef]
Gallet, M. , Nawratil, G. , and Schicho, J. , 2015, “ Möbius Photogrammetry,” J. Geometry, 106(3), pp. 421–439. [CrossRef]
Gallet, M. , Nawratil, G. , Schicho, J. , and Selig, J. , 2016, “ Mobile Icosapods,” Adv. Appl. Math. (in press).
Duporcq, E. , 1901, “ Sur un Remarquable Déplacement à Deux Paramétres,” Bull. Soc. Math. France, 29, pp. 1–4.
Röschel, O. , and Mick, S. , 1998, “ Characterisation of Architecturally Shaky Platforms,” Advances in Robot Kinematics: Analysis and Control, J. Lenarcic , and M. Husty , eds., Kluwer, Dortrecht, The Netherlands, pp. 465–474.
Karger, A. , 2003, “ Architecture Singular Planar Parallel Manipulators,” Mech. Mach. Theory, 38(11), pp. 1149–1164. [CrossRef]
Gallet, M. , Nawratil, G. , and Schicho, J. , 2015, “ Erratum to: Möbius Photogrammetry,” J. Geometry, 106(3), pp. 441–442. [CrossRef]
Husty, M. L. , 1996, “ An Algorithm for Solving the Direct Kinematics of General Stewart–Gough Platforms,” Mech. Mach. Theory, 31(4), pp. 365–380. [CrossRef]
Gallet, M. , Nawratil, G. , and Schicho, J. , 2015, “ Bond Theory for Pentapods and Hexapods,” J. Geometry, 106(2), pp. 211–228. [CrossRef]
Gallet, M. , Nawratil, G. , and Schicho, J. , 2015, “ Liaison Linkages,” J. Symbolic Comput. (in press).
Nawratil, G. , 2014, “ Introducing the Theory of Bonds for Stewart Gough Platforms With Self-Motions,” ASME J. Mech. Rob., 6(1), p. 011004. [CrossRef]
Hartshorne, R. , 1962, “ Complete Intersections and Connectedness,” Am. J. Math., 84(3), pp. 497–508. [CrossRef]
Karger, A. , 1998, “ Architecture Singular Parallel Manipulators,” Advances in Robot Kinematics: Analysis and Control, J. Lenarcic , and M. Husty , eds., Kluwer, Dortrecht, The Netherlands, pp. 445–454.
Karger, A. , 2008, “ Architecturally Singular Non-Planar Parallel Manipulators,” Mech. Mach. Theory, 43(3), pp. 335–346. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Illustration of Duporcq's complete quadrilaterals

Grahic Jump Location
Fig. 2

The photographic map sends any direction vector parallel to a line through 2 (but not 3) of the points Mi, Mj to the unique point in the Möbius picture on the line Lij of the quintic surface P5. In the base configuration above, green (g) is sent to L25, orange (o) is sent to L24, yellow (y) is sent to L15, and pink (p) is sent to L14. It is not clear whether the directions blue (b) and metallic (m) are being sent; later, we will show that b is sent to L12 and m is sent to L45.

Grahic Jump Location
Fig. 3

The two possible reconstructions of the platform configuration from the Möbius picture, under the additional assumption that m3 is the intersection of lines m1m2 and m4m5. Note that the line m2m5 must have direction g, the line m2m4 must have direction o, the line m1m5 must have direction y and the line m1m4 must have direction p. The left configuration coincides with the base configuration. We will see later that the right configuration is not compatible because the lines through m1,m2,m3 and m3,m4,m5, respectively, do not have the correct directions.

Grahic Jump Location
Fig. 4

The two possible reconstructions of the platform configuration from the Möbius picture, under the additional assumption that m3 is the intersection of lines m1m5 and m2m4. Here, the directions of lines m1m2, m4m5, m1m4, and m2m5 are fixed to b, m, p, and g, respectively. We will later see that the left configuration is not compatible. The right configuration leads to a Duporcq pentapod.

Grahic Jump Location
Fig. 5

The two possible reconstructions of the platform configuration from the Möbius picture, under the additional assumption that m3 is the intersection of lines m1m4 and m2m5. Here, the directions of lines m1m2, m4m5, m1m5, and m2m4 are fixed to b, m, y, and o, respectively. The left configuration is not compatible. The right configuration leads to a Duporcq pentapod.

Grahic Jump Location
Fig. 6

Projection along the black direction leads to one-dimensional configurations that are similar. This shows that the pentapod with such base/platform configuration has a similarity bond.

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