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Research Papers

Pentapods With Mobility 2

[+] 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

In the remainder of the article, the word pentapod denotes a five-legged manipulator with noncollinear platform points and noncollinear base points; exceptions are noted explicitly.

A pentapod (hexapod) is called architecturally singular, if in any pose of the platform, the rank of its Jacobian matrix J is less than five (six), which is equivalent with the statement that the carrier lines of the five (six) legs belong to a linear congruence of lines (linear line complex). This equivalence is easy to see, as J is composed of the Plücker coordinates of these five (six) lines (cf. Ref. [7]).

This theorem is originally stated for hexapods but it also holds for pentapods, as its proof is also valid for five-legged manipulators.

This can always be done if the given pentapod is not architecturally singular.

One can also exclude k = 1 as it equals the congruent case, but the following calculation also holds for this case.

It can easily be verified that it is impossible that Λ1 is also independent of f0 and f2.

It can easily be verified that it is impossible that Λ1 is also independent of f0 and f1.

This can easily be checked by computing the resultant G1,3 of G1 and G3 with respect to e0, which is of degree 8 in e¯3. Then the resultant of G1,3 and H¯3 (or K¯) with respect to e¯3 does not contain the factor V.

1After a possible necessary renumbering of anchor points and exchange of the platform and the base.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received June 3, 2014; final manuscript received September 10, 2014; published online December 4, 2014. Assoc. Editor: J.M. Selig.

J. Mechanisms Robotics 7(3), 031016 (Aug 01, 2015) (11 pages) Paper No: JMR-14-1122; doi: 10.1115/1.4028934 History: Received June 03, 2014; Revised September 10, 2014; Online December 04, 2014

In this paper, we give a full classification of all pentapods with mobility 2, where neither all platform anchor points nor all base anchor points are located on a line. Therefore, this paper solves the famous Borel–Bricard problem for two-dimensional motions beside the excluded case of five collinear points with spherical trajectories. But even for this special case, we present three new types as a side-result. Based on our study of pentapods, we also give a complete list of all nonarchitecturally singular hexapods with two-dimensional self-motions.

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References

Borel, E., 1908, “Mémoire sur les Déplacements à Trajectoires Sphériques,” Mémoire Présenteés par divers savants à l'Académie des Sciences de l'Institut National de France, 33(1), pp. 1–128.
Bricard, R., 1906, “Mémoire sur les Déplacements à Trajectoires Sphériques,” J. Éc. Polytech., 11(2), pp. 1–96.
Husty, M., 2000, “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 (HMM 2000), Cassino, Italy, May 11–13, pp. 163–171. [CrossRef]
Gallet, M., Nawratil, G., and Schicho, J., “Möbius Photogrammetry,” e-print arXiv:1408.6716v1. http://arXiv:1408.6716
Nawratil, G., 2014, “On Stewart Gough Manipulators With Multidimensional Self-Motions,” Comput. Aided Geom. Des., 31(7–8), pp. 582–594. http://arxiv.org/abs/1408.6716
Nawratil, G., and Schicho, J., “Self-Motions of Pentapods With Linear Platform,” e-print arXiv:1407.6126v1. http://arXiv:1407.6126
Pottmann, H., and Wallner, J., 2001, Computational Line Geometry, Springer, Berlin, Germany.
Karger, A., 2008, “Architecturally Singular Non-Planar Parallel Manipulators,” Mech. Mach. Theory, 43(3), pp. 335–346. [CrossRef]
Nawratil, G., 2012, “Comments on ‘Architectural Singularities of a Class of Pentapods',” Mech. Mach. Theory, 57(1), pp. 139. [CrossRef]
Nawratil, G., 2012, “Self-Motions of Planar Projective Stewart Gough Platforms,” Latest Advances in Robot Kinematics, J.Lenarcic and M.Husty, eds., Springer, Dordrecht, The Netherlands, pp. 27–34.
Nawratil, G., 2014, “Introducing the Theory of Bonds for Stewart Gough Platforms With Self-Motions,” ASME J. Mech. Rob., 6(1), p. 011004. [CrossRef]
Hegedüs, G., Schicho, J., and Schröcker, H.-P., 2012, “Bond Theory and Closed 5R Linkages,” Latest Advances in Robot Kinematics, J.Lenarcic and M.Husty, eds., Springer, Dordrecht, The Netherlands, pp. 221–228.
Hegedüs, G., Schicho, J., and Schröcker, H.-P., 2013, “The Theory of Bonds: A New Method for the Analysis of Linkages,” Mech. Mach. Theory, 70, pp. 407–424. [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., 2014, “Bond Theory for Pentapods and Hexapods,” J. Geom. (in press). [CrossRef]
Nawratil, G., 2014, “Congruent Stewart Gough Platforms With Non-Translational Self-Motions,” 16th International Conference on Geometry and Graphics, Innsbruck, Austria, Aug. 4–8, H.-P.Schröcker and M.Husty, eds., Innsbruck University, Innsbruck Austria, pp. 204–215.
Nawratil, G., 2013, “On Equiform Stewart Gough Platforms With Self-Motions,” J. Geom. Graphics, 17(2), pp. 163–175.
Chasles, M., 1861, “Sur les Six Droites qui Peuvent étre les Directions de Six Forces en Équilibre,” C. R. Acad. Sci., 52, pp. 1094–1104.
Duporcq, E., 1898, “Sur la Correspondance Quadratique et Rationnelle de Deux Figures Planes et sur un Déplacement Remarquable,” C. R. Acad. Sci., 126, pp. 1405–1406.
Koenigs, G., 1897, Leçons de Cinématique (avec notes par M. G. Darboux), A. Hermann, Paris.
Mannheim, A., 1894, Principes et Développements de Géométrie Cinématique, Gauthier-Villars, Paris.
Duporcq, E., 1898, “Sur le Déplacement le Plus Général D'une Droite Dont Tous les Points Décrivent des Trajectoires Sphériques,” J. Math. Pures Appl., 4(5), pp. 121–136.
Borras, J., Thomas, F., and Torras, C., 2010, “Singularity-Invariant Leg Rearrangements in Stewart-Gough Platforms,” Advances in Robot Kinematics: Motion in Man and Machine, J.Lenarcic and M. M.Stanisic, eds., Springer, Dordrecht, The Netherlands, pp. 421–428.

Figures

Grahic Jump Location
Fig. 1

Sketch of a pentapod with planar platform and planar base, which is referred as planar pentapod. Moreover, this planar pentapod has a two-dimensional self-motion due to its special geometric design (cf. item 2 of Theorem 3).

Grahic Jump Location
Fig. 4

(a) Sketch of a three-legged spherical 3dof RPR manipulator. (b) Self-motion of type (I). (c) Self-motion of type (II). (d) Two-legged spherical manipulator ( = spherical 4-bar mechanism).

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