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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

and Applied Mathematics,
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

## Abstract

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

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Gallet, M., Nawratil, G., and Schicho, J., “Möbius Photogrammetry,” e-print arXiv:1408.6716v1.
Nawratil, G., 2014, “On Stewart Gough Manipulators With Multidimensional Self-Motions,” Comput. Aided Geom. Des., 31(7–8), pp. 582–594.
Nawratil, G., and Schicho, J., “Self-Motions of Pentapods With Linear Platform,” e-print arXiv:1407.6126v1.
Pottmann, H., and Wallner, J., 2001, Computational Line Geometry, Springer, Berlin, Germany.
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Nawratil, G., 2012, “Comments on ‘Architectural Singularities of a Class of Pentapods',” Mech. Mach. Theory, 57(1), pp. 139.
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.
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.
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Gallet, M., Nawratil, G., and Schicho, J., 2014, “Bond Theory for Pentapods and Hexapods,” J. Geom. (in press).
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.
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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

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).

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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