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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

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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