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Research Papers

Self-Motions of TSSM Manipulators With Two Parallel Rotary Axes

[+] Author and Article Information
Georg Nawratil

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

J. Mechanisms Robotics 3(3), 031007 (Aug 10, 2011) (10 pages) doi:10.1115/1.4004030 History: Received August 17, 2010; Revised March 28, 2011; Published August 10, 2011; Online August 10, 2011

In this paper, we determine all nontrivial self-motions of triangular symmetric simplified manipulators (TSSMs) with two parallel rotary axes which equal the determination of all flexible octahedra where one vertex is an ideal point. This study also closes the classification of these motions for the whole set of parallel manipulators of TSSM type. Our approach is based on Kokotsakis meshes and the reducible compositions of spherical coupler motions with a spherical coupler component.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

TSSM with parallel rotary axes r1,r2. Moreover, the substitution of l1 by p1 and q1 is illustrated

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Figure 2

A Kokotsakis mesh is a polyhedral structure consisting of a n-sided central polygon Σ0∈E3 surrounded by a belt of polygons in the following way: Each side Ii0 of Σ0 is shared by an adjacent polygon Σi, and the relative motion between cyclically consecutive neighbor polygons is a spherical coupler motion. Here, the Kokotsakis mesh for n=3 is given, which determines an octahedron.

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Figure 3

Composition of the two spherical four-bars I10A1B1I20 and I20A2B2I30 with spherical side lengths αi,βi,γi,δi, i=1,2 (Courtesy of H. Stachel)

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Figure 4

Oriented sides of the central triangle Σ0 with the oriented enclosed angles δi

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Figure 5

Planar four-bar mechanism with driving arm a, follower b, coupler c and base d

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Figure 6

Schematic sketch of the octahedron V1,…,V6. The dihedral angles are denoted by ϕi,ψi,χi,κi with i=1,2,3.

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Figure 7

The two cases implied by the conditions V2V4¯=V1V5¯ and V4V5¯=V1V2¯

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Figure 8

The carrier plane of both flat poses V1,V2,V4,V5 and V1′,V2′,V4′,V5′ of the prism is spanned by [V1,V5,V6]

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