The Ninth Homotopy Class of Spatial 3R Serial Regional Manipulators

[+] Author and Article Information
Davide Paganelli

DIEM, Università degli Studi di Bologna, Viale Risorgimento 2, 40136 Bologna, Italydavide.paganelli@mail.ing.unibo.it

J. Mech. Des 129(4), 445-448 (Apr 26, 2006) (4 pages) doi:10.1115/1.2437805 History: Received September 23, 2005; Revised April 26, 2006

Singularities form surfaces in the jointspace of a serial manipulator. Paï and Leu (Paï and Leu, 1992, IEEE Trans. Rob. Autom., 8, pp. 545–559) introduced the important notion of generic manipulator, the singularity surfaces of which are smooth and do not intersect with each other. Burdick (Burdick, 1995, J. Mech. Mach. Theor., 30, pp. 71–89) proposed a homotopy-based classification method for generic 3R manipulators. Through this classification method, it was stated in Wenger, 1998, J. Mech. Des., 120, pp. 327–332 that there exist exactly eight classes of generic 3R manipulators. A counterexample to this classification is provided: a generic 3R manipulator belonging to none of the eight classes identified in (Wenger, 1998, J. Mech. Des., 120, pp. 327–332) is presented. The weak point of the proof given in (J. Mech. Des., 120, pp. 327–332) is highlighted. The counterexample proves the existence of at least nine homotopy classes of generic 3R manipulators. The paper points out two peculiar properties of the manipulator proposed as a counterexample, which are not featured by any manipulator belonging to the eight homotopy classes so far discovered. Eventually, it is proven in this paper that at most four branches of the singularity curve can coexist in the jointspace of a generic 3R manipulator and therefore at most eleven homotopy classes are possible.

Copyright © 2007 by American Society of Mechanical Engineers
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Grahic Jump Location
Figure 1

Examples of different classes of branches

Grahic Jump Location
Figure 2

Three-dimensional view of manipulator M, defined by DH parameters: a1=1.178, a2=0.339, a3=1.0, d2=0.32, d3=0.67, α1=1.55rad, and α2=−1.124rad

Grahic Jump Location
Figure 3

(a) The singularity curve in the jointspace of M:M belongs to the ninth homotopy class 2(0,0)+2(1,0); and (b) the image of the singularity curve in the workspace

Grahic Jump Location
Figure 4

An arbitrary number of (0,0) branches can appear alone or together with two (1,0) branches, without exceeding four intersections with any θ3 generator, nor two intersections with any θ2 generator

Grahic Jump Location
Figure 5

The points where the tangent to the singularity curve is parallel to the θ2 generator are marked with a “×”. Each (0,0) or (1,0) branch contains at least two of these points.




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