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Research Papers

A Distance Geometry Approach to the Singularity Analysis of 3R Robots

[+] Author and Article Information
Federico Thomas

Institut de Robòtica i Informàtica Industrial,
CSIC-UPC,
Llorens Artigas 4-6,
Barcelona 08028, Spain
e-mail: fthomas@iri.upc.edu

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received June 23, 2014; final manuscript received December 23, 2014; published online August 18, 2015. Assoc. Editor: Yuefa Fang.

J. Mechanisms Robotics 8(1), 011001 (Aug 18, 2015) (11 pages) Paper No: JMR-14-1146; doi: 10.1115/1.4029500 History: Received June 23, 2014

This paper shows how the computation of the singularity locus of a 3R robot can be reduced to the analysis of the relative position of two coplanar ellipses. Since the relative position of two conics is a projective invariant and the basic projective geometric invariants are determinants, it is not surprising that, using distance geometry, the computation of the singularity locus of a 3R robot can be fully expressed in terms of determinants. Geometric invariants have the benefit of simplifying symbolic manipulations. This paper shows how their use leads to a simpler characterization, compared to previous approaches, of the cusps and nodes in the singularity loci of 3R robots.

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Figures

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Fig. 1

A 3R robot and associated notation

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Fig. 3

Left: the equivalent bar-and-joint framework associated with the 3R robot in Fig. 1. Right: this framework can be split by the plane defined by P3, P4, and P7 into two subassemblies, each containing a tetrahedron (shown as a shaded volume) and a triangle.

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Fig. 4

The Denavit–Hartenberg parameters of the robot used as an example and its schematic representation including orthogonal sections of its singularity locus in the robot's workspace

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Fig. 5

Center: plot of the curves defined by det(A)=0 (in gray) and Δ = 0 (in green). These curves segment the plane into regions where the spatial relationship between A and B is the same. We are only interested in the region where A is a real ellipse (that is, the region where det(A)≤0). Left column: spatial relationships between A (in red) and B (in blue) associated with different regions of this plane. Right column: spatial relationships between A and B in different points of the singularity locus.

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Fig. 6

Shaded depth map of  log (abs(Δ(s1,7,s2,7))). The robot's singularity locus appears as valleys of this map. The two points marked with white dots correspond to configurations unreachable by the robot where A and B have a double contact in the complex domain.

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Fig. 7

What it seemed to be a higher order singularity in Fig. 5, it is revealed to be a node close to meet two cusps

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Fig. 8

The singularity locus shown in Fig. 5 (center) mapped onto the robot's workspace (ρ,z). The two singularities at ρ = 0 correspond to the two tangencies between the curves defined by Δ = 0 and det(A)=0 in the distance space (s1,7, s2,7).

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Fig. 9

Plot of δ2 = 0 (in red) and δ3 = 0 (in green). Observe how both curves intersect at the cusps of the singularity locus (light gray).

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Fig. 10

Left: plot of the line (in red) and ellipse (in green) resulting from substituting λ1 = -2.3439 in Eqs. (33) and (34), respectively. Right: the same for λ2 = -0.5461. Observe in the first case the intersection points do not lie in the singularity locus represented in light gray.

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