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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