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Research Papers

# On the Manifold Property of the Set of Singularities of Kinematic Mappings: Genericity Conditions

[+] Author and Article Information
Andreas Müller

Chair of Mechanics and Robotics, University Duisburg-Essen, 47057 Duisburg, Germanyandreas-mueller@uni-due.de

J. Mechanisms Robotics 4(1), 011006 (Feb 03, 2012) (9 pages) doi:10.1115/1.4005524 History: Received May 21, 2010; Revised October 25, 2011; Published February 03, 2012; Online February 03, 2012

## Abstract

Whether the singularities of a kinematic mapping constitute smoothmanifolds is an important question with significance to mechanism design and robot control. It is thus obvious to ask if this is generically so. In a preceding paper, two kinematically meaningful genericity concepts have been introduced. In this paper, geometric conditions for the manifold property of families of kinematic mappings are addressed, and a sufficient condition is presented. This condition involves the joint screws and screws representing feasible link geometries specific to a class of kinematic mappings. It admits to establish genericity of the manifold property for given classes of kinematic mappings. This is a step forward to prove that singularities of kinematic mappings form generically smooth manifolds. As an example it is shown that the singularities of 3-DOF forward kinematic mappings form generically smooth manifolds. Restricting this condition to a particular geometry allows to check whether singularities of a given kinematic mapping form smooth manifolds.

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## Figures

Figure 1

Sets of singularities in the plane. The varieties Σ1 and Σ2 are one-dimensional smooth manifolds, and Σ3 is a zero-dimensional smooth manifold (an isolated singularity). Σ4 and Σ5 are not manifolds since they possess singularities themselves (the bifurcation points and the cusp point, respectively).

Figure 2

(a) The mapping f:R→R2 is not transversal to the submanifold Z, (b) f is transversal to Z

Figure 3

6R chain in a corank 2 singularity

Figure 4

6R chain in a corank 3 singularity

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