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Research Papers

# On the Manifold Property of the Set of Singularities of Kinematic Mappings: Modeling, Classification, and Genericity

[+] Author and Article Information
Andreas Müller

University Duisburg-Essen, Duisburg 47057, Germanyandreas-mueller@uni-due.de

The term “kinematic mapping” has traditionally been assigned to the mapping from the $SE(3)$ to the kinematic image space (1-2). With a slight abuse of notation, we use this term to refer to the mapping $f:Vn→SE(3)$.

The co-dimension of an $m$-dimensional submanifold $M$ in an $n$-dimensional manifold $N$ is $n−m$.

Although the transformations in $La$ and $Ga$ are relative to reference frames, all statements are coordinate invariant by conjugation.

J. Mechanisms Robotics 3(1), 011006 (Nov 30, 2010) (8 pages) doi:10.1115/1.4002695 History: Received October 12, 2009; Revised September 20, 2010; Published November 30, 2010; Online November 30, 2010

## Abstract

It is commonly assumed that the singularities of kinematic mappings constitute generically smooth manifolds but this has not yet been proven. Moreover, before this assumption can be verified, the concept of genericity needs to be clarified. In this paper, two different notions of generic properties of kinematic mappings are discussed. One accounts for the stability of the manifold property with respect to small changes in the geometry of a mechanism while the other concerns the likelihood that a mechanism possesses smooth manifolds of singularities. Singularities forming smooth manifolds is the condition for singularity-free motion of overconstrained mechanisms but also has consequences for reliable control of serial manipulators. As basis for establishing genericity, a formulation of the kinematic mapping is presented that takes into account the type of joints and feasible link geometries. The continuous transition between link geometries defines a deformation of a kinematic mapping. All mappings obtained in this way constitute a class of kinematic mappings. A basic characteristic of a kinematic chain is its motion space. An explicit expression for the motion spaces of individual as well as classes of kinematic mappings is given. The actual conditions for genericity will be addressed in a forthcoming publication.

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## Figures

Figure 1

(a) A 3R regional manipulator with special (nongeneric) geometry, which is not cuspidal, i.e., it does not possess manifolds of singularities that are mapped to cusp curves in workspace, as in (b)

Figure 2

(a) Overconstrained Bricard mechanism and (b) underconstrained 7R mechanism

Figure 3

Description of the relative configuration Ra of adjacent links. Ma represents the link geometry and Na represents the joint displacement.

Figure 4

Kinematics in presence of a deformation Aa from the nominal link geometry Ma

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