0
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:VnSE(3).

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

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

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.

FIGURES IN THIS ARTICLE
<>
Copyright © 2011 by American Society of Mechanical Engineers
Your Session has timed out. Please sign back in to continue.

References

Figures

Grahic Jump Location
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)

Grahic Jump Location
Figure 2

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

Grahic Jump Location
Figure 3

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

Grahic Jump Location
Figure 4

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

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In