Distance Metrics on the Rigid-Body Motions with Applications to Mechanism Design

In this article we examine the problem of designing a mechanism whose tool frame comes closest to reaching a set of desired goal frames. The basic mathematical question we address is characterizing the set of distance metrics in SE(3), the Euclidean group of rigid-body motions. Using Lie theory, we...

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Bibliographic Details
Published in:Journal of mechanical design (1990) 1995-03, Vol.117 (1), p.48-54
Main Author: Park, F. C.
Format: Article
Language:English
Subjects:
Exact sciences and technology
Fundamental areas of phenomenology (including applications)
Physics
Solid dynamics (ballistics, collision, multibody system, stabilization...)
Solid mechanics
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Summary:In this article we examine the problem of designing a mechanism whose tool frame comes closest to reaching a set of desired goal frames. The basic mathematical question we address is characterizing the set of distance metrics in SE(3), the Euclidean group of rigid-body motions. Using Lie theory, we show that no bi-invariant distance metric (i.e., one that is invariant under both left and right translations) exists in SE(3), and that because physical space does not have a natural length scale, any distance metric in SE(3) will ultimately depend on a choice of length scale. We show how to construct left- and right-invariant distance metrics in SE(3), and suggest a particular left-invariant distance metric parametrized by length scale that is useful for kinematic applications. Ways of including engineering considerations into the choice of length scale are suggested, and applications of this distance metric to the design and positioning of certain planar and spherical mechanisms are given.
ISSN:1050-0472
1528-9001
DOI:10.1115/1.2826116