Rotations of 4nπ and the Kinematic Design of Parallel Manipulators
Identifiers and Pagination:Year: 2010
First Page: 86
Last Page: 92
Publisher Id: TOMEJ-4-86
Article History:Received Date: 8/5/2010
Revision Received Date: 18/6/2010
Acceptance Date: 29/6/2010
Electronic publication date: 30/12/2010
Collection year: 2010
open-access license: This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International Public License (CC-BY 4.0), a copy of which is available at: (https://creativecommons.org/licenses/by/4.0/legalcode). This license permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
For parallel manipulator systems a fundamental distinction is drawn between displacement and motion. The former is a spatial relation adequately modelled by standard vector and matrix algebra. The latter is a spatial trajectory whose ‘history’also requires modelling. Quaternion, spinor and Clifford algebra representations are utilised for this purpose - specifically for rigid body finite rotation in 3D space. Each involves rotation half-angles and hence exhibits apparently counter-intuitive features, notably that rotations of 0 and 2π are not equivalent, whereas rotations of 0 and 4π are equivalent. In general, rotations of 4nπ are not equivalent to rotations of (4n+2)π, where n is any integer. These representations have real physical manifestations, demonstrated here for parallel manipulator designs, adapted from a mechanical model devised by Dirac.