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Applications of the 4D Geometric Algebra to Dimensional Mobility Criteria of Delassus-Parallelogram and Bennett Paradoxical Linkages

[+] Author Affiliations
Chung-Ching Lee

National Kaohsiung University of Applied Sciences, Kaohsiung, Taiwan

Paper No. DETC2015-46667, pp. V05CT08A052; 8 pages
doi:10.1115/DETC2015-46667
From:
  • ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
  • Volume 5C: 39th Mechanisms and Robotics Conference
  • Boston, Massachusetts, USA, August 2–5, 2015
  • Conference Sponsors: Design Engineering Division, Computers and Information in Engineering Division
  • ISBN: 978-0-7918-5714-4
  • Copyright © 2015 by ASME

abstract

Geometric algebra is also termed Clifford-Grassmann algebra or hypercomplex number. It allows studying space geometric problems in an easy and compact way. Transforming three-dimensional (3D) Euclidean geometric entities to actual elements of four-dimensional (4D) geometric algebra (abbreviated to g4) through a methodical approach of geometric algebra, one can describe motion displacements as even elements of g4. This article relies on the combined rotation and translation in g4 to establish the dimensional constraints of two non-exceptional overconstrained paradoxical linkages. Firstly, fundamentals of geometric algebra are recalled. Then, the single finite rotation and the composition of two successive finite rotations are introduced. After that, a general rigid-body motion in g4 is revealed for a possible application in exploring paradoxical chains using the geometric algebra. Finally, the metric or dimensional mobility criteria of Delassus-parallelogram four-screw and Bennett four-revolute paradoxical linkages are algebraically verified.

Copyright © 2015 by ASME

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