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Dimensional Mobility Criteria of Planar 6T-9R Paradoxical Chains

[+] Author Affiliations
Chung-Ching Lee, Jeng-Hsiung Lee

National Kaohsiung University of Applied Sciences, Kaohsiung, Taiwan

Po-Chih Lee

National Taiwan University, Taipei, Taiwan

Paper No. DETC2012-70916, pp. 1579-1586; 8 pages
  • ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
  • Volume 4: 36th Mechanisms and Robotics Conference, Parts A and B
  • Chicago, Illinois, USA, August 12–15, 2012
  • Conference Sponsors: Design Engineering Division, Computers and Information in Engineering Division
  • ISBN: 978-0-7918-4503-5
  • Copyright © 2012 by ASME


Focusing on the structural type of 6 ternary links and 9 revolute pairs (denoted 6T-9R), which stems from the commercialized Expanda-Triangles construction toy, we conduct in-depth study on the mobility constraints of the generalized planar 6T-9R paradoxical chains. According to Grübler criterion, the degree of freedom (DoF) of the chains of this type is minus 3 and they should have no constrained motion. However, due to the special dimensional constraints (Euclidean metric), some of such paradoxical mechanisms can still have the full-cycle mobility. To identify the dimensional constraints of mobility algebraically, we adopt a simple direct algebraic elimination approach to attain an eliminant of sixth-order polynomial in one variable only. The identity of polynomial whose coefficients are merely function of link lengths and structural angles produces six necessary dimensional constraints for movable 6T-9R paradoxical chain. One new and two existing paradoxical chains are revealed and their numerical examples are illustrated to show the correctness and validity of the proposed dimensional constraints. A novel generalized form of planar 6-bar paradoxical chain is disclosed too.

Copyright © 2012 by ASME



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