, Volume 51, Issue 12, pp 3211–3225 | Cite as

Synthesis, optimization and experimental validation of reactionless two-DOF parallel mechanisms using counter-mechanisms

  • Thierry Laliberté
  • Clément GosselinEmail author
50th Anniversary of Meccanica


The synthesis of reactionless mechanisms generally involves the force balancing of a mechanism using countermasses followed by the dynamic balancing of the inertia using counter-rotations. This approach requires that the force balanced mechanism have a constant equivalent moment of inertia for any configuration. Two of the main drawbacks of reactionless mechanisms are a significant increase in mass and actuation inertia. In this paper, a counter-mechanism is introduced in order to dynamically balance a force balanced two-DOF mechanism with variable inertia, which is expected to reduce the aforementioned drawbacks. The conditions for which the counter-mechanism matches the moment of inertia of the main mechanism for any configuration are derived. Then, in order to fairly compare the use of a counter-mechanism instead of counter-rotations, the balancing strategies are optimized regarding the addition of mass and actuation inertia using Lagrange multipliers. The results are optimal rules for the design of reactionless mechanisms and optimal mass-inertia curves which are akin to Pareto curves. These results allow to demonstrate the advantages of counter-mechanisms over counter-rotations, especially regarding added actuation inertia and compactness. Also, the significant influence of the radius of gyration of the counter-inertias on the optimal mass-inertia curves is revealed. Finally, examples of reactionless mechanisms are presented and prototypes are tested in order to validate the concepts.


Dynamic balancing Parallel mechanism Pantograph Reactionless mechanism Planar mechanism 



The authors gratefully acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC) (Grant No. 89715) and the Canada Research Chair Program.

Supplementary material

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Supplementary material 2 (mp4 24265 KB)


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Département de Génie MécaniqueUniversité LavalQuébecCanada

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