Dynamic balancing of planar mechanisms using toric geometry. (English) Zbl 1169.14034

Summary: A mechanism is statically balanced if for any motion, it does not apply forces on the base. Moreover, if it does not apply torques on the base, the mechanism is said to be dynamically balanced. In this paper, a new method for determining the complete set of dynamically balanced planar four-bar mechanisms is presented. Using complex variables to model the kinematics of the mechanism, the static and dynamic balancing constraints are written as algebraic equations over complex variables and joint angular velocities. After elimination of the joint angular velocity variables, the problem is formulated as a problem of factorization of Laurent polynomials. Using tools from toric geometry including toric polynomial division, necessary and sufficient conditions for static and dynamic balancing of planar four-bar mechanisms are derived.


14M25 Toric varieties, Newton polyhedra, Okounkov bodies
52B99 Polytopes and polyhedra
14Q99 Computational aspects in algebraic geometry
Full Text: DOI arXiv


[1] Aubry, P.; Lazard, D.; Moreno Maza, M., On the theories of triangular sets, Journal of Symbolic Computation, 28, 105-124 (1999) · Zbl 0943.12003
[2] Berkof, R. S.; Lowen, G. G., A new method for completely force balancing simple linkage, Journal of Engineering for Industry, 21-26 (1969)
[3] Berkof, R. S., Complete force and moment balancing of inline four-bar linkages, Mechanism and Machine Theory, 8, 397-410 (1973)
[4] Bernshtein, D., The number of roots of a system of equations, Functional Analysis and its Applications, 183-185 (1975)
[5] Bottema, O.; Roth, B., Theoretical Kinematics (1990), Dover · Zbl 0747.70001
[6] Ebert-Uphoff, I.; Gosselin, C. M.; Laliberté, T., Static balancing of spatial parallel platform mechanisms-revisited, Journal of Mechanical Design, 122, 43-51 (2000)
[8] Gosselin, C. M.; Vollmer, F.; Côté, G.; Wu, Y., Synthesis and design of reactionless three-degree-of-freedom parallel mechanisms, IEEE Transactions on Robotics and Automation (2004)
[9] Lazard, D.; Rouillier, F., Solving parametric polynomial systems, Journal of Symbolic Computation, 42, 636-667 (2007) · Zbl 1156.14044
[10] Montes, A., A new algorithm for discussing Gröbner bases with parameters, Journal of Symbolic Computation, 33, 183-208 (2002) · Zbl 1068.13016
[12] Ostrowski, A. M., Über die bedeutung der theorie des konvexen polyeder für die formale algebra, Jahresbericht der Deutschen Mathematiker Vereinigung, 20, 98-99 (1921) · JFM 48.0106.01
[13] Ostrowski, A. M., On multiplication and factorization of polynomials, i. lexicographic ordering and extreme aggregates of terms, Aequationes Mathematicae, 13, 201-228 (1975) · Zbl 0319.13004
[16] Schost, E., Computing parametric geometric resolutions, Applicable Algebra in Engineering, Communication and Computing, 13, 5, 349-393 (2003) · Zbl 1058.68123
[18] Wang, D., Elimimation Methods (2001), Springer
[19] Weispfenning, V., Comprehensive Gröbner bases, Journal of Symbolic Computation, 14, 1-29 (1992) · Zbl 0784.13013
[20] Weispfenning, V., Solving parametric polynomial equations and inequalities by symbolic algorithms, (Proc. of the Workshop Computer Algebra in Science and Engineering, Bielefeld, August 1994 (1995), World Scientific: World Scientific Singapore), 163-179
[21] Wu, Y.; Gosselin, C. M., Synthesis of reactionless spatial 3-dof and 6-dof mechanisms without separate counter-rotations, The International Journal of Robotics Research, 23, 6, 625-642 (2004)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.