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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.

MSC:

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
52B99 Polytopes and polyhedra
14Q99 Computational aspects in algebraic geometry
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[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)
[7] Gosselin, C.M., 1997. Note sur l’équilibrage de Berkof et Lowen. In: Canadian Congress of Applied Mechanics, CANCAM 97, pp. 497-498
[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
[11] Moroz, G., 2006. Complexity of the resolution of parametric systems of equations and inequations. In: Proceedings of the ISSAC’06 Conference, pp. 246-253 · Zbl 1356.14050
[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
[14] Ricard, R., Gosselin, C.M., 2000. On the development of reactionless parallel manipulators. In: Proceedings of ASME Design Engineering Technical Conferences
[15] Salem, F.A., Gao, S., Lauder, A.G.B., 2004. Factoring polynomial via polytopes. In: ISSAC 2004, pp. 4-11 · Zbl 1088.68183
[16] Schost, E., Computing parametric geometric resolutions, Applicable algebra in engineering, communication and computing, 13, 5, 349-393, (2003) · Zbl 1058.68123
[17] Yoshida, K., Hashizume, K., Abiko, S., 2001. Zero reaction maneuver: Flight validation with ETS-VII space robot and extension to kinematically redundant arm. In: Proceedings of IEEE International Conference on Robotics and Automation 2001
[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, (), 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)
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