Precision-point synthesis problems for design of four-bar linkages have typically been formulated using two approaches. The exclusive use of path-points is known as “path synthesis,” whereas the use of poses, i.e., path-points with orientation, is called “rigid-body guidance” or the “Burmester problem.” We consider the family of “Alt–Burmester” synthesis problems, in which some combination of path-points and poses is specified, with the extreme cases corresponding to the classical problems. The Alt–Burmester problems that have, in general, a finite number of solutions include Burmester's original five-pose problem and also Alt's problem for nine path-points. The elimination of one path-point increases the dimension of the solution set by one, while the elimination of a pose increases it by two. Using techniques from numerical algebraic geometry, we tabulate the dimension and degree of all problems in this Alt–Burmester family, and provide more details concerning all the zero- and one-dimensional cases.
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August 2016
Research-Article
The Complete Solution of Alt–Burmester Synthesis Problems for Four-Bar Linkages
Daniel A. Brake,
Daniel A. Brake
Department of Applied and
Computational Mathematics and Statistics,
University of Notre Dame,
Notre Dame, IN 46556
e-mail: dbrake@nd.edu
Computational Mathematics and Statistics,
University of Notre Dame,
Notre Dame, IN 46556
e-mail: dbrake@nd.edu
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Jonathan D. Hauenstein,
Jonathan D. Hauenstein
Department of Applied and
Computational Mathematics and Statistics,
University of Notre Dame,
Notre Dame, IN 46556
e-mail: hauenstein@nd.edu
Computational Mathematics and Statistics,
University of Notre Dame,
Notre Dame, IN 46556
e-mail: hauenstein@nd.edu
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Andrew P. Murray,
Andrew P. Murray
Department of Mechanical Engineering,
University of Dayton,
Dayton, OH 45469
e-mail: murray@udayton.edu
University of Dayton,
Dayton, OH 45469
e-mail: murray@udayton.edu
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David H. Myszka,
David H. Myszka
Department of Mechanical Engineering,
University of Dayton,
Dayton, OH 45469
e-mail: dmyszka@udayton.edu
University of Dayton,
Dayton, OH 45469
e-mail: dmyszka@udayton.edu
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Charles W. Wampler
Charles W. Wampler
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Daniel A. Brake
Department of Applied and
Computational Mathematics and Statistics,
University of Notre Dame,
Notre Dame, IN 46556
e-mail: dbrake@nd.edu
Computational Mathematics and Statistics,
University of Notre Dame,
Notre Dame, IN 46556
e-mail: dbrake@nd.edu
Jonathan D. Hauenstein
Department of Applied and
Computational Mathematics and Statistics,
University of Notre Dame,
Notre Dame, IN 46556
e-mail: hauenstein@nd.edu
Computational Mathematics and Statistics,
University of Notre Dame,
Notre Dame, IN 46556
e-mail: hauenstein@nd.edu
Andrew P. Murray
Department of Mechanical Engineering,
University of Dayton,
Dayton, OH 45469
e-mail: murray@udayton.edu
University of Dayton,
Dayton, OH 45469
e-mail: murray@udayton.edu
David H. Myszka
Department of Mechanical Engineering,
University of Dayton,
Dayton, OH 45469
e-mail: dmyszka@udayton.edu
University of Dayton,
Dayton, OH 45469
e-mail: dmyszka@udayton.edu
Charles W. Wampler
Manuscript received July 27, 2015; final manuscript received March 1, 2016; published online April 19, 2016. Assoc. Editor: Qiaode Jeffrey Ge.
J. Mechanisms Robotics. Aug 2016, 8(4): 041018 (8 pages)
Published Online: April 19, 2016
Article history
Received:
July 27, 2015
Revised:
March 1, 2016
Citation
Brake, D. A., Hauenstein, J. D., Murray, A. P., Myszka, D. H., and Wampler, C. W. (April 19, 2016). "The Complete Solution of Alt–Burmester Synthesis Problems for Four-Bar Linkages." ASME. J. Mechanisms Robotics. August 2016; 8(4): 041018. https://doi.org/10.1115/1.4033251
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