Research Papers

The Complete Solution of Alt–Burmester Synthesis Problems for Four-Bar Linkages

[+] Author and Article Information
Daniel A. Brake

Department of Applied and
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

Andrew P. Murray

Department of Mechanical Engineering,
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

Charles W. Wampler

General Motors R&D Center,
Warren, MI 48090-9055
e-mail: charles.w.wampler@gm.com

Manuscript received July 27, 2015; final manuscript received March 1, 2016; published online April 19, 2016. Assoc. Editor: Qiaode Jeffrey Ge.

J. Mechanisms Robotics 8(4), 041018 (Apr 19, 2016) (8 pages) Paper No: JMR-15-1207; doi: 10.1115/1.4033251 History: Received July 27, 2015; Revised March 01, 2016

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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Grahic Jump Location
Fig. 1

A planar four-bar linkage, achieving a desired pose consisting of target point and orientation. The distal links of two RR dyads are rigidly joined by the coupler.

Grahic Jump Location
Fig. 2

Vector diagram of a four-bar linkage at the jth pose. The squares and circle indicate precision poses and a path-point, respectively.

Grahic Jump Location
Fig. 3

(a) The solution set for the (4, 2) Alt–Burmester problem in Table 5. (b) Demonstration mechanism corresponding to solution G1, G2.

Grahic Jump Location
Fig. 5

Plots of the (3, 3) Alt–Burmester problem from Table 7, projected onto the center point of dyad 1. (a) Far out view, showing 12 asymptotes to infinity, (b) zoom showing how the asymptotes transition into the curve, (c) the apparent crossings of the curve in this close-up correspond to the pole triangle in (d), and (d) the pole triangle.

Grahic Jump Location
Fig. 4

A projection of the solution curve for the (4, 1) Alt–Burmester example problem. There is an interval where no mechanism may be constructed.



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