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

Abstract

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.

Probabilistic Saturations and Alt’s Problem

“…, 6 for various primes as reported in Section 5. In particular, since the values of g i (X, C 8 ) match the results in Table 2, we have utilized symbolic methods to confirm numerical homotopy continuation computations in [12,40].…”

“…In particular, they showed that there are 1442 distinct four-bar coupler curves which pass through nine general points which, together with Roberts cognates [35], yields 4326 distinct four-bar linkages. Although this computation has been repeatedly confirmed using various homotopy continuation methods, e.g., [5,12,21,22,34,38], these numerical computations do not preclude the existence of additional solutions. In fact, one of the distinct four-bar linkages was missed by the homotopy continuation solver in [40] but was reconstructed using the cognate formula.…”

“…generically already has degree 5 = g 1 (X, C 2 ), we can simplify the presentation by just considering I 1 (Θ, Λ) equal to the ideal in (12 .…”

Numerical Computation of Galois Groups

“…Alt [1] proposed synthesis problems based on path generation, specifying positions along a curve. The synthesis problem consisting of some poses and some positions was called an Alt-Burmester problem in [35] with the complete solution to all Alt-Burmester problems described in [6]. We compute the Galois group for four of the Alt-Burmester problems having finitely many solutions.…”

“…Specifying M poses and N = 10 − 2M positions, there will generically be finitely many linkages that take on the given poses and whose apex can pass through the given positions in its motion. Following [6] in isotropic coordinates, the M -pose and N -position Alt-Burmester problem is described by the following parameters: positions:…”