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

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.