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

Abstract

Precisionpoint synthesis problems for design of fourbar linkages have typically been formulated using two approaches. The exclusive use of pathpoints is known as “path synthesis,â€‌ whereas the use of poses, i.e., pathpoints with orientation, is called “rigidbody guidanceâ€‌ or the “Burmester problem.â€‌ We consider the family of “Alt–Burmesterâ€‌ synthesis problems, in which some combination of pathpoints 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 fivepose problem and also Alt's problem for nine pathpoints. The elimination of one pathpoint 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 zeroand onedimensional cases.