A Solution Technique for Equations Arising in the Design and Geometric analysis
of Mechanisms
Stephen Nelson
Master of Science, December 1994
ABSTRACT
The "7R Mechanism"
is known in the mechanism community as "the Mount Everest " of mechanism
problems. It was first solved in 1986 by H.Y. Lee and C.G. Liang. In1993 Blaise
Morton and Michael Elgersma, at Honeywell Technology Center, developed a general
approach to the mechanism geometry problem which they applied to the 7R problem.
At the time, the Honeywell researchers thought they had been the first to solve
this problem, being unaware of the earlier solution by Lee and Liang. Though
the Honeywell solution did not lead to the first solution of 7R, it does appear
to be more suitable for generalization to more complex mechanism types. In particular,
it could lead to the first technique for solving the geometry problem for multi-loop
spatial mechanisms. The technique calls for the determination of the common
roots of certain types of multi-variable polynomials, called "multi-affine
polynomials", which we define below. Although Morton and Elgersma were
able to solve the systems of multi-affine polynomial equations arising from
the 7R mechanism, a general algorithm for finding the common roots of an arbitrary
set of multi-affine polynomials does not exist. Since the technique used to
generate the kinematic geometrical equations in the analysis of the 7R mechanism
can bee used to generate these equations for the analysis of any type of spatial
mechanism, and since the solution of these equations involves the solution of
systems of multi-affine polynomial equations, the need to develop methods to
solve systems of multi-affine equations becomes obvious.
Research supported by the Minnesota Center for Industrial
Mathematics (MCIM)
