**Advances in Polynomial Continuation**

** for Solving Problems in Kinematics **

*Andrew J. Sommese*,
*Jan Verschelde*, and
*Charles W. Wampler*

#### Abstract:

For many mechanical systems, including nearly all robotic manipulators,
the set of possible configurations that the links may assume can be
described by a system of polynomial equations.
Thus, solving such systems is central to many problems in analyzing
the motion of a mechanism or in designing a mechanism to achieve a
desired motion.
This paper describes techniques, based on polynomial continuation,
for numerically solving such systems.
Whereas in the past, these techniques were focused on
finding isolated roots, we now address the treatment
of systems having higher-dimensional solution sets.
Special attention is given to cases of exceptional mechanisms,
which have an higher degree of freedom of motion than predicted
by their mobility. In fact, such mechanisms often have several
disjoint assembly modes, and the degree of freedomof motion is not
necessarily the same in each mode. Our algorithms identify all such
assembly modes, determine their dimension and degree, and give sample
points on each.
*ASME Journal of Mechanical Design* 126(2):262-268, 2004.