Advances in Polynomial Continuation for Solving Problems in Kinematics

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 freedom of 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. Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, U.S.A. E-mail: [email protected] URL: http://www.nd.edu/ ̃sommese Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 South Morgan (M/C 249), Chicago, IL 60607-7045, U.S.A. E-mail: [email protected] [email protected] URL: http://www.math.uic.edu/ ̃jan General Motors Research Laboratories, Enterprise Systems Lab, Mail Code 480-106-359, 30500 Mound Road, Warren, MI 48090-9055, U.S.A. E-mail: [email protected]