A coarse grained parallel homotopy for mechanism design

Polynomial systems of equations arise frequently in many scientific and eng ineering applications such as solid modeling, kinematics, and robotics and chemical process design. Locally convergent iterative solution methods may diverge or fail to find all possible solutions to a system of polynomial eq uations. Recently, homotopy algorithms have been proposed for solving polyn omial equations, These methods are globally convergent from any arbitrary s tarting point, can reliably compute all possible solutions, and are inheren tly parallel in nature, For problems arising in mechanism design, it is sho wn in this paper that through a use of m-homogenization and by defining aux iliary equations in addition to the design equations, the number of homotop y paths to be tracked land the associated computational effort) to obtain a ll possible solutions can be reduced significantly. Further computation tim e gains are realized by using the data-parallel nature of these methods, wh ich involved implementing them on Connection Machines CM-2/5, Numerical exa mples dealing with the synthesis of a slider-crank mechanism for six and ei ght finitely and multiply separated precision positions are presented.