Patterned Bootstrap: A New Method That Gives Efficiency for Some Precision Position Synthesis Problems

Abstract

This paper presents a heuristic global convergence method, termed as patterned bootstrap (PB), for solving systems of equations. In the PB method, multiple directions starting from a given point are searched. A number of intermediate underdetermined systems are selected and solved sequentially using classic globally convergence methods. Numerical experiments demonstrate that the PB method outperforms Levenberg-Marquardt method on solving a number of challenging synthesis problems in no more than 18 variables. On the other hand, Levenberg-Marquardt method normally outperforms the PB method on solving several systems of equations in 30 variables which are derived from the five precision-position motion generation problem of spatial RRR manipulators. In the paper, tunneling functions are also introduced to exclude degenerated solution sets in several synthesis problems. The research reveals that appropriate numerical methods and synthesis equations can be chosen for obtaining most solutions efficiently and provide a complete solution set of a precision position synthesis problem within a domain of interest.