Closed-form displacement and coupler curve analysis of planar multi-loop mechanisms using Gröbner bases

The displacement analysis problem for planar and spatial mechanisms can be written as a system of algebraic equations in particular as a system of multi-variate polynomial equations. Elimination theory based on resultants and polynomial continuation are some of the methods which have been used to solve this problem. This paper explores an alternate approach, based on Gröbner bases, to solve the displacement analysis problem for planar mechanisms. It is shown that the reduced set of generators obtained using Buchbergerʹs algorithm for Gröbner bases not only yields the input–output polynomial for the mechanism, but also provides comprehensive information on the number of closures and the relationships between various links of the mechanism. Numerical examples illustrating the applicability of Gröbner bases to displacement analysis of 10- and 12-link mechanisms, and determination of coupler curve equation for an 8-link mechanism are presented. It is seen that even though the Gröbner bases method is versatile enough to handle finitely solvable as well as over-constrained systems of equations, it can run into computational problems due to rapidly growing numerical coefficients and/or the set of generators. The examples presented show how these difficulties can be overcome by artificially decoupling complex mechanisms to help facilitate their closed-form analysis.