Improving the Method for Kinematic Analysis of Mechanisms That Was Based on Parametric Polynomial System With Gröbner Cover

Abstract

A polynomial system that contains parameters is termed a parametric polynomial system (PPS). We had previously proposed a method of kinematic analysis of mechanisms based on PPS with Gröbner cover, where the parameters are used to express link lengths, displacements of active joints, and so on. Calculating Gröbner cover of PPS that expresses kinematic constraints, and interpreting the segments of the parameter space that are generated by Gröbner cover, it is possible to gain an insight for comprehensively understanding kinematic properties of mechanisms characterized by the parameters. In this study, certain improvements to the method were made to enhance its practical application. The validity check of the segments in the real domain using quantifier elimination provides an automatic reliable check even for a large number of segments. The evaluation of the solution spaces in the segments using primary decomposition facilitates the kinematic interpretation of the complex solution spaces. The active joint selection based on the variable order in Gröbner cover enables the analyses without explicitly specifying active joints. Moreover, the alternative algebraic formulation of kinematic problems based on a homogeneous transformation matrix provides further insight regarding the mechanisms containing zero-length links. The effectiveness of these improvements was verified by the analyses of the configurations of 3RPR mechanism and five-bar linkage.