Kinematic Analysis of Mechanisms based on Parametric Polynomial System: Basic Concept and Application to Planar Mechanisms

[+] Author and Article Information
Keisuke Arikawa

Member of ASME, Department of Mechanical Engineering, Kanagawa Institute of Technology, Atsugi, Kanagawa, 243-0292, Japan

1Corresponding author.

ASME doi:10.1115/1.4042475 History: Received October 13, 2018; Revised January 04, 2019


Many kinematic problems in mechanisms can be represented by polynomial systems. By algebraically analyzing the polynomial systems, we can obtain the kinematic properties of the mechanisms. Among these algebraic methods, approaches based on Groebner Bases are effective. Usually, the analyses are performed for specific mechanisms; however, we often encounter phenomena for which, even within the same class of mechanisms, the kinematic properties differ significantly. In this research, we consider the cases where the parameters are included in the polynomial systems. The parameters are used to express link lengths, displacements of active joints, and hand positions, and so on. By analyzing a parametric polynomial system (PPS), we intend to comprehensively analyze the kinematic properties of mechanisms represented by these parameters. In the proposed method, we first express the kinematic constraints in the form of PPS. Subsequently, by calculating the Groebner Cover of the PPS, we obtain the segmentation of the parameter space and valid Groebner Bases for each segment. Finally, we interpret the meanings of the segments and their corresponding Groebner Bases. We analyzed planar four- and five-bar linkages and five-bar truss structures using the proposed method. We confirmed that it was possible to enumerate the assembly and working modes and to identify the geometrical conditions that enable overconstrained motions.

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