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

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.

Kinematic Analysis of Mechanisms Based on Parametric Polynomial System: Basic Concept of a Method Using Gröbner Cover and Its Application to Planar Mechanisms

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 Gröbner 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, 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 Gröbner cover of the PPS, we obtain the segmentation of the parameter space and valid Gröbner bases for each segment. Finally, we interpret the meaning of the segments and their corresponding Gröbner 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.

Kinematic Analysis of Mechanisms Based on Parametric Polynomial System

Many kinematic problems of mechanisms can be expressed in the form of polynomial systems. Gröbner Bases computation is effective for algebraically analyzing such systems. In this research, we discuss the cases in which the parameters are included in the polynomial systems. The parameters are used to express the link lengths, the displacements of active joints, hand positions, and so on. By calculating Gröbner Cover of the parametric polynomial system that expresses kinematic constraints, we obtain segmentation of the parameter space and valid Gröbner Bases for each segment. In the application examples, we use planar linkages to interpret the meanings of the algebraic equations that define the segments and the Gröbner Bases. Using these interpretations, we confirmed that it was possible to enumerate the assembly and working modes and to identify the geometrical conditions that enable overconstrained motions.