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

2019 ◽  
Vol 11 (2) ◽  
Author(s):  
Keisuke Arikawa
Keyword(s):  
Basic Concept ◽  
Polynomial System ◽  
Gröbner Bases ◽  
Polynomial Systems ◽  
Grobner Bases ◽  
Truss Structures ◽  
Planar Mechanisms ◽  
Kinematic Constraints ◽  
Parametric Polynomial System ◽  
Kinematic Properties

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

2018 ◽  
Author(s):  
Keisuke Arikawa
Keyword(s):  
Parameter Space ◽  
Kinematic Analysis ◽  
Polynomial System ◽  
Gröbner Bases ◽  
Polynomial Systems ◽  
Grobner Bases ◽  
Algebraic Equations ◽  
Kinematic Constraints ◽  
Parametric Polynomial System ◽  
Planar Linkages

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.

scholarly journals A Signature-Based Algorithm for Computing Gröbner Bases over Principal Ideal Domains

2019 ◽  
Vol 14 (2) ◽  
pp. 515-530
Author(s):  
Maria Francis ◽  
Thibaut Verron
Keyword(s):  
Principal Ideal ◽  
Gröbner Bases ◽  
General Setting ◽  
Polynomial Systems ◽  
Regular Sequence ◽  
Grobner Bases ◽  
Integral Domains ◽  
Unique Factorization Domain ◽  
Univariate Polynomials ◽  
Principal Ideal Domains

AbstractSignature-based algorithms have become a standard approach for Gröbner basis computations for polynomial systems over fields, but how to extend these techniques to coefficients in general rings is not yet as well understood. In this paper, we present a proof-of-concept signature-based algorithm for computing Gröbner bases over commutative integral domains. It is adapted from a general version of Möller’s algorithm (J Symb Comput 6(2–3), 345–359, 1988) which considers reductions by multiple polynomials at each step. This algorithm performs reductions with non-decreasing signatures, and in particular, signature drops do not occur. When the coefficients are from a principal ideal domain (e.g. the ring of integers or the ring of univariate polynomials over a field), we prove correctness and termination of the algorithm, and we show how to use signature properties to implement classic signature-based criteria to eliminate some redundant reductions. In particular, if the input is a regular sequence, the algorithm operates without any reduction to 0. We have written a toy implementation of the algorithm in Magma. Early experimental results suggest that the algorithm might even be correct and terminate in a more general setting, for polynomials over a unique factorization domain (e.g. the ring of multivariate polynomials over a field or a PID).

scholarly journals An extension of Buchberger’s criteria for Gröbner basis decision

2010 ◽  
Vol 13 ◽  
pp. 111-129
Author(s):  
John Perry
Keyword(s):  
Gröbner Basis ◽  
Gröbner Bases ◽  
Characterization Theorem ◽  
Polynomial Systems ◽  
Polynomial Ideal ◽  
Grobner Bases ◽  
Grobner Basis ◽  
Case Scenario ◽  
Worst Case ◽  
New Criterion

AbstractTwo fundamental questions in the theory of Gröbner bases are decision (‘Is a basisGof a polynomial ideal a Gröbner basis?’) and transformation (‘If it is not, how do we transform it into a Gröbner basis?’) This paper considers the first question. It is well known thatGis a Gröbner basis if and only if a certain set of polynomials (theS-polynomials) satisfy a certain property. In general there arem(m−1)/2 of these, wheremis the number of polynomials inG, but criteria due to Buchberger and others often allow one to consider a smaller number. This paper presents two original results. The first is a new characterization theorem for Gröbner bases that makes use of a new criterion that extends Buchberger’s criteria. The second is the identification of a class of polynomial systemsGfor which the new criterion has dramatic impact, reducing the worst-case scenario fromm(m−1)/2 S-polynomials tom−1.

scholarly journals SHARPER COMPLEXITY BOUNDS FOR ZERO-DIMENSIONAL GRÖBNER BASES AND POLYNOMIAL SYSTEM SOLVING

2011 ◽  
Vol 21 (05) ◽  
pp. 703-713 ◽  
Author(s):  
AMIR HASHEMI ◽  
DANIEL LAZARD
Keyword(s):  
Polynomial System ◽  
Gröbner Bases ◽  
Mean Value ◽  
Arithmetic Mean ◽  
Grobner Bases ◽  
Maximal Degree ◽  
Complexity Bounds ◽  
Polynomial System Solving ◽  
Polynomial Ideals ◽  
Dense Representation

The main purpose of this paper is to improve the bound of complexity of the well-known algorithms on polynomial ideals having complexities polynomial in dn, where d is the maximal degree of input polynomials and n is the number of variables. Instead of this bound, we present the more accurate bound max {S, Dn} where S is the size of the input polynomials in dense representation, and D is the arithmetic mean value of the degrees of input polynomials.

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

2019 ◽  
Author(s):  
Keisuke Arikawa
Keyword(s):  
Kinematic Analysis ◽  
Polynomial System ◽  
Quantifier Elimination ◽  
Variable Order ◽  
Parametric Polynomial System ◽  
Validity Check ◽  
Homogeneous Transformation Matrix ◽  
Number Of Segments ◽  
Kinematic Properties ◽  
Active Joints

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.

scholarly journals Gröbner bases for polynomial systems with parameters

2010 ◽  
Vol 45 (12) ◽  
pp. 1391-1425 ◽  
Author(s):  
Antonio Montes ◽  
Michael Wibmer
Keyword(s):  
Gröbner Bases ◽  
Polynomial Systems ◽  
Grobner Bases

scholarly journals Approximate Gröbner Bases, Overdetermined Polynomial Systems, and Approximate GCDs

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Daniel Lichtblau
Keyword(s):  
Relative Error ◽  
Gröbner Bases ◽  
Approximate Solutions ◽  
Polynomial Systems ◽  
Grobner Bases ◽  
Polynomial Equations ◽  
Secondary Feature ◽  
Approximate Polynomial ◽  
Polynomial Gcd ◽  
Systems Of Polynomial Equations

We discuss computation of Gröbner bases using approximate arithmetic for coefficients. We show how certain considerations of tolerance, corresponding roughly to absolute and relative error from numeric computation, allow us to obtain good approximate solutions to problems that are overdetermined. We provide examples of solving overdetermined systems of polynomial equations. As a secondary feature we show handling of approximate polynomial GCD computations, using benchmarks from the literature.

Displacement Analysis of Multi-Loop Mechanisms Using Gröbner Bases

1998 ◽  
Author(s):  
A. K. Dhingra ◽  
A. N. Almadi ◽  
D. Kohli
Keyword(s):  
Gröbner Bases ◽  
Grobner Bases ◽  
Algebraic Equations ◽  
Multivariate Polynomial ◽  
Planar Mechanisms ◽  
Analysis Problem ◽  
Displacement Analysis ◽  
Comprehensive Information ◽  
Curve Equation

Abstract The displacement analysis problem for planar mechanisms can be written as a system of algebraic equations, in particular as a system of multivariate 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 the Buchberger’s 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 8-link mechanisms are presented.

Constraint-based reasoning via Grobner Bases

1997 ◽  
Vol 11 (1) ◽  
pp. 5-15 ◽  
Author(s):  
Sivand Lakmazaheri
Keyword(s):  
Constraint Satisfaction ◽  
Deductive Reasoning ◽  
Gröbner Bases ◽  
Nonlinear Problems ◽  
Grobner Bases ◽  
Truss Structures ◽  
Nonlinear Constraints ◽  
Geometry Configuration ◽  
Linear And Nonlinear ◽  
Algebraic Constraints

AbstractConstraint-based reasoning is a problem-solving approach based on deductive reasoning. In this approach, a problem is modeled in terms of hypotheses and conclusion constraints, and it is solved via constraint satisfaction. The ability to handle linear and nonlinear algebraic constraints is essential for successful application of constraint-based reasoning in engineering. Due to the scarcity of algebraic techniques for satisfying nonlinear constraints, little attention has been paid to the use of constraint-based reasoning for solving nonlinear problems. This paper examines the use of the Grobner Bases method for satisfying nonlinear constraints in the context of constraint-based reasoning. After a brief introduction to the Grobner Bases method and its role in constraint-based reasoning, two examples are presented. The first example illustrates the use of Grobner bases, in the context of constraint-based reasoning, for reasoning about the behavior of beams. The second example illustrates the geometry configuration of truss structures via constraint-based reasoning.

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