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.

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.

scholarly journals Amplitude Independent Frequency Synchroniser for a Cubic Planar Polynomial System

2012 ◽  
Vol 7 (2) ◽  
Author(s):  
Islam Boussaada
Keyword(s):  
New York ◽  
Initial Conditions ◽  
Polynomial System ◽  
Polynomial Systems ◽  
Algebraic Equations ◽  
Compact Region ◽  
Synchronization Problem ◽  
The Family ◽  
Cubic System ◽  
Isochronous Centers

The problem of local linearizability of the planar linear center perturbed by cubic non- linearities in all generalities on the system parameters (14 parameters) is far from being solved. The synchronization problem (as noted in Pikovsky, A., Rosenblum, M., and Kurths, J., 2003, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Science Series, Cambridge University Press, UK, and Blekhman, I. I., 1988, Synchronisation in Science and Technology, ASME Press Translations, New York) consists in bringing appropriate modifications on a given system to obtain a desired dynamic. The desired phase portrait along this paper contains a compact region around a singular point at the origin in which lie periodic orbits with the same period (independently from the chosen initial conditions). In this paper, starting from a five parameters non isochronous Chouikha cubic system (Chouikha, A. R., 2007, “Isochronous Centers of Lienard Type Equations and Applications,” J. Math. Anal. Appl., 331, pp. 358–376) we identify all possible monomial perturbations of degree d ∈ {2, 3} insuring local linearizability of the perturbed system. The necessary conditions are obtained by the Normal Forms method. These conditions are real algebraic equations (multivariate polynomials) in the parameters of the studied ordinary differential system. The efficient algorithm FGb (J. C. Faugère, “FGb Salsa Software,” http://fgbrs.lip6.fr) for computing the Gröbner basis is used. For the family studied in this paper, an exhaustive list of possible parameters values insuring local linearizability is established. All the found cases are already known in the literature but the contexts are different since our object is the synchronisation rather than the classification. This paper can be seen as a direct continuation of several new works concerned with the hinting of cubic isochronous centers, (in particular Bardet, M., and Boussaada, I., 2011, “Compexity Reduction of C-algorithm,” App. Math. Comp., in press; Boussaada, I., Chouikha, A. R., and Strelcyn, J.-M., 2011, “Isochronicity Conditions for some Planar Polynomial Systems,” Bull. Sci. Math, 135(1), pp. 89–112; Bardet, M., Boussaada, I., Chouikha, A. R., and Strelcyn, J.-M., 2011, “Isochronicity Conditions for some Planar Polynomial Systems,” Bull. Sci. Math, 135(2), pp. 230–249; and furthermore, it can be considered as an adaptation of a qualitative theory method to a synchronization problem.

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.

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