An Analysis of Inverse Kinematics of Robot Manipulators using Grobner Basis

1997 ◽  
Vol 9 (5) ◽  
pp. 324-331
Author(s):  
Toshimi Shimizu ◽  
◽  
Haruhisa Kawasaki
Keyword(s):  
Computer Algebra ◽  
Inverse Kinematics ◽  
Gröbner Basis ◽  
Robot Manipulators ◽  
Symbolic Analysis ◽  
New Method ◽  
Polynomial Equations ◽  
Trigonometric Functions ◽  
Grobner Basis ◽  
Multivariate Polynomial

This paper presents a new method for solving the inverse kinematics of robot manipulators symbolically using computer algebra. The kinematics equations, including the trigonometric functions of joint displacements, are expressed as multivariate polynomial equations by transforming these functions into variables. The multivariate polynomial equations can be solved by evaluating their reduced Grobner basis. The properties for efficient evaluation of the reduced Grobner basis and the inverse kinematics of a robot, whose last three joint axes intersect at a point, are shown. This procedure is implemented using Maple V and built into ROSAM (Robot Symbolic Analysis, by Maple) that is a robot analysis library made by our group. An analysis example of a structurechanged PUMA type robot is given.

scholarly journals Algebraic Cryptanalysis Scheme of AES-256 Using Gröbner Basis

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Kaixin Zhao ◽  
Jie Cui ◽  
Zhiqiang Xie
Keyword(s):  
Gröbner Basis ◽  
Variable Order ◽  
Polynomial Equations ◽  
Construction Process ◽  
Grobner Basis ◽  
Multivariate Polynomial ◽  
Algebraic Cryptanalysis ◽  
Term Order ◽  
Basis Construction ◽  
Depth Study

The zero-dimensional Gröbner basis construction is a crucial step in Gröbner basis cryptanalysis on AES-256. In this paper, after performing an in-depth study on the linear transformation and the system of multivariate polynomial equations of AES-256, the zero-dimensional Gröbner basis construction method is proposed by choosing suitable term order and variable order. After giving a detailed construction process of the zero-dimensional Gröbner basis, the necessary theoretical proof is presented. Based on this, an algebraic cryptanalysis scheme of AES-256 using Gröbner basis is proposed. Analysis shows that the complexity of our scheme is lower than that of the exhaustive attack.

scholarly journals Reduction of Feynman integrals in the parametric representation II: reduction of tensor integrals

2021 ◽  
Vol 81 (3) ◽  
Author(s):  
Wen Chen
Keyword(s):  
Gröbner Basis ◽  
Parametric Representation ◽  
Polynomial Equations ◽  
Grobner Basis ◽  
Spacetime Dimension ◽  
Integration By Parts ◽  
Feynman Integrals ◽  
Parametric Integrals ◽  
Parametric Integration

AbstractIn a recent paper by the author (Chen in JHEP 02:115, 2020), the reduction of Feynman integrals in the parametric representation was considered. Tensor integrals were directly parametrized by using a generator method. The resulting parametric integrals were reduced by constructing and solving parametric integration-by-parts (IBP) identities. In this paper, we furthermore show that polynomial equations for the operators that generate tensor integrals can be derived. Based on these equations, two methods to reduce tensor integrals are developed. In the first method, by introducing some auxiliary parameters, tensor integrals are parametrized without shifting the spacetime dimension. The resulting parametric integrals can be reduced by using the standard IBP method. In the second method, tensor integrals are (partially) reduced by using the technique of Gröbner basis combined with the application of symbolic rules. The unreduced integrals can further be reduced by solving parametric IBP identities.

scholarly journals Incremental column-wise verification of arithmetic circuits using computer algebra

2019 ◽  
Vol 56 (1-3) ◽  
pp. 22-54 ◽  
Author(s):  
Daniela Kaufmann ◽  
Armin Biere ◽  
Manuel Kauers
Keyword(s):  
Computer Algebra ◽  
Gröbner Basis ◽  
Computation Time ◽  
Arithmetic Circuits ◽  
Grobner Basis ◽  
Word Level ◽  
Half Adder ◽  
Soundness And Completeness

AbstractVerifying arithmetic circuits and most prominently multiplier circuits is an important problem which in practice still requires substantial manual effort. The currently most effective approach uses polynomial reasoning over pseudo boolean polynomials. In this approach a word-level specification is reduced by a Gröbner basis which is implied by the gate-level representation of the circuit. This reduction returns zero if and only if the circuit is correct. We give a rigorous formalization of this approach including soundness and completeness arguments. Furthermore we present a novel incremental column-wise technique to verify gate-level multipliers. This approach is further improved by extracting full- and half-adder constraints in the circuit which allows to rewrite and reduce the Gröbner basis. We also present a new technical theorem which allows to rewrite local parts of the Gröbner basis. Optimizing the Gröbner basis reduces computation time substantially. In addition we extend these algebraic techniques to verify the equivalence of bit-level multipliers without using a word-level specification. Our experiments show that regular multipliers can be verified efficiently by using off-the-shelf computer algebra tools, while more complex and optimized multipliers require more sophisticated techniques. We discuss in detail our complete verification approach including all optimizations.

Symbolic Analysis of Robot Base Parameter Set Using Grobner-Basis

1998 ◽  
Vol 10 (6) ◽  
pp. 475-481 ◽  
Author(s):  
Harushisa Kawasaki ◽  
◽  
Toshimi Shimizu
Keyword(s):  
Gröbner Basis ◽  
Closed Loop ◽  
Linear Independence ◽  
Symbolic Analysis ◽  
Ideal Theory ◽  
Robot Dynamics ◽  
Grobner Basis ◽  
Base Parameters ◽  
Constrained Equations ◽  
Analysis System

We analyzed base parameters for closed-loop robots using robot symbolic analysis based on the completion procedure in polynomial ideal theory. The robot dynamics regressor is represented as a matrix of multivariate polynomials and reduced to normal form based on Buchberger's algorithm by constructing reduced Grobner basis from kinematic constrained equations. The linear independence of the reduced regressor's column vectors is studied by Gauss-Jordan elimination. Original dynamic parameters are regrouped and some eliminated, depending on results. This omits the need to solve kinematic constrained equations explicitly, deriving all base parameters systematically in theory. An example is shown using robot symbolic analysis system: ROSAM 11.

scholarly journals A note on multivariate polynomial division and Gröbner bases

2015 ◽  
Vol 97 (111) ◽  
pp. 43-48
Author(s):  
Aleksandar Lipkovski ◽  
Samira Zeada
Keyword(s):  
Gröbner Basis ◽  
Gröbner Bases ◽  
Heuristic Approach ◽  
Grobner Bases ◽  
Multivariate Polynomials ◽  
Grobner Basis ◽  
Multivariate Polynomial ◽  
Leading Term ◽  
Monomial Ordering

We first present purely combinatorial proofs of two facts: the well-known fact that a monomial ordering must be a well ordering, and the fact (obtained earlier by Buchberger, but not widely known) that the division procedure in the ring of multivariate polynomials over a field terminates even if the division term is not the leading term, but is freely chosen. The latter is then used to introduce a previously unnoted, seemingly weaker, criterion for an ideal basis to be Grobner, and to suggest a new heuristic approach to Grobner basis computations.

scholarly journals Grobner Basis Over Ideals of Multivariate Polynomial Ring: قواعد جروبنر فوق مثاليَّات الحدوديَّات بأكثر من متغير

2019 ◽  
Vol 3 (2) ◽  
Author(s):  
Khaled Suleiman Al-Akla
Keyword(s):  
Polynomial Ring ◽  
Gröbner Basis ◽  
Main Axis ◽  
Structural Approach ◽  
Algebraic Equations ◽  
Grobner Basis ◽  
Multivariate Polynomial ◽  
Multiple Variables ◽  
Definition Of

Grobner basis are considered one of the modern mathematical tools which has become of interest for the researchers in all fields of mathematics. Grobner basis are generally polynomials with multiple variables that has certain characteristics. it's includes two main axis:                                                                            1- The first axis we have presented the definition of Grobner basis and their properties. 2- The second axis we have studied some applications of Grobner basis, and we give some examples about its. The goal of these paper is to identify Grobner basis and some algorithms related to how to find them and talked about the most important applications, including: the issue of belonging and the issue of containment, and to reach our goal to follow the analytical and structural approach, we defined these basis and we have many results, The Grosvenor we obtained is not alone in general and to be single, some additional conditions must be set on these basis, and we conclude that Grobner basis have many applications in the solutions of algebraic equations in more than one transformer and in many fields.

scholarly journals An Algorithm to Compute the H-Bases for Ideals of Subalgebras

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Rabia ◽  
Muhammad Ahsan Binyamin ◽  
Nazia Jabeen ◽  
Adnan Aslam ◽  
Kraidi Anoh Yannick
Keyword(s):  
Polynomial Ring ◽  
Gröbner Basis ◽  
Polynomial Rings ◽  
Grobner Basis ◽  
Finitely Generated ◽  
Computational Algebra ◽  
Important Ingredient ◽  
Multivariate Polynomial

The concept of H-bases, introduced long ago by Macauly, has become an important ingredient for the treatment of various problems in computational algebra. The concept of H-bases is for ideals in polynomial rings, which allows an investigation of multivariate polynomial spaces degree by degree. Similarly, we have the analogue of H-bases for subalgebras, termed as SH-bases. In this paper, we present an analogue of H-bases for finitely generated ideals in a given subalgebra of a polynomial ring, and we call them “HSG-bases.” We present their connection to the SAGBI-Gröbner basis concept, characterize HSG-basis, and show how to construct them.

Resolution of a system of fuzzy polynomial equations using the Gröbner basis

2013 ◽  
Vol 220 ◽  
pp. 541-558 ◽  
Author(s):  
Ali Abbasi Molai ◽  
Abdolali Basiri ◽  
Sajjad Rahmany
Keyword(s):  
Gröbner Basis ◽  
Polynomial Equations ◽  
Grobner Basis
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