arXiv subjects

Akira Terui

Publications and source records attributed to Akira Terui.

At least 19 recordsLinked to original sources

arXiv2026

Trajectory Planning and Certification for 3-DOF Robot Manipulators Using Real Quantifier Elimination Based on Comprehensive Gr\"obner Systems

We propose an algorithm and its implementation for trajectory planning and certification for 3-DOF robot manipulators. The method uses Real Quantifier Elimination (QE) based on Comprehensive Gr\"obner Systems (CGS), also known as the CGS-QE method. The main advantage of the proposed method is its efficiency in trajectory planning and solution certification. This efficiency comes from the effective use of the CGS. First, for trajectory planning, we solve the inverse kinematics problem at each point along the trajectory via Gr\"obner basis computation. This usually requires recalculating the Gr\"obner basis at every point, which is time-consuming. We avoid this by computing the CGS for a parametric system. Here, the end-effector coordinates are parameters. This approach streamlines the algorithm. Second, for solution certification, the CGS-QE method certifies that an inverse kinematics solution exists at any point along the end-effector's trajectory. Our method also certifies solutions for trajectories composed of line segments and cubic natural splines. The algorithm is implemented within the computer algebra system Risa/Asir.

cs.RO↗

arXiv2025

An Exact Algorithm for Computing the Structure of Jordan Blocks

An efficient method is proposed for computing the structure of Jordan blocks of a matrix of integers or rational numbers by exact computation. We have given a method for computing Jordan chains of a matrix with exact computation. However, for deriving just the structure of Jordan chains, the algorithm can be reduced to increase its efficiency. We propose a modification of the algorithm for that purpose. Results of numerical experiments are given.

cs.SC↗

arXiv2025

Inverse Kinematics for a 6-Degree-of-Freedom Robot Manipulator Using Comprehensive Gr\"obner Systems

We propose an effective method for solving the inverse kinematic problem of a specific model of 6-degree-of-freedom (6-DOF) robot manipulator using computer algebra. It is known that when the rotation axes of three consecutive rotational joints of a manipulator intersect at a single point, the inverse kinematics problem can be divided into determining position and orientation. We extend this method to more general manipulators in which the rotational axes of two consecutive joints intersect. This extension broadens the class of 6-DOF manipulators for which the inverse kinematics problem can be solved, and is expected to enable more efficient solutions. The inverse kinematic problem is solved using the Comprehensive Gr\"obner System (CGS) with joint parameters of the robot appearing as parameters in the coefficients to prevent repetitive calculations of the Gr\"obner bases. The effectiveness of the proposed method is shown by experiments.

cs.RO↗

arXiv2025

An Effective Trajectory Planning and an Optimized Path Planning for a 6-Degree-of-Freedom Robot Manipulator

An effective method for optimizing path planning for a specific model of a 6-degree-of-freedom (6-DOF) robot manipulator is presented as part of the motion planning of the manipulator using computer algebra. We assume that we are given a path in the form of a set of line segments that the end-effector should follow. We also assume that we have a method to solve the inverse kinematic problem of the manipulator at each via-point of the trajectory. The proposed method consists of three steps. First, we calculate the feasible region of the manipulator under a specific configuration of the end-effector. Next, we aim to find a trajectory on the line segments and a sequence of joint configurations the manipulator should follow to move the end-effector along the specified trajectory. Finally, we find the optimal combination of solutions to the inverse kinematic problem at each via-point along the trajectory by reducing the problem to a shortest-path problem of the graph and applying Dijkstra's algorithm. We show the effectiveness of the proposed method by experiments.

cs.RO↗

arXiv2024

An Optimized Path Planning of Manipulator Using Spline Curves and Real Quantifier Elimination Based on Comprehensive Gr\"obner Systems

This paper presents an advanced method for addressing the inverse kinematics and optimal path planning challenges in robot manipulators. The inverse kinematics problem involves determining the joint angles for a given position and orientation of the end-effector. Furthermore, the path planning problem seeks a trajectory between two points. Traditional approaches in computer algebra have utilized Gr\"obner basis computations to solve these problems, offering a global solution but at a high computational cost. To overcome the issue, the present authors have proposed a novel approach that employs the Comprehensive Gr\"obner System (CGS) and CGS-based quantifier elimination (CGS-QE) methods to efficiently solve the inverse kinematics problem and certify the existence of solutions for trajectory planning. This paper extends these methods by incorporating smooth curves via cubic spline interpolation for path planning and optimizing joint configurations using shortest path algorithms to minimize the sum of joint configurations along a trajectory. This approach significantly enhances the manipulator's ability to navigate complex paths and optimize movement sequences.

cs.RO↗

arXiv2023

Inverse kinematics and path planning of manipulator using real quantifier elimination based on Comprehensive Gröbner Systems

Methods for inverse kinematics computation and path planning of a three degree-of-freedom (DOF) manipulator using the algorithm for quantifier elimination based on Comprehensive Gröbner Systems (CGS), called CGS-QE method, are proposed. The first method for solving the inverse kinematics problem employs counting the real roots of a system of polynomial equations to verify the solution's existence. In the second method for trajectory planning of the manipulator, the use of CGS guarantees the existence of an inverse kinematics solution. Moreover, it makes the algorithm more efficient by preventing repeated computation of Gröbner basis. In the third method for path planning of the manipulator, for a path of the motion given as a function of a parameter, the CGS-QE method verifies the whole path's feasibility. Computational examples and an experiment are provided to illustrate the effectiveness of the proposed methods.

cs.RO↗

arXiv2023

A Design and an Implementation of an Inverse Kinematics Computation in Robotics Using Real Quantifier Elimination based on Comprehensive Gröbner Systems

The solution and implementation of the inverse kinematics computation of a three degree-of-freedom (DOF) robot manipulator using an algorithm for real quantifier elimination with Comprehensive Gröbner Systems (CGS) are presented. The method enables us to verify if the given parameters are feasible before solving the inverse kinematics problem. Furthermore, pre-computation of CGS and substituting parameters in the CGS with the given values avoids the repetitive computation of Gröbner basis. Experimental results compared with our previous implementation are shown.

cs.RO↗

arXiv2023

Computer-assisted proofs of "Kariya's theorem" with computer algebra

We demonstrate computer-assisted proofs of "Kariya's theorem," a theorem in elementary geometry, with computer algebra. In the proof of geometry theorem with computer algebra, vertices of geometric figures that are subjects for the proof are expressed as variables. The variables are classified into two classes: arbitrarily given points and the points defined from the former points by constraints. We show proofs of Kariya's theorem with two formulations according to two ways for giving the arbitrary points: one is called "vertex formulation," and the other is called "incenter formulation," with two methods: one is Gröbner basis computation, and the other is Wu's method. Furthermore, we show computer-assisted proofs of the property that the point so-called "Kariya point" is located on the hyperbola so-called "Feuerbach's hyperbola", with two formulations and two methods.

cs.SC↗

arXiv2022

Exact Algorithms for Computing Generalized Eigenspaces of Matrices via Jordan-Krylov Basis

An effective exact method is proposed for computing generalized eigenspaces of a matrix of integers or rational numbers. Keys of our approach are the use of minimal annihilating polynomials and the concept of the Jourdan-Krylov basis. A new method, called Jordan-Krylov elimination, is introduced to design an algorithm for computing Jordan-Krylov basis. The resulting algorithm outputs generalized eigenspaces as a form of Jordan chains. Notably, in the output, components of generalized eigenvectors are expressed as polynomials in the associated eigenvalue as a variable.

math.RA↗

arXiv2022

The GPGCD Algorithm with the Bézout Matrix for Multiple Univariate Polynomials

We propose a modification of the GPGCD algorithm, which has been presented in our previous research, for calculating approximate greatest common divisor (GCD) of more than 2 univariate polynomials with real coefficients and a given degree. In transferring the approximate GCD problem to a constrained minimization problem, different from the original GPGCD algorithm for multiple polynomials which uses the Sylvester subresultant matrix, the proposed algorithm uses the Bézout matrix. Experiments show that the proposed algorithm is more efficient than the original GPGCD algorithm for multiple polynomials with maintaining almost the same accuracy for most of the cases.

math.AC↗

arXiv2020

Solving System of Nonlinear Equations with the Genetic Algorithm and Newton's Method

An implementation and an application of the combination of the genetic algorithm and Newton's method for solving a system of nonlinear equations is presented. The method first uses the advantage of the robustness of the genetic algorithm for guessing the rough location of the roots, then it uses the advantage of a good rate of convergence of Newton's method. An effective application of the method for the positioning problem of multiple small rovers proposed for the use in asteroid exploration is shown.

math.NA↗

arXiv2018

Fast Algorithms for Computing Eigenvectors of Matrices via Pseudo Annihilating Polynomials

An efficient algorithm for computing eigenvectors of a matrix of integers by exact computation is proposed. The components of calculated eigenvectors are expressed as polynomials in the eigenvalue to which the eigenvector is associated, as a variable. The algorithm, in principle, utilizes the minimal annihilating polynomials for eliminating redundant calculations. Furthermore, in the actual computation, the algorithm computes candidates of eigenvectors by utilizing pseudo annihilating polynomials and verifies their correctness. The experimental results show that our algorithms have better performance compared to conventional methods.

math.NA↗

arXiv2018

Fast Algorithm for Calculating the Minimal Annihilating Polynomials of Matrices via Pseudo Annihilating Polynomials

Minimal annihilating polynomials are very useful in a wide variety of algorithms in exact linear algebra. A new efficient method is proposed for calculating the minimal annihilating polynomials for all the unit vectors, for a square matrix over a field of characteristic zero. Key ideas of the proposed method are the concept of pseudo annihilating polynomial and the use of binary splitting technique. Efficiency of the resulting algorithms is shown by arithmetic time complexity analysis.

math.AC↗

arXiv2016

GPGCD: An iterative method for calculating approximate GCD of univariate polynomials

We present an iterative algorithm for calculating approximate greatest common divisor (GCD) of univariate polynomials with the real or the complex coefficients. For a given pair of polynomials and a degree, our algorithm finds a pair of polynomials which has a GCD of the given degree and whose coefficients are perturbed from those in the original inputs, making the perturbations as small as possible, along with the GCD. The problem of approximate GCD is transfered to a constrained minimization problem, then solved with the so-called modified Newton method, which is a generalization of the gradient-projection method, by searching the solution iteratively. We demonstrate that, in some test cases, our algorithm calculates approximate GCD with perturbations as small as those calculated by a method based on the structured total least norm (STLN) method and the UVGCD method, while our method runs significantly faster than theirs by approximately up to 30 or 10 times, respectively, compared with their implementation. We also show that our algorithm properly handles some ill-conditioned polynomials which have a GCD with small or large leading coefficient.

math.AC↗

arXiv2010

GPGCD, an Iterative Method for Calculating Approximate GCD of Univariate Polynomials, with the Complex Coefficients

We present an extension of our GPGCD method, an iterative method for calculating approximate greatest common divisor (GCD) of univariate polynomials, to polynomials with the complex coefficients. For a given pair of polynomials and a degree, our algorithm finds a pair of polynomials which has a GCD of the given degree and whose coefficients are perturbed from those in the original inputs, making the perturbations as small as possible, along with the GCD. In our GPGCD method, the problem of approximate GCD is transfered to a constrained minimization problem, then solved with a so-called modified Newton method, which is a generalization of the gradient-projection method, by searching the solution iteratively. While our original method is designed for polynomials with the real coefficients, we extend it to accept polynomials with the complex coefficients in this paper.

math.AC↗

arXiv2010

GPGCD, an Iterative Method for Calculating Approximate GCD, for Multiple Univariate Polynomials

We present an extension of our GPGCD method, an iterative method for calculating approximate greatest common divisor (GCD) of univariate polynomials, to multiple polynomial inputs. For a given pair of polynomials and a degree, our algorithm finds a pair of polynomials which has a GCD of the given degree and whose coefficients are perturbed from those in the original inputs, making the perturbations as small as possible, along with the GCD. In our GPGCD method, the problem of approximate GCD is transferred to a constrained minimization problem, then solved with the so-called modified Newton method, which is a generalization of the gradient-projection method, by searching the solution iteratively. In this paper, we extend our method to accept more than two polynomials with the real coefficients as an input.

math.AC↗

arXiv2008

Subresultants in Recursive Polynomial Remainder Sequence

We introduce concepts of "recursive polynomial remainder sequence (PRS)" and "recursive subresultant," and investigate their properties. In calculating PRS, if there exists the GCD (greatest common divisor) of initial polynomials, we calculate "recursively" with new PRS for the GCD and its derivative, until a constant is derived. We call such a PRS a recursive PRS. We define recursive subresultants to be determinants representing the coefficients in recursive PRS by coefficients of initial polynomials. Finally, we discuss usage of recursive subresultants in approximate algebraic computation, which motivates the present work.

math.AC↗

arXiv2008

Recursive Polynomial Remainder Sequence and the Nested Subresultants

We give two new expressions of subresultants, nested subresultant and reduced nested subresultant, for the recursive polynomial remainder sequence (PRS) which has been introduced by the author. The reduced nested subresultant reduces the size of the subresultant matrix drastically compared with the recursive subresultant proposed by the authors before, hence it is much more useful for investigation of the recursive PRS. Finally, we discuss usage of the reduced nested subresultant in approximate algebraic computation, which motivates the present work.

math.AC↗