Analysis of computational efficiency for the solution of inverse kinematics problem of anthropomorphic robots using Gröbner bases theory

This article presents an analysis of computational efficiency to solve the inverse kinematics problem of anthropomorphic robots. Two approaches are investigated: the first approach uses Paul’s method applied to the matrix obtained by the Denavit–Hartenberg algorithm and the second approach uses Gröbner bases theory. With each approach, the problem of inverse kinematics for an anthropomorphic robot will be solved. When comparing each method, this article will demonstrate that the method using Gröbner bases theory is more computationally efficient.

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A Solution of the Inverse Kinematics Problem for a 7-Degrees-of-Freedom Serial Redundant Manipulator Using Gröbner Bases Theory

Mathematical Problems in Engineering ◽

10.1155/2021/6680687 ◽

2021 ◽

Vol 2021 ◽

pp. 1-14

Author(s):

Sérgio Ricardo Xavier da Silva ◽

Leizer Schnitman ◽

Vitalino Cesca Filho

Keyword(s):

Inverse Kinematics ◽

Degrees Of Freedom ◽

Gröbner Bases ◽

Grobner Bases ◽

Operational Performance ◽

Redundant Manipulators ◽

Redundant Manipulator ◽

All Solutions ◽

Inverse Kinematics Problem ◽

Algebraic Computing

This article presents a solution of the inverse kinematics problem of 7-degrees-of-freedom serial redundant manipulators. A 7-degrees-of-freedom (7-DoF) redundant manipulator can avoid obstacles and thus improve operational performance. However, its inverse kinematics is difficult to solve since it has one more DoF than that necessary for reaching the whole workspace, which causes infinite solutions. In this article, Gröbner bases theory is proposed to solve the inverse kinematics. First, the Denavit–Hartenberg model for the manipulator is established. Second, different joint configurations are obtained using Gröbner bases theory. All solutions are confirmed with the aid of algebraic computing software, confirming that this method is accurate and easy to be implemented.

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Inverse Kinematics of General 4R2P, 3R3P, 4R1C, 2R2C, and 3C Serial Manipulators

22nd Biennial Mechanisms Conference: Robotics, Spatial Mechanisms, and Mechanical Systems ◽

10.1115/detc1992-0207 ◽

1992 ◽

Author(s):

Dilip Kohli ◽

Michael Osvatic

Keyword(s):

Inverse Kinematics ◽

Inverse Kinematic ◽

Serial Manipulators ◽

Linearly Independent ◽

The Matrix ◽

Half Angle ◽

Inverse Kinematics Problem ◽

Joint Variable ◽

General Geometry ◽

Solution Methodology

Abstract This paper presents a solution to the inverse kinematics problem for 4R2P, 3R3P, 4R1C, 2R2C and 3C manipulators of general geometry. The method used to solve these is based on a technique recently presented by the authors for solving the inverse kinematics of general 6R and 5R1P manipulators. In the 6R and 5R1P cases, the method initially starts using 14 linearly independent equations where as for the 4R2P, 3R3P, 4R1C, 2R2C and 3C manipulator only 3, 6, 7 or 10 linearly independent equations are required, depending on the case. Through the use of a linearization and dialytic elimination method all 4R2P, 3R3P, 4R1C, 2R2C and 3C cases are reduced to equating to zero the determinant of a matrix whose elements are linear in the tangent of a half angle of a joint variable. The size of this matrix is (8 × 8) for all 4R2P manipulators, (2 × 2) for all 3R3P and 3C manipulators, (16 × 16) for 4R1C manipulators, (4 × 4) for RCRC and CRCR manipulators and (8 × 8) for the remaining 2R2C manipulators providing 8th, 2nd, 16th, 4th and 8th degree inverse kinematic polynomial respectively. Thus, the determinant equated to zero gives us the characteristic equation of the degree expected. The unique form of the matrix allows us to obtain the solution by solving an eigenvalue problem. Many variations of the 4R2P, 3R3P, 4R1C, 2R2C and 3C manipulators are presented and the solution methodology is illustrated by several numerical examples.

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A Design and an Implementation of an Inverse Kinematics Computation in Robotics Using Gröbner Bases

Lecture Notes in Computer Science - Mathematical Software – ICMS 2020 ◽

10.1007/978-3-030-52200-1_1 ◽

2020 ◽

pp. 3-13

Author(s):

Noriyuki Horigome ◽

Akira Terui ◽

Masahiko Mikawa

Keyword(s):

Inverse Kinematics ◽

Gröbner Bases ◽

Grobner Bases

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A modified parallel F4 algorithm for shared and distributed memory architectures

10.29007/98fh ◽

2018 ◽

Author(s):

Severin Neumann

Keyword(s):

Distributed Memory ◽

Gröbner Bases ◽

Parallel Architectures ◽

Grobner Bases ◽

Communication Overhead ◽

Memory Consumption ◽

The Matrix ◽

Complex Procedure ◽

Speed Up ◽

Memory Architectures

In applications of symbolic computation an often required but complex procedure is the computation of Gröbner bases and hence it is obviousto realize parallel algorithms to compute them. There are parallel flavours of the F4 algorithm using the special structure of the occurring matricesto speed up the reduction. In this paper we start from this and present modifications allowing efficient computations of Gröbner bases on parallel architecturesusing shared as well as distributed memory. To achieve this we concentrate on one objective: reducing the memory consumption and avoiding communication overhead.We remove unrequired steps of the reduction, split the columns of the matrix in blocks for distribution and review the effectiveness of the SIMPLIFY function.Finally we provide benchmarks with up to 256 distributed threads of an implementation which will be available at https://github.com/svrnm/parallelGBC.

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