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2020 | OriginalPaper | Chapter

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

Authors : Noriyuki Horigome, Akira Terui, Masahiko Mikawa

Published in: Mathematical Software – ICMS 2020

Publisher: Springer International Publishing

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Abstract

The solution and a portable implementation of the inverse kinematics computation of a 3 degree-of-freedom (DOF) robot manipulator using Gröbner bases are presented. The main system was written Python with computer algebra system SymPy. Gröbner bases are computed with computer algebra system Risa/Asir, called from Python via OpenXM infrastructure for communicating mathematical software systems. For solving a system of algebraic equations, several solvers (both symbolic and numerical) are used from Python, and their performance has been compared. Experimental results with different solvers for solving a system of algebraic equations are shown.

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LEGO and MINDSTORMS are trademarks of the LEGO Group.

We have computed the comprehensive Gröbner system on Risa/Asir using an implementation by Nabeshima [13].

As the initial values, \((c_1,s_1,c_2,s_2,c_3,s_3)=(1,1,1,1,1,1)\) were given.

The solver may not need a Gröbner basis of lex order as an input, but it might be better to compute beforehand for faster computation.

We have tested the method with other initial values. With the initial values (1, 0, 1, 0, 1, 0), the number of test cases in which approximate roots do not converge was the same as the test cases with initial values (1, 1, 1, 1, 1, 1). With initial values (0, 1, 0, 1, 0, 1), the approximate roots have never converged to the roots.

We have also applied the multivariate numerical solver to the original system of equations with initial values (1, 1, 1, 1, 1, 1), (0, 0, 0, 0, 0, 0), (1, 0, 1, 0, 1, 0) and (0, 1, 0, 1, 0, 1), and found that none of the initial values converge to true roots in all the test cases.

At this time Risa/Asir is invoked by sending the command in the text form, with the waiting time (approximately 1.5 s) for synchronizing output data is set. We expect that this process becomes more efficient by using appropriate API for sending/receiving commands and data.

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Title: A Design and an Implementation of an Inverse Kinematics Computation in Robotics Using Gröbner Bases
Authors: Noriyuki Horigome
Akira Terui
Masahiko Mikawa
Publisher: Springer International Publishing
Book: Mathematical Software – ICMS 2020
Print ISBN: 978-3-030-52199-8

Electronic ISBN: 978-3-030-52200-1

Copyright Year: 2020
DOI: https://doi.org/10.1007/978-3-030-52200-1_1

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