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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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Footnotes
1
LEGO and MINDSTORMS are trademarks of the LEGO Group.
 
2
We have computed the comprehensive Gröbner system on Risa/Asir using an implementation by Nabeshima [13].
 
3
As the initial values, \((c_1,s_1,c_2,s_2,c_3,s_3)=(1,1,1,1,1,1)\) were given.
 
4
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.
 
5
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.
 
6
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.
 
7
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.
 
Literature
6.
go back to reference Johansson, F.: mpmath developers: mpmath: a Python library for arbitrary-precision floating-point arithmetic (version 1.1.0) (2018). http://mpmath.org/. Accessed 20 Mar 2020 Johansson, F.: mpmath developers: mpmath: a Python library for arbitrary-precision floating-point arithmetic (version 1.1.0) (2018). http://​mpmath.​org/​. Accessed 20 Mar 2020
10.
go back to reference Maekawa, M., Noro, M., Ohara, K., Takayama, N., Tamura, K.: The design and implementation of OpenXM-RFC 100 and 101. In: Shirayanagi, K., Yokoyama, K. (eds.) Computer Mathematics: Proceedings of the Fifth Asian Symposium on Computer Mathematics (ASCM 2001), pp. 102–111. World Scientific (2001). https://doi.org/10.1142/9789812799661_0011 Maekawa, M., Noro, M., Ohara, K., Takayama, N., Tamura, K.: The design and implementation of OpenXM-RFC 100 and 101. In: Shirayanagi, K., Yokoyama, K. (eds.) Computer Mathematics: Proceedings of the Fifth Asian Symposium on Computer Mathematics (ASCM 2001), pp. 102–111. World Scientific (2001). https://​doi.​org/​10.​1142/​9789812799661_​0011
13.
go back to reference Nabeshima, K.: An implementation of GCS algorithm for Risa/Asir. Private communication (2012) Nabeshima, K.: An implementation of GCS algorithm for Risa/Asir. Private communication (2012)
Metadata
Title
A Design and an Implementation of an Inverse Kinematics Computation in Robotics Using Gröbner Bases
Authors
Noriyuki Horigome
Akira Terui
Masahiko Mikawa
Copyright Year
2020
DOI
https://doi.org/10.1007/978-3-030-52200-1_1

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