Construction of Generalized Coordinates’ Basis Functions in Lagrangian Dynamics of Flat Manipulators

Bagautdinov Ildar Nyrgaiazovich; Pavlov Alexander Ivanovich; Zhuravlev Evgeny Alekseevich; Bogdanov Evgeny Nikolaevich

doi:doi:10.11648/j.acm.20140304.20

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Construction of Generalized Coordinates’ Basis Functions in Lagrangian Dynamics of Flat Manipulators

Bagautdinov Ildar Nyrgaiazovich, Pavlov Alexander Ivanovich, Zhuravlev Evgeny Alekseevich, Bogdanov Evgeny Nikolaevich

Published in Applied and Computational Mathematics (Volume 3, Issue 4)

Received: 13 August 2014 Accepted: 29 August 2014 Published: 20 September 2014

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Abstract

Second order Lagrange equations are used for describing dynamics of planar mechanism with rotation joints. For calculating kinetic energy of the links local coordinates of velocity vectors are used as well as recursive matrix transformations. Kinetic energy quadratic form coefficients are represented by linear combinations of seven independent trigonometric functions of generalized coordinates, i.e. basis functions. A number of these functions are connected to number of links by quadratic dependence. Constant coefficients in expansions in basic functions are determined from linear equation systems, representing kinetic energy of the mechanism in its several nonrecurring configurations with non-zero values for one or two generalized velocities. The resulting system of dynamics differential equations is integrated numerically with Runge-Kutta method in software environment Mathcad. Efficiency of the proposed method of creating and solving dynamic equations is demonstrated by example of numerical solution the direct dynamic problem of three-link mechanism.

Published in	Applied and Computational Mathematics (Volume 3, Issue 4)
DOI	10.11648/j.acm.20140304.20
Page(s)	186-190
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2014. Published by Science Publishing Group

Keywords

Flat Multilink Mechanism, Lagrange Equations, Basis Functions, Direct Dynamic Problem

References

[1]	Belousov I. R. Calculation of the robot manipulator dynamic equations. IAM RAS preprints №45, 2002.- 28p.
[2]	Bosyakov S.M. Kinematic and dynamic modeling of mechanical systems. — Minsk: BSU, 2011.- 260 p.
[3]	Zhuravlev E.A. Application of basic functions for description of manipulator dynamics. // Research. Technologies. Innovations.: collection of articles edited by Ivanov V.A. — Yoshkar-Ola: Mari STU, 2011.- P.107-110.
[4]	Course of theoretical mechanics/ edited by Kolesnickov K.S. — M.: Published by Bauman Moscow STU, 2005.- 736p.
[5]	Walker M. W., Orin D. E., Efficient Dynamic Computer Simulation of Robotic Mechanisms, Trans. ASME, J. Dynamic Systems, Measurement &Control, vol. 104, 1982, pp. 205-211.

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APA Style

Bagautdinov Ildar Nyrgaiazovich, Pavlov Alexander Ivanovich, Zhuravlev Evgeny Alekseevich, Bogdanov Evgeny Nikolaevich. (2014). Construction of Generalized Coordinates’ Basis Functions in Lagrangian Dynamics of Flat Manipulators. Applied and Computational Mathematics, 3(4), 186-190. https://doi.org/10.11648/j.acm.20140304.20

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Bagautdinov Ildar Nyrgaiazovich; Pavlov Alexander Ivanovich; Zhuravlev Evgeny Alekseevich; Bogdanov Evgeny Nikolaevich. Construction of Generalized Coordinates’ Basis Functions in Lagrangian Dynamics of Flat Manipulators. Appl. Comput. Math. 2014, 3(4), 186-190. doi: 10.11648/j.acm.20140304.20

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AMA Style

Bagautdinov Ildar Nyrgaiazovich, Pavlov Alexander Ivanovich, Zhuravlev Evgeny Alekseevich, Bogdanov Evgeny Nikolaevich. Construction of Generalized Coordinates’ Basis Functions in Lagrangian Dynamics of Flat Manipulators. Appl Comput Math. 2014;3(4):186-190. doi: 10.11648/j.acm.20140304.20

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@article{10.11648/j.acm.20140304.20,
  author = {Bagautdinov Ildar Nyrgaiazovich and Pavlov Alexander Ivanovich and Zhuravlev Evgeny Alekseevich and Bogdanov Evgeny Nikolaevich},
  title = {Construction of Generalized Coordinates’ Basis Functions in Lagrangian Dynamics of Flat Manipulators},
  journal = {Applied and Computational Mathematics},
  volume = {3},
  number = {4},
  pages = {186-190},
  doi = {10.11648/j.acm.20140304.20},
  url = {https://doi.org/10.11648/j.acm.20140304.20},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140304.20},
  abstract = {Second order Lagrange equations are used for describing dynamics of planar mechanism with rotation joints. For calculating kinetic energy of the links local coordinates of velocity vectors are used as well as recursive matrix transformations. Kinetic energy quadratic form coefficients are represented by linear combinations of seven independent trigonometric functions of generalized coordinates, i.e. basis functions. A number of these functions are connected to number of links by quadratic dependence. Constant coefficients in expansions in basic functions are determined from linear equation systems, representing kinetic energy of the mechanism in its several nonrecurring configurations with non-zero values for one or two generalized velocities. The resulting system of dynamics differential equations is integrated numerically with Runge-Kutta method in software environment Mathcad. Efficiency of the proposed method of creating and solving dynamic equations is demonstrated by example of numerical solution the direct dynamic problem of three-link mechanism.},
 year = {2014}
}

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TY  - JOUR
T1  - Construction of Generalized Coordinates’ Basis Functions in Lagrangian Dynamics of Flat Manipulators
AU  - Bagautdinov Ildar Nyrgaiazovich
AU  - Pavlov Alexander Ivanovich
AU  - Zhuravlev Evgeny Alekseevich
AU  - Bogdanov Evgeny Nikolaevich
Y1  - 2014/09/20
PY  - 2014
N1  - https://doi.org/10.11648/j.acm.20140304.20
DO  - 10.11648/j.acm.20140304.20
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 186
EP  - 190
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20140304.20
AB  - Second order Lagrange equations are used for describing dynamics of planar mechanism with rotation joints. For calculating kinetic energy of the links local coordinates of velocity vectors are used as well as recursive matrix transformations. Kinetic energy quadratic form coefficients are represented by linear combinations of seven independent trigonometric functions of generalized coordinates, i.e. basis functions. A number of these functions are connected to number of links by quadratic dependence. Constant coefficients in expansions in basic functions are determined from linear equation systems, representing kinetic energy of the mechanism in its several nonrecurring configurations with non-zero values for one or two generalized velocities. The resulting system of dynamics differential equations is integrated numerically with Runge-Kutta method in software environment Mathcad. Efficiency of the proposed method of creating and solving dynamic equations is demonstrated by example of numerical solution the direct dynamic problem of three-link mechanism.
VL  - 3
IS  - 4
ER  -

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Author Information

Bagautdinov Ildar Nyrgaiazovich

Faculty of Mechanics and Machine Building of Volga State University of Technology, Yoshkar-Ola, Russia
Pavlov Alexander Ivanovich

Professor of Syktyvkar State University, Syktyvkar, Russia
Zhuravlev Evgeny Alekseevich

Faculty of Mechanics and Machine Building of Volga State University of Technology, Yoshkar-Ola, Russia
Bogdanov Evgeny Nikolaevich

Faculty of Mechanics and Machine Building of Volga State University of Technology, Yoshkar-Ola, Russia

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Plain Text BibTeX RIS

APA Style

Bagautdinov Ildar Nyrgaiazovich, Pavlov Alexander Ivanovich, Zhuravlev Evgeny Alekseevich, Bogdanov Evgeny Nikolaevich. (2014). Construction of Generalized Coordinates’ Basis Functions in Lagrangian Dynamics of Flat Manipulators. Applied and Computational Mathematics, 3(4), 186-190. https://doi.org/10.11648/j.acm.20140304.20

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ACS Style

Bagautdinov Ildar Nyrgaiazovich; Pavlov Alexander Ivanovich; Zhuravlev Evgeny Alekseevich; Bogdanov Evgeny Nikolaevich. Construction of Generalized Coordinates’ Basis Functions in Lagrangian Dynamics of Flat Manipulators. Appl. Comput. Math. 2014, 3(4), 186-190. doi: 10.11648/j.acm.20140304.20

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AMA Style

Bagautdinov Ildar Nyrgaiazovich, Pavlov Alexander Ivanovich, Zhuravlev Evgeny Alekseevich, Bogdanov Evgeny Nikolaevich. Construction of Generalized Coordinates’ Basis Functions in Lagrangian Dynamics of Flat Manipulators. Appl Comput Math. 2014;3(4):186-190. doi: 10.11648/j.acm.20140304.20

Copy | Download

@article{10.11648/j.acm.20140304.20,
  author = {Bagautdinov Ildar Nyrgaiazovich and Pavlov Alexander Ivanovich and Zhuravlev Evgeny Alekseevich and Bogdanov Evgeny Nikolaevich},
  title = {Construction of Generalized Coordinates’ Basis Functions in Lagrangian Dynamics of Flat Manipulators},
  journal = {Applied and Computational Mathematics},
  volume = {3},
  number = {4},
  pages = {186-190},
  doi = {10.11648/j.acm.20140304.20},
  url = {https://doi.org/10.11648/j.acm.20140304.20},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140304.20},
  abstract = {Second order Lagrange equations are used for describing dynamics of planar mechanism with rotation joints. For calculating kinetic energy of the links local coordinates of velocity vectors are used as well as recursive matrix transformations. Kinetic energy quadratic form coefficients are represented by linear combinations of seven independent trigonometric functions of generalized coordinates, i.e. basis functions. A number of these functions are connected to number of links by quadratic dependence. Constant coefficients in expansions in basic functions are determined from linear equation systems, representing kinetic energy of the mechanism in its several nonrecurring configurations with non-zero values for one or two generalized velocities. The resulting system of dynamics differential equations is integrated numerically with Runge-Kutta method in software environment Mathcad. Efficiency of the proposed method of creating and solving dynamic equations is demonstrated by example of numerical solution the direct dynamic problem of three-link mechanism.},
 year = {2014}
}

Copy | Download

TY  - JOUR
T1  - Construction of Generalized Coordinates’ Basis Functions in Lagrangian Dynamics of Flat Manipulators
AU  - Bagautdinov Ildar Nyrgaiazovich
AU  - Pavlov Alexander Ivanovich
AU  - Zhuravlev Evgeny Alekseevich
AU  - Bogdanov Evgeny Nikolaevich
Y1  - 2014/09/20
PY  - 2014
N1  - https://doi.org/10.11648/j.acm.20140304.20
DO  - 10.11648/j.acm.20140304.20
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 186
EP  - 190
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20140304.20
AB  - Second order Lagrange equations are used for describing dynamics of planar mechanism with rotation joints. For calculating kinetic energy of the links local coordinates of velocity vectors are used as well as recursive matrix transformations. Kinetic energy quadratic form coefficients are represented by linear combinations of seven independent trigonometric functions of generalized coordinates, i.e. basis functions. A number of these functions are connected to number of links by quadratic dependence. Constant coefficients in expansions in basic functions are determined from linear equation systems, representing kinetic energy of the mechanism in its several nonrecurring configurations with non-zero values for one or two generalized velocities. The resulting system of dynamics differential equations is integrated numerically with Runge-Kutta method in software environment Mathcad. Efficiency of the proposed method of creating and solving dynamic equations is demonstrated by example of numerical solution the direct dynamic problem of three-link mechanism.
VL  - 3
IS  - 4
ER  -

Copy | Download