Automatic differential kinematics of serial manipulator robots through dual numbers

Dual Numbers are an extension of real numbers known for its capability of performing exact automatic differentiation of one-valued functions theoretically without error approximation. Also, Differential Kinematics of robots involves the computation of the Jacobian, which is a matrix of partial deriv...

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Autores:
Orbegoso Moreno, Luis Antonio
Valverde Ramírez, Edgar David
Tipo de recurso:
Article of journal
Fecha de publicación:
2024
Institución:
Universidad Tecnológica de Bolívar
Repositorio:
Repositorio Institucional UTB
Idioma:
eng
OAI Identifier:
oai:repositorio.utb.edu.co:20.500.12585/13548
Acceso en línea:
https://doi.org/10.32397/tesea.vol5.n2.625
Palabra clave:
Dual numbers
Jacobian calculation
Robotics kinematics
Computational efficiency
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openAccess
License
Luis Antonio Orbegoso Moreno, Edgar David Valverde Ramírez - 2024
id UTB2_34e62c74bbd4a0a8c4ff62dbc7c01c5a
oai_identifier_str oai:repositorio.utb.edu.co:20.500.12585/13548
network_acronym_str UTB2
network_name_str Repositorio Institucional UTB
repository_id_str
dc.title.spa.fl_str_mv Automatic differential kinematics of serial manipulator robots through dual numbers
dc.title.translated.spa.fl_str_mv Automatic differential kinematics of serial manipulator robots through dual numbers
title Automatic differential kinematics of serial manipulator robots through dual numbers
spellingShingle Automatic differential kinematics of serial manipulator robots through dual numbers
Dual numbers
Jacobian calculation
Robotics kinematics
Computational efficiency
title_short Automatic differential kinematics of serial manipulator robots through dual numbers
title_full Automatic differential kinematics of serial manipulator robots through dual numbers
title_fullStr Automatic differential kinematics of serial manipulator robots through dual numbers
title_full_unstemmed Automatic differential kinematics of serial manipulator robots through dual numbers
title_sort Automatic differential kinematics of serial manipulator robots through dual numbers
dc.creator.fl_str_mv Orbegoso Moreno, Luis Antonio
Valverde Ramírez, Edgar David
dc.contributor.author.eng.fl_str_mv Orbegoso Moreno, Luis Antonio
Valverde Ramírez, Edgar David
dc.subject.eng.fl_str_mv Dual numbers
Jacobian calculation
Robotics kinematics
Computational efficiency
topic Dual numbers
Jacobian calculation
Robotics kinematics
Computational efficiency
description Dual Numbers are an extension of real numbers known for its capability of performing exact automatic differentiation of one-valued functions theoretically without error approximation. Also, Differential Kinematics of robots involves the computation of the Jacobian, which is a matrix of partial derivatives of the Forward Kinematic equations with respect to the robot’s joints. Thus, to perform the automatic calculation of the Jacobian matrix, this paper presents an extension of dual numbers composed of a scalar real part and a vector dual part, where the real part represents the angular value of the robot joint, and the dual part represents the direction of the corresponding partial derivative for each joint. The presented method was implemented in Matlab through Object Orientes Programming (OOP), and the results for calculating the Jacobian of the KUKA KR 500 robot model for 1000 random postures were subsequently compared in terms of execution time and Mean Squared Error (MSE) with other conventional methods: the geometric method, the symbolic method, and the finite difference method. The results showed a significant improvement in the computing time for calculating the Jacobian of the robotic model compared to the other methods, as well as a minimum MSE having as reference the numerical value of the symbolic calculations.
publishDate 2024
dc.date.accessioned.none.fl_str_mv 2024-12-24 00:00:00
dc.date.available.none.fl_str_mv 2024-12-24 00:00:00
dc.date.issued.none.fl_str_mv 2024-12-24
dc.type.spa.fl_str_mv Artículo de revista
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dc.type.driver.eng.fl_str_mv info:eu-repo/semantics/article
dc.type.coar.eng.fl_str_mv http://purl.org/coar/resource_type/c_6501
dc.type.local.eng.fl_str_mv Journal article
dc.type.content.eng.fl_str_mv Text
dc.type.version.eng.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.coarversion.eng.fl_str_mv http://purl.org/coar/version/c_970fb48d4fbd8a85
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dc.identifier.url.none.fl_str_mv https://doi.org/10.32397/tesea.vol5.n2.625
dc.identifier.doi.none.fl_str_mv 10.32397/tesea.vol5.n2.625
dc.identifier.eissn.none.fl_str_mv 2745-0120
url https://doi.org/10.32397/tesea.vol5.n2.625
identifier_str_mv 10.32397/tesea.vol5.n2.625
2745-0120
dc.language.iso.eng.fl_str_mv eng
language eng
dc.relation.references.eng.fl_str_mv V. Brodsky and M. Shoham. Dual numbers representation of rigid body dynamics. Mechanism and Machine Theory, 34(5):693–718, 1999. [2] Clifford. Preliminary sketch of biquaternions. Proceedings of The London Mathematical Society, pages 381–395, 1871. [3] Neil T Dantam. Robust and efficient forward, differential, and inverse kinematics using dual quaternions. The International Journal of Robotics Research, 40(10-11):1087–1105, 2021. [4] You-Liang Gu and J. Luh. Dual-number transformation and its applications to robotics. IEEE Journal on Robotics and Automation, 3(6):615–623, 1987. [5] Dan Piponi. Automatic differentiation, c++ templates, and photogrammetry. Journal of Graphics Tools, 9, 01 2004. [6] Hannes Sommer, Cédric Pradalier, and Paul Timothy Furgale. Automatic differentiation on differentiable manifolds as a tool for robotics. In International Symposium of Robotics Research, 2013. [7] Avraham Cohen and Moshe Shoham. Application of hyper-dual numbers to multi-body kinematics. Journal of Mechanisms and Robotics, 8, 05 2015. [8] Avraham Cohen and Moshe Shoham. Application of hyper-dual numbers to rigid bodies equations of motion. Mechanism and Machine Theory, 111:76–84, 2017. [9] David González Sánchez. Dual numbers and automatic differentiation to efficiently compute velocities and accelerations. Acta Applicandae Mathematicae, July 2020. [10] David Eager, Ann-Marie Pendrill, and Nina Reistad. Beyond velocity and acceleration: jerk, snap and higher derivatives. European Journal of Physics, 37(6):065008, oct 2016. [11] A. Espinosa-Romero R. Peón-Escalante and F. Peñuñuri. Higher order kinematic formulas and its numerical computation employing dual numbers. Mechanics Based Design of Structures and Machines, 0(0):1–16, 2023. [12] Jan Brinker, Michael Lorenz, Sami Charaf Eddine, and Burkhard Corves. Analytical derivation and application of the jacobian matrix of parallel kinematic manipulators. 11 2015. [13] Jessica Villalobos, Irma Y. Sanchez, and Fernando Martell. Singularity analysis and complete methods to compute the inverse kinematics for a 6-dof ur/tm-type robot. Robotics, 11(6), 2022. [14] Jesse Haviland and Peter Corke. A systematic approach to computing the manipulator jacobian and hessian using the elementary transform sequence. ArXiv, abs/2010.08696, 2020. [15] Avantsa V.S.S. Somasundar and G. Yedukondalu. Robotic path planning and simulation by jacobian inverse for industrial applications. Procedia Computer Science, 133:338–347, 2018. International Conference on Robotics and Smart Manufacturing (RoSMa2018). [16] W. Kandasamy and Florentin Smarandache. Dual Numbers. 01 2014. [17] Nicolas Behr, Giuseppe Dattoli, Ambra Lattanzi, and Silvia Licciardi. Dual Numbers and Operational Umbral Methods. Axioms, 8(3):77, 7 2019. [18] Philipp Rehner and Gernot Bauer. Application of Generalized (Hyper-) Dual Numbers in Equation of State Modeling. Frontiers in chemical engineering, 3, 10 2021. [19] Bruno Siciliano, Lorenzo Sciavicco, Luigi Villani, and Giuseppe Oriolo. Robotics: Modelling, Planning and Control. Springer Publishing Company, Incorporated, 1st edition, 2008. [20] Balaguer C. Barrientos A., Peñín F. and Aracil R. Fundamentos de Robótica. McGraw Hill, 2007. [21] Soeren Laue. On the Equivalence of Automatic and Symbolic Differentiation. arXiv (Cornell University), 1 2019. [22] Vasily E. Tarasov. Exact Finite-Difference Calculus: Beyond Set of Entire Functions. Mathematics, 12(7):972, 3 2024. [23] Tˇ rešˇ nák Adam. Forces Acting on the Robot during Grinding. ˇ Ceské vysoké uˇ cení technické v Praze, 2017.
dc.relation.ispartofjournal.eng.fl_str_mv Transactions on Energy Systems and Engineering Applications
dc.relation.citationvolume.eng.fl_str_mv 5
dc.relation.citationstartpage.none.fl_str_mv 1
dc.relation.citationendpage.none.fl_str_mv 17
dc.relation.bitstream.none.fl_str_mv https://revistas.utb.edu.co/tesea/article/download/625/422
dc.relation.citationedition.eng.fl_str_mv Núm. 2 , Año 2024 : Transactions on Energy Systems and Engineering Applications
dc.relation.citationissue.eng.fl_str_mv 2
dc.rights.eng.fl_str_mv Luis Antonio Orbegoso Moreno, Edgar David Valverde Ramírez - 2024
dc.rights.uri.eng.fl_str_mv https://creativecommons.org/licenses/by/4.0
dc.rights.accessrights.eng.fl_str_mv info:eu-repo/semantics/openAccess
dc.rights.creativecommons.eng.fl_str_mv This work is licensed under a Creative Commons Attribution 4.0 International License.
dc.rights.coar.eng.fl_str_mv http://purl.org/coar/access_right/c_abf2
rights_invalid_str_mv Luis Antonio Orbegoso Moreno, Edgar David Valverde Ramírez - 2024
https://creativecommons.org/licenses/by/4.0
This work is licensed under a Creative Commons Attribution 4.0 International License.
http://purl.org/coar/access_right/c_abf2
eu_rights_str_mv openAccess
dc.format.mimetype.eng.fl_str_mv application/pdf
dc.publisher.eng.fl_str_mv Universidad Tecnológica de Bolívar
dc.source.eng.fl_str_mv https://revistas.utb.edu.co/tesea/article/view/625
institution Universidad Tecnológica de Bolívar
repository.name.fl_str_mv Repositorio Digital Universidad Tecnológica de Bolívar
repository.mail.fl_str_mv bdigital@metabiblioteca.com
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spelling Orbegoso Moreno, Luis AntonioValverde Ramírez, Edgar David2024-12-24 00:00:002024-12-24 00:00:002024-12-24Dual Numbers are an extension of real numbers known for its capability of performing exact automatic differentiation of one-valued functions theoretically without error approximation. Also, Differential Kinematics of robots involves the computation of the Jacobian, which is a matrix of partial derivatives of the Forward Kinematic equations with respect to the robot’s joints. Thus, to perform the automatic calculation of the Jacobian matrix, this paper presents an extension of dual numbers composed of a scalar real part and a vector dual part, where the real part represents the angular value of the robot joint, and the dual part represents the direction of the corresponding partial derivative for each joint. The presented method was implemented in Matlab through Object Orientes Programming (OOP), and the results for calculating the Jacobian of the KUKA KR 500 robot model for 1000 random postures were subsequently compared in terms of execution time and Mean Squared Error (MSE) with other conventional methods: the geometric method, the symbolic method, and the finite difference method. The results showed a significant improvement in the computing time for calculating the Jacobian of the robotic model compared to the other methods, as well as a minimum MSE having as reference the numerical value of the symbolic calculations.application/pdfengUniversidad Tecnológica de BolívarLuis Antonio Orbegoso Moreno, Edgar David Valverde Ramírez - 2024https://creativecommons.org/licenses/by/4.0info:eu-repo/semantics/openAccessThis work is licensed under a Creative Commons Attribution 4.0 International License.http://purl.org/coar/access_right/c_abf2https://revistas.utb.edu.co/tesea/article/view/625Dual numbersJacobian calculationRobotics kinematicsComputational efficiencyAutomatic differential kinematics of serial manipulator robots through dual numbersAutomatic differential kinematics of serial manipulator robots through dual numbersArtículo de revistainfo:eu-repo/semantics/articlehttp://purl.org/coar/resource_type/c_6501http://purl.org/coar/resource_type/c_2df8fbb1Journal articleTextinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/version/c_970fb48d4fbd8a85https://doi.org/10.32397/tesea.vol5.n2.62510.32397/tesea.vol5.n2.6252745-0120V. Brodsky and M. Shoham. Dual numbers representation of rigid body dynamics. Mechanism and Machine Theory, 34(5):693–718, 1999. [2] Clifford. Preliminary sketch of biquaternions. Proceedings of The London Mathematical Society, pages 381–395, 1871. [3] Neil T Dantam. Robust and efficient forward, differential, and inverse kinematics using dual quaternions. The International Journal of Robotics Research, 40(10-11):1087–1105, 2021. [4] You-Liang Gu and J. Luh. Dual-number transformation and its applications to robotics. IEEE Journal on Robotics and Automation, 3(6):615–623, 1987. [5] Dan Piponi. Automatic differentiation, c++ templates, and photogrammetry. Journal of Graphics Tools, 9, 01 2004. [6] Hannes Sommer, Cédric Pradalier, and Paul Timothy Furgale. Automatic differentiation on differentiable manifolds as a tool for robotics. In International Symposium of Robotics Research, 2013. [7] Avraham Cohen and Moshe Shoham. Application of hyper-dual numbers to multi-body kinematics. Journal of Mechanisms and Robotics, 8, 05 2015. [8] Avraham Cohen and Moshe Shoham. Application of hyper-dual numbers to rigid bodies equations of motion. Mechanism and Machine Theory, 111:76–84, 2017. [9] David González Sánchez. Dual numbers and automatic differentiation to efficiently compute velocities and accelerations. Acta Applicandae Mathematicae, July 2020. [10] David Eager, Ann-Marie Pendrill, and Nina Reistad. Beyond velocity and acceleration: jerk, snap and higher derivatives. European Journal of Physics, 37(6):065008, oct 2016. [11] A. Espinosa-Romero R. Peón-Escalante and F. Peñuñuri. Higher order kinematic formulas and its numerical computation employing dual numbers. Mechanics Based Design of Structures and Machines, 0(0):1–16, 2023. [12] Jan Brinker, Michael Lorenz, Sami Charaf Eddine, and Burkhard Corves. Analytical derivation and application of the jacobian matrix of parallel kinematic manipulators. 11 2015. [13] Jessica Villalobos, Irma Y. Sanchez, and Fernando Martell. Singularity analysis and complete methods to compute the inverse kinematics for a 6-dof ur/tm-type robot. Robotics, 11(6), 2022. [14] Jesse Haviland and Peter Corke. A systematic approach to computing the manipulator jacobian and hessian using the elementary transform sequence. ArXiv, abs/2010.08696, 2020. [15] Avantsa V.S.S. Somasundar and G. Yedukondalu. Robotic path planning and simulation by jacobian inverse for industrial applications. Procedia Computer Science, 133:338–347, 2018. International Conference on Robotics and Smart Manufacturing (RoSMa2018). [16] W. Kandasamy and Florentin Smarandache. Dual Numbers. 01 2014. [17] Nicolas Behr, Giuseppe Dattoli, Ambra Lattanzi, and Silvia Licciardi. Dual Numbers and Operational Umbral Methods. Axioms, 8(3):77, 7 2019. [18] Philipp Rehner and Gernot Bauer. Application of Generalized (Hyper-) Dual Numbers in Equation of State Modeling. Frontiers in chemical engineering, 3, 10 2021. [19] Bruno Siciliano, Lorenzo Sciavicco, Luigi Villani, and Giuseppe Oriolo. Robotics: Modelling, Planning and Control. Springer Publishing Company, Incorporated, 1st edition, 2008. [20] Balaguer C. Barrientos A., Peñín F. and Aracil R. Fundamentos de Robótica. McGraw Hill, 2007. [21] Soeren Laue. On the Equivalence of Automatic and Symbolic Differentiation. arXiv (Cornell University), 1 2019. [22] Vasily E. Tarasov. Exact Finite-Difference Calculus: Beyond Set of Entire Functions. Mathematics, 12(7):972, 3 2024. [23] Tˇ rešˇ nák Adam. Forces Acting on the Robot during Grinding. ˇ Ceské vysoké uˇ cení technické v Praze, 2017.Transactions on Energy Systems and Engineering Applications5117https://revistas.utb.edu.co/tesea/article/download/625/422Núm. 2 , Año 2024 : Transactions on Energy Systems and Engineering Applications220.500.12585/13548oai:repositorio.utb.edu.co:20.500.12585/135482025-09-16 09:15:15.096https://creativecommons.org/licenses/by/4.0Luis Antonio Orbegoso Moreno, Edgar David Valverde Ramírez - 2024metadata.onlyhttps://repositorio.utb.edu.coRepositorio Digital Universidad Tecnológica de Bolívarbdigital@metabiblioteca.com

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