Differentiation-free taylor approximation of finite motion in closed loop kinematics

J.J. de Jong; A. Müller; J. van Dijk; J.L. Herder

doi:10.1007/978-3-319-60867-9_66

Differentiation-free taylor approximation of finite motion in closed loop kinematics

J.J. de Jong^*, A. Müller, J. van Dijk, J.L. Herder

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

1 Citation (Scopus)

125 Downloads (Pure)

Abstract

Higher-order derivatives of kinematic mappings give insight into the motion characteristics of complex mechanisms. Screw theory and its associated Lie group theory have been used to find these derivatives of loop closure equations up to an arbitrary order. However this has not been extended to the higher-order derivatives of finite motion as given by the inverse or forward kinematic model of closed loop mechanisms. In this paper, a recursive algorithm is presented, consisting solely of matrix multiplications, which is capable of giving these higher-order derivatives of kinematic models of closed loop linkages. It depends on a simplified representation of the higher-order derivatives of an open chain. From these higher-order derivatives a Taylor expansion of a finite motion becomes available. The evaluation of this method on a Taylor approximation (up to 5th order) of the inverse kinematic model of a 5-bar mechanism shows a good approximation in a large part of workspace around the evaluation point.

Original language	English
Title of host publication	Computational Kinematics
Subtitle of host publication	Proceedings of the 7th International Workshop on Computational Kinematics, 2017
Editors	Said Zeghloul, Lotfi Romdhane, Med Amine Laribi
Place of Publication	Cham
Publisher	Springer
Pages	577-584
Number of pages	8
ISBN (Electronic)	978-3-319-60867-9
ISBN (Print)	9783319608662
DOIs	https://doi.org/10.1007/978-3-319-60867-9_66
Publication status	Published - 2018
Event	7th IFToMM International Workshop on Computational Kinematics, CK 2017 - Futuroscope, Poitiers, France Duration: 22 May 2017 → 24 May 2017 Conference number: 7 https://iftomm-ck2017.sciencesconf.org/

Publication series

Name	Mechanisms and Machine Science
Publisher	Springer
Volume	50
ISSN (Print)	2211-0984
ISSN (Electronic)	2211-0992

Conference

Conference	7th IFToMM International Workshop on Computational Kinematics, CK 2017
Abbreviated title	CK
Country	France
City	Poitiers
Period	22/05/17 → 24/05/17
Internet address	https://iftomm-ck2017.sciencesconf.org/

Keywords

5-bar mechanism
Higher-order kinematics
Screw theory
Taylor approximation

Access to Document

10.1007/978-3-319-60867-9_66

DeJong2017differentiation_freeAccepted author version, 162 KB

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de Jong, J. J., Müller, A., van Dijk, J., & Herder, J. L. (2018). Differentiation-free taylor approximation of finite motion in closed loop kinematics. In S. Zeghloul, L. Romdhane, & M. Amine Laribi (Eds.), Computational Kinematics: Proceedings of the 7th International Workshop on Computational Kinematics, 2017 (pp. 577-584). (Mechanisms and Machine Science; Vol. 50). Cham: Springer. https://doi.org/10.1007/978-3-319-60867-9_66

de Jong, J.J. ; Müller, A. ; van Dijk, J. ; Herder, J.L. / Differentiation-free taylor approximation of finite motion in closed loop kinematics. Computational Kinematics: Proceedings of the 7th International Workshop on Computational Kinematics, 2017. editor / Said Zeghloul ; Lotfi Romdhane ; Med Amine Laribi. Cham : Springer, 2018. pp. 577-584 (Mechanisms and Machine Science).

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abstract = "Higher-order derivatives of kinematic mappings give insight into the motion characteristics of complex mechanisms. Screw theory and its associated Lie group theory have been used to find these derivatives of loop closure equations up to an arbitrary order. However this has not been extended to the higher-order derivatives of finite motion as given by the inverse or forward kinematic model of closed loop mechanisms. In this paper, a recursive algorithm is presented, consisting solely of matrix multiplications, which is capable of giving these higher-order derivatives of kinematic models of closed loop linkages. It depends on a simplified representation of the higher-order derivatives of an open chain. From these higher-order derivatives a Taylor expansion of a finite motion becomes available. The evaluation of this method on a Taylor approximation (up to 5th order) of the inverse kinematic model of a 5-bar mechanism shows a good approximation in a large part of workspace around the evaluation point.",

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author = "{de Jong}, J.J. and A. M{\"u}ller and {van Dijk}, J. and J.L. Herder",

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de Jong, JJ, Müller, A, van Dijk, J & Herder, JL 2018, Differentiation-free taylor approximation of finite motion in closed loop kinematics. in S Zeghloul, L Romdhane & M Amine Laribi (eds), Computational Kinematics: Proceedings of the 7th International Workshop on Computational Kinematics, 2017. Mechanisms and Machine Science, vol. 50, Springer, Cham, pp. 577-584, 7th IFToMM International Workshop on Computational Kinematics, CK 2017, Poitiers, France, 22/05/17. https://doi.org/10.1007/978-3-319-60867-9_66

Differentiation-free taylor approximation of finite motion in closed loop kinematics. / de Jong, J.J.; Müller, A.; van Dijk, J.; Herder, J.L.

Computational Kinematics: Proceedings of the 7th International Workshop on Computational Kinematics, 2017. ed. / Said Zeghloul; Lotfi Romdhane; Med Amine Laribi. Cham : Springer, 2018. p. 577-584 (Mechanisms and Machine Science; Vol. 50).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

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N2 - Higher-order derivatives of kinematic mappings give insight into the motion characteristics of complex mechanisms. Screw theory and its associated Lie group theory have been used to find these derivatives of loop closure equations up to an arbitrary order. However this has not been extended to the higher-order derivatives of finite motion as given by the inverse or forward kinematic model of closed loop mechanisms. In this paper, a recursive algorithm is presented, consisting solely of matrix multiplications, which is capable of giving these higher-order derivatives of kinematic models of closed loop linkages. It depends on a simplified representation of the higher-order derivatives of an open chain. From these higher-order derivatives a Taylor expansion of a finite motion becomes available. The evaluation of this method on a Taylor approximation (up to 5th order) of the inverse kinematic model of a 5-bar mechanism shows a good approximation in a large part of workspace around the evaluation point.

AB - Higher-order derivatives of kinematic mappings give insight into the motion characteristics of complex mechanisms. Screw theory and its associated Lie group theory have been used to find these derivatives of loop closure equations up to an arbitrary order. However this has not been extended to the higher-order derivatives of finite motion as given by the inverse or forward kinematic model of closed loop mechanisms. In this paper, a recursive algorithm is presented, consisting solely of matrix multiplications, which is capable of giving these higher-order derivatives of kinematic models of closed loop linkages. It depends on a simplified representation of the higher-order derivatives of an open chain. From these higher-order derivatives a Taylor expansion of a finite motion becomes available. The evaluation of this method on a Taylor approximation (up to 5th order) of the inverse kinematic model of a 5-bar mechanism shows a good approximation in a large part of workspace around the evaluation point.

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de Jong JJ, Müller A, van Dijk J, Herder JL. Differentiation-free taylor approximation of finite motion in closed loop kinematics. In Zeghloul S, Romdhane L, Amine Laribi M, editors, Computational Kinematics: Proceedings of the 7th International Workshop on Computational Kinematics, 2017. Cham: Springer. 2018. p. 577-584. (Mechanisms and Machine Science). https://doi.org/10.1007/978-3-319-60867-9_66