https://orcid.org/0000-0002-1500-3584
Abstract
In this paper, we explore a geometric algebra-based approach to model the dynamics of a geometrically exact beam with large deflections. The configuration space of the beam is parametrized with geometric algebra , which indicates the rotation and translation fields are interpolated in a coupled manner. We derive the dynamic equations by applying the Lagrangian method and employing a full discretization approach. This methodology is well-suited for modeling the large deflection deformations of the beam while preserving the symplectic structure in long-term simulations. Finally, we validate the proposed approach through some numerical examples, demonstrating its computational efficiency compared to matrix-based methods.
Issue Section:
Research Papers
References
1.
Simo
, J. C.
, 1985
, “A Finite Strain Beam Formulation. the Three-Dimensional Dynamic Problem. Part I
,” Comput. Methods Appl. Mech. Eng.
, 49
(1
), pp. 55
–70
.10.1016/0045-7825(85)90050-72.
Simo
, J. C.
, and Vu-Quoc
, L.
, 1986
, “On the Dynamics of Flexible Beams Under Large Overall Motions—The Plane Case: Part II
,” ASME J. Appl. Mech.
, 53
(4
), pp. 855
–863
. 1210.1115/1.31718713.
Simo
, J. C.
, and Vu-Quoc
, L.
, 1986
, “On the Dynamics of Flexible Beams Under Large Overall Motions—The Plane Case: Part I
,” ASME J. Appl. Mech.
, 53
(4
), pp. 849
–854
.10.1115/1.31718704.
Antman
, S. S.
, 2013
, Nonlinear Problems of Elasticity
, Springer
, New York
.5.
Rucker
, D. C.
, and Webster
, R. J.
III, 2011
, “Statics and Dynamics of Continuum Robots With General Tendon Routing and External Loading
,” IEEE Trans. Rob.
, 27
(6
), pp. 1033
–1044
.10.1109/TRO.2011.21604696.
Trivedi
, D.
, Lotfi
, A.
, and Rahn
, C. D.
, 2008
, “Geometrically Exact Models for Soft Robotic Manipulators
,” IEEE Trans. Rob.
, 24
(4
), pp. 773
–780
.10.1109/TRO.2008.9249237.
Tummers
, M.
, Lebastard
, V.
, Boyer
, F.
, Troccaz
, J.
, Rosa
, B.
, and Chikhaoui
, M. T.
, 2023
, “Cosserat Rod Modeling of Continuum Robots From Newtonian and Lagrangian Perspectives
,” IEEE Trans. Rob.
, 39
(3
), pp. 2360
–2378
.10.1109/TRO.2023.32381718.
Rucker
, D. C.
, Jones
, B. A.
, and Webster
, R. J.
, III, 2010
, “A Geometrically Exact Model for Externally Loaded Concentric-Tube Continuum Robots
,” IEEE Trans. Rob.
, 26
(5
), pp. 769
–780
.10.1109/TRO.2010.20625709.
Till
, J.
, Aloi
, V.
, and Rucker
, C.
, 2019
, “Real-Time Dynamics of Soft and Continuum Robots Based on Cosserat Rod Models
,” Int. J. Rob. Res.
, 38
(6
), pp. 723
–746
.10.1177/027836491984226910.
Janabi-Sharifi
, F.
, Jalali
, A.
, and Walker
, I. D.
, 2021
, “Cosserat Rod-Based Dynamic Modeling of Tendon-Driven Continuum Robots: A Tutorial
,” IEEE Access
, 9
, pp. 68703
–68719
.10.1109/ACCESS.2021.307718611.
Sonneville
, V.
, Cardona
, A.
, and Brüls
, O.
, 2014
, “Geometrically Exact Beam Finite Element Formulated on the Special Euclidean Group SE (3)
,” Comput. Methods Appl. Mech. Eng.
, 268
, pp. 451
–474
.10.1016/j.cma.2013.10.00812.
Demoures
, F.
, Gay-Balmaz
, F.
, Leyendecker
, S.
, Ober-Blöbaum
, S.
, Ratiu
, T. S.
, and Weinand
, Y.
, 2015
, “Discrete Variational Lie Group Formulation of Geometrically Exact Beam Dynamics
,” Numer. Math.
, 130
(1
), pp. 73
–123
.10.1007/s00211-014-0659-413.
Chadha
, M.
, and Todd
, M. D.
, 2017
, “An Introductory Treatise on Reduced Balance Laws of Cosserat Beams
,” Int. J. Solids Struct.
, 126-127
, pp. 54
–73
.10.1016/j.ijsolstr.2017.07.02814.
Boyer
, F.
, Lebastard
, V.
, Candelier
, F.
, Renda
, F.
, and Alamir
, M.
, 2023
, “Statics and Dynamics of Continuum Robots Based on Cosserat Rods and Optimal Control Theories
,” IEEE Trans. Rob.
, 39
(2
), pp. 1544
–1562
.10.1109/TRO.2022.322611215.
Brüls
, O.
, Cardona
, A.
, and Arnold
, M.
, 2012
, “Lie Group Generalized- Time Integration of Constrained Flexible Multibody Systems
,” Mech. Mach. Theory
, 48
, pp. 121
–137
.10.1016/j.mechmachtheory.2011.07.01716.
Wenger
, T.
, Ober-Blöbaum
, S.
, and Leyendecker
, S.
, 2017
, “Construction and Analysis of Higher Order Variational Integrators for Dynamical Systems With Holonomic Constraints
,” Adv. Comput. Math.
, 43
(5
), pp. 1163
–1195
.10.1007/s10444-017-9520-517.
Hall
, J.
, and Leok
, M.
, 2017
, “Lie Group Spectral Variational Integrators
,” Found. Comput. Math.
, 17
(1
), pp. 199
–257
. feb10.1007/s10208-015-9287-318.
Leitz
, T.
, and Leyendecker
, S.
, 2018
, “Galerkin Lie-Group Variational Integrators Based on Unit Quaternion Interpolation
,” Comput. Methods Appl. Mech. Eng.
, 338
, pp. 333
–361
.10.1016/j.cma.2018.04.02219.
Leitz
, T.
, Sato Martín de Almagro
, R. T.
, and Leyendecker
, S.
, 2021
, “Multisymplectic Galerkin Lie Group Variational Integrators for Geometrically Exact Beam Dynamics Based on Unit Dual Quaternion Interpolation—No Shear Locking
,” Comput. Methods Appl. Mech. Eng.
, 374
, p. 113475
.10.1016/j.cma.2020.11347520.
Chen
, J.
, Huang
, Z.
, and Tian
, Q.
, 2022
, “A Multisymplectic Lie Algebra Variational Integrator for Flexible Multibody Dynamics on the Special Euclidean Group SE (3)
,” Mech. Mach. Theory
, 174
, p. 104918
.10.1016/j.mechmachtheory.2022.10491821.
Ren
, H.
, Fan
, W.
, and Zhu
, W.
, 2018
, “An Accurate and Robust Geometrically Exact Curved Beam Formulation for Multibody Dynamic Analysis
,” ASME J. Vib. Acoust.
, 140
(1
), p. 011012
.10.1115/1.403751322.
Sonneville
, V.
, Brüls
, O.
, and Bauchau
, O. A.
, 2017
, “Interpolation Schemes for Geometrically Exact Beams: A Motion Approach
,” Int. J. Numer. Methods Eng.
, 112
(9
), pp. 1129
–1153
.10.1002/nme.554823.
Hante
, S.
, Tumiotto
, D.
, and Arnold
, M.
, 2022
, “A Lie Group Variational Integration Approach to the Full Discretization of a Constrained Geometrically Exact Cosserat Beam Model
,” Multibody Syst. Dyn.
, 54
(1
), pp. 97
–123
.10.1007/s11044-021-09807-824.
Gunn
, C. G.
, 2017
, “Doing Euclidean Plane Geometry Using Projective Geometric Algebra
,” Adv. Appl. Clifford Algebras
, 27
(2
), pp. 1203
–1232
.10.1007/s00006-016-0731-525.
Gunn
, C.
, 2017
, “Geometric Algebras for Euclidean Geometry
,” Adv. Appl. Clifford Algebras
, 27
(1
), pp. 185
–208
.10.1007/s00006-016-0647-026.
Dorst
, L.
, De Keninck
, S.
, 2020
, “A Guided Tour to the Plane-Based Geometric Algebra PGA
,” University of Amsterdam, Amsterdam, The Netherlands, accessed Mar. 14, 2024, https://bivector. net/PGA4CS. html27.
Hestenes
, D.
, and Sobczyk
, G.
, 1984
, Clifford Algebra to Geometric Calculus
, Springer
, Dordrecht, Netherlands
.28.
Sun
, G.
, and Ding
, Y.
, 2023
, “High-Order Inverse Dynamics of Serial Robots Based on Projective Geometric Algebra
,” Multibody Syst. Dyn.
, 59
(3
), pp. 337
–362
.10.1007/s11044-023-09915-729.
Kanatani
, K.
, 2015
, Understanding Geometric Algebra: Hamilton, Grassmann, and Clifford for Computer Vision and Graphics
, 1st ed., A K Peters/CRC Press
, New York
.30.
Müller
, A.
, 2017
, “Coordinate Mappings for Rigid Body Motions
,” ASME J. Comput. Nonlinear Dyn.
, 12
(2
), p. 021010
.10.1115/1.403473031.
Marsden
, J. E.
, and West
, M.
, 2001
, “Discrete Mechanics and Variational Integrators
,” Acta Numer.
, 10
, pp. 357
–514
.10.1017/S096249290100006X32.
Demoures
, F.
, Gay-Balmaz
, F.
, Kobilarov
, M.
, and Ratiu
, T. S.
, 2014
, “Multisymplectic Lie Group Variational Integrator for a Geometrically Exact Beam in r3
,” Commun. Nonlinear Sci. Numer. Simul.
, 19
(10
), pp. 3492
–3512
.10.1016/j.cnsns.2014.02.03233.
Crisfield
, M. A.
, and Jelenić
, G.
, 1999
, “Objectivity of Strain Measures in the Geometrically Exact Three-Dimensional Beam Theory and Its Finite-Element Implementation
,” Proc. R. Soc. A Math. Phys. Eng. Sci.
, 455(1983), pp. 1125
–1147
.10.1098/rspa.1999.035234.
Sauer
, T.
, 2014
, Numerical Analysis
, 2nd ed., Pearson Education
, Edinburgh
.35.
Broyden
, C. G.
, 1965
, “A Class of Methods for Solving Nonlinear Simultaneous Equations
,” Math. Comput.
, 19
(92
), pp. 577
–593
.10.1090/S0025-5718-1965-0198670-636.
Brüls
, O.
, and Cardona
, A.
, 2010
, “On the Use of Lie Group Time Integrators in Multibody Dynamics
,” ASME J. Comput. Nonlinear Dyn.
, 5
(3
), p. 031002
.10.1115/1.400137037.
Müller
, A.
, 2018
, “Screw and Lie Group Theory in Multibody Dynamics: Recursive Algorithms and Equations of Motion of Tree-Topology Systems
,” Multibody Syst. Dyn.
, 42
(2
), pp. 219
–248
.10.1007/s11044-017-9583-638.
Müller
, A.
, 2021
, “Review of the Exponential and Cayley Map on SE (3) as Relevant for Lie Group Integration of the Generalized Poisson Equation and Flexible Multibody Systems
,” Proc. R. Soc. A
, 477
(2253
), p. 20210303
.10.1098/rspa.2021.030339.
Iserles
, A.
, 1984
, “Solving Linear Ordinary Differential Equations by Exponentials of Iterated Commutators
,” Numer. Math.
, 45
(2
), pp. 183
–199
.10.1007/BF0138946440.
Bullo
, F.
, and Murray
, R. M.
, 1995
, “Proportional Derivative (PD) Control on the Euclidean Group
,” California Institute of Technology
, California
, Report No. Report No. Caltech/CDS 95
–010
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