Variational Lie Group Formulation of Geometrically Exact Beam Dynamics: Synchronous and Asynchronous Integration

Leitz, Thomas; Ober‐Blöbaum, Sina; Leyendecker, Sigrid

doi:10.1007/978-3-319-07260-9_8

Cited by 5 publications

(6 citation statements)

References 52 publications

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“…Lie groups are well researched theoretically [36]. They also have been applied in numerical integrations [14], and especially in the description of mechanical systems [4,18,26] and beams in particular [7,8,20,33]. Note that for Lie groups, we will use the notations of Brüls et al [4][5][6].…”

Section: A Lie Group Structured Configuration Spacementioning

confidence: 99%

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A Lie group variational integration approach to the full discretization of a constrained geometrically exact Cosserat beam model

2021

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In this paper, we will consider a geometrically exact Cosserat beam model taking into account the industrial challenges. The beam is represented by a framed curve, which we parametrize in the configuration space $\mathbb{S}^{3}\ltimes \mathbb{R}^{3}$ S 3 ⋉ R 3 with semi-direct product Lie group structure, where $\mathbb{S}^{3}$ S 3 is the set of unit quaternions. Velocities and angular velocities with respect to the body-fixed frame are given as the velocity vector of the configuration. We introduce internal constraints, where the rigid cross sections have to remain perpendicular to the center line to reduce the full Cosserat beam model to a Kirchhoff beam model. We derive the equations of motion by Hamilton’s principle with an augmented Lagrangian. In order to fully discretize the beam model in space and time, we only consider piecewise interpolated configurations in the variational principle. This leads, after approximating the action integral with second order, to the discrete equations of motion. Here, it is notable that we allow the Lagrange multipliers to be discontinuous in time in order to respect the derivatives of the constraint equations, also known as hidden constraints. In the last part, we will test our numerical scheme on two benchmark problems that show that there is no shear locking observable in the discretized beam model and that the errors are observed to decrease with second order with the spatial step size and the time step size.

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Section: A Lie Group Structured Configuration Spacementioning

confidence: 99%

“…In order to do this, we will heavily rely on the toolbox of variational integrators [24]. A similar approach was used in [8,9,20].…”

Section: The Fully Discretized Equations Of Motionmentioning

confidence: 99%

A Lie group variational integration approach to the full discretization of a constrained geometrically exact Cosserat beam model

2021

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“…Also, it allows the usage of different time steps at different points in a given finite element for the geometry of soft manipulators. These properties are investigated in previous work (e.g., [35,38,39]), while the main focus of this paper is the experimental validation of the method on magnetically-actuated soft continuum manipulators.…”

Section: Plos Onementioning

confidence: 99%

“…It should be noted that the evaluation of Lagrangian at midpoints of nodes is employed. Other evaluations of the Lagrangian depending on a different number or combinations of nodes are possible (see [38]).…”

Section: Modelingmentioning

confidence: 99%

Dynamic modeling of soft continuum manipulators using lie group variational integration

et al. 2020

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This paper presents the derivation and experimental validation of algorithms for modeling and estimation of soft continuum manipulators using Lie group variational integration. Existing approaches are generally limited to static and quasi-static analyses, and are not sufficiently validated for dynamic motion. However, in several applications, models need to consider the dynamical behavior of the continuum manipulators. The proposed modeling and estimation formulation is obtained from a discrete variational principle, and therefore grants outstanding conservation properties to the continuum mechanical model. The main contribution of this article is the experimental validation of the dynamic model of soft continuum manipulators, including external torques and forces (e.g., generated by magnetic fields, friction, and the gravity), by carrying out different experiments with metal rods and polymerbased soft rods. To consider dissipative forces in the validation process, distributed estimation filters are proposed. The experimental and numerical tests also illustrate the algorithm's performance on a magnetically-actuated soft continuum manipulator. The model demonstrates good agreement with dynamic experiments in estimating the tip position of a Polydimethylsiloxane (PDMS) rod. The experimental results show an average absolute error and maximum error in tip position estimation of 0.13 mm and 0.58 mm, respectively, for a manipulator length of 60.55 mm.

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“…an inertial frame, and the orientations are parameterized using rotation matrices, elements of the special orthogonal group SO (3). A similar model with an extension to asynchronous integration was presented by Leitz et al [50]. Structurally similar multisymplectic integrators have been derived by Demoures et al [51,52] and recently Chen et al [53] and Carré and Bensoam [54] in the covariant mechanics framework, resulting in integrators conserving a symplectic form both in time and space.…”

Section: Background and Related Literaturementioning

confidence: 94%

Relative-Kinematic Formulation of Geometrically Exact Beam Dynamics based on Lie-Group Variational Integrators

Herrmann,

Kotyczka

2024

Preprint

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Geometrically exact beam models can accurately describe the statics and dynamics of elastic, slender beams undergoing large deformations. This work introduces a variational integrator for such models based on a relative-kinematic formulation, in which positions and velocities of nodes in the discretized beam are expressed relative to the respective preceding nodes in the kinematic chain. The resulting model is especially suited for applications in robotics and control, since it has a minimum number of states and possesses the same structure as rigid mechanical systems, simplifying the combined modeling of rigid-flexible systems. Moreover, this approach allows to specifically select the modeled deformation modes, avoiding numerical issues associated with stiff, high-frequency modes and greatly increasing numerical efficiency. The proposed model is derived fully variational in the discrete mechanics framework and under consistent consideration of the underlying Lie group structure, which translates into beneficial numerical properties. We provide a detailed treatment of the inclusion of dissipation and propose two solver strategies for time integration, including a linear-time solver based on recursive rigid-body dynamics algorithms. The model is validated and analyzed in detailed simulation studies, which underline its efficiency for the simulation of stiff and slender beams. In the case of light dissipation, a Kirchhoff model in our relative-kinematic formulation achieved computation times between 22 and 219 times faster than the compared Simo-Reissner beam model from literature at the same error level.

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Variational Lie Group Formulation of Geometrically Exact Beam Dynamics: Synchronous and Asynchronous Integration

Cited by 5 publications

References 52 publications

A Lie group variational integration approach to the full discretization of a constrained geometrically exact Cosserat beam model

A Lie group variational integration approach to the full discretization of a constrained geometrically exact Cosserat beam model

Dynamic modeling of soft continuum manipulators using lie group variational integration

Relative-Kinematic Formulation of Geometrically Exact Beam Dynamics based on Lie-Group Variational Integrators

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