Lie group integration for constrained generalized Hamiltonian system with dissipation by projection method

2004 ◽  
Vol 25 (4) ◽  
pp. 424-429 ◽  
Author(s):  
Zhang Su-ying ◽  
Deng Zi-chen
Keyword(s):  
Hamiltonian System ◽  
Projection Method ◽  
Lie Group ◽  
Group Integration ◽  
Generalized Hamiltonian System ◽  
Lie Group Integration

Singularity-Free Lie Group Integration and Geometrically Consistent Evaluation of Multibody System Models described in terms of Standard Absolute Coordinates

2021 ◽  
Author(s):  
Andreas Mueller
Keyword(s):  
Rigid Body ◽  
Lie Group ◽  
Time Integration ◽  
Integration Methods ◽  
Product Group ◽  
Local Coordinates ◽  
Group Integration ◽  
The Absolute ◽  
Lie Group Integration ◽  
Absolute Coordinates

Abstract A classical approach to the MBS modeling is to use absolute coordinates, i.e. a set of (possibly redundant) coordinates that describe the absolute position and orientation of the individual bodies w.r.t. to an inertial frame (IFR). A well-known problem for the time integration of the equations of motion (EOM) is the lack of a singularity-free parameterization of spatial motions, which is usually tackled by using unit quaternions. Lie group integration methods were proposed as alternative approach to the singularity-free time integration. Lie group integration methods, operating directly on the configuration space Lie group, are incompatible with standard formulations of the EOM, and cannot be implemented in existing MBS simulation codes without a major restructuring. A framework for interfacing Lie group integrators to standard EOM formulations is presented in this paper. It allows describing MBS in terms of various absolute coordinates and at the same using Lie group integration schemes. The direct product group SO(3)xR3; and the semidirect product group SE(3) are use for representing rigid body motions. The key element of this method is the local-global transitions (LGT) transition map, which facilitates the update of (global) absolute coordinates in terms of the (local) coordinates on the Lie group. This LGT map is specific to the absolute coordinates, the local coordinates on the Lie group, and the Lie group used to represent rigid body configurations. This embedding of Lie group integration methods allows for interfacing with standard vector space integration methods.

Singularity-Free Lie Group Integration of Multibody System Models Described in Absolute Coordinates

2021 ◽  
Author(s):  
Andreas Müller
Keyword(s):  
Rigid Body ◽  
Lie Group ◽  
Time Integration ◽  
Integration Scheme ◽  
Integration Methods ◽  
Product Group ◽  
Local Coordinates ◽  
Group Integration ◽  
Lie Group Integration ◽  
Absolute Coordinates

Abstract A classical approach to the modeling of multibody systems (MBS) is to use absolute coordinates, i.e. a set of (redundant) coordinates that describe the absolute position and orientation of the individual bodies w.r.t. to an inertial frame (IFR). A well-known problem for the time integration of the equations of motion (EOM) is the lack of a singularity-free parameterization of spatial motions, which is usually tackled by using unit quaternions. Lie group integration methods were proposed as alternative approach to the singularity-free time integration. Lie group methods are inherently coordinate free and thus incompatible with any absolute coordinate description. In this paper, an integration scheme is proposed that allows describing MBS in terms of arbitrary absolute coordinates and at the same using Lie group integration schemes, which allows for singularity-free time integration. Moreover, the direct product group SO (3) × ℝ3 as well as the semidirect product group SE (3) can be use for representing rigid body motions, which is beneficial for constraint satisfaction. The crucial step of this method, which renders the underlying Lie group integration scheme applicable to EOM in absolute coordinates, is the update of the (global) absolute coordinates in terms of the (local) coordinates on the Lie group by means of a local-global transitions (LGT) transition map. This LGT map depends on the used absolute coordinates and the local coordinates on the Lie group, but also on the Lie group itself used to represent rigid body configurations (respectively the deformation field of flexible bodies), i.e. the geometry of spatial frame motions. The Lie group formulation is thus embedded, which allows interfacing with standard vector space integration methods.

scholarly journals Is There an Optimal Choice of Configuration Space for Lie Group Integration Schemes Applied to Constrained MBS?

2013 ◽  
Author(s):  
Andreas Müller ◽  
Zdravko Terze
Keyword(s):  
Numerical Integration ◽  
Configuration Space ◽  
Lie Group ◽  
Optimal Choice ◽  
Rigid Bodies ◽  
Integration Scheme ◽  
Integration Schemes ◽  
Numerical Integration Scheme ◽  
Group Integration ◽  
Lie Group Integration

Recently various numerical integration schemes have been proposed for numerically simulating the dynamics of constrained multibody systems (MBS) operating. These integration schemes operate directly on the MBS configuration space considered as a Lie group. For discrete spatial mechanical systems there are two Lie group that can be used as configuration space: SE(3) and SO(3) × ℝ3. Since the performance of the numerical integration scheme clearly depends on the underlying configuration space it is important to analyze the effect of using either variant. For constrained MBS a crucial aspect is the constraint satisfaction. In this paper the constraint violation observed for the two variants are investigated. It is concluded that the SE(3) formulation outperforms the SO(3) × ℝ3 formulation if the absolute motions of the rigid bodies, as part of a constrained MBS, belong to a motion subgroup. In all other cases both formulations are equivalent. In the latter cases the SO(3) × ℝ3 formulation should be used since the SE(3) formulation is numerically more complex, however.

Lie-group integration method for constrained multibody systems in state space

2014 ◽  
Vol 34 (3) ◽  
pp. 275-305 ◽  
Author(s):  
Zdravko Terze ◽  
Andreas Müller ◽  
Dario Zlatar
Keyword(s):  
State Space ◽  
Lie Group ◽  
Multibody Systems ◽  
Integration Method ◽  
Constrained Multibody Systems ◽  
Group Integration ◽  
Lie Group Integration

scholarly journals 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

2021 ◽  
Vol 477 (2253) ◽  
pp. 20210303
Author(s):  
Andreas Müller
Keyword(s):  
Closed Form ◽  
Lie Group ◽  
Multibody Systems ◽  
Directional Derivative ◽  
Flexible Multibody Systems ◽  
Cayley Map ◽  
Integration Schemes ◽  
Group Integration ◽  
Flexible Multibody ◽  
Lie Group Integration

The exponential and Cayley maps on SE(3) are the prevailing coordinate maps used in Lie group integration schemes for rigid body and flexible body systems. Such geometric integrators are the Munthe–Kaas and generalized- α schemes, which involve the differential and its directional derivative of the respective coordinate map. Relevant closed form expressions, which were reported over the last two decades, are scattered in the literature, and some are reported without proof. This paper provides a reference summarizing all relevant closed-form relations along with the relevant proofs, including the right-trivialized differential of the exponential and Cayley map and their directional derivatives (resembling the Hessian). The latter gives rise to an implicit generalized- α scheme for rigid/flexible multibody systems in terms of the Cayley map with improved computational efficiency.

scholarly journals Asynchronous variational Lie group integration for geometrically exact beam dynamics

PAMM ◽  
2013 ◽  
Vol 13 (1) ◽  
pp. 45-46 ◽  
Author(s):  
Francois Demoures ◽  
Francois Gay-Balmaz ◽  
Thomas Leitz ◽  
Sigrid Leyendecker ◽  
Sina Ober-Blöbaum ◽  
...  
Keyword(s):  
Lie Group ◽  
Beam Dynamics ◽  
Geometrically Exact Beam ◽  
Group Integration ◽  
Lie Group Integration

A new n-dimensional conservative chaos based on Generalized Hamiltonian System and its’ applications in image encryption

2022 ◽  
Vol 154 ◽  
pp. 111693
Author(s):  
Xilin Liu ◽  
Xiaojun Tong ◽  
Zhu Wang ◽  
Miao Zhang
Keyword(s):  
Hamiltonian System ◽  
Image Encryption ◽  
Generalized Hamiltonian System
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