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.

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.

A Constraint Stabilization Method for Time Integration of Constrained Multibody Systems in Lie Group Setting

2014 ◽  
Author(s):  
Andreas Müller ◽  
Zdravko Terze
Keyword(s):  
Vector Space ◽  
Lie Group ◽  
Multibody Systems ◽  
Differential Algebraic Equations ◽  
Iteration Step ◽  
Stabilization Method ◽  
Constrained Multibody Systems ◽  
Integration Schemes ◽  
Local Coordinates ◽  
Classical Vector

The stabilization of geometric constraints is vital for an accurate numerical solution of the differential-algebraic equations (DAE) governing the dynamics of constrained multibody systems (MBS). Although this has been a central topic in numerical MBS dynamics using classical vector space formulations, it has not yet been sufficiently addressed when using Lie group formulations. A straightforward approach is to impose constraints directly on the Lie group elements that represent the MBS motion, which requires additional constraints accounting for the invariants of the Lie group. On the other hand, most numerical Lie group integration schemes introduce local coordinates within the integration step, and it is natural to perform the stabilization in terms of these local coordinates. Such a formulation is presented in this paper for index 1 formulation. The stabilization method is applicable to general coordinate mappings (canonical coordinates, Cayley-Rodriguez, Study) on the MBS configuration space Lie group. The stabilization scheme resembles the well-known vectors space projection and pseudo-inverse method consisting in an iterative procedure. A numerical example is presented and it is shown that the Lie group stabilization scheme converges normally within one iteration step, like the scheme in the vector space formulation.

scholarly journals Numerical Flight Vehicle Forward Dynamics With State-Space Lie-Group Integration Scheme

2013 ◽  
Author(s):  
Zdravko Terze ◽  
Milan Vrdoljak ◽  
Dario Zlatar
Keyword(s):  
State Space ◽  
Lie Group ◽  
Rigid Bodies ◽  
Integration Scheme ◽  
Flight Vehicle ◽  
Group Settings ◽  
Integration Schemes ◽  
Angular Velocities ◽  
Algebraic Constraints ◽  
Lie Group Integration

Dynamic simulation procedures of flight vehicle maneuvers need robust and efficient integration methods in order to allow for reliable simulation missions. Derivation of such integration schemes in Lie-group settings is especially efficient since the coordinate-free Lie-group dynamical models operate directly on SO(3) rotational matrices and angular velocities, avoiding local rotation parameters and artificial algebraic constraints as well as kinematical differential equations. In the adopted modeling approach, a state-space of the flight vehicle (modeled as a multi-body system comprising rigid bodies) is modeled as a Lie-group. The numerical algorithm is demonstrated and tested within the framework of the characteristic case study of the aircraft 3D maneuver.

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