Coordinate Mappings for Rigid Body Motions

2016 ◽  
Vol 12 (2) ◽  
Author(s):  
Andreas Müller
Keyword(s):  
Rigid Body ◽  
Lie Group ◽  
Equations Of Motion ◽  
Time Integration ◽  
Exponential Mapping ◽  
Group Setting ◽  
Dual Quaternions ◽  
Integration Schemes ◽  
Rigid Body Motions ◽  
Lie Group Integration

Geometric methods have become increasingly accepted in computational multibody system (MBS) dynamics. This includes the kinematic and dynamic modeling as well as the time integration of the equations of motion. In particular, the observation that rigid body motions form a Lie group motivated the application of Lie group integration schemes, such as the Munthe-Kaas method. Also established vector space integration schemes tailored for structural and MBS dynamics were adopted to the Lie group setting, such as the generalized α integration method. Common to all is the use of coordinate mappings on the Lie group SE(3) of Euclidean motions. In terms of canonical coordinates (screw coordinates), this is the exponential mapping. Rigid body velocities (twists) are determined by its right-trivialized differential, denoted dexp. These concepts have, however, not yet been discussed in compact and concise form, which is the contribution of this paper with particular focus on the computational aspects. Rigid body motions can also be represented by dual quaternions, that form the Lie group Sp̂(1), and the corresponding dynamics formulations have recently found a renewed attention. The relevant coordinate mappings for dual quaternions are presented and related to the SE(3) representation. This relation gives rise to a novel closed form of the dexp mapping on SE(3). In addition to the canonical parameterization via the exponential mapping, the noncanonical parameterization via the Cayley mapping is presented.

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.

MULTIBODY DYNAMIC MODELING AND FLUTTER ANALYSIS OF A FLEXIBLE SLENDER VEHICLE

2012 ◽  
Vol 12 (06) ◽  
pp. 1250049 ◽  
Author(s):  
A. RASTI ◽  
S. A. FAZELZADEH
Keyword(s):  
Differential Equations ◽  
Rigid Body ◽  
Dynamic Modeling ◽  
Equations Of Motion ◽  
The Body ◽  
Elastic Deformations ◽  
Strip Theory ◽  
Multibody Dynamic ◽  
Flutter Analysis ◽  
Rigid Body Motions

In this paper, multibody dynamic modeling and flutter analysis of a flexible slender vehicle are investigated. The method is a comprehensive procedure based on the hybrid equations of motion in terms of quasi-coordinates. The equations consist of ordinary differential equations for the rigid body motions of the vehicle and partial differential equations for the elastic deformations of the flexible components of the vehicle. These equations are naturally nonlinear, but to avoid high nonlinearity of equations the elastic displacements are assumed to be small so that the equations of motion can be linearized. For the aeroelastic analysis a perturbation approach is used, by which the problem is divided into a nonlinear flight dynamics problem for quasi-rigid flight vehicle and a linear extended aeroelasticity problem for the elastic deformations and perturbations in the rigid body motions. In this manner, the trim values that are obtained from the first problem are used as an input to the second problem. The body of the vehicle is modeled with a uniform free–free beam and the aeroelastic forces are derived from the strip theory. The effect of some crucial geometric and physical parameters and the acting forces on the flutter speed and frequency of the vehicle are investigated.

CONJUGATION IN THE DISPLACEMENT GROUP AND MOBILITY IN MECHANISMS

2009 ◽  
Vol 33 (2) ◽  
pp. 163-174 ◽  
Author(s):  
Jacques M. Hervé
Keyword(s):  
Rigid Body ◽  
Lie Group ◽  
Algebraic Structure ◽  
Frame Of Reference ◽  
Mathematical Tool ◽  
Simple Matter ◽  
Rigid Body Motions ◽  
Displacement Group

The paper deals with the Lie group algebraic structure of the set of Euclidean displacements, which represent rigid-body motions. We begin by looking for a representation of a displacement, which is independent of the choice of a frame of reference. Then, it is a simple matter to prove that displacement subgroups may be invariant by conjugation. This mathematical tool is suitable for solving special problems of mobility in mechanisms.

On the Nonlinear Intrinsic Dynamics of Doubly Curved Shells

1981 ◽  
Vol 48 (4) ◽  
pp. 909-914 ◽  
Author(s):  
A. Libai
Keyword(s):  
Equations Of Motion ◽  
Time Integration ◽  
Constant Load ◽  
Field Equations ◽  
Integration Schemes ◽  
Doubly Curved Shells ◽  
External Reference ◽  
Time Integration Schemes ◽  
Intrinsic Form ◽  
Doubly Curved

The field equations of motion and compatibility for the nonlinear dynamics of doubly curved shells are recast in an intrinsic form, in terms of the metric and curvature functions of their reference surfaces. For appropriate input, the motion of the shell is described without the need for an external reference coordinate system or the use of vector quantities such as position, velocity, and acceleration. The equations are shown to be readily applicable to time integration schemes. Such cases, as the (spatially) constant load problem and the inextensional dynamics problem, are also considered. The need for further work in the area is emphasized.

Lie group methods for rigid body dynamics and time integration on manifolds

2003 ◽  
Vol 192 (3-4) ◽  
pp. 421-438 ◽  
Author(s):  
Elena Celledoni ◽  
Brynjulf Owren
Keyword(s):  
Rigid Body ◽  
Lie Group ◽  
Time Integration ◽  
Rigid Body Dynamics ◽  
Group Methods ◽  
Body Dynamics

The Use of Vector Techniques in Variational Problems

1986 ◽  
Vol 108 (2) ◽  
pp. 141-145 ◽  
Author(s):  
L. J. Everett ◽  
M. McDermott
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Variational Problems ◽  
Double Pendulum ◽  
Coordinate Transformations ◽  
Elastic Deformations ◽  
Variational Techniques ◽  
Convenient Means ◽  
Moving Coordinate ◽  
Rigid Body Motions

A convenient means for applying vector mathematics to variational problems is presented. The total and relative variations of a vector are defined and results which follow from these definitions are developed and proved. These results are then used to express the variation of a functional using vector techniques rather than the classical scalar or matrix techniques. The simple problems of deriving equations of motion for a rigid body and for a rigid double pendulum are presented as examples of the technique. The key advantages of the method are that (1) it allows the investigator who is familiar and proficient with vector techniques to apply these skills to variational problems and (2) it greatly simplifies the application of variational techniques to problems which include both rigid body motions and elastic deformations. This is accomplished by providing the techniques necessary for computing the variation of a vector defined in a moving coordinate system without using coordinate transformations.

Choice of Riemannian Metrics for Rigid Body Kinematics

1996 ◽  
Author(s):  
Miloš Žefran ◽  
Vijay Kumar ◽  
Christopher Croke
Keyword(s):  
Rigid Body ◽  
Lie Group ◽  
Riemannian Metrics ◽  
Riemannian Metric ◽  
Three Dimensions ◽  
Euclidean Group ◽  
Screw Motion ◽  
Screw Motions ◽  
Rigid Body Motions ◽  
Two Parameter

Abstract The set of spatial rigid body motions forms a Lie group known as the special Euclidean group in three dimensions, SE(3). Chasles’s theorem states that there exists a screw motion between two arbitrary elements of SE(3). In this paper we investigate whether there exist a Riemannian metric whose geodesics are screw motions. We prove that no Riemannian metric with such geodesics exists and we show that the metrics whose geodesics are screw motions form a two-parameter family of semi-Riemannian metrics.

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.

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