Partial Bi-Invariance of SE(3) Metrics1

2015 ◽  
Vol 15 (1) ◽  
Author(s):  
Gregory S. Chirikjian
Keyword(s):  
Rigid Body ◽  
Reference Frames ◽  
Distance Functions ◽  
Frobenius Norm ◽  
Transformation Matrices ◽  
Invariance Properties ◽  
Full Picture ◽  
Weighted Frobenius Norm ◽  
Rigid Body Motions ◽  
The Difference

In a flurry of articles in the mid to late 1990s, various metrics for the group of rigid-body motions, SE(3), were introduced for measuring distance between any two reference frames or rigid-body motions. During this time, it was shown that one can choose a smooth distance function that is invariant under either all left shifts or all right shifts, but not both. For example, if one defines the distance between two reference frames to be an appropriately weighted Frobenius norm of the difference of the corresponding homogeneous transformation matrices, this will be invariant under left shifts by arbitrary rigid-body motions. However, this is not the full picture—other invariance properties exist. Though the Frobenius norm is not invariant under right shifts by arbitrary rigid-body motions, for an appropriate weighting it is invariant under right shifts by pure rotations. This is also true for metrics based on the Lie-theoretic logarithm. This paper goes further to investigate the full invariance properties of distance functions on SE(3), clarifying the full subsets of motions under which both left and right invariance is possible.

Partial Bi-Invariance of SE(3) Metrics

2014 ◽  
Author(s):  
Gregory S. Chirikjian
Keyword(s):  
Rigid Body ◽  
Reference Frames ◽  
Distance Functions ◽  
Frobenius Norm ◽  
Transformation Matrices ◽  
Invariance Properties ◽  
Full Picture ◽  
Weighted Frobenius Norm ◽  
Rigid Body Motions ◽  
The Difference

In a flurry of articles in the mid to late 1990s, various metrics for the group of rigid-body motions, SE(3), were introduced for measuring distance between any two reference frames or rigid-body motions. During this time it was shown that one can choose a smooth distance function that is invariant under either all left shifts or all right shifts, but not both. For example, if one defines the distance between two reference frames to be an appropriately weighted Frobenius norm of the difference of the corresponding homogeneous transformation matrices, this will be invariant under left shifts by arbitrary rigid-body motions. However, this is not the full picture — other invariance properties exist. Though the Frobenius norm is not invariant under right shifts by arbitrary rigid-body motions, for an appropriate weighting it is invariant under right shifts by pure rotations. This is also true for metrics based on the Lie-theoretic logarithm. This paper goes further to investigate the full invariance properties of distance functions on SE(3), clarifying the full subsets of motions under which both left and right invariance is possible.

scholarly journals The Concept of Multiple Impacts of Renewable Energy Sources: A Critical Review

Energies ◽  
2021 ◽  
Vol 14 (11) ◽  
pp. 3183
Author(s):  
Michaela Makešová ◽  
Michaela Valentová
Keyword(s):  
Renewable Energy ◽  
Energy Policy ◽  
Renewable Energy Sources ◽  
Energy Sources ◽  
The European Union ◽  
Multiple Impacts ◽  
Energy Field ◽  
Full Picture ◽  
The Difference ◽  
Social Environmental

Reaching climate neutrality by 2050 is one of the main long-term objectives of the European Union climate and energy policy, and renewable energy sources (RES) are integral parts of this transition. RES development results in many effects, direct and indirect, linked to each other, societal, local and individual, i.e., “multiple impacts of RES” (MI RES). These effects need to be carefully assessed and evaluated to obtain the full picture of energy field transformation and its context, and enable further development of RES. Nevertheless, the MI RES concept is often presented misleadingly and its scope varies throughout the literature. This paper provides a literature overview of the methodologies of this concept and presents a new concept of MI RES, respecting the difference between effects resulting from the implementation of RES and ultimate multiple impacts. We have summarized the effects into four groups: economic, social, environmental, and technical, which all lead to group of ultimate multiple impacts. Finally, we provide the complex overview of all MI RES and present the framework, which is used to analyze the multiple impacts and effects of RES and to show how the RES development leads and contributes to these impacts and effects. The concept is recommended to be considered in designing a robust energy policy by decision-makers.

Analytical models for the rigid body motions of skew bridges

1987 ◽  
Vol 15 (8) ◽  
pp. 923-944 ◽  
Author(s):  
Emmanuel A. Maragakis ◽  
Paul C. Jennings
Keyword(s):  
Rigid Body ◽  
Analytical Models ◽  
Skew Bridges ◽  
Rigid Body Motions

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.

Eigenanalysis of the Rigid Body Motion Effect on Elastic Systems

1983 ◽  
Vol 105 (4) ◽  
pp. 461-466 ◽  
Author(s):  
A. Maher ◽  
A. L. Schlack
Keyword(s):  
Eigenvalue Problem ◽  
Rigid Body ◽  
Free Vibration Analysis ◽  
Plane Motion ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Computer Calculations ◽  
Motion Effect ◽  
Ritz Procedure ◽  
The Difference

In this paper, the influence of rigid body motion on the behavior of a vibrating elastic system is treated by the development of a difference eigenvalue problem. The maximum possible changes in eigenfrequencies due to removal of constraints are obtained by the employment of the bound approach [1, 2]. As an application to a structural system the Rayleigh-Ritz procedure is employed for constructing the difference eigenvalue problem. Discussion of the use of the method for various types of engineering problems is outlined. An example of a free vibration analysis of a simply supported beam in plane motion with a nonuniform mass and elasticity distribution is solved. A comparison between computer calculations and previously published results is presented.

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