scholarly journals Characterizing isoclinic matrices and the Cayley factorization

2016 ◽  
Vol 230 (18) ◽  
pp. 3267-3273
Author(s):  
Matthew Hunt ◽  
Glen Mullineux ◽  
Robert J Cripps ◽  
Ben Cross
Keyword(s):  
Rigid Body ◽  
Symmetric Matrix ◽  
Dimensional Space ◽  
Geometric Properties ◽  
Underlying Theory ◽  
Scalar Multiple ◽  
Cayley Factorization ◽  
Rigid Body Motions ◽  
Dual Forms ◽  
Skew Symmetric Matrix

Quaternions, particularly the double and dual forms, are important for the representation rotations and more general rigid-body motions. The Cayley factorization allows a real orthogonal 4 × 4 matrix to be expressed as the product of two isoclinic matrices and this is a key part of the underlying theory and a useful tool in applications. An isoclinic matrix is defined in terms of its representation of a rotation in four-dimensional space. This paper looks at characterizing such a matrix as the sum of a skew symmetric matrix and a scalar multiple of the identity whose product with its own transpose is diagonal. This removes the need to deal with its geometric properties and provides a means for showing the existence of the Cayley factorization.

scholarly journals On Symmetric, Orthogonal, and Skew-Symmetric Matrices

1953 ◽  
Vol 10 (1) ◽  
pp. 37-44 ◽  
Author(s):  
P. L. Hsu
Keyword(s):  
Symmetric Matrix ◽  
Dimensional Space ◽  
Diagonal Matrix ◽  
Symmetric Matrices ◽  
Point Function ◽  
Characteristic Roots ◽  
Specific Order ◽  
Set Of Points ◽  
Skew Symmetric Matrix

Introduction and Notation. In this paper all the scalars are real and all matrices are, if not stated to be otherwise, p-rowed square matrices. The diagonal and superdiagonal elements of a symmetric matrix, and the superdiagonal elements of a skew-symmetric matrix, will be called the distinct elements of the respective matrices. Σ will denote both the set of all symmetric matrices and the ½p(p + 1)-dimensional space whose coordinates are the distinct elements arranged in some specific order. K will denote both the set of all skew-symmetric matrices and the ½p(p – 1)-dimensional space whose coordinates are the distinct elements arranged in some specific order. Any sub-set of Σ(K) will mean both the sub-set of symmetric (skew-symmetric) matrices and the set of points of Σ(K). Any point function defined in Σ(K) will be written as a function of a symmetric (skew-symmetric) matrix. Dα will denote the diagonal matrix whose diagonal elements are α1, α2, …, αp. The characteristic roots of a symmetric matrix will be called its roots.

Principles of Transference in Theoretical Kinematics

2015 ◽  
Author(s):  
Jin Seob Kim ◽  
Gregory S. Chirikjian
Keyword(s):  
Rigid Body ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Rigid Body Motions ◽  
Pass Through ◽  
Three Dimensional Space

The concept of symmetrical parameterizations of rigid-body motions was introduced in a paper in the Yang Symposium in 2014 in the context of biomolecular docking applications. These parameters are symmetrical in the sense that they “look the same” in a motion and in its inverse. Here we examine properties of special symmetrical parameterizations of rotations and full rigid-body motions in the plane and three-dimensional space. In particular, it is shown how special kinds of motions can “pass through”, or transfer, from left to right. And conjugations of rotation or homogeneous transforms can transfer to the symmetrical parameters.

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.

scholarly journals A Constraint Solver for Flexible Protein Model

2013 ◽  
Vol 48 ◽  
pp. 953-1000 ◽  
Author(s):  
F. Campeotto ◽  
A. Dal Palù ◽  
A. Dovier ◽  
F. Fioretto ◽  
E. Pontelli
Keyword(s):  
Structure Prediction ◽  
Dimensional Space ◽  
Protein Structures ◽  
Empirical Evaluation ◽  
Geometric Constraints ◽  
Conformational Space ◽  
Loop Modeling ◽  
Geometric Properties ◽  
Constraint Solver ◽  
Multi Body

This paper proposes the formalization and implementation of a novel class of constraints aimed at modeling problems related to placement of multi-body systems in the 3-dimensional space. Each multi-body is a system composed of body elements, connected by joint relationships and constrained by geometric properties. The emphasis of this investigation is the use of multi-body systems to model native conformations of protein structures---where each body represents an entity of the protein (e.g., an amino acid, a small peptide) and the geometric constraints are related to the spatial properties of the composing atoms. The paper explores the use of the proposed class of constraints to support a variety of different structural analysis of proteins, such as loop modeling and structure prediction. The declarative nature of a constraint-based encoding provides elaboration tolerance and the ability to make use of any additional knowledge in the analysis studies. The filtering capabilities of the proposed constraints also allow to control the number of representative solutions that are withdrawn from the conformational space of the protein, by means of criteria driven by uniform distribution sampling principles. In this scenario it is possible to select the desired degree of precision and/or number of solutions. The filtering component automatically excludes configurations that violate the spatial and geometric properties of the composing multi-body system. The paper illustrates the implementation of a constraint solver based on the multi-body perspective and its empirical evaluation on protein structure analysis problems.

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