Author: bugman
Date: Tue Sep 29 19:06:50 2015
New Revision: 27899
URL: http://svn.gna.org/viewcvs/relax?rev=27899&view=rev
Log:
Removed some duplicated text in the frame order models chapter of the manual.
This is duplicated from the frame order analysis chapter.
Modified:
branches/frame_order_cleanup/docs/latex/frame_order/models.tex
Modified: branches/frame_order_cleanup/docs/latex/frame_order/models.tex
URL:
http://svn.gna.org/viewcvs/relax/branches/frame_order_cleanup/docs/latex/frame_order/models.tex?rev=27899&r1=27898&r2=27899&view=diff
==============================================================================
--- branches/frame_order_cleanup/docs/latex/frame_order/models.tex
(original)
+++ branches/frame_order_cleanup/docs/latex/frame_order/models.tex Tue
Sep 29 19:06:50 2015
@@ -27,129 +27,6 @@
\end{enumerate}
For a basic introduction to the frame order concept and the modelling of the
tilt and torsion components, see Section~\ref{sect: Frame order modelling}.
-
-
-
-% Rigid body motions for a two domain system.
-%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
-\section{Rigid body motions for a two domain system}
-
-
-
-
-% Ball and socket joint.
-%-----------------------
-
-\subsection{Ball and socket joint}
-
-For a molecule consisting of two rigid bodies with pivoted inter-domain or
inter-segment motions, the most natural mechanical description of the motion
would be that of the spherical joint.
-This is also known as the ball and socket joint.
-The mechanical system consists of a single pivot point and three rotational
degrees of freedom.
-
-
-
-% Tilt and torsion angles.
-%-------------------------
-
-\subsection{Tilt and torsion angles from robotics}
-
-To describe the motional mechanics of a ball and socket joint, the Euler
angle system is a poor representation as the angles do not correspond to the
mechanical modes of motion.
-In the field of robotics, many different orientation parameter sets have
been developed for describing three degree-of-freedom joint systems.
-For the spherical joint description of intra-molecular rigid body motions,
an angle system for describing symmetrical spherical parallel mechanisms
(SPMs), a parallel manipulator, was found to be ideal.
-This is the `tilt-and-torsion' angle system \citep{Huang99,BonevGosselin06}.
-These angles were derived independently a number of times to model human
joint mechanics, originally as the `halfplane-deviation-twist' angles
\citep{Korein85}, and then as the `tilt/twist' angles \citep{Crawford99}.
-
-In the tilt-and-torsion angle system, the rigid body is first tilted by the
angle $\theta$ about the horizontal axis $a$.
-The axis $a$, which lies in the xy-plane, is defined by the azimuthal angle
$\phi$ (the angle is between the rotated z' axis projection onto the xy-plane
and the x-axis).
-The tilt component is hence defined by both $\theta$ and $\phi$.
-Finally the domain is rotated about the z' axis by the torsion angle
$\sigma$.
-The resultant rotation matrix is
-\begin{subequations}
-\begin{align}
- R(\theta, \phi, \sigma)
- &= R_z(\phi)R_y(\theta)R_z(\sigma-\phi) , \\
- &= R_{zyz}(\sigma-\phi, \theta, \phi) , \\
- &= \begin{pmatrix}
- \mathrm{c}_\phi \mathrm{c}_\theta \mathrm{c}_{\sigma-\phi} -
\mathrm{s}_\phi \mathrm{s}_{\sigma-\phi} & -\mathrm{c}_\phi \mathrm{c}_\theta
\mathrm{s}_{\sigma-\phi} - \mathrm{s}_\phi \mathrm{c}_{\sigma-\phi} &
\mathrm{c}_\phi \mathrm{s}_\theta \\
- \mathrm{s}_\phi \mathrm{c}_\theta \mathrm{c}_{\sigma-\phi} +
\mathrm{c}_\phi \mathrm{s}_{\sigma-\phi} & -\mathrm{s}_\phi \mathrm{c}_\theta
\mathrm{s}_{\sigma-\phi} + \mathrm{c}_\phi \mathrm{c}_{\sigma-\phi} &
\mathrm{s}_\phi \mathrm{s}_\theta \\
- -\mathrm{s}_\theta \mathrm{c}_{\sigma-\phi}
& \mathrm{s}_\theta
\mathrm{s}_{\sigma-\phi}
& \mathrm{c}_\theta \\
- \end{pmatrix} , \label{eq: R torsion-tilt}
-\end{align}
-\end{subequations}
-
-where $\mathrm{c}_\eta = \cos(\eta)$ and $\mathrm{s}_\eta = \sin(\eta)$ and
$R_{zyz}$ is the Euler rotation in $zyz$ axis rotation notation where
-\begin{subequations}
-\begin{align}
- \alpha &= \sigma - \phi , \\
- \beta &= \theta , \\
- \gamma &= \phi .
-\end{align}
-\end{subequations}
-
-As $\sigma = \alpha + \gamma$, it can be seen that both these Euler angles
influence the torsion angle, demonstrating the problem with this
parameterisation.
-
-%A distinct advantage of the tilt-torsion angles for describing molecular
domain motions is that singularities are avoided.
-
-
-
-% Torsion angle restriction.
-\paragraph{Modelling torsion}
-
-An advantage of this angle system is that the tilt and torsion components
can be treated separately in the modelling of domain motions.
-The simplest model for the torsion angle would be the restriction
-\begin{equation} \label{eq: torsion angle restriction}
- -\conesmax \le \sigma \le \conesmax .
-\end{equation}
-
-The angle can be completely restricted as $\conesmax = 0$ to create
torsionless models.
-In this case, the tilt and torsion rotation matrix simplifies to
-\begin{subequations}
-\begin{align}
- R(\theta, \phi, 0)
- &= R_z(\phi)R_y(\theta)R_z(-\phi) , \\
- &= R_{zyz}(-\phi, \theta, \phi) , \\
- &= \begin{pmatrix}
- \mathrm{c}^2_\phi \mathrm{c}_\theta + \mathrm{s}^2_\phi
& \mathrm{c}_\phi \mathrm{s}_\phi \mathrm{c}_\theta -
\mathrm{c}_{\phi} \mathrm{s}_\phi & \mathrm{c}_\phi \mathrm{s}_\theta \\
- \mathrm{c}_{\phi} \mathrm{s}_\phi \mathrm{c}_\theta -
\mathrm{c}_\phi \mathrm{s}_{\phi} & \mathrm{s}^2_\phi \mathrm{c}_\theta +
\mathrm{c}^2_\phi & \mathrm{s}_\phi
\mathrm{s}_\theta \\
- -\mathrm{c}_{\phi} \mathrm{s}_\theta
& -\mathrm{s}_{\phi} \mathrm{s}_\theta
& \mathrm{c}_\theta \\
- \end{pmatrix} . \label{eq: R matrix torsionless}
-\end{align}
-\end{subequations}
-
-
-
-% Modelling tilt.
-\paragraph{Modelling tilt}
-
-The tilt angles $\theta$ and $\phi$ are related to spherical angles, hence
the modelling of this component relates to a distribution on the surface of a
sphere.
-At the simplest level, this can be modelled as both isotropic and
anisotropic cones of uniform distribution.
-
-
-
-
-% Model list.
-%------------
-
-\subsection{Model list}
-
-For the modelling of the ordering of the motional frame, the tilt and
torsion angle system will be used together with uniform distributions of
rigid body position.
-For the torsion angle $\conesmax$, this can be modelled as being rigid
($\conesmax = 0$), being a free rotor ($\conesmax = \pi$), or as having a
torsional restriction ($0 < \conesmax < \pi$).
-For the $\theta$ and $\phi$ angles of the tilt component, the rigid body
motion can be modelled as being rigid ($\theta = 0$), as moving in an
isotropic cone, or moving anisotropically in a pseudo-elliptic cone.
-Both single and double modes of motion have been modelled.
-The total list of models so far implemented are:
-\begin{enumerate}
- \item Rigid
- \item Rotor
- \item Free rotor
- \item Isotropic cone
- \item Isotropic cone, torsionless
- \item Isotropic cone, free rotor
- \item Pseudo-ellipse
- \item Pseudo-ellipse, torsionless
- \item Pseudo-ellipse, free rotor
- \item Double rotor
-\end{enumerate}
r27899 - /branches/frame_order_cleanup/docs/latex/frame_order/models.tex