variables.tex

\chapter{Variables of the QR with load system}\label{ch:variables}
	
\begin{tabularx}{\linewidth}{lll}
		$ \boldsymbol{\eta}_1=\mathbf{x}$&$ =\begin{bmatrix}\!x&\!\!y&\!\!z\end{bmatrix}^T\in \mathbb{R}^3 $ & Position of  the origin of $ \{\mathcal{B}\} $ w.r.t. $ \{\mathcal{I}\} $\\
		$ \boldsymbol{\eta}_2$&$= \begin{bmatrix}\!\psi &\!\!\theta &\!\!\phi\end{bmatrix}^T \in\mathbb{R}^3 $ &  Angles of roll, pitch, yaw.\\
		&&Parametrizes locally the orientation of $ \{\mathcal{B}\}  $ w.r.t. $ \{\mathcal{I}\} $\\	
		$ \frac{d\boldsymbol{\eta}_1}{dt}=\mathbf{v}$&$= \begin{bmatrix}\!\dot{x}&\!\!\dot{y}&\!\!\dot{z}\end{bmatrix}^T\in \mathbb{R}^3$ & Velocity of  the origin of $ \{\mathcal{B}\} $ w.r.t. $ \{\mathcal{I}\} $\\	
		$ \frac{d\boldsymbol{\eta}_2}{dt}={\boldsymbol{\omega}}$&$= \begin{bmatrix}\!\dot{\psi}&\!\!\dot{\theta}&\!\!\dot{\phi}\end{bmatrix}^T\in \mathbb{R}^3$ & Angular velocity of  the origin of $ \{\mathcal{B}\} $ w.r.t. $ \{\mathcal{I}\} $\\			
		$ \boldsymbol{\nu}_1=\mathbf{v}^B$&$= \begin{bmatrix}\!u&\!\! v& \!\!w\end{bmatrix}^T \in\mathbb{R}^3  $ & Linear velocity of the origin of $ \{\mathcal{B}\} $ w.r.t. $ \{\mathcal{I}\} $ in $ \{\mathcal{B}\} $\\
		&&(i.e., body-fixed linear velocity)\\
		$ \boldsymbol{\nu}_2=\boldsymbol{\omega}^B=\mathbf{\Omega}$&$= \begin{bmatrix}\!p &\!\!q&\!\! r\end{bmatrix}^T \in\mathbb{R}^3$ & Angular velocity of $ \{\mathcal{B}\}$ w.r.t. $ \{\mathcal{I}\} $ in $ \{\mathcal{B}\} $\\
		&&(i.e., body-fixed angular velocity)\\
		$ R(\boldsymbol{\eta}_2)=R$&$\in SO(3) $ & The rotation matrix from $ \{ \mathcal{B} \} $ to $ \{ \mathcal{I} \} $		\\		
%		$ v^B $&$ \in\mathbb{R}^3 $&Body velocity in $ \{\mathcal{B}\} $\\
%		$ \omega^B$&$ \in\mathbb{R}^3$&Angular velocity in $ \{\mathcal{B}\} $\\
%		$ {x}_q$&$ =\begin{bmatrix}\!x&\!\!y&\!\!z\end{bmatrix}^T $ & Position of  the origin of $ \{\mathcal{B}\} $ w.r.t. $ \{\mathcal{I}\} $\\
%		$ {v}_q$&$= \begin{bmatrix}\!\dot{x}&\!\!\dot{y}&\!\!\dot{z}\end{bmatrix}^T $ & Velocity of  the origin of $ \{\mathcal{B}\} $ w.r.t. $ \{\mathcal{I}\} $\\
%		$ \eta_2$&$= \begin{bmatrix}\!\phi &\!\!\theta &\!\!\psi\end{bmatrix}^T  $ &  Angles of roll, pitch, yaw. Orientation of $ \{\mathcal{B}\}  $ w.r.t. $ \{\mathcal{I}\} $\\	
%		$ v$&$= \begin{bmatrix}\!u&\!\! v& \!\!w\end{bmatrix}^T   $ & Body-fixed linear velocity of the origin of $ \{\mathcal{B}\} $ w.r.t. $ \{\mathcal{I}\} $\\
%		$ \mathbf{\omega}$&$= \begin{bmatrix}\!p &\!\!q&\!\! r\end{bmatrix}^T$ & Body-fixed angular velocity of $ \{\mathcal{B}\}$ w.r.t. $ \{\mathcal{I}\} $\\
%		$ \Omega$&$  $&Angular velocity in $ \{\mathcal{B}\} $\\

		$ \boldsymbol{\eta} $ &$= \begin{bmatrix}\!x&\!\!y&\!\!z&\!\!\psi&\!\!\theta&\!\!\phi\end{bmatrix}^T $&  Generalized coordinates of QR w.r.t. $ \{\mathcal{I}\} $ \\
		$ \boldsymbol{\nu} $&$= \begin{bmatrix}\!u&\!\!v&\!\!w&\!\!p&\!\!q&\!\!r\end{bmatrix}^T $ & Vector of linear and angular velocities\\
		$ \mathbf{r}$&&Position vector of \acs{cog} of QR in $ \{\mathcal{I}\} $\\
		$ \mathbf{r}_{G}$&$=\begin{bmatrix}\!x_G&\!\!y_G&\!\!z_G\end{bmatrix}^T  \in\mathbb{R}^3  $ & Distance from the origin of $ \{\mathcal{B}\} $ to \acs{cog} of QR\\
		$ \mathbf{p}$ & $ \in S^2 $ & Unit vector from QR to load in $ \{\mathcal{I}\} $\\
		$ \mathbf{x}_L, \mathbf{v}_L $ & $ \in \mathbb{R}^3 $ & Position and velocity vectors of suspended load in $ \{\mathcal{I}\} $\\
		$ \boldsymbol{\omega}_L $ & $ \in \mathbb{R}^3 $ & Angular velocity of the suspended load in $ \{\mathcal{I}\} $\\
		$ m, m_L $&$ \in \mathbb{R}$ & Mass of the QR and load \\
		$ l_L $&$ \in\mathbb{R} $&Length of suspension cable\\
		$ T $&$ \in\mathbb{R} $&Magnitude of tension in cable\\
		
		$ l $ && Distance from \a{cog} to the center of each rotor\\
		$\mathbb{I}$&$\in\mathbb{R}^{3x3}  $    & QR inertia matrix of the QR with respect to $ \{\mathcal{B}\}   $              \\


		$ b, d $ && Thrust factor, drag factor\\
		$ {I}_R$ && Rotor inertia \\
		$ \Omega_i $ && Angular velocity of rotor $ i $ around its axis, $ i=\{1,2,3,4\} $.\\	
		$ f_i $&$ \in\mathbb{R} $&Thrust generated by the $ i $-th rotor along the $ \vec{b}_3 $ axis\\	
		$ f $ &$ =\sum_{i=1}^{4}f_i \in\mathbb{R} $& Total thrust magnitude\\
		$ f^B $&$ =\begin{bmatrix} \!0 &\!\! 0&\!\! f \end{bmatrix}^T $     & External forces acting on QR with respect to $ \{\mathcal{B}\}   $                \\
%		$ \tau^B $&$=\begin{bmatrix} \!\tau_x &\!\! \tau_y&\!\! \tau_z \end{bmatrix}^T $     & External moments acting on QR with respect to $ \{\mathcal{B}\}   $                  \\
		$ \boldsymbol{\tau}=M$&$=\begin{bmatrix} \!\tau_x &\!\! \tau_y&\!\! \tau_z \end{bmatrix}^T $     & Generalized torques vector in $ \{\mathcal{B}\}   $                  \\
		$ \boldsymbol{\tau}_L$&$ =\begin{bmatrix}\!F_H&\!\!T_H\end{bmatrix}^T$ &Vector of forces and torques that the load exerts on QR \\

		$ \{\mathbf{c}_1,\mathbf{c}_2,\mathbf{c}_3\} $     && Unit vectors without frame of reference\\
		$ \{\mathbf{b}_1,\mathbf{b}_2,\mathbf{b}_3\} $ && Unit vectors along the axes of $ \{\mathcal{B}\} $\\
		$ \{\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3\} $ && Unit vectors along the axes of $ \{\mathcal{I}\} $\\

\end{tabularx}