Fix Lev's math

content/irisa/feature-estimation.tex

@@ -4,6 +4,6 @@
   \begin{itemize}
     \item Estimation of the normal to the plane containing the tether in the world frame
     \item Rejection of the noise on the estimate of $\alpha$ by considering the plane vertical~(parallel to the gravity) 
-    \item New set of cable points obtained by projecting this plane into the camera frame for better depth estimation for the image points $\mathbf{p}_{im}=(x,y,1)$: $z_i = -\frac{^cd}{^c\mathbf{n}^T\mathbf{p}_{im}}$ where $^c\mathbf{n}$ is the projected normal and $^cd$ - the distance to the plane from the camera.
+    \item New set of cable points obtained by projecting this plane into the camera frame for better depth estimation for the image points $\vect{p}_{\msubt{im}} = (x, y, 1)$: $z_i = -\frac{\prescript{c}{}{d}}{\transpose{\vect[c]{n}} \vect{p}_{\msubt{im}}}$ where $\vect[c]{n}$ is the projected normal and $\prescript{c}{}{d}$ isthe distance to the plane from the camera
   \end{itemize}
 \end{exampleblock}

content/irisa/interaction-matrix.tex

 %!TEX root=../../draft.tex
 %!TEX file=content/irisa/interaction-matrix.tex
-\begin{block}{Interaction matrix}
+\begin{block}{Interaction Matrix}
   \begin{itemize}
-    \item For controlling both attachment points of the cable $\mathbf{L}_\mathbf{s} = \begin{bmatrix} -\mathbf{M}^{-1} , & \mathbf{M}^{-1} \end{bmatrix}$
-    \item For controlling only one point when the other is static $\mathbf{L}_\mathbf{s}=-\mathbf{M}^{-1}$ .
-    \item $\mathbf{M}
+    \item For controlling both attachment points of the cable $\matr{L}_{\vect{s}} = \begin{bmatrix} -\matr{M}^{-1} , & \matr{M}^{-1} \end{bmatrix}$
+    \item For controlling only one point when the other is static $\matr{L}_{\vect{s}} = -\matr{M}^{-1}$
+    \item $\matr{M}
       =
       \begin{bmatrix}
-      k_1 c\alpha && k_2 c\alpha && -D s\alpha \\
-      k_1 s\alpha && k_2 s\alpha && D c\alpha \\
-      D^2+ t_2 k_1 && D+ t_2 k_2 && 0
-      \end{bmatrix}$ with $k_i = f(\mathbf{s},D,L)$ where $L$ is the cable length needed as the only {\it a priori} knowledge about the cable
-    \item Necessary condition to do the task is keeping the rank 3 of $\mathbf{L}_\mathbf{s}$ and $\mathbf{M}$ 
-    \item The rank is les then 3 if and only if the tether is taut horizontally or vertically which we avoid by never specifying such desired shape. 
+        k_1 \mc{c}_{\alpha}
+          & k_2 \mc{c}_{\alpha}
+          & -D \mc{s}_{\alpha}
+          \\
+        k_1 \mc{s}_{\alpha}
+          & k_2 \mc{s}_{\alpha}
+          & D \mc{c}_{\alpha}
+          \\
+        D^2 + t_2 k_1
+          & D + t_2 k_2
+          & 0
+          \\
+      \end{bmatrix}$ with $k_i = f(\vect{s},D,L)$ where $L$ is the cable length needed as the only \textit{a priori} knowledge about the cable
+    \item Necessary condition to do the task is keeping the rank 3 of $\matr{L}_{\vect{s}}$ and $\matr{M}$
+    \item The rank is less than 3 iff the tether is taut horizontally or vertically which we avoid by never specifying such desired shape.
   \end{itemize}
 \end{block}

content/irisa/methodology.tex

@@ -20,9 +20,9 @@
 
   \begin{itemize}
     \item Choice of parabola curve to model the suspended tether cable shape subject to gravity deformation
-    \item Proposed visual features extracted from RGB-D image: $\mathbf{s} = (a,b,\alpha)$
-    \item Visual error to minimize: $\mathbf{e}=\mathbf{s}-\mathbf{s}^*$
-    \item Control law with exponential decrease of the visual error over time: $\mathbf{v}_m = -\lambda\mathbf{L}_{\mathbf{s}}\mathbf{e}$ with control gain$\lambda>0$ and interaction matrix $\mathbf{L}_{\mathbf{s}}$ relating the velocities of the cable attachment points with the velocities of the visual features
+    \item Proposed visual features extracted from RGB-D image: $\vect{s} = (a, b, \alpha)$
+    \item Visual error to minimize: $\vect{e} = \vect{s} - \vect{s}^{\ast}$
+    \item Control law with exponential decrease of the visual error over time: $\vect{v}_m = -\lambda \vect{L}_{\vect{s}} \vect{e}$ with control gain$\lambda>0$ and interaction matrix $\vect{L}_{\vect{s}}$ relating the velocities of the cable attachment points with the velocities of the visual features
     \item Secondary control task of the quadrotor yaw $\psi$ for observing the tether with an onboard RGB-D sensor while performing the cable shaping task
     \item Quadrotor tilt motion and secondary control task are not deteriorating shape control task while the cable is attached in quadrotor center of thrust otherwise both are considered as an external perturbation for the shape controller~(actually)
   \end{itemize}