[AMR] updated first 4.5 sections

electives/amr/parts/01_kinematics/00_intro.tex

@@ -41,17 +41,17 @@ $\begin{smallmatrix}
         \varphi = \arcsin\left( R_{31} \div \sqrt{1 - R_{31}^2} \right)\\
     \end{smallmatrix}$
 $[\vec{n}]^\times = \begin{bmatrix}
-        0    & -a_3 & a_2  \\
-        a_3  & 0    & -a_1 \\
-        -a_2 & a_1  & 0
+        0    & -n_3 & n_2  \\
+        n_3  & 0    & -n_1 \\
+        -n_2 & n_1  & 0
     \end{bmatrix}$
 
-For pitch axis $= \pm 90\deg$, Gimbal Lock, Jacobian singular.
+For pitch axis $\theta = \pm 90\deg$, \bi{Gimbal Lock}, Jacobian \bi{singular}.
 
 \shortdefinition[Angle-Axis]
-$\vec{\alpha} = \alpha \vec{n}$ ($\vec{n}$ normal); Convert to rot. mat\\
+$\vec{\alpha} = \alpha \vec{n}$ ($\vec{n}$ normal); To \bi{rotation matrix}:\\
 $\mat{R}(\alpha, \vec{n}) = \mat{I}_3 + \sin(\alpha)[\vec{n}]^\times + (1 - \cos(\alpha))([\vec{n}]^\times)^2$\\
-To quat: $\vec{q} = [\vec{n}, \alpha]$
+To \bi{quaternion}: $\vec{q} = [\vec{n}, \alpha]$
 
 
 \shortdefinition[Quaternions] $q = q_w + q_x i + q_y j + q_z k$ with\\
@@ -68,14 +68,14 @@ $\vec{q} = \begin{bmatrix}
         \end{bmatrix}$
 }
 
-\bi{To Rot Mat} {\scriptsize
+\bi{To Rot Mat} {
     $R_{AB} = \mat{I}_3 + 2a(\vec{q}_{AB}) [\vec{v}(\vec{q}_{AB})]^\times + 2([\vec{v}(\vec{q}_{AB})]^\times)^2$
 }
 
 
 \shortdefinition[Transf. M]
 {\scriptsize
-    $\mat{T}_{AC} = \mat{T}_{AB} \mat{T}_{BC}$ R.-Handed typ; Aero: Left-H\\
+    $\mat{T}_{AC} = \mat{T}_{AB} \mat{T}_{BC}$ Right-Handed typically; Aero: Left-H\\
     $\mat{T}_{AB} = \begin{bmatrix}
             \mat{R}_{AB}        & {_A}\vec{t}_B \\
             \mat{0}_{1\times 3} & 1