[AMR] Notes on tasks, improved layouts

electives/amr/parts/01_kinematics/01_forward.tex

@@ -10,7 +10,8 @@ $ \begin{bmatrix}
 }
 Workspace $W$: $\theta_1, \theta_2 \in [-\pi, \pi]$.
 Similar for $n$R sys (more angles, more lengths).
-For 2D move in 3D space, last dim is sum of angles (or equiv). \bi{Jacobian}: see \ref{sec:ms-lin}
+For 2D move in 3D space, last dim is sum of angles (or equiv).
+The Jacobian of this $nD$ position is typically computed ($nD$ pos is a vec).
+\bi{Jacobian}: see \ref{sec:ms-lin}.
 
-\shortdefinition[Singularity] Loss of deg of Freed. $\det(\mat{J}(\vec{\theta})) = 0$
-% TODO: Determinant computation
+\shortdefinition[Singularity] Loss of deg of Freedom $\det(\mat{J}(\vec{\theta})) = 0$

electives/amr/parts/01_kinematics/02_inverse.tex

@@ -2,7 +2,7 @@
 \begin{wrapfigure}[10]{r}{0.35\columnwidth}
     \includegraphics[width=0.35\columnwidth]{assets/inverse-kinematics.png}
 \end{wrapfigure}
-\bi{Option}: Solve Forward Kinematics for angles.\\
+\bi{Option}: Solve FK for angles.\\
 \bi{Better}: Law of cosine with polar coordinates. Compute angle using cosine rule,
 $\theta_1 = \varphi \pm \alpha$, $\theta_2 = \pm(\pi - \beta)$
 (Positive for {\color{ForestGreen} Elbow Down}, Neg. for {\color{red} Elbow Up})
@@ -10,5 +10,5 @@ $\theta_1 = \varphi \pm \alpha$, $\theta_2 = \pm(\pi - \beta)$
 \bi{Extension to 6R}:
 1. Waist: spherical coords (2 sol.)\\
 2. 2 sols from 2R for shoulder + elbow\\
-3. Solve for wrist joints (no influence on pos)
+3. Solve for wrist joints (no infl. on pos)
 

electives/amr/parts/01_kinematics/05_rigid-body-dynamics.tex

@@ -9,3 +9,11 @@
 ${_B} \vec{v}_{CoM}$ vel. of CoM, ${_B}\omega$ rot. speed; both w.r.t. inert. frame
 
     {\scriptsize Legged robots don't need to maintain stability!}
+
+To determine dynamics:
+\bi{(1)} Define control inputs (e.g. wheel speeds),
+\bi{(2)} Assumptions / Simplifications about state \& inputs
+\bi{(3)} List forces, torques, etc needed,
+\bi{(4)} Express as func of states / inp
+\bi{(5)} Reduce to the CoM
+\bi{(6)} Plug into Newton-Euler eq combined with rigid body kinematics

electives/amr/parts/01_kinematics/06_wheeled-robot.tex

@@ -19,6 +19,7 @@ To compute $\vec{c}$ in $\vec{c} \cdot {_B}\vec{v}_{WB} = \omega$
 $c_i = [\sin(\alpha + \beta) \div r, -\cos(\alpha + \beta) \div r, -\cos(\beta) \cdot l \div r]$
 
 
+\newpage
 \bi{Maneuverability}
 \begin{itemize}
     \item Deg. of Mobility: $\delta_m = 3 - $\#constrained directions
@@ -26,7 +27,6 @@ $c_i = [\sin(\alpha + \beta) \div r, -\cos(\alpha + \beta) \div r, -\cos(\beta)
     \item Deg. of Maneuverability: $\delta_M = \delta_m + \delta_s$
 \end{itemize}
 
-\newpage
 \shade{ForestGreen}{Wheel Configurations}
 
 \includegraphics[width=1\columnwidth]{assets/wheel-config.png}
@@ -64,3 +64,10 @@ $\vec{b}_s = {\scriptsize \begin{bmatrix}
 \textit{Turning}: $\Omega = (\omega_l r_l) / R_l\! =\! (\omega_r r_r) / R_r$, $R\! =\! v / \Omega$, $\Delta \theta\! =\! \Omega \Delta t$
 
 \textbf{Discretized}: $\vec{x}_k = \vec{x}_{k - 1} + b_i$ with $i \in \{s, t\}$, respectively
+
+\textbf{Swedish Wheels}: Have 3 DoF, typ. (3 pcs) equally spaced on circle.
+Typ. Param. each: $\alpha, \beta, \gamma$ ($z$, $x$, $y$ axes, oriented along $x$, $z$ up)
+
+\textbf{Wheel rot speed from movement} Typ. comp: $J \cdot \mat{R}(\theta) \cdot \vec{x} = \dot{\varphi}$,
+with $\mat{R}(\theta) \cdot \vec{x}$ transform from base frame to robot frame.
+% TODO: What's the second Jacobian from?? (See 2017 exam, question B.4)