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[AMR] Some layout updates for smaller font sizes

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janishutz
2026-07-29 11:10:17 +02:00
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6 changed files with 55 additions and 54 deletions
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electives/amr/assets/inverse-kinematics.png
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electives/amr/autonomous-mobile-robots-cheatsheet.pdf
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electives/amr/parts/01_kinematics/00_intro.tex
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@@ -7,25 +7,25 @@ $_{\color{blue}\fbox{W}}\,\vec{t}\,_{\color{red}\fbox{B}} = \, _{\color{blue}\fb
\shortdefinition[State vector] $x_R$: $x$, $v$ of rob in $W$, pos of sensors \shortdefinition[State vector] $x_R$: $x$, $v$ of rob in $W$, pos of sensors
\shortdefinition[Rot. Mat.] \shortdefinition[Rot. Mat.]
$\mat{R}_{z}(\psi)$ (Yaw),
$\mat{R}_y(\theta)$ (Pitch),
$\mat{R}_x(\varphi)$ (Roll)\\
{\scriptsize {\scriptsize
$\mat{R}_{z}(\psi)$ (Yaw), $\begin{bmatrix}
$\mat{R}_y(\theta)$ (Pitch), c(\psi) & -s(\psi) & 0 \\
$\mat{R}_x(\varphi)$ (Roll)\\ s(\psi) & c(\psi) & 0 \\
$\begin{bmatrix} 0 & 0 & 1
c(\psi) & -s(\psi) & 0 \\ \end{bmatrix};
s(\psi) & c(\psi) & 0 \\ \begin{bmatrix}
0 & 0 & 1 c(\theta) & 0 & s(\theta) \\
\end{bmatrix}; 0 & 1 & 0 \\
\begin{bmatrix} -s(\theta) & 0 & c(\theta) \\
c(\theta) & 0 & s(\theta) \\ \end{bmatrix};
0 & 1 & 0 \\ \begin{bmatrix}
-s(\theta) & 0 & c(\theta) \\ 1 & 0 & 0 \\
\end{bmatrix}; 0 & c(\varphi) & -s(\varphi) \\
\begin{bmatrix} 0 & s(\varphi) & c(\varphi)
1 & 0 & 0 \\ \end{bmatrix}$
0 & c(\varphi) & -s(\varphi) \\
0 & s(\varphi) & c(\varphi)
\end{bmatrix}$
} }
\shortremark Application: ${_W} \vec{a} = \mat{R}_{WB} {_B} \vec{a}$ \shortremark Application: ${_W} \vec{a} = \mat{R}_{WB} {_B} \vec{a}$
@@ -36,17 +36,17 @@ $_{\color{blue}\fbox{W}}\,\vec{t}\,_{\color{red}\fbox{B}} = \, _{\color{blue}\fb
\shortdefinition[Euler Ang] (Tait-Brian) $\mat{R}_{EB} = \mat{R}_z(\psi) \cdot \mat{R}_y(\theta) \cdot \mat{R}_x(\varphi)$. \shortdefinition[Euler Ang] (Tait-Brian) $\mat{R}_{EB} = \mat{R}_z(\psi) \cdot \mat{R}_y(\theta) \cdot \mat{R}_x(\varphi)$.
$\begin{smallmatrix} $\begin{smallmatrix}
\psi = \arcsin\left( R_{21} \div \sqrt{1 - R_{31}^2} \right)\\ \psi = \arcsin\left( R_{21} \div \sqrt{1 - R_{31}^2} \right)\\
\theta = \arcsin(-R_{31})\\ \theta = \arcsin(-R_{31})\\
\varphi = \arcsin\left( R_{31} \div \sqrt{1 - R_{31}^2} \right)\\ \varphi = \arcsin\left( R_{31} \div \sqrt{1 - R_{31}^2} \right)\\
\end{smallmatrix}$ \end{smallmatrix}$
{\scriptsize $[\vec{n}]^\times = \begin{bmatrix}
$[\vec{n}]^\times = \begin{bmatrix} 0 & -a_3 & a_2 \\
0 & -a_3 & a_2 \\ a_3 & 0 & -a_1 \\
a_3 & 0 & -a_1 \\ -a_2 & a_1 & 0
-a_2 & a_1 & 0 \end{bmatrix}$
\end{bmatrix}$
} For pitch axis $= \pm 90\deg$, Gimbal Lock, Jacobian singular.
\shortdefinition[Angle-Axis] \shortdefinition[Angle-Axis]
$\vec{\alpha} = \alpha \vec{n}$ ($\vec{n}$ normal); Convert to rot. mat\\ $\vec{\alpha} = \alpha \vec{n}$ ($\vec{n}$ normal); Convert to rot. mat\\
@@ -58,31 +58,31 @@ To quat: $\vec{q} = [\vec{n}, \alpha]$
$i^2 = j^2 = k^2 = -1$, ($ij = -ji = k$, same for $jk$ and $ki$). $i^2 = j^2 = k^2 = -1$, ($ij = -ji = k$, same for $jk$ and $ki$).
$\vec{q} = \begin{bmatrix} $\vec{q} = \begin{bmatrix}
\vec{v}(\vec{q}), a(\vec{q}) \vec{v}(\vec{q}), a(\vec{q})
\end{bmatrix}^\top$, $a(\vec{q}) = q_w$, Add like $\C$ (by ``groups'') \end{bmatrix}^\top$, $a(\vec{q}) = q_w$, Add like $\C$ (by ``groups'')
\bi{Mult} {\scriptsize \bi{Mult} {\scriptsize
$\vec{q} \otimes \vec{p} = \begin{bmatrix} $\vec{q} \otimes \vec{p} = \begin{bmatrix}
a(\vec{q}) \vec{v}(\vec{p}) + a(\vec{p}) + \vec{v}(\vec{q}) \times \vec{v}(\vec{p}) \\ a(\vec{q}) \vec{v}(\vec{p}) + a(\vec{p}) + \vec{v}(\vec{q}) \times \vec{v}(\vec{p}) \\
a(\vec{q}) a(\vec{p}) - \vec{v}(\vec{q})^\top \vec{v}(\vec{p}) a(\vec{q}) a(\vec{p}) - \vec{v}(\vec{q})^\top \vec{v}(\vec{p})
\end{bmatrix}$ \end{bmatrix}$
} }
\bi{To Rot Mat} {\scriptsize \bi{To Rot Mat} {\scriptsize
$R_{AB} = \mat{I}_3 + 2a(\vec{q}_{AB}) [\vec{v}(\vec{q}_{AB})]^\times + 2([\vec{v}(\vec{q}_{AB})]^\times)^2$ $R_{AB} = \mat{I}_3 + 2a(\vec{q}_{AB}) [\vec{v}(\vec{q}_{AB})]^\times + 2([\vec{v}(\vec{q}_{AB})]^\times)^2$
} }
\shortdefinition[Transf. M] \shortdefinition[Transf. M]
{\scriptsize {\scriptsize
$\mat{T}_{AC} = \mat{T}_{AB} \mat{T}_{BC}$ R.-Handed typ; Aero: Left-H\\ $\mat{T}_{AC} = \mat{T}_{AB} \mat{T}_{BC}$ R.-Handed typ; Aero: Left-H\\
$\mat{T}_{AB} = \begin{bmatrix} $\mat{T}_{AB} = \begin{bmatrix}
\mat{R}_{AB} & {_A}\vec{t}_B \\ \mat{R}_{AB} & {_A}\vec{t}_B \\
\mat{0}_{1\times 3} & 1 \mat{0}_{1\times 3} & 1
\end{bmatrix}; \end{bmatrix};
\mat{T}_{BA} = \mat{T}_{AB}^{-1} = \mat{T}_{BA} = \mat{T}_{AB}^{-1} =
\begin{bmatrix} \begin{bmatrix}
\mat{R}_{AB}^\top & -\mat{R}_{AB}^\top {_A}\vec{t}_B \\ \mat{R}_{AB}^\top & -\mat{R}_{AB}^\top {_A}\vec{t}_B \\
\mat{0}_{1 \times 3} & 1 \mat{0}_{1 \times 3} & 1
\end{bmatrix}$ \end{bmatrix}$
} }
electives/amr/parts/01_kinematics/02_inverse.tex
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@@ -1,12 +1,11 @@
\subsection{Inverse Kinematics (IK)} \subsection{Inverse Kinematics (IK)}
\begin{wrapfigure}[10]{r}{0.4\columnwidth} \begin{wrapfigure}[10]{r}{0.35\columnwidth}
\includegraphics[width=0.4\columnwidth]{assets/inverse-kinematics.png} \includegraphics[width=0.35\columnwidth]{assets/inverse-kinematics.png}
\end{wrapfigure} \end{wrapfigure}
\bi{Option}: Solve Forward Kinematics for angles.\\ \bi{Option}: Solve Forward Kinematics for angles.\\
\bi{Better}: Law of cosine with polar coordinates. Compute angle using cosine rule,\\ \bi{Better}: Law of cosine with polar coordinates. Compute angle using cosine rule,
$\theta_1 = \varphi \pm \alpha$, $\theta_2 = \pm(\pi - \beta)$ $\theta_1 = \varphi \pm \alpha$, $\theta_2 = \pm(\pi - \beta)$
(Positive for {\color{ForestGreen} Elbow Down}, Neg. for {\color{red} Elbow Up})
(Positive for {\color{ForestGreen} Elbow Down}, Negative for {\color{red} Elbow Up})
\bi{Extension to 6R}: \bi{Extension to 6R}:
1. Waist: spherical coords (2 sol.)\\ 1. Waist: spherical coords (2 sol.)\\
electives/amr/parts/01_kinematics/03_temporal-models.tex
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@@ -7,7 +7,7 @@ $\mat{H}$ is meas., $\mat{F}_C$ system, $\mat{G}$ input gain, $\vec{w}$ process
both zero-mean \bi{Gaussian White Noise Process}. both zero-mean \bi{Gaussian White Noise Process}.
To \bi{discretize}, integrate from $t_{k - 1}$ to $t_k$:\\ To \bi{discretize}, integrate from $t_{k - 1}$ to $t_k$:\\
$\vec{x}_k = \vec{f}(\vec{x}_{k - 1}, \vec{u}_k, \vec{w}_k)$ $\vec{x}_k = \vec{f}(\vec{x}_{k - 1}, \vec{u}_k, \vec{w}_k)$;
$\vec{z}_k = \vec{h}(\vec{x}_k) + \vec{v}_k$, $\vec{z}_k = \vec{h}(\vec{x}_k) + \vec{v}_k$,
\bi{linearised}:\\ \bi{linearised}:\\
$\delta \vec{x}_k = \vec{f}(\vec{\overline{x}}, \vec{\overline{u}}) + \mat{F} \delta \vec{x}_{k - 1} + \mat{G}_k \delta \vec{u}_k + \mat{L}_k \vec{w}_k$; $\delta \vec{x}_k = \vec{f}(\vec{\overline{x}}, \vec{\overline{u}}) + \mat{F} \delta \vec{x}_{k - 1} + \mat{G}_k \delta \vec{u}_k + \mat{L}_k \vec{w}_k$;
electives/amr/parts/01_kinematics/05_rigid-body-dynamics.tex
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@@ -2,8 +2,10 @@
\label{sec:rigid-body-dynamics} \label{sec:rigid-body-dynamics}
\shortdefinition[Newton II] For fin. body w/ mass $m$ and inertia mat. $I$, with force $\vec{F}$ and torque $\vec{T}$ on \bi{Centre of Mass} (CoM), expressed in body frame: \shortdefinition[Newton II] For fin. body w/ mass $m$ and inertia mat. $I$, with force $\vec{F}$ and torque $\vec{T}$ on \bi{Centre of Mass} (CoM), expressed in body frame:
\begin{align*} \begin{align*}
{_B}\vec{F} &= \sum {_B}\vec{F}_i = m({_B} \vec{\dot{v}}_{CoM}) + m_B \vec{\omega} \times {_B}\vec{v}_{CoM} \\ {_B}\vec{F} & = \sum {_B}\vec{F}_i = m({_B} \vec{\dot{v}}_{CoM}) + m_B \vec{\omega} \times {_B}\vec{v}_{CoM} \\
{_B}\vec{T} &= \sum {_B}\vec{T}_i = \mat{I}({_B} \vec{\dot{\omega}}) + {_B} \vec{\omega} \times \mat{I}_B\vec{\omega} {_B}\vec{T} & = \sum {_B}\vec{T}_i = \mat{I}({_B} \vec{\dot{\omega}}) + {_B} \vec{\omega} \times \mat{I}_B\vec{\omega}
\end{align*} \end{align*}
${_B} \vec{v}_{CoM}$ vel. of CoM, ${_B}\omega$ rot. speed; both w.r.t. inert. frame ${_B} \vec{v}_{CoM}$ vel. of CoM, ${_B}\omega$ rot. speed; both w.r.t. inert. frame
{\scriptsize Legged robots don't need to maintain stability!}