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