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[AMR] Fix typos, improve space usage some more
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@@ -1,6 +1,6 @@
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\subsection{Rigid Body Dynamics}
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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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\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), 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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@@ -6,10 +6,12 @@
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\bi{Wheel constraints} $v_i = \omega_i r_i$ ($r_i$ constraints)
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\begin{itemize}
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\item \textit{Driving straight} all $\vec{v}$ equal
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\item \textit{Driving straight} all $\vec{v}$ equal (ICR: R.Cent.)
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\item \textit{Turning} Wheel axis must intersect the \bi{Instant Centre of Rotation} (ICR) of vehicle,
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speeds: $v_i \div R_i = \Omega$ ($R_i$ = dist. wheel-ICR; $\Omega$: vehicle rotation rate (around ICR))
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\end{itemize}
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To compute ICR, use $v_i \div R_i = \Omega$ and similarity.
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Below: $\alpha$, $l$ pos in frame, $\beta$ rot at that pos ($z$-ax).
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To compute $\vec{c}$ in $\vec{c} \cdot {_B}\vec{v}_{WB} = \omega$
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(For multiple wheels, construct mat. from this)
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