\subsection{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), in body frame:
\begin{align*}
    {_B}\vec{F} & = m({_B} \vec{\dot{v}}_{CoM}) + m ({_B}\vec{\omega} \times {_B}\vec{v}_{CoM})       \\
    {_B}\vec{T} & = \mat{I}({_B} \vec{\dot{\omega}}) + {_B} \vec{\omega} \times \mat{I}_B\vec{\omega}
\end{align*}

${_B} \vec{v}_{CoM}$ vel. of CoM, ${_B}\omega$ rot. speed; both w.r.t. inert. frame

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

\subsection{Manipulator Velocity Kinematics}
\label{sec:manipulator-velocity-kinematics}
Differentiate FK $x = f(\vec{\theta})$, with $\dot{\vec{x}} = [{_W}\vec{v}, {_W} \vec{\omega}]^\top = \mat{J}(\vec{\theta})\dot{\vec{\theta}}$,
with $\mat{J} = \frac{\partial f}{\partial \vec{\theta}}$ manipulator Jacobian. Then $\dot{\vec{\theta}} = J^{-1} \vec{\dot{x}}$ or $\vec{\dot{\theta}} = J^+ \vec{\dot{x}}$,
if over-actuated, with $J^+ = J^\top(J J^\top)^{-1}$ (Moore Penrose-Inverse)


\newpage
\subsection{Legged robots}
{\scriptsize Legged robots don't need to maintain static stability!}

\shortdefinition[Static Walking] 3 or more legs on ground, stable if frozen, safe, slow, inefficient.

\shortdefinition[Dynamic Walking] $<$ 3 legs on ground, falls when not moving, fast, efficient, hard.