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[AMR] Reduce font size, some additions

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janishutz
2026-07-18 11:03:37 +02:00
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electives/amr/parts/01_kinematics/01_forward.tex
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@@ -1,15 +1,16 @@
\subsection{Forward Kinematics (FK)}
$\mat{T}_{WB_n}(\vec{\theta}) = \mat{T}_{WB_0} \mat{T}_{B_0B_1}(\theta_1) \cdots \mat{T}_{B_{n - 1}B_n}(\theta_n)$.\\
For 2R system:
${_W}\vec{t}_{WE} =$
$\mat{T}_{WB_n}(\vec{\theta}) = \mat{T}_{WB_0} \mat{T}_{B_0B_1}(\theta_1) \cdots \mat{T}_{B_{n - 1}B_n}(\theta_n)$.
For 2R system: ${_W}\vec{t}_{WE} =$
{\scriptsize
$ \begin{bmatrix}
L_1 \cos(\theta_1) + L_2 \cos(\theta_1 + \theta_2) \\
L_1 \sin(\theta_1) + L_2 \sin(\theta_1 + \theta_2)
\end{bmatrix}$\\
}
Similar for $n$R sys (more angles). Wspc $W$ $\theta_1, \theta_2 \in [-\pi, \pi]$.
Last dim may be sum of angles. Jacobian: see \ref{sec:ms-lin}
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}
\shortdefinition[Singularity] Loss of deg of Freed. $\det(\mat{J}(\vec{\theta})) = 0$
% TODO: Determinant computation