#+TITLE: Stewart Platform - Static Analysis
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* Coupling
What causes the coupling from $F_i$ to $X_i$ ?

#+begin_src latex :file coupling.pdf :post pdf2svg(file=*this*, ext="png") :exports both
  \begin{tikzpicture}
    \node[block] (Jt) at (0, 0) {$J^{-T}$};
    \node[block, right= of Jt] (G) {$G$};
    \node[block, right= of G] (J) {$J^{-1}$};

    \draw[->] ($(Jt.west)+(-0.8, 0)$) -- (Jt.west) node[above left]{$F_i$};
    \draw[->] (Jt.east) -- (G.west) node[above left]{$\tau_i$};
    \draw[->] (G.east) -- (J.west) node[above left]{$q_i$};
    \draw[->] (J.east) -- ++(0.8, 0) node[above left]{$X_i$};
  \end{tikzpicture}
#+end_src

#+name: fig:block_diag_coupling
#+caption: Block diagram to control an hexapod
#+RESULTS:
[[file:figs/coupling.png]]

There is no coupling from $F_i$ to $X_j$ if $J^{-1} G J^{-T}$ is diagonal.

If $G$ is diagonal (cubic configuration), then $J^{-1} G J^{-T} = G J^{-1} J^{-T} = G (J^{T} J)^{-1} = G K^{-1}$

Thus, the system is uncoupled if $G$ and $K$ are diagonal.