#+TITLE: Stewart Platform - Static Analysis
:DRAWER:
#+STARTUP: overview
#+LANGUAGE: en
#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas
#+HTML_LINK_HOME: ./index.html
#+HTML_LINK_UP: ./index.html
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#+PROPERTY: header-args:matlab :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments org
#+PROPERTY: header-args:matlab+ :exports both
#+PROPERTY: header-args:matlab+ :results none
#+PROPERTY: header-args:matlab+ :eval no-export
#+PROPERTY: header-args:matlab+ :noweb yes
#+PROPERTY: header-args:matlab+ :mkdirp yes
#+PROPERTY: header-args:matlab+ :output-dir figs
#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/thesis/latex/}{config.tex}")
#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
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#+PROPERTY: header-args:latex+ :eval no-export
#+PROPERTY: header-args:latex+ :exports results
#+PROPERTY: header-args:latex+ :mkdirp yes
#+PROPERTY: header-args:latex+ :output-dir figs
#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
:END:
* 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.