#+TITLE: Decoupling Control of Parallel Manipulators
:DRAWER:
#+LANGUAGE: en
#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas

#+PROPERTY: header-args:latex  :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/tikz/org/}{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
#+PROPERTY: header-args:latex+ :results file raw replace
#+PROPERTY: header-args:latex+ :buffer no
#+PROPERTY: header-args:latex+ :tangle no
#+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:

#+latex: \clearpage


#+begin_src latex :file detail_control_decoupling_control_jacobian.pdf
\begin{tikzpicture}
  \node[block] (G) {$\bm{G}_{\{\mathcal{L}\}}$};
  \node[block, left=0.6 of G] (Jt) {$\bm{J}_{\{O\}}^{-\intercal}$};
  \node[block, right=0.6 of G] (Ja) {$\bm{J}_{\{O\}}^{-1}$};

  % Connections and labels
  \draw[<-] (Jt.west) -- ++(-1.4, 0) node[above right]{$\bm{\mathcal{F}}_{\{O\}}$};
  \draw[->] (Jt.east) -- (G.west)  node[above left]{$\bm{\tau}$};
  \draw[->] (G.east) -- (Ja.west)  node[above left]{$\bm{\mathcal{L}}$};
  \draw[->] (Ja.east) -- ++( 1.4, 0)  node[above left]{$\bm{\mathcal{X}}_{\{O\}}$};

  \begin{scope}[on background layer]
    \node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=4pt] (Gx) {};
    \node[above] at (Gx.north) {$\bm{G}_{\{O\}}$};
  \end{scope}
\end{tikzpicture}
#+end_src

#+RESULTS:
[[file:figs/detail_control_decoupling_control_jacobian.png]]


#+begin_src latex :file detail_control_decoupling_modal.pdf
\begin{tikzpicture}
  \node[block] (G) {$\bm{G}_{\{\mathcal{L}\}}$};
  \node[block, left=0.6 of G] (Jt) {$\bm{J}_{\{M\}}^{-\intercal}$};
  \node[block, left=1.2 of Jt] (Bm) {$\bm{\Phi}^{-\intercal}$};
  \node[block, right=0.6 of G] (J) {$\bm{J}_{\{M\}}^{-1}$};
  \node[block, right=1.2 of J] (Cm) {$\bm{\Phi}^{-1}$};

  % Connections and labels
  \draw[<-] (Bm.west) -- ++(-1.0, 0) node[above right]{$\bm{\tau}_m$};
  \draw[->] (Bm.east) -- (Jt.west)  node[above left]{$\bm{\mathcal{F}}_{\{M\}}$};
  \draw[->] (Jt.east) -- (G.west)  node[above left]{$\bm{\tau}$};
  \draw[->] (G.east) -- (J.west)  node[above left]{$\bm{\mathcal{L}}$};
  \draw[->] (J.east) -- (Cm.west)  node[above left]{$\bm{\mathcal{X}}_{\{M\}}$};
  \draw[->] (Cm.east) -- ++( 1.0, 0)  node[above left]{$\bm{\mathcal{X}}_m$};

  \begin{scope}[on background layer]
    \node[fit={(Bm.south west) (Cm.north east)}, fill=black!10!white, draw, dashed, inner sep=4pt] (Gm) {};
    \node[above] at (Gm.north) {$\bm{G}_m$};
  \end{scope}
\end{tikzpicture}
#+end_src

#+RESULTS:
[[file:figs/detail_control_decoupling_modal.png]]

#+begin_src latex :file detail_control_decoupling_svd.pdf
\begin{tikzpicture}
  \node[block] (G) {$\bm{G}_{\{\mathcal{L}\}}$};
  \node[block, left=0.6 of G.west] (V) {$\bm{V}^{-\intercal}$};
  \node[block, right=0.6 of G.east] (U) {$\bm{U}^{-1}$};

  % Connections and labels
  \draw[<-] (V.west) -- ++(-0.8, 0) node[above right]{$\bm{u}$};
  \draw[->] (V.east) -- (G.west) node[above left]{$\bm{\tau}$};
  \draw[->] (G.east) -- (U.west) node[above left]{$\bm{\mathcal{L}}$};
  \draw[->] (U.east) -- ++( 0.8, 0) node[above left]{$\bm{y}$};

  \begin{scope}[on background layer]
    \node[fit={(V.south west) (G.north-|U.east)}, fill=black!10!white, draw, dashed, inner sep=4pt] (Gsvd) {};
    \node[above] at (Gsvd.north) {$\bm{G}_{\text{SVD}}$};
  \end{scope}
\end{tikzpicture}
#+end_src

#+RESULTS:
[[file:figs/detail_control_decoupling_svd.png]]