49 lines
1.8 KiB
Org Mode
49 lines
1.8 KiB
Org Mode
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#+TITLE: Stewart Platform - Static Analysis
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:DRAWER:
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#+HTML_LINK_HOME: ./index.html
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#+HTML_LINK_UP: ./index.html
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
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#+HTML_HEAD: <script src="./js/jquery.min.js"></script>
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#+HTML_HEAD: <script src="./js/bootstrap.min.js"></script>
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#+HTML_HEAD: <script src="./js/jquery.stickytableheaders.min.js"></script>
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#+HTML_HEAD: <script src="./js/readtheorg.js"></script>
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#+PROPERTY: header-args:matlab :session *MATLAB*
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#+PROPERTY: header-args:matlab+ :comments org
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#+PROPERTY: header-args:matlab+ :exports both
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#+PROPERTY: header-args:matlab+ :results none
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#+PROPERTY: header-args:matlab+ :eval no-export
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#+PROPERTY: header-args:matlab+ :noweb yes
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#+PROPERTY: header-args:matlab+ :mkdirp yes
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#+PROPERTY: header-args:matlab+ :output-dir figs
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:END:
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* Coupling
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What causes the coupling from $F_i$ to $X_i$ ?
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#+begin_src latex :file coupling.pdf :post pdf2svg(file=*this*, ext="png") :exports both
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\begin{tikzpicture}
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\node[block] (Jt) at (0, 0) {$J^{-T}$};
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\node[block, right= of Jt] (G) {$G$};
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\node[block, right= of G] (J) {$J^{-1}$};
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\draw[->] ($(Jt.west)+(-0.8, 0)$) -- (Jt.west) node[above left]{$F_i$};
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\draw[->] (Jt.east) -- (G.west) node[above left]{$\tau_i$};
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\draw[->] (G.east) -- (J.west) node[above left]{$q_i$};
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\draw[->] (J.east) -- ++(0.8, 0) node[above left]{$X_i$};
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\end{tikzpicture}
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#+end_src
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#+name: fig:block_diag_coupling
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#+caption: Block diagram to control an hexapod
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#+RESULTS:
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[[file:figs/coupling.png]]
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There is no coupling from $F_i$ to $X_j$ if $J^{-1} G J^{-T}$ is diagonal.
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If $G$ is diagonal (cubic configuration), then $J^{-1} G J^{-T} = G J^{-1} J^{-T} = G (J^{T} J)^{-1} = G K^{-1}$
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Thus, the system is uncoupled if $G$ and $K$ are diagonal.
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