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<?xml version="1.0" encoding="utf-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
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<title>Active Damping of Rotating Platforms using Integral Force Feedback - Tikz Figures</title>
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<h1 class="title">Active Damping of Rotating Platforms using Integral Force Feedback - Tikz Figures</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#orgd15976a">1. X-Y Rotating Positioning Platform</a></li>
<li><a href="#orgfba88c8">2. X-Y Rotating Positioning Platform</a></li>
<li><a href="#org732f401">3. Decentralized Integral Force Feedback</a></li>
<li><a href="#org1dd642f">4. Springs in parallel</a></li>
</ul>
</div>
</div>
<p>
Configuration file is accessible <a href="config.html">here</a>.
</p>
<div id="outline-container-orgd15976a" class="outline-2">
<h2 id="orgd15976a"><span class="section-number-2">1</span> X-Y Rotating Positioning Platform</h2>
<div class="outline-text-2" id="text-1">
<div class="org-src-container">
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
<span class="org-comment">% Angle</span>
<span class="org-font-latex-sedate">\def\thetau</span>{25}
<span class="org-comment">% Rotational Stage</span>
<span class="org-font-latex-sedate">\draw</span>[fill=black!60!white] (0, 0) circle (4.3);
<span class="org-font-latex-sedate">\draw</span>[fill=black!40!white] (0, 0) circle (3.8);
<span class="org-comment">% Label</span>
<span class="org-font-latex-sedate">\node</span>[anchor=north west, rotate=<span class="org-font-latex-sedate">\thetau</span>] at (-2.5, 2.5) {<span class="org-font-latex-sedate"><span class="org-keyword">\small</span></span><span class="org-type"> Rotating Stage</span>};
<span class="org-comment">% Rotating Scope</span>
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[rotate=<span class="org-font-latex-sedate">\thetau</span>]
<span class="org-comment">% Rotating Frame</span>
<span class="org-font-latex-sedate">\draw</span>[fill=black!20!white] (-2.6, -2.6) rectangle (2.6, 2.6);
<span class="org-comment">% Label</span>
<span class="org-font-latex-sedate">\node</span>[anchor=north west, rotate=<span class="org-font-latex-sedate">\thetau</span>] at (-2.6, 2.6) {<span class="org-font-latex-sedate"><span class="org-keyword">\small</span></span><span class="org-type"> Suspended Platform</span>};
<span class="org-comment">% Mass</span>
<span class="org-font-latex-sedate">\draw</span>[fill=white] (-1, -1) rectangle (1, 1);
<span class="org-comment">% Label</span>
<span class="org-font-latex-sedate">\node</span>[anchor=south west, rotate=<span class="org-font-latex-sedate">\thetau</span>] at (-1, -1) {<span class="org-font-latex-sedate"><span class="org-keyword">\small</span></span><span class="org-type"> Payload</span>};
<span class="org-comment">% Attached Points</span>
<span class="org-font-latex-sedate">\node</span>[] at (-1, 0){<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\bullet</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\draw</span>[] (-1, 0) -- ++(-0.2, 0) coordinate(cu);
<span class="org-font-latex-sedate">\draw</span>[] (<span class="org-font-latex-math">$(cu) + (0, -0.8)$</span>) coordinate(actu) -- (<span class="org-font-latex-math">$(cu) + (0, 0.8)$</span>) coordinate(ku);
<span class="org-font-latex-sedate">\node</span>[] at (0, -1){<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\bullet</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\draw</span>[] (0, -1) -- ++(0, -0.2) coordinate(cv);
<span class="org-font-latex-sedate">\draw</span>[] (<span class="org-font-latex-math">$(cv) + (-0.8, 0)$</span>)coordinate(kv) -- (<span class="org-font-latex-math">$(cv) + (0.8, 0)$</span>) coordinate(actv);
<span class="org-comment">% Spring and Actuator for U</span>
<span class="org-font-latex-sedate">\draw</span>[actuator={0.6}{0.2}] (actu) -- node[above=0.1, rotate=<span class="org-font-latex-sedate">\thetau</span>]{<span class="org-font-latex-math">$F</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">u$</span>} (actu-|-2.6,0);
<span class="org-font-latex-sedate">\draw</span>[spring=0.2] (ku) -- node[above=0.1, rotate=<span class="org-font-latex-sedate">\thetau</span>]{<span class="org-font-latex-math">$k$</span>} (ku-|-2.6,0);
<span class="org-font-latex-sedate">\draw</span>[damper={8}{8}] (cu) -- node[above left=0.2 and -0.1, rotate=<span class="org-font-latex-sedate">\thetau</span>]{<span class="org-font-latex-math">$c$</span>} (cu-|-2.6,0);
<span class="org-font-latex-sedate">\draw</span>[actuator={0.6}{0.2}] (actv) -- node[left, rotate=<span class="org-font-latex-sedate">\thetau</span>]{<span class="org-font-latex-math">$F</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">v$</span>} (actv|-0,-2.6);
<span class="org-font-latex-sedate">\draw</span>[spring=0.2] (kv) -- node[left, rotate=<span class="org-font-latex-sedate">\thetau</span>]{<span class="org-font-latex-math">$k$</span>} (kv|-0,-2.6);
<span class="org-font-latex-sedate">\draw</span>[damper={8}{8}] (cv) -- node[left=0.1, rotate=<span class="org-font-latex-sedate">\thetau</span>]{<span class="org-font-latex-math">$c$</span>} (cv|-0,-2.6);
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
<span class="org-comment">% Inertial Frame</span>
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (-4, -4) -- ++(2, 0) node[below]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\vec</span></span><span class="org-font-latex-math">{i}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">x$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (-4, -4) -- ++(0, 2) node[left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\vec</span></span><span class="org-font-latex-math">{i}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">y$</span>};
<span class="org-font-latex-sedate">\draw</span>[fill, color=black] (-4, -4) circle (0.06);
<span class="org-font-latex-sedate">\node</span>[draw, circle, inner sep=0pt, minimum size=0.3cm, label=left:<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\vec</span></span><span class="org-font-latex-math">{i}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">z$</span>] at (-4, -4){};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (0, 0) node[above left, rotate=<span class="org-font-latex-sedate">\thetau</span>]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\vec</span></span><span class="org-font-latex-math">{i}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">w$</span>} -- ++(<span class="org-font-latex-sedate">\thetau</span>:2) node[above, rotate=<span class="org-font-latex-sedate">\thetau</span>]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\vec</span></span><span class="org-font-latex-math">{i}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">u$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (0, 0) -- ++(<span class="org-font-latex-sedate">\thetau</span>+90:2) node[left, rotate=<span class="org-font-latex-sedate">\thetau</span>]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\vec</span></span><span class="org-font-latex-math">{i}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">v$</span>};
<span class="org-font-latex-sedate">\draw</span>[fill, color=black] (0,0) circle (0.06);
<span class="org-font-latex-sedate">\node</span>[draw, circle, inner sep=0pt, minimum size=0.3cm] at (0, 0){};
<span class="org-font-latex-sedate">\draw</span>[dashed] (0, 0) -- ++(2, 0);
<span class="org-font-latex-sedate">\draw</span>[] (1.5, 0) arc (0:<span class="org-font-latex-sedate">\thetau</span>:1.5) node[midway, right]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\theta</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (3.5, 0) arc (0:40:3.5) node[midway, left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\Omega</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
</pre>
</div>
<div id="orge4c1dec" class="figure">
<p><img src="figs/system.png" alt="system.png" />
</p>
</div>
</div>
</div>
<div id="outline-container-orgfba88c8" class="outline-2">
<h2 id="orgfba88c8"><span class="section-number-2">2</span> X-Y Rotating Positioning Platform</h2>
<div class="outline-text-2" id="text-2">
<div class="org-src-container">
<pre class="src src-latex"><span class="org-font-latex-sedate">\tikzset</span>{block/.default={0.8cm}{0.8cm}}
<span class="org-font-latex-sedate">\tikzset</span>{addb/.append style={scale=0.7}}
<span class="org-font-latex-sedate">\tikzset</span>{node distance=0.6}
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
<span class="org-font-latex-sedate">\node</span>[block={1.8cm}{2.2cm}] (G) {<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\bm</span></span><span class="org-font-latex-math">{G}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">f$</span>};
<span class="org-comment">% Inputs of the controllers</span>
<span class="org-font-latex-sedate">\coordinate</span>[] (output1) at (<span class="org-font-latex-math">$(G.south east)!0.75!(G.north east)$</span>);
<span class="org-font-latex-sedate">\coordinate</span>[] (output2) at (<span class="org-font-latex-math">$(G.south east)!0.25!(G.north east)$</span>);
<span class="org-font-latex-sedate">\coordinate</span>[] (input1) at (<span class="org-font-latex-math">$(G.south west)!0.75!(G.north west)$</span>);
<span class="org-font-latex-sedate">\coordinate</span>[] (input2) at (<span class="org-font-latex-math">$(G.south west)!0.25!(G.north west)$</span>);
<span class="org-font-latex-sedate">\node</span>[block, left=1.8 of input1] (K1) {<span class="org-font-latex-math">$K</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">F$</span>};
<span class="org-font-latex-sedate">\node</span>[block] (K2) at (<span class="org-font-latex-math">$(K1.east|-input2)+(0.6, 0)$</span>) {<span class="org-font-latex-math">$K</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">F$</span>};
<span class="org-comment">% Connections and labels</span>
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (K1.east) -- (input1)node[above left]{<span class="org-font-latex-math">$F</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">u$</span>}node[below left]{<span class="org-font-latex-math">$-$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (K2.east) -- (input2)node[above left]{<span class="org-font-latex-math">$F</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">v$</span>}node[below left]{<span class="org-font-latex-math">$-$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (output1) -- ++(0.8, 0) node[above left]{<span class="org-font-latex-math">$f</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">u$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (output2) -- ++(0.8, 0) node[above left]{<span class="org-font-latex-math">$f</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">v$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (<span class="org-font-latex-math">$(output1)+(0.2, 0)$</span>)node[branch]{} -- ++(0, 1.2) -| (<span class="org-font-latex-math">$(K1.west) + (-0.8, 0)$</span>)coordinate(start) -- (K1.west);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (<span class="org-font-latex-math">$(output2)+(0.2, 0)$</span>)node[branch]{} -- ++(0, -1.2) -| (start|-K2) -- (K2.west);
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[on background layer]
<span class="org-font-latex-sedate">\node</span>[fit={(K1.north west) (K2.south east)}, inner sep=6pt, draw, dashed, fill=black!20!white] (K) {};
<span class="org-font-latex-sedate">\node</span>[below left] at (K.north east) {<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\bm</span></span><span class="org-font-latex-math">{K}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">F$</span>};
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
</pre>
</div>
<div id="orgd28fad3" class="figure">
<p><img src="figs/control_diagram_iff.png" alt="control_diagram_iff.png" />
</p>
</div>
</div>
</div>
<div id="outline-container-org732f401" class="outline-2">
<h2 id="org732f401"><span class="section-number-2">3</span> Decentralized Integral Force Feedback</h2>
<div class="outline-text-2" id="text-3">
<div class="org-src-container">
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
<span class="org-comment">% Angle</span>
<span class="org-font-latex-sedate">\def\thetau</span>{25}
<span class="org-comment">% Rotational Stage</span>
<span class="org-font-latex-sedate">\draw</span>[fill=black!60!white] (0, 0) circle (4.3);
<span class="org-font-latex-sedate">\draw</span>[fill=black!40!white] (0, 0) circle (3.8);
<span class="org-comment">% Label</span>
<span class="org-font-latex-sedate">\node</span>[anchor=north west, rotate=<span class="org-font-latex-sedate">\thetau</span>] at (-2.5, 2.5) {<span class="org-font-latex-sedate"><span class="org-keyword">\small</span></span><span class="org-type"> Rotating Stage</span>};
<span class="org-comment">% Rotating Scope</span>
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[rotate=<span class="org-font-latex-sedate">\thetau</span>]
<span class="org-comment">% Rotating Frame</span>
<span class="org-font-latex-sedate">\draw</span>[fill=black!20!white] (-2.6, -2.6) rectangle (2.6, 2.6);
<span class="org-comment">% Label</span>
<span class="org-font-latex-sedate">\node</span>[anchor=north west, rotate=<span class="org-font-latex-sedate">\thetau</span>] at (-2.6, 2.6) {<span class="org-font-latex-sedate"><span class="org-keyword">\small</span></span><span class="org-type"> Suspended Platform</span>};
<span class="org-comment">% Mass</span>
<span class="org-font-latex-sedate">\draw</span>[fill=white] (-1, -1) rectangle (1, 1);
<span class="org-comment">% Label</span>
<span class="org-font-latex-sedate">\node</span>[anchor=south west, rotate=<span class="org-font-latex-sedate">\thetau</span>] at (-1, -1) {<span class="org-font-latex-sedate"><span class="org-keyword">\small</span></span><span class="org-type"> Payload</span>};
<span class="org-comment">% Attached Points</span>
<span class="org-font-latex-sedate">\node</span>[] at (-1, 0){<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\bullet</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\draw</span>[] (-1, 0) -- ++(-0.2, 0) coordinate(au);
<span class="org-font-latex-sedate">\node</span>[] at (0, -1){<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\bullet</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\draw</span>[] (0, -1) -- ++(0, -0.2) coordinate(av);
<span class="org-comment">% Force Sensors</span>
<span class="org-font-latex-sedate">\draw</span>[fill=white] (<span class="org-font-latex-math">$(au) + (-0.2, -0.5)$</span>) rectangle (<span class="org-font-latex-math">$(au) + (0, 0.5)$</span>);
<span class="org-font-latex-sedate">\draw</span>[] (<span class="org-font-latex-math">$(au) + (-0.2, -0.5)$</span>)coordinate(actu) -- (<span class="org-font-latex-math">$(au) + (0, 0.5)$</span>);
<span class="org-font-latex-sedate">\draw</span>[] (<span class="org-font-latex-math">$(au) + (-0.2, 0.5)$</span>)coordinate(ku) -- (<span class="org-font-latex-math">$(au) + (0, -0.5)$</span>);
<span class="org-font-latex-sedate">\draw</span>[fill=white] (<span class="org-font-latex-math">$(av) + (-0.5, -0.2)$</span>) rectangle (<span class="org-font-latex-math">$(av) + (0.5, 0)$</span>);
<span class="org-font-latex-sedate">\draw</span>[] (<span class="org-font-latex-math">$(av) + ( 0.5, -0.2)$</span>)coordinate(actv) -- (<span class="org-font-latex-math">$(av) + (-0.5, 0)$</span>);
<span class="org-font-latex-sedate">\draw</span>[] (<span class="org-font-latex-math">$(av) + (-0.5, -0.2)$</span>)coordinate(kv) -- (<span class="org-font-latex-math">$(av) + ( 0.5, 0)$</span>);
<span class="org-comment">% Spring and Actuator for U</span>
<span class="org-font-latex-sedate">\draw</span>[actuator={0.6}{0.2}] (actu) -- coordinate[midway](actumid) (actu-|-2.6,0);
<span class="org-font-latex-sedate">\draw</span>[spring=0.2] (ku) -- node[above=0.1, rotate=<span class="org-font-latex-sedate">\thetau</span>]{<span class="org-font-latex-math">$k$</span>} (ku-|-2.6,0);
<span class="org-comment">% </span><span class="org-font-latex-sedate"><span class="org-comment">\draw</span></span><span class="org-comment">[actuator={0.6}{0.2}] (actv) -- node[right, rotate=</span><span class="org-font-latex-sedate"><span class="org-comment">\thetau</span></span><span class="org-comment">]{$F_v$} (actv|-0,-2.6);</span>
<span class="org-font-latex-sedate">\draw</span>[actuator={0.6}{0.2}] (actv) -- coordinate[midway](actvmid) (actv|-0,-2.6);
<span class="org-font-latex-sedate">\draw</span>[spring=0.2] (kv) -- node[left, rotate=<span class="org-font-latex-sedate">\thetau</span>]{<span class="org-font-latex-math">$k$</span>} (kv|-0,-2.6);
<span class="org-font-latex-sedate">\node</span>[block={0.8cm}{0.6cm}, rotate=<span class="org-font-latex-sedate">\thetau</span>] (Ku) at (<span class="org-font-latex-math">$(actumid) + (0, -1.2)$</span>) {<span class="org-font-latex-math">$K_{F}$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (<span class="org-font-latex-math">$(au) + (-0.1, -0.5)$</span>) |- (Ku.east) node[below right, rotate=<span class="org-font-latex-sedate">\thetau</span>]{<span class="org-font-latex-math">$f_{u}$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (Ku.north) -- (<span class="org-font-latex-math">$(actumid) + (0, -0.1)$</span>) node[below left, rotate=<span class="org-font-latex-sedate">\thetau</span>]{<span class="org-font-latex-math">$F_u$</span>} node[below right, rotate=<span class="org-font-latex-sedate">\thetau</span>]{<span class="org-font-latex-math">$-$</span>};
<span class="org-font-latex-sedate">\node</span>[block={0.8cm}{0.6cm}, rotate=<span class="org-font-latex-sedate">\thetau</span>] (Kv) at (<span class="org-font-latex-math">$(actvmid) + (1.2, 0)$</span>) {<span class="org-font-latex-math">$K_{F}$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (<span class="org-font-latex-math">$(av) + (0.5, -0.1)$</span>) -| (Kv.north) node[above right, rotate=<span class="org-font-latex-sedate">\thetau</span>]{<span class="org-font-latex-math">$f_{v}$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (Kv.west) -- (<span class="org-font-latex-math">$(actvmid) + (0.1, 0)$</span>) node[below right, rotate=<span class="org-font-latex-sedate">\thetau</span>]{<span class="org-font-latex-math">$F_v$</span>} node[above right, rotate=<span class="org-font-latex-sedate">\thetau</span>]{<span class="org-font-latex-math">$-$</span>};
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
<span class="org-comment">% Inertial Frame</span>
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (-4, -4) -- ++(2, 0) node[below]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\vec</span></span><span class="org-font-latex-math">{i}_x$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (-4, -4) -- ++(0, 2) node[left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\vec</span></span><span class="org-font-latex-math">{i}_y$</span>};
<span class="org-font-latex-sedate">\draw</span>[fill, color=black] (-4, -4) circle (0.06);
<span class="org-font-latex-sedate">\node</span>[draw, circle, inner sep=0pt, minimum size=0.3cm, label=left:<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\vec</span></span><span class="org-font-latex-math">{i}_z$</span>] at (-4, -4){};
<span class="org-font-latex-sedate">\node</span>[draw, circle, inner sep=0pt, minimum size=0.3cm] at (0, 0){};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (0, 0) node[above left, rotate=<span class="org-font-latex-sedate">\thetau</span>]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\vec</span></span><span class="org-font-latex-math">{i}_w$</span>} -- ++(<span class="org-font-latex-sedate">\thetau</span>:2) node[above, rotate=<span class="org-font-latex-sedate">\thetau</span>]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\vec</span></span><span class="org-font-latex-math">{i}_u$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (0, 0) -- ++(<span class="org-font-latex-sedate">\thetau</span>+90:2) node[left, rotate=<span class="org-font-latex-sedate">\thetau</span>]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\vec</span></span><span class="org-font-latex-math">{i}_v$</span>};
<span class="org-font-latex-sedate">\draw</span>[dashed] (0, 0) -- ++(2, 0);
<span class="org-font-latex-sedate">\draw</span>[] (1.5, 0) arc (0:<span class="org-font-latex-sedate">\thetau</span>:1.5) node[midway, right]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\theta</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\node</span>[] at (0,0) {<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\bullet</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (3.5, 0) arc (0:40:3.5) node[midway, left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\Omega</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
</pre>
</div>
<div id="orgfc98ac7" class="figure">
<p><img src="figs/system_iff.png" alt="system_iff.png" />
</p>
</div>
</div>
</div>
<div id="outline-container-org1dd642f" class="outline-2">
<h2 id="org1dd642f"><span class="section-number-2">4</span> Springs in parallel</h2>
<div class="outline-text-2" id="text-4">
<div class="org-src-container">
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
<span class="org-comment">% Angle</span>
<span class="org-font-latex-sedate">\def\thetau</span>{25}
<span class="org-comment">% Rotational Stage</span>
<span class="org-font-latex-sedate">\draw</span>[fill=black!60!white] (0, 0) circle (4.3);
<span class="org-font-latex-sedate">\draw</span>[fill=black!40!white] (0, 0) circle (3.8);
<span class="org-comment">% Label</span>
<span class="org-font-latex-sedate">\node</span>[anchor=north west, rotate=<span class="org-font-latex-sedate">\thetau</span>] at (-2.5, 2.5) {<span class="org-font-latex-sedate"><span class="org-keyword">\small</span></span><span class="org-type"> Rotating Stage</span>};
<span class="org-comment">% Rotating Scope</span>
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[rotate=<span class="org-font-latex-sedate">\thetau</span>]
<span class="org-comment">% Rotating Frame</span>
<span class="org-font-latex-sedate">\draw</span>[fill=black!20!white] (-2.6, -2.6) rectangle (2.6, 2.6);
<span class="org-comment">% Label</span>
<span class="org-font-latex-sedate">\node</span>[anchor=north west, rotate=<span class="org-font-latex-sedate">\thetau</span>] at (-2.6, 2.6) {<span class="org-font-latex-sedate"><span class="org-keyword">\small</span></span><span class="org-type"> Suspended Platform</span>};
<span class="org-comment">% Mass</span>
<span class="org-font-latex-sedate">\draw</span>[fill=white] (-1, -1) rectangle (1, 1);
<span class="org-comment">% Label</span>
<span class="org-font-latex-sedate">\node</span>[anchor=south west, rotate=<span class="org-font-latex-sedate">\thetau</span>] at (-1, -1) {<span class="org-font-latex-sedate"><span class="org-keyword">\small</span></span><span class="org-type"> Payload</span>};
<span class="org-comment">% Attached Points</span>
<span class="org-font-latex-sedate">\draw</span>[] (-1, 0) -- ++(-0.2, 0) coordinate(au);
<span class="org-font-latex-sedate">\draw</span>[] (0, -1) -- ++(0, -0.2) coordinate(av);
<span class="org-comment">% Force Sensors</span>
<span class="org-font-latex-sedate">\draw</span>[fill=white] (<span class="org-font-latex-math">$(au) + (-0.2, -0.5)$</span>) rectangle (<span class="org-font-latex-math">$(au) + (0, 0.5)$</span>);
<span class="org-font-latex-sedate">\draw</span>[] (<span class="org-font-latex-math">$(au) + (-0.2, -0.5)$</span>)coordinate(actu) -- (<span class="org-font-latex-math">$(au) + (0, 0.5)$</span>);
<span class="org-font-latex-sedate">\draw</span>[] (<span class="org-font-latex-math">$(au) + (-0.2, 0.5)$</span>)coordinate(ku) -- (<span class="org-font-latex-math">$(au) + (0, -0.5)$</span>);
<span class="org-font-latex-sedate">\node</span>[below=0.1, rotate=<span class="org-font-latex-sedate">\thetau</span>] at (<span class="org-font-latex-math">$(au) + (-0.1, -0.5)$</span>) {<span class="org-font-latex-math">$f</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">{u}$</span>}
<span class="org-font-latex-sedate">\draw</span>[fill=white] (<span class="org-font-latex-math">$(av) + (-0.5, -0.2)$</span>) rectangle (<span class="org-font-latex-math">$(av) + (0.5, 0)$</span>);
<span class="org-font-latex-sedate">\draw</span>[] (<span class="org-font-latex-math">$(av) + ( 0.5, -0.2)$</span>)coordinate(actv) -- (<span class="org-font-latex-math">$(av) + (-0.5, 0)$</span>);
<span class="org-font-latex-sedate">\draw</span>[] (<span class="org-font-latex-math">$(av) + (-0.5, -0.2)$</span>)coordinate(kv) -- (<span class="org-font-latex-math">$(av) + ( 0.5, 0)$</span>) ;
<span class="org-font-latex-sedate">\node</span>[right=0.1, rotate=<span class="org-font-latex-sedate">\thetau</span>] at (<span class="org-font-latex-math">$(av) + (0.5, -0.1)$</span>) {<span class="org-font-latex-math">$f</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">{v}$</span>}
<span class="org-comment">% Spring and Actuator for U</span>
<span class="org-font-latex-sedate">\draw</span>[actuator={0.6}{0.2}] (actu) -- node[below=0.1, rotate=<span class="org-font-latex-sedate">\thetau</span>]{<span class="org-font-latex-math">$F</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">u$</span>} (actu-|-2.6,0);
<span class="org-font-latex-sedate">\draw</span>[spring=0.2] (ku) -- node[below=0.1, rotate=<span class="org-font-latex-sedate">\thetau</span>]{<span class="org-font-latex-math">$k</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">a$</span>} (ku-|-2.6,0);
<span class="org-font-latex-sedate">\draw</span>[spring=0.2] (-1, 0.8) -- node[above=0.1, rotate=<span class="org-font-latex-sedate">\thetau</span>]{<span class="org-font-latex-math">$k</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">p$</span>} (-1, 0.8-|-2.6,0);
<span class="org-font-latex-sedate">\draw</span>[actuator={0.6}{0.2}] (actv) -- node[right=0.1, rotate=<span class="org-font-latex-sedate">\thetau</span>]{<span class="org-font-latex-math">$F</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">v$</span>} (actv|-0,-2.6);
<span class="org-font-latex-sedate">\draw</span>[spring=0.2] (kv) -- node[right=0.1, rotate=<span class="org-font-latex-sedate">\thetau</span>]{<span class="org-font-latex-math">$k</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">a$</span>} (kv|-0,-2.6);
<span class="org-font-latex-sedate">\draw</span>[spring=0.2] (-0.8, -1) -- node[left=0.1, rotate=<span class="org-font-latex-sedate">\thetau</span>]{<span class="org-font-latex-math">$k</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">p$</span>} (-0.8, -1|-0,-2.6);
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
<span class="org-comment">% Inertial Frame</span>
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (-4, -4) -- ++(2, 0) node[below]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\vec</span></span><span class="org-font-latex-math">{i}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">x$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (-4, -4) -- ++(0, 2) node[left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\vec</span></span><span class="org-font-latex-math">{i}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">y$</span>};
<span class="org-font-latex-sedate">\draw</span>[fill, color=black] (-4, -4) circle (0.06);
<span class="org-font-latex-sedate">\node</span>[draw, circle, inner sep=0pt, minimum size=0.3cm, label=left:<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\vec</span></span><span class="org-font-latex-math">{i}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">z$</span>] at (-4, -4){};
<span class="org-font-latex-sedate">\node</span>[draw, circle, inner sep=0pt, minimum size=0.3cm] at (0, 0){};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (0, 0) node[above left, rotate=<span class="org-font-latex-sedate">\thetau</span>]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\vec</span></span><span class="org-font-latex-math">{i}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">w$</span>} -- ++(<span class="org-font-latex-sedate">\thetau</span>:2) node[above, rotate=<span class="org-font-latex-sedate">\thetau</span>]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\vec</span></span><span class="org-font-latex-math">{i}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">u$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (0, 0) -- ++(<span class="org-font-latex-sedate">\thetau</span>+90:2) node[left, rotate=<span class="org-font-latex-sedate">\thetau</span>]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\vec</span></span><span class="org-font-latex-math">{i}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">v$</span>};
<span class="org-font-latex-sedate">\draw</span>[dashed] (0, 0) -- ++(2, 0);
<span class="org-font-latex-sedate">\draw</span>[] (1.5, 0) arc (0:<span class="org-font-latex-sedate">\thetau</span>:1.5) node[midway, right]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\theta</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\node</span>[] at (0,0) {<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\bullet</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (3.5, 0) arc (0:40:3.5) node[midway, left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\Omega</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
</pre>
</div>
<div id="orgc785e50" class="figure">
<p><img src="figs/system_parallel_springs.png" alt="system_parallel_springs.png" />
</p>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Thomas Dehaeze</p>
<p class="date">Created: 2020-11-12 jeu. 10:40</p>
</div>
</body>
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