From 48b2d364186a37d8bffb8b37b8f47bbfb2a06bfd Mon Sep 17 00:00:00 2001
From: Thomas Dehaeze <dehaeze.thomas@gmail.com>
Date: Tue, 7 Apr 2020 16:19:15 +0200
Subject: [PATCH] Correct wrong sentence about mechanical impedance

---
 docs/uncertainty_payload.html | 41 ++++++++++++++++++-----------------
 org/uncertainty_payload.org   |  9 ++++----
 2 files changed, 25 insertions(+), 25 deletions(-)
diff --git a/docs/uncertainty_payload.html b/docs/uncertainty_payload.html
index 3ba1358..996eb62 100644
--- a/docs/uncertainty_payload.html
+++ b/docs/uncertainty_payload.html
@@ -4,7 +4,7 @@
 "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
 <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
 <head>
-<!-- 2020-04-01 mer. 16:14 -->
+<!-- 2020-04-07 mar. 16:17 -->
 <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
 <meta name="viewport" content="width=device-width, initial-scale=1" />
 <title>Effect of Uncertainty on the payload&rsquo;s dynamics on the isolation platform dynamics</title>
@@ -227,7 +227,9 @@
 </script>
 <script>
       MathJax = {
-          tex: { macros: {
+          tex: {
+          tags: 'ams',
+          macros: {
                   bm: ["\\boldsymbol{#1}",1],
                   }
               }
@@ -249,17 +251,17 @@
 <ul>
 <li><a href="#orgcc5f0ec">1. Simple Introductory Example</a>
 <ul>
-<li><a href="#org6264842">1.1. Equations of motion</a></li>
+<li><a href="#org1f20d62">1.1. Equations of motion</a></li>
 <li><a href="#org4efccbf">1.2. Initialization of the payload dynamics</a></li>
 <li><a href="#orgb400ca3">1.3. Initialization of the isolation platform</a></li>
 <li><a href="#orgd0dd88b">1.4. Comparison</a></li>
-<li><a href="#org1b051ce">1.5. Conclusion</a></li>
+<li><a href="#org1637b13">1.5. Conclusion</a></li>
 </ul>
 </li>
 <li><a href="#org1f8e63e">2. Generalization to arbitrary dynamics</a>
 <ul>
 <li><a href="#orgc4fa63e">2.1. Introduction</a></li>
-<li><a href="#org35ac80d">2.2. Equations of motion</a></li>
+<li><a href="#orgd6da9a7">2.2. Equations of motion</a></li>
 <li><a href="#orge217a33">2.3. Impedance \(G^\prime(s)\) of a mass-spring-damper payload</a></li>
 <li><a href="#org0ee44da">2.4. First Analytical analysis</a></li>
 <li><a href="#orgfe81c1c">2.5. Impedance of the Payload and Dynamical Uncertainty</a></li>
@@ -272,7 +274,7 @@
 <li><a href="#org9086831">2.8.3. Effect of the platform&rsquo;s mass \(m\)</a></li>
 </ul>
 </li>
-<li><a href="#org43f33dc">2.9. Conclusion</a></li>
+<li><a href="#org3a1ebf1">2.9. Conclusion</a></li>
 </ul>
 </li>
 </ul>
@@ -326,8 +328,8 @@ The goal is to stabilize \(x\) using \(F\) in spite of uncertainty on the payloa
 </div>
 </div>
 
-<div id="outline-container-org6264842" class="outline-3">
-<h3 id="org6264842"><span class="section-number-3">1.1</span> Equations of motion</h3>
+<div id="outline-container-org1f20d62" class="outline-3">
+<h3 id="org1f20d62"><span class="section-number-3">1.1</span> Equations of motion</h3>
 <div class="outline-text-3" id="text-1-1">
 <p>
 If we write the equation of motion of the system in Figure <a href="#orgaa77a57">1</a>, we obtain:
@@ -434,8 +436,8 @@ The obtained dynamics from \(F\) to \(x\) for the three isolation platform are s
 </div>
 </div>
 
-<div id="outline-container-org1b051ce" class="outline-3">
-<h3 id="org1b051ce"><span class="section-number-3">1.5</span> Conclusion</h3>
+<div id="outline-container-org1637b13" class="outline-3">
+<h3 id="org1637b13"><span class="section-number-3">1.5</span> Conclusion</h3>
 <div class="outline-text-3" id="text-1-5">
 <div class="important">
 <p>
@@ -458,12 +460,12 @@ The stiff platform dynamics does not seems to depend on the dynamics of the payl
 <h3 id="orgc4fa63e"><span class="section-number-3">2.1</span> Introduction</h3>
 <div class="outline-text-3" id="text-2-1">
 <p>
-Let&rsquo;s now consider a general payload described by its <b>impedance</b> \(G^\prime(s) = \frac{x}{F^\prime}\) as shown in Figure <a href="#orgb54b79a">4</a>.
+Let&rsquo;s now consider a general payload described by its <b>impedance</b> \(G^\prime(s) = \frac{F^\prime}{x}\) as shown in Figure <a href="#orgb54b79a">4</a>.
 </p>
 
 <div class="note">
 <p>
-Note here that we use the term <i>impedance</i>, however, the mechanical impedance is usually defined as the ratio of the velocity over the force \(\dot{x}/F^\prime\). We should refer to <i>resistance</i> instead of <i>impedance</i>.
+Note here that we use the term <i>impedance</i>, however, the mechanical impedance is usually defined as the ratio of the force over the velocity \(F^\prime/\dot{x}\). We should refer to <i>resistance</i> instead of <i>impedance</i>.
 </p>
 
 </div>
@@ -487,8 +489,8 @@ Now let&rsquo;s consider the system consisting of a mass-spring-system (the isol
 </div>
 </div>
 
-<div id="outline-container-org35ac80d" class="outline-3">
-<h3 id="org35ac80d"><span class="section-number-3">2.2</span> Equations of motion</h3>
+<div id="outline-container-orgd6da9a7" class="outline-3">
+<h3 id="orgd6da9a7"><span class="section-number-3">2.2</span> Equations of motion</h3>
 <div class="outline-text-3" id="text-2-2">
 <p>
 We have to following equations of motion:
@@ -536,12 +538,11 @@ By eliminating \(x^\prime\) of the equations, we obtain:
 </p>
 <div class="important">
 \begin{equation}
-\label{orgae0b162}
-  G^\prime(s) = \frac{F^\prime}{x} = \frac{m^\prime s^2 (c^\prime s + k)}{m^\prime s^2 + c^\prime s + k^\prime}
+  G^\prime(s) = \frac{F^\prime}{x} = \frac{m^\prime s^2 (c^\prime s + k)}{m^\prime s^2 + c^\prime s + k^\prime} \label{eq:impedance_mass_spring_damper}
 \end{equation}
 
 <p>
-The impedance of a 1dof mass-spring-damper system is described by Eq. \eqref{orgae0b162}.
+The impedance of a 1dof mass-spring-damper system is described by Eq. \eqref{eq:impedance_mass_spring_damper}.
 </p>
 
 </div>
@@ -945,8 +946,8 @@ Let&rsquo;s fix \(k = 10^7\ [N/m]\), \(\xi = \frac{c}{2\sqrt{km}} = 0.1\) and se
 </div>
 </div>
 
-<div id="outline-container-org43f33dc" class="outline-3">
-<h3 id="org43f33dc"><span class="section-number-3">2.9</span> Conclusion</h3>
+<div id="outline-container-org3a1ebf1" class="outline-3">
+<h3 id="org3a1ebf1"><span class="section-number-3">2.9</span> Conclusion</h3>
 <div class="outline-text-3" id="text-2-9">
 <div class="important">
 <p>
@@ -970,7 +971,7 @@ In that case, maximizing the stiffness of the payload is a good idea.
 </div>
 <div id="postamble" class="status">
 <p class="author">Author: Dehaeze Thomas</p>
-<p class="date">Created: 2020-04-01 mer. 16:14</p>
+<p class="date">Created: 2020-04-07 mar. 16:17</p>
 </div>
 </body>
 </html>
diff --git a/org/uncertainty_payload.org b/org/uncertainty_payload.org
index 3eefe19..cc59e19 100644
--- a/org/uncertainty_payload.org
+++ b/org/uncertainty_payload.org
@@ -337,10 +337,10 @@ The obtained dynamics from $F$ to $x$ for the three isolation platform are shown
 * Generalization to arbitrary dynamics
 <<sec:arbitrary_dynamics>>
 ** Introduction
-Let's now consider a general payload described by its *impedance* $G^\prime(s) = \frac{x}{F^\prime}$ as shown in Figure [[fig:general_payload_impedance]].
+Let's now consider a general payload described by its *impedance* $G^\prime(s) = \frac{F^\prime}{x}$ as shown in Figure [[fig:general_payload_impedance]].
 
 #+begin_note
-Note here that we use the term /impedance/, however, the mechanical impedance is usually defined as the ratio of the velocity over the force $\dot{x}/F^\prime$. We should refer to /resistance/ instead of /impedance/.
+Note here that we use the term /impedance/, however, the mechanical impedance is usually defined as the ratio of the force over the velocity $F^\prime/\dot{x}$. We should refer to /resistance/ instead of /impedance/.
 #+end_note
 
 #+begin_src latex :file general_payload_impedance.pdf
@@ -455,12 +455,11 @@ In order to verify that the formula is correct, let's take the same mass-spring-
 
 By eliminating $x^\prime$ of the equations, we obtain:
 #+begin_important
-#+name: eq:impedance_mass_spring_damper
 \begin{equation}
-  G^\prime(s) = \frac{F^\prime}{x} = \frac{m^\prime s^2 (c^\prime s + k)}{m^\prime s^2 + c^\prime s + k^\prime}
+  G^\prime(s) = \frac{F^\prime}{x} = \frac{m^\prime s^2 (c^\prime s + k)}{m^\prime s^2 + c^\prime s + k^\prime} \label{eq:impedance_mass_spring_damper}
 \end{equation}
 
-The impedance of a 1dof mass-spring-damper system is described by Eq. [[eq:impedance_mass_spring_damper]].
+The impedance of a 1dof mass-spring-damper system is described by Eq. eqref:eq:impedance_mass_spring_damper.
 #+end_important
 
 And we obtain
