Initial Commit

2020-11-10 12:55:51 +01:00
commit bd06f66a2a
51 changed files with 9764 additions and 0 deletions
--- a/css/custom.css
+++ b/css/custom.css
@@ -0,0 +1,49 @@
+.figure p{
+    text-align: center;
+}
+
+.figure img{
+    max-width:100%;
+    display: block;
+    margin: auto;
+}
+
+table {
+    margin-left: auto;
+    margin-right: auto;
+}
+
+.org-src-container > pre.src:before {
+  display: inline;
+  position: absolute;
+  color: #808080;
+  background-color: white;
+  top: -10px;
+  left: 10px;
+  padding: 0px 4px;
+  border: 1px solid #d0d0d0;
+  font-size: 80%;
+}
+.org-src-container > pre {
+  margin-top: 1.5em;
+  position: relative;
+  overflow: visible;
+}
+.org-src-container > pre > code.src:before {
+  display: inline;
+  position: absolute;
+  color: #808080;
+  background-color: white;
+  top: -10px;
+  left: 10px;
+  padding: 0px 4px;
+  border: 1px solid #d0d0d0;
+  font-size: 80%;
+}
+.org-src-container > pre.src-emacs-lisp:before { content: 'Emacs Lisp'; }
+.org-src-container > pre.src-elisp:before { content: 'Emacs Lisp'; }
+.org-src-container > pre.src-sh:before { content: 'shell'; }
+.org-src-container > pre.src-bash:before { content: 'bash'; }
+.org-src-container > pre.src-org:before { content: 'Org mode'; }
+.org-src-container > pre.src-python:before { content: 'Python'; }
+.org-src-container > pre.src-matlab:before { content: 'Matlab'; }
--- a/css/htmlize.css
+++ b/css/htmlize.css
@@ -0,0 +1,145 @@
+.org-bold { /* bold */ font-weight: bold; }
+.org-bold-italic { /* bold-italic */ font-weight: bold; font-style: italic; }
+.org-buffer-menu-buffer { /* buffer-menu-buffer */ font-weight: bold; }
+.org-builtin { /* font-lock-builtin-face */ color: #7a378b; }
+.org-button { /* button */ text-decoration: underline; }
+.org-calendar-today { /* calendar-today */ text-decoration: underline; }
+.org-change-log-acknowledgement { /* change-log-acknowledgement */ color: #b22222; }
+.org-change-log-conditionals { /* change-log-conditionals */ color: #a0522d; }
+.org-change-log-date { /* change-log-date */ color: #8b2252; }
+.org-change-log-email { /* change-log-email */ color: #a0522d; }
+.org-change-log-file { /* change-log-file */ color: #0000ff; }
+.org-change-log-function { /* change-log-function */ color: #a0522d; }
+.org-change-log-list { /* change-log-list */ color: #a020f0; }
+.org-change-log-name { /* change-log-name */ color: #008b8b; }
+.org-comint-highlight-input { /* comint-highlight-input */ font-weight: bold; }
+.org-comint-highlight-prompt { /* comint-highlight-prompt */ color: #00008b; }
+.org-comment { /* font-lock-comment-face */ color: #999988; font-style: italic; }
+.org-comment-delimiter { /* font-lock-comment-delimiter-face */ color: #999988; font-style: italic; }
+.org-completions-annotations { /* completions-annotations */ font-style: italic; }
+.org-completions-common-part { /* completions-common-part */ color: #000000; background-color: #ffffff; }
+.org-completions-first-difference { /* completions-first-difference */ font-weight: bold; }
+.org-constant { /* font-lock-constant-face */ color: #008b8b; }
+.org-diary { /* diary */ color: #ff0000; }
+.org-diff-context { /* diff-context */ color: #7f7f7f; }
+.org-diff-file-header { /* diff-file-header */ background-color: #b3b3b3; font-weight: bold; }
+.org-diff-function { /* diff-function */ background-color: #cccccc; }
+.org-diff-header { /* diff-header */ background-color: #cccccc; }
+.org-diff-hunk-header { /* diff-hunk-header */ background-color: #cccccc; }
+.org-diff-index { /* diff-index */ background-color: #b3b3b3; font-weight: bold; }
+.org-diff-nonexistent { /* diff-nonexistent */ background-color: #b3b3b3; font-weight: bold; }
+.org-diff-refine-change { /* diff-refine-change */ background-color: #d9d9d9; }
+.org-dired-directory { /* dired-directory */ color: #0000ff; }
+.org-dired-flagged { /* dired-flagged */ color: #ff0000; font-weight: bold; }
+.org-dired-header { /* dired-header */ color: #228b22; }
+.org-dired-ignored { /* dired-ignored */ color: #7f7f7f; }
+.org-dired-mark { /* dired-mark */ color: #008b8b; }
+.org-dired-marked { /* dired-marked */ color: #ff0000; font-weight: bold; }
+.org-dired-perm-write { /* dired-perm-write */ color: #b22222; }
+.org-dired-symlink { /* dired-symlink */ color: #a020f0; }
+.org-dired-warning { /* dired-warning */ color: #ff0000; font-weight: bold; }
+.org-doc { /* font-lock-doc-face */ color: #8b2252; }
+.org-escape-glyph { /* escape-glyph */ color: #a52a2a; }
+.org-file-name-shadow { /* file-name-shadow */ color: #7f7f7f; }
+.org-flyspell-duplicate { /* flyspell-duplicate */ color: #cdad00; font-weight: bold; text-decoration: underline; }
+.org-flyspell-incorrect { /* flyspell-incorrect */ color: #ff4500; font-weight: bold; text-decoration: underline; }
+.org-fringe { /* fringe */ background-color: #f2f2f2; }
+.org-function-name { /* font-lock-function-name-face */ color: teal; }
+.org-header-line { /* header-line */ color: #333333; background-color: #e5e5e5; }
+.org-help-argument-name { /* help-argument-name */ font-style: italic; }
+.org-highlight { /* highlight */ background-color: #b4eeb4; }
+.org-holiday { /* holiday */ background-color: #ffc0cb; }
+.org-isearch { /* isearch */ color: #b0e2ff; background-color: #cd00cd; }
+.org-isearch-fail { /* isearch-fail */ background-color: #ffc1c1; }
+.org-italic { /* italic */ font-style: italic; }
+.org-keyword { /* font-lock-keyword-face */ color: #0086b3; }
+.org-lazy-highlight { /* lazy-highlight */ background-color: #afeeee; }
+.org-link { /* link */ color: #0000ff; text-decoration: underline; }
+.org-link-visited { /* link-visited */ color: #8b008b; text-decoration: underline; }
+.org-log-edit-header { /* log-edit-header */ color: #a020f0; }
+.org-log-edit-summary { /* log-edit-summary */ color: #0000ff; }
+.org-log-edit-unknown-header { /* log-edit-unknown-header */ color: #b22222; }
+.org-match { /* match */ background-color: #ffff00; }
+.org-next-error { /* next-error */ background-color: #eedc82; }
+.org-nobreak-space { /* nobreak-space */ color: #a52a2a; text-decoration: underline; }
+.org-org-archived { /* org-archived */ color: #7f7f7f; }
+.org-org-block { /* org-block */ color: #7f7f7f; }
+.org-org-block-begin-line { /* org-block-begin-line */ color: #b22222; }
+.org-org-block-end-line { /* org-block-end-line */ color: #b22222; }
+.org-org-checkbox { /* org-checkbox */ font-weight: bold; }
+.org-org-checkbox-statistics-done { /* org-checkbox-statistics-done */ color: #228b22; font-weight: bold; }
+.org-org-checkbox-statistics-todo { /* org-checkbox-statistics-todo */ color: #ff0000; font-weight: bold; }
+.org-org-clock-overlay { /* org-clock-overlay */ background-color: #ffff00; }
+.org-org-code { /* org-code */ color: #7f7f7f; }
+.org-org-column { /* org-column */ background-color: #e5e5e5; }
+.org-org-column-title { /* org-column-title */ background-color: #e5e5e5; font-weight: bold; text-decoration: underline; }
+.org-org-date { /* org-date */ color: #a020f0; text-decoration: underline; }
+.org-org-document-info { /* org-document-info */ color: #191970; }
+.org-org-document-info-keyword { /* org-document-info-keyword */ color: #7f7f7f; }
+.org-org-document-title { /* org-document-title */ color: #191970; font-size: 144%; font-weight: bold; }
+.org-org-done { /* org-done */ color: #228b22; font-weight: bold; }
+.org-org-drawer { /* org-drawer */ color: #0000ff; }
+.org-org-ellipsis { /* org-ellipsis */ color: #b8860b; text-decoration: underline; }
+.org-org-footnote { /* org-footnote */ color: #a020f0; text-decoration: underline; }
+.org-org-formula { /* org-formula */ color: #b22222; }
+.org-org-headline-done { /* org-headline-done */ color: #bc8f8f; }
+.org-org-hide { /* org-hide */ color: #ffffff; }
+.org-org-latex-and-export-specials { /* org-latex-and-export-specials */ color: #8b4513; }
+.org-org-level-1 { /* org-level-1 */ color: #0000ff; }
+.org-org-level-2 { /* org-level-2 */ color: #a0522d; }
+.org-org-level-3 { /* org-level-3 */ color: #a020f0; }
+.org-org-level-4 { /* org-level-4 */ color: #b22222; }
+.org-org-level-5 { /* org-level-5 */ color: #228b22; }
+.org-org-level-6 { /* org-level-6 */ color: #008b8b; }
+.org-org-level-7 { /* org-level-7 */ color: #7a378b; }
+.org-org-level-8 { /* org-level-8 */ color: #8b2252; }
+.org-org-link { /* org-link */ color: #0000ff; text-decoration: underline; }
+.org-org-meta-line { /* org-meta-line */ color: #b22222; }
+.org-org-mode-line-clock { /* org-mode-line-clock */ color: #000000; background-color: #bfbfbf; }
+.org-org-mode-line-clock-overrun { /* org-mode-line-clock-overrun */ color: #000000; background-color: #ff0000; }
+.org-org-quote { /* org-quote */ color: #7f7f7f; }
+.org-org-scheduled { /* org-scheduled */ color: #006400; }
+.org-org-scheduled-previously { /* org-scheduled-previously */ color: #b22222; }
+.org-org-scheduled-today { /* org-scheduled-today */ color: #006400; }
+.org-org-sexp-date { /* org-sexp-date */ color: #a020f0; }
+.org-org-special-keyword { /* org-special-keyword */ color: #a020f0; }
+.org-org-table { /* org-table */ color: #0000ff; }
+.org-org-tag { /* org-tag */ font-weight: bold; }
+.org-org-target { /* org-target */ text-decoration: underline; }
+.org-org-time-grid { /* org-time-grid */ color: #b8860b; }
+.org-org-todo { /* org-todo */ color: #ff0000; font-weight: bold; }
+.org-org-upcoming-deadline { /* org-upcoming-deadline */ color: #b22222; }
+.org-org-verbatim { /* org-verbatim */ color: #7f7f7f; }
+.org-org-verse { /* org-verse */ color: #7f7f7f; }
+.org-org-warning { /* org-warning */ color: #ff0000; font-weight: bold; }
+.org-outline-1 { /* outline-1 */ color: #0000ff; }
+.org-outline-2 { /* outline-2 */ color: #a0522d; }
+.org-outline-3 { /* outline-3 */ color: #a020f0; }
+.org-outline-4 { /* outline-4 */ color: #b22222; }
+.org-outline-5 { /* outline-5 */ color: #228b22; }
+.org-outline-6 { /* outline-6 */ color: #008b8b; }
+.org-outline-7 { /* outline-7 */ color: #7a378b; }
+.org-outline-8 { /* outline-8 */ color: #8b2252; }
+.org-preprocessor { /* font-lock-preprocessor-face */ color: #7a378b; }
+.org-query-replace { /* query-replace */ color: #b0e2ff; background-color: #cd00cd; }
+.org-regexp-grouping-backslash { /* font-lock-regexp-grouping-backslash */ font-weight: bold; }
+.org-regexp-grouping-construct { /* font-lock-regexp-grouping-construct */ font-weight: bold; }
+.org-region { /* region */ background-color: #eedc82; }
+.org-secondary-selection { /* secondary-selection */ background-color: #ffff00; }
+.org-shadow { /* shadow */ color: #7f7f7f; }
+.org-show-paren-match { /* show-paren-match */ background-color: #40e0d0; }
+.org-show-paren-mismatch { /* show-paren-mismatch */ color: #ffffff; background-color: #a020f0; }
+.org-string { /* font-lock-string-face */ color: #dd1144; }
+.org-tool-bar { /* tool-bar */ color: #000000; background-color: #bfbfbf; }
+.org-tooltip { /* tooltip */ color: #000000; background-color: #ffffe0; }
+.org-trailing-whitespace { /* trailing-whitespace */ background-color: #ff0000; }
+.org-type { /* font-lock-type-face */ color: #228b22; }
+.org-underline { /* underline */ text-decoration: underline; }
+.org-variable-name { /* font-lock-variable-name-face */ color: teal; }
+.org-warning { /* font-lock-warning-face */ color: #ff0000; font-weight: bold; }
+.org-widget-button { /* widget-button */ font-weight: bold; }
+.org-widget-button-pressed { /* widget-button-pressed */ color: #ff0000; }
+.org-widget-documentation { /* widget-documentation */ color: #006400; }
+.org-widget-field { /* widget-field */ background-color: #d9d9d9; }
+.org-widget-inactive { /* widget-inactive */ color: #7f7f7f; }
+.org-widget-single-line-field { /* widget-single-line-field */ background-color: #d9d9d9; }
--- a/css/readtheorg.css
+++ b/css/readtheorg.css
--- a/figs/IMG_20201023_153905.jpg
+++ b/figs/IMG_20201023_153905.jpg
--- a/figs/IMG_20201023_153914.jpg
+++ b/figs/IMG_20201023_153914.jpg
--- a/figs/exp_setup_schematic.pdf
+++ b/figs/exp_setup_schematic.pdf
--- a/figs/exp_setup_schematic.png
+++ b/figs/exp_setup_schematic.png
--- a/figs/exp_setup_schematic.svg
+++ b/figs/exp_setup_schematic.svg
--- a/figs/force_sen_steps_time_domain.pdf
+++ b/figs/force_sen_steps_time_domain.pdf
--- a/figs/force_sen_steps_time_domain.png
+++ b/figs/force_sen_steps_time_domain.png
--- a/figs/force_sen_steps_time_domain_par_R.pdf
+++ b/figs/force_sen_steps_time_domain_par_R.pdf
--- a/figs/force_sen_steps_time_domain_par_R.png
+++ b/figs/force_sen_steps_time_domain_par_R.png
--- a/figs/force_sensor_linear_relation.pdf
+++ b/figs/force_sensor_linear_relation.pdf
--- a/figs/force_sensor_linear_relation.png
+++ b/figs/force_sensor_linear_relation.png
--- a/figs/force_sensor_model_electronics.png
+++ b/figs/force_sensor_model_electronics.png
--- a/figs/force_sensor_model_electronics.svg
+++ b/figs/force_sensor_model_electronics.svg
--- a/figs/force_sensor_model_electronics_without_R.png
+++ b/figs/force_sensor_model_electronics_without_R.png
--- a/figs/force_sensor_model_electronics_without_R.svg
+++ b/figs/force_sensor_model_electronics_without_R.svg
--- a/figs/force_sensor_setup.pdf
+++ b/figs/force_sensor_setup.pdf
--- a/figs/force_sensor_setup.png
+++ b/figs/force_sensor_setup.png
--- a/figs/force_sensor_sin_charge.pdf
+++ b/figs/force_sensor_sin_charge.pdf
--- a/figs/force_sensor_sin_charge.png
+++ b/figs/force_sensor_sin_charge.png
--- a/figs/force_sensor_sin_encoder.pdf
+++ b/figs/force_sensor_sin_encoder.pdf
--- a/figs/force_sensor_sin_encoder.png
+++ b/figs/force_sensor_sin_encoder.png
--- a/figs/force_sensor_sin_stack.pdf
+++ b/figs/force_sensor_sin_stack.pdf
--- a/figs/force_sensor_sin_stack.png
+++ b/figs/force_sensor_sin_stack.png
--- a/figs/force_sensor_sin_u.pdf
+++ b/figs/force_sensor_sin_u.pdf
--- a/figs/force_sensor_sin_u.png
+++ b/figs/force_sensor_sin_u.png
--- a/figs/stiffness_force_sensor_bode.pdf
+++ b/figs/stiffness_force_sensor_bode.pdf
--- a/figs/stiffness_force_sensor_bode.png
+++ b/figs/stiffness_force_sensor_bode.png
--- a/figs/stiffness_force_sensor_bode_zoom.pdf
+++ b/figs/stiffness_force_sensor_bode_zoom.pdf
--- a/figs/stiffness_force_sensor_bode_zoom.png
+++ b/figs/stiffness_force_sensor_bode_zoom.png
--- a/figs/stiffness_force_sensor_coherence.pdf
+++ b/figs/stiffness_force_sensor_coherence.pdf
--- a/figs/stiffness_force_sensor_coherence.png
+++ b/figs/stiffness_force_sensor_coherence.png
--- a/index.html
+++ b/index.html
@@ -0,0 +1,637 @@
+<?xml version="1.0" encoding="utf-8"?>
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
+<head>
+<!-- 2020-11-10 mar. 12:55 -->
+<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
+<title>Piezoelectric Force Sensor - Test Bench</title>
+<meta name="generator" content="Org mode" />
+<meta name="author" content="Dehaeze Thomas" />
+<link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
+<link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
+<link rel="stylesheet" type="text/css" href="./css/custom.css"/>
+<script type="text/javascript" src="./js/jquery.min.js"></script>
+<script type="text/javascript" src="./js/bootstrap.min.js"></script>
+<script type="text/javascript" src="./js/readtheorg.js"></script>
+<script>MathJax = {
+          tex: {
+            tags: 'ams',
+            macros: {bm: ["\\boldsymbol{#1}",1],}
+            }
+          };
+          </script>
+          <script type="text/javascript" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
+</head>
+<body>
+<div id="org-div-home-and-up">
+ <a accesskey="h" href="../index.html"> UP </a>
+ |
+ <a accesskey="H" href="../index.html"> HOME </a>
+</div><div id="content">
+<h1 class="title">Piezoelectric Force Sensor - Test Bench</h1>
+<div id="table-of-contents">
+<h2>Table of Contents</h2>
+<div id="text-table-of-contents">
+<ul>
+<li><a href="#orga1465ad">1. Change of Stiffness due to Sensors stack being open/closed circuit</a>
+<ul>
+<li><a href="#orgd924c73">1.1. Load Data</a></li>
+<li><a href="#org59cc20a">1.2. Transfer Functions</a></li>
+</ul>
+</li>
+<li><a href="#org76a1832">2. Generated Number of Charge / Voltage</a>
+<ul>
+<li><a href="#org1fa991d">2.1. Steps</a></li>
+<li><a href="#org5e9eb44">2.2. Add Parallel Resistor</a></li>
+<li><a href="#org15676e1">2.3. Sinus</a></li>
+</ul>
+</li>
+</ul>
+</div>
+</div>
+
+<p>
+In this document is studied how a piezoelectric stack can be used to measured the force.
+</p>
+
+<ul class="org-ul">
+<li>Section <a href="#org887b61a">1</a>: the effect of the input impedance of the electronics connected to the force sensor stack on the stiffness of the stack is studied</li>
+<li>Section <a href="#org2b5f630">2</a>:</li>
+</ul>
+
+<div id="outline-container-orga1465ad" class="outline-2">
+<h2 id="orga1465ad"><span class="section-number-2">1</span> Change of Stiffness due to Sensors stack being open/closed circuit</h2>
+<div class="outline-text-2" id="text-1">
+<p>
+<a id="org887b61a"></a>
+</p>
+</div>
+
+<div id="outline-container-orgd924c73" class="outline-3">
+<h3 id="orgd924c73"><span class="section-number-3">1.1</span> Load Data</h3>
+<div class="outline-text-3" id="text-1-1">
+<div class="org-src-container">
+<pre class="src src-matlab">oc = load(<span class="org-string">'identification_open_circuit.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'encoder'</span>, <span class="org-string">'u'</span>);
+sc = load(<span class="org-string">'identification_short_circuit.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'encoder'</span>, <span class="org-string">'u'</span>);
+</pre>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org59cc20a" class="outline-3">
+<h3 id="org59cc20a"><span class="section-number-3">1.2</span> Transfer Functions</h3>
+<div class="outline-text-3" id="text-1-2">
+<div class="org-src-container">
+<pre class="src src-matlab">Ts = 1e<span class="org-type">-</span>4; <span class="org-comment">% Sampling Time [s]</span>
+win = hann(ceil(10<span class="org-type">/</span>Ts));
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">[tf_oc_est, f] = tfestimate(oc.u, oc.encoder, win, [], [], 1<span class="org-type">/</span>Ts);
+[co_oc_est, <span class="org-type">~</span>] = mscohere(  oc.u, oc.encoder, win, [], [], 1<span class="org-type">/</span>Ts);
+
+[tf_sc_est, <span class="org-type">~</span>] = tfestimate(sc.u, sc.encoder, win, [], [], 1<span class="org-type">/</span>Ts);
+[co_sc_est, <span class="org-type">~</span>] = mscohere(  sc.u, sc.encoder, win, [], [], 1<span class="org-type">/</span>Ts);
+</pre>
+</div>
+
+
+<div id="org3c75143" class="figure">
+<p><img src="figs/stiffness_force_sensor_coherence.png" alt="stiffness_force_sensor_coherence.png" />
+</p>
+</div>
+
+
+
+<div id="org4424b1c" class="figure">
+<p><img src="figs/stiffness_force_sensor_bode.png" alt="stiffness_force_sensor_bode.png" />
+</p>
+</div>
+
+
+<div id="org216fcc3" class="figure">
+<p><img src="figs/stiffness_force_sensor_bode_zoom.png" alt="stiffness_force_sensor_bode_zoom.png" />
+</p>
+<p><span class="figure-number">Figure 3: </span>Zoom on the change of resonance</p>
+</div>
+
+<div class="important" id="org9ea3712">
+<p>
+The change of resonance frequency / stiffness is very small and is not important here.
+</p>
+
+</div>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org76a1832" class="outline-2">
+<h2 id="org76a1832"><span class="section-number-2">2</span> Generated Number of Charge / Voltage</h2>
+<div class="outline-text-2" id="text-2">
+<p>
+<a id="org2b5f630"></a>
+</p>
+<p>
+Two stacks are used as actuator (in parallel) and one stack is used as sensor.
+</p>
+
+<p>
+The amplifier gain is 20V/V (Cedrat LA75B).
+</p>
+</div>
+
+<div id="outline-container-org1fa991d" class="outline-3">
+<h3 id="org1fa991d"><span class="section-number-3">2.1</span> Steps</h3>
+<div class="outline-text-3" id="text-2-1">
+<div class="org-src-container">
+<pre class="src src-matlab">load(<span class="org-string">'force_sensor_steps.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'encoder'</span>, <span class="org-string">'u'</span>, <span class="org-string">'v'</span>);
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab"><span class="org-type">figure</span>;
+tiledlayout(2, 1, <span class="org-string">'TileSpacing'</span>, <span class="org-string">'None'</span>, <span class="org-string">'Padding'</span>, <span class="org-string">'None'</span>);
+nexttile;
+plot(t, v);
+xlabel(<span class="org-string">'Time [s]'</span>); ylabel(<span class="org-string">'Measured voltage [V]'</span>);
+nexttile;
+plot(t, u);
+xlabel(<span class="org-string">'Time [s]'</span>); ylabel(<span class="org-string">'Actuator Voltage [V]'</span>);
+</pre>
+</div>
+
+
+<div id="orgf889803" class="figure">
+<p><img src="figs/force_sen_steps_time_domain.png" alt="force_sen_steps_time_domain.png" />
+</p>
+<p><span class="figure-number">Figure 4: </span>Time domain signal during the 3 actuator voltage steps</p>
+</div>
+
+<p>
+Three steps are performed at the following time intervals:
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">t_s = [ 2.5, 23;
+       23.8, 35;
+       35.8, 50];
+</pre>
+</div>
+
+<p>
+Fit function:
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">f = @(b,x) b(1)<span class="org-type">.*</span>exp(b(2)<span class="org-type">.*</span>x) <span class="org-type">+</span> b(3);
+</pre>
+</div>
+
+<p>
+We are interested by the <code>b(2)</code> term, which is the time constant of the exponential.
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">tau = zeros(size(t_s, 1),1);
+V0  = zeros(size(t_s, 1),1);
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab"><span class="org-keyword">for</span> <span class="org-variable-name">t_i</span> = <span class="org-constant">1:size(t_s, 1)</span>
+    t_cur = t(t_s(t_i, 1) <span class="org-type">&lt;</span> t <span class="org-type">&amp;</span> t <span class="org-type">&lt;</span> t_s(t_i, 2));
+    t_cur = t_cur <span class="org-type">-</span> t_cur(1);
+    y_cur = v(t_s(t_i, 1) <span class="org-type">&lt;</span> t <span class="org-type">&amp;</span> t <span class="org-type">&lt;</span> t_s(t_i, 2));
+
+    nrmrsd = @(b) norm(y_cur <span class="org-type">-</span> f(b,t_cur)); <span class="org-comment">% Residual Norm Cost Function</span>
+    B0 = [0.5, <span class="org-type">-</span>0.15, 2.2];                        <span class="org-comment">% Choose Appropriate Initial Estimates</span>
+    [B,rnrm] = fminsearch(nrmrsd, B0);     <span class="org-comment">% Estimate Parameters &#8216;B&#8217;</span>
+
+    tau(t_i) = 1<span class="org-type">/</span>B(2);
+    V0(t_i)  = B(3);
+<span class="org-keyword">end</span>
+</pre>
+</div>
+
+<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
+
+
+<colgroup>
+<col  class="org-right" />
+
+<col  class="org-right" />
+</colgroup>
+<thead>
+<tr>
+<th scope="col" class="org-right">\(tau\) [s]</th>
+<th scope="col" class="org-right">\(V_0\) [V]</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="org-right">6.47</td>
+<td class="org-right">2.26</td>
+</tr>
+
+<tr>
+<td class="org-right">6.76</td>
+<td class="org-right">2.26</td>
+</tr>
+
+<tr>
+<td class="org-right">6.49</td>
+<td class="org-right">2.25</td>
+</tr>
+</tbody>
+</table>
+
+<p>
+With the capacitance being \(C = 4.4 \mu F\), the internal impedance of the Speedgoat ADC can be computed as follows:
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">Cp = 4.4e<span class="org-type">-</span>6; <span class="org-comment">% [F]</span>
+Rin = abs(mean(tau))<span class="org-type">/</span>Cp;
+</pre>
+</div>
+
+<pre class="example">
+1494100.0
+</pre>
+
+
+<p>
+The input impedance of the Speedgoat&rsquo;s ADC should then be close to \(1.5\,M\Omega\) (specified at \(1\,M\Omega\)).
+</p>
+
+<div class="important" id="org572654b">
+<p>
+How can we explain the voltage offset?
+</p>
+
+</div>
+
+<p>
+As shown in Figure <a href="#org7c2c57f">5</a> (taken from (<a href="#citeproc_bib_item_1">Reza and Andrew 2006</a>)), an input voltage offset is due to the input bias current \(i_n\).
+</p>
+
+
+<div id="org7c2c57f" class="figure">
+<p><img src="figs/force_sensor_model_electronics.png" alt="force_sensor_model_electronics.png" />
+</p>
+<p><span class="figure-number">Figure 5: </span>Model of a piezoelectric transducer (left) and instrumentation amplifier (right)</p>
+</div>
+
+<p>
+The estimated input bias current is then:
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">in = mean(V0)<span class="org-type">/</span>Rin;
+</pre>
+</div>
+
+<pre class="example">
+1.5119e-06
+</pre>
+
+
+<p>
+An additional resistor in parallel with \(R_{in}\) would have two effects:
+</p>
+<ul class="org-ul">
+<li>reduce the input voltage offset
+\[ V_{off} = \frac{R_a R_{in}}{R_a + R_{in}} i_n \]</li>
+<li>increase the high pass corner frequency \(f_c\)
+\[ C_p \frac{R_{in}R_a}{R_{in} + R_a} = \tau_c = \frac{1}{f_c} \]
+\[ R_a = \frac{R_i}{f_c C_p R_i - 1} \]</li>
+</ul>
+
+
+<p>
+If we allow the high pass corner frequency to be equals to 3Hz:
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">fc = 3;
+Ra = Rin<span class="org-type">/</span>(fc<span class="org-type">*</span>Cp<span class="org-type">*</span>Rin <span class="org-type">-</span> 1);
+</pre>
+</div>
+
+<pre class="example">
+79804
+</pre>
+
+
+<p>
+With this parallel resistance value, the voltage offset would be:
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">V_offset = Ra<span class="org-type">*</span>Rin<span class="org-type">/</span>(Ra <span class="org-type">+</span> Rin) <span class="org-type">*</span> in;
+</pre>
+</div>
+
+<pre class="example">
+0.11454
+</pre>
+
+
+<p>
+Which is much more acceptable.
+</p>
+</div>
+</div>
+
+<div id="outline-container-org5e9eb44" class="outline-3">
+<h3 id="org5e9eb44"><span class="section-number-3">2.2</span> Add Parallel Resistor</h3>
+<div class="outline-text-3" id="text-2-2">
+<p>
+A resistor \(R_p \approx 100\,k\Omega\) is added in parallel with the force sensor as shown in Figure <a href="#org1fac5a7">6</a>.
+</p>
+
+
+<div id="org1fac5a7" class="figure">
+<p><img src="figs/force_sensor_model_electronics_without_R.png" alt="force_sensor_model_electronics_without_R.png" />
+</p>
+<p><span class="figure-number">Figure 6: </span>Model of a piezoelectric transducer (left) and instrumentation amplifier (right) with added resistor \(R_p\)</p>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">load(<span class="org-string">'force_sensor_steps_R_82k7.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'encoder'</span>, <span class="org-string">'u'</span>, <span class="org-string">'v'</span>);
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab"><span class="org-type">figure</span>;
+tiledlayout(2, 1, <span class="org-string">'TileSpacing'</span>, <span class="org-string">'None'</span>, <span class="org-string">'Padding'</span>, <span class="org-string">'None'</span>);
+nexttile;
+plot(t, v);
+xlabel(<span class="org-string">'Time [s]'</span>); ylabel(<span class="org-string">'Measured voltage [V]'</span>);
+nexttile;
+plot(t, u);
+xlabel(<span class="org-string">'Time [s]'</span>); ylabel(<span class="org-string">'Actuator Voltage [V]'</span>);
+</pre>
+</div>
+
+
+<div id="org29964b5" class="figure">
+<p><img src="figs/force_sen_steps_time_domain_par_R.png" alt="force_sen_steps_time_domain_par_R.png" />
+</p>
+<p><span class="figure-number">Figure 7: </span>Time domain signal during the actuator voltage steps</p>
+</div>
+
+<p>
+Three steps are performed at the following time intervals:
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">t_s = [1.9,  6;
+       8.5, 13;
+      15.5, 21;
+      22.6, 26;
+      30.0, 36;
+      37.5, 41;
+      46.2, 49.5]
+</pre>
+</div>
+
+<p>
+Fit function:
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">f = @(b,x) b(1)<span class="org-type">.*</span>exp(b(2)<span class="org-type">.*</span>x) <span class="org-type">+</span> b(3);
+</pre>
+</div>
+
+<p>
+We are interested by the <code>b(2)</code> term, which is the time constant of the exponential.
+</p>
+
+<div class="org-src-container">
+<pre class="src src-matlab">tau = zeros(size(t_s, 1),1);
+V0  = zeros(size(t_s, 1),1);
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab"><span class="org-keyword">for</span> <span class="org-variable-name">t_i</span> = <span class="org-constant">1:size(t_s, 1)</span>
+    t_cur = t(t_s(t_i, 1) <span class="org-type">&lt;</span> t <span class="org-type">&amp;</span> t <span class="org-type">&lt;</span> t_s(t_i, 2));
+    t_cur = t_cur <span class="org-type">-</span> t_cur(1);
+    y_cur = v(t_s(t_i, 1) <span class="org-type">&lt;</span> t <span class="org-type">&amp;</span> t <span class="org-type">&lt;</span> t_s(t_i, 2));
+
+    nrmrsd = @(b) norm(y_cur <span class="org-type">-</span> f(b,t_cur)); <span class="org-comment">% Residual Norm Cost Function</span>
+    B0 = [0.5, <span class="org-type">-</span>0.2, 0.2];                        <span class="org-comment">% Choose Appropriate Initial Estimates</span>
+    [B,rnrm] = fminsearch(nrmrsd, B0);     <span class="org-comment">% Estimate Parameters &#8216;B&#8217;</span>
+
+    tau(t_i) = 1<span class="org-type">/</span>B(2);
+    V0(t_i)  = B(3);
+<span class="org-keyword">end</span>
+</pre>
+</div>
+
+<p>
+And indeed, we obtain a much smaller offset voltage and a much faster time constant.
+</p>
+
+<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
+
+
+<colgroup>
+<col  class="org-right" />
+
+<col  class="org-right" />
+</colgroup>
+<thead>
+<tr>
+<th scope="col" class="org-right">\(tau\) [s]</th>
+<th scope="col" class="org-right">\(V_0\) [V]</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="org-right">0.43</td>
+<td class="org-right">0.15</td>
+</tr>
+
+<tr>
+<td class="org-right">0.45</td>
+<td class="org-right">0.16</td>
+</tr>
+
+<tr>
+<td class="org-right">0.43</td>
+<td class="org-right">0.15</td>
+</tr>
+
+<tr>
+<td class="org-right">0.43</td>
+<td class="org-right">0.15</td>
+</tr>
+
+<tr>
+<td class="org-right">0.45</td>
+<td class="org-right">0.15</td>
+</tr>
+
+<tr>
+<td class="org-right">0.46</td>
+<td class="org-right">0.16</td>
+</tr>
+
+<tr>
+<td class="org-right">0.48</td>
+<td class="org-right">0.16</td>
+</tr>
+</tbody>
+</table>
+
+<p>
+Knowing the capacitance value, we can estimate the value of the added resistor (neglecting the input impedance of \(\approx 1\,M\Omega\)):
+</p>
+
+<div class="org-src-container">
+<pre class="src src-matlab">Cp = 4.4e<span class="org-type">-</span>6; <span class="org-comment">% [F]</span>
+Rin = abs(mean(tau))<span class="org-type">/</span>Cp;
+</pre>
+</div>
+
+<pre class="example">
+101200.0
+</pre>
+
+
+<p>
+And we can verify that the bias current estimation stays the same:
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">in = mean(V0)<span class="org-type">/</span>Rin;
+</pre>
+</div>
+
+<pre class="example">
+1.5305e-06
+</pre>
+
+
+<p>
+This validates the model of the ADC and the effectiveness of the added resistor.
+</p>
+</div>
+</div>
+
+<div id="outline-container-org15676e1" class="outline-3">
+<h3 id="org15676e1"><span class="section-number-3">2.3</span> Sinus</h3>
+<div class="outline-text-3" id="text-2-3">
+<div class="org-src-container">
+<pre class="src src-matlab">load(<span class="org-string">'force_sensor_sin.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'encoder'</span>, <span class="org-string">'u'</span>, <span class="org-string">'v'</span>);
+
+u       = u(t<span class="org-type">&gt;</span>25);
+v       = v(t<span class="org-type">&gt;</span>25);
+encoder = encoder(t<span class="org-type">&gt;</span>25) <span class="org-type">-</span> mean(encoder(t<span class="org-type">&gt;</span>25));
+t       = t(t<span class="org-type">&gt;</span>25);
+</pre>
+</div>
+
+<p>
+The driving voltage is a sinus at 0.5Hz centered on 3V and with an amplitude of 3V (Figure <a href="#org1fbf89d">8</a>).
+</p>
+
+
+<div id="org1fbf89d" class="figure">
+<p><img src="figs/force_sensor_sin_u.png" alt="force_sensor_sin_u.png" />
+</p>
+<p><span class="figure-number">Figure 8: </span>Driving Voltage</p>
+</div>
+
+<p>
+The full stroke as measured by the encoder is:
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">max(encoder)<span class="org-type">-</span>min(encoder)
+</pre>
+</div>
+
+<pre class="example">
+5.005e-05
+</pre>
+
+
+<p>
+Its signal is shown in Figure <a href="#org1d74efa">9</a>.
+</p>
+
+
+<div id="org1d74efa" class="figure">
+<p><img src="figs/force_sensor_sin_encoder.png" alt="force_sensor_sin_encoder.png" />
+</p>
+<p><span class="figure-number">Figure 9: </span>Encoder measurement</p>
+</div>
+
+<p>
+The generated voltage by the stack is shown in Figure
+</p>
+
+
+<div id="org077a6d7" class="figure">
+<p><img src="figs/force_sensor_sin_stack.png" alt="force_sensor_sin_stack.png" />
+</p>
+<p><span class="figure-number">Figure 10: </span>Voltage measured on the stack used as a sensor</p>
+</div>
+
+<p>
+The capacitance of the stack is
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">Cp = 4.4e<span class="org-type">-</span>6; <span class="org-comment">% [F]</span>
+</pre>
+</div>
+
+<p>
+The corresponding generated charge is then shown in Figure <a href="#org4baf062">11</a>.
+</p>
+
+<div id="org4baf062" class="figure">
+<p><img src="figs/force_sensor_sin_charge.png" alt="force_sensor_sin_charge.png" />
+</p>
+<p><span class="figure-number">Figure 11: </span>Generated Charge</p>
+</div>
+
+
+<p>
+The relation between the generated voltage and the measured displacement is almost linear as shown in Figure <a href="#org8b9df34">12</a>.
+</p>
+
+<div class="org-src-container">
+<pre class="src src-matlab">b1 = encoder<span class="org-type">\</span>(v<span class="org-type">-</span>mean(v));
+</pre>
+</div>
+
+
+<div id="org8b9df34" class="figure">
+<p><img src="figs/force_sensor_linear_relation.png" alt="force_sensor_linear_relation.png" />
+</p>
+<p><span class="figure-number">Figure 12: </span>Almost linear relation between the relative displacement and the generated voltage</p>
+</div>
+
+<p>
+With a 16bits ADC, the resolution will then be equals to (in [nm]):
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">abs((20<span class="org-type">/</span>2<span class="org-type">^</span>16)<span class="org-type">/</span>(b1<span class="org-type">/</span>1e9))
+</pre>
+</div>
+
+<pre class="example">
+3.9838
+</pre>
+
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><h2 class='citeproc-org-bib-h2'>Bibliography</h2>
+<div class="csl-bib-body">
+  <div class="csl-entry"><a name="citeproc_bib_item_1"></a>Reza, Moheimani, and Fleming Andrew. 2006. <i>Piezoelectric Transducers for Vibration Control and Damping</i>. London: Springer.</div>
+</div>
+</div>
+</div>
+</div>
+</div>
+<div id="postamble" class="status">
+<p class="author">Author: Dehaeze Thomas</p>
+<p class="date">Created: 2020-11-10 mar. 12:55</p>
+</div>
+</body>
+</html>
--- a/index.org
+++ b/index.org
@@ -0,0 +1,568 @@
+#+TITLE: Piezoelectric Force Sensor - Test Bench
+:DRAWER:
+#+LANGUAGE: en
+#+EMAIL: dehaeze.thomas@gmail.com
+#+AUTHOR: Dehaeze Thomas
+
+#+HTML_LINK_HOME: ../index.html
+#+HTML_LINK_UP:   ../index.html
+
+#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
+#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
+#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/custom.css"/>
+#+HTML_HEAD: <script type="text/javascript" src="./js/jquery.min.js"></script>
+#+HTML_HEAD: <script type="text/javascript" src="./js/bootstrap.min.js"></script>
+#+HTML_HEAD: <script type="text/javascript" src="./js/readtheorg.js"></script>
+
+#+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 raw replace :buffer no
+#+PROPERTY: header-args:latex+ :eval no-export
+#+PROPERTY: header-args:latex+ :exports both
+#+PROPERTY: header-args:latex+ :mkdirp yes
+#+PROPERTY: header-args:latex+ :output-dir figs
+#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
+
+#+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
+:END:
+
+* Introduction                                                        :ignore:
+In this document is studied how a piezoelectric stack can be used to measured the force.
+
+- Section [[sec:open_closed_circuit]]: the effect of the input impedance of the electronics connected to the force sensor stack on the stiffness of the stack is studied
+- Section [[sec:charge_voltage_estimation]]:
+
+* Change of Stiffness due to Sensors stack being open/closed circuit
+:PROPERTIES:
+:header-args:matlab+: :tangle matlab/open_closed_circuit.m
+:END:
+<<sec:open_closed_circuit>>
+
+** Introduction                                                      :ignore:
+
+** Matlab Init                                             :noexport:ignore:
+#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
+  <<matlab-dir>>
+#+end_src
+
+#+begin_src matlab :exports none :results silent :noweb yes
+  <<matlab-init>>
+#+end_src
+
+#+begin_src matlab :tangle no
+  addpath('./matlab/mat/');
+#+end_src
+
+#+begin_src matlab :eval no
+  addpath('./mat/');
+#+end_src
+
+** Load Data
+#+begin_src matlab
+  oc = load('identification_open_circuit.mat', 't', 'encoder', 'u');
+  sc = load('identification_short_circuit.mat', 't', 'encoder', 'u');
+#+end_src
+
+** Transfer Functions
+#+begin_src matlab
+  Ts = 1e-4; % Sampling Time [s]
+  win = hann(ceil(10/Ts));
+#+end_src
+
+#+begin_src matlab
+  [tf_oc_est, f] = tfestimate(oc.u, oc.encoder, win, [], [], 1/Ts);
+  [co_oc_est, ~] = mscohere(  oc.u, oc.encoder, win, [], [], 1/Ts);
+
+  [tf_sc_est, ~] = tfestimate(sc.u, sc.encoder, win, [], [], 1/Ts);
+  [co_sc_est, ~] = mscohere(  sc.u, sc.encoder, win, [], [], 1/Ts);
+#+end_src
+
+#+begin_src matlab :exports none
+  figure;
+  hold on;
+  plot(f, co_oc_est, '-')
+  plot(f, co_sc_est, '-')
+  set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
+  ylabel('Coherence'); xlabel('Frequency [Hz]');
+  hold off;
+  xlim([0.5, 5e3]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/stiffness_force_sensor_coherence.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:stiffness_force_sensor_coherence
+#+caption:
+#+RESULTS:
+[[file:figs/stiffness_force_sensor_coherence.png]]
+
+
+#+begin_src matlab :exports none
+  figure;
+  tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+  ax1 = nexttile;
+  hold on;
+  plot(f, abs(tf_oc_est), '-', 'DisplayName', 'Open-Circuit')
+  plot(f, abs(tf_sc_est), '-', 'DisplayName', 'Short-Circuit')
+  set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
+  ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
+  hold off;
+  ylim([1e-7, 3e-4]);
+  legend('location', 'southwest');
+
+  ax2 = nexttile;
+  hold on;
+  plot(f, 180/pi*angle(tf_oc_est), '-')
+  plot(f, 180/pi*angle(tf_sc_est), '-')
+  set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
+  ylabel('Phase'); xlabel('Frequency [Hz]');
+  hold off;
+  yticks(-360:90:360);
+  axis padded 'auto x'
+
+  linkaxes([ax1,ax2], 'x');
+  xlim([0.5, 5e3]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/stiffness_force_sensor_bode.pdf', 'width', 'wide', 'height', 'tall');
+#+end_src
+
+#+name: fig:stiffness_force_sensor_bode
+#+caption:
+#+RESULTS:
+[[file:figs/stiffness_force_sensor_bode.png]]
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  xlim([180, 280]);
+  exportFig('figs/stiffness_force_sensor_bode_zoom.pdf', 'width', 'small', 'height', 'tall');
+#+end_src
+
+#+name: fig:stiffness_force_sensor_bode_zoom
+#+caption: Zoom on the change of resonance
+#+RESULTS:
+[[file:figs/stiffness_force_sensor_bode_zoom.png]]
+
+#+begin_important
+  The change of resonance frequency / stiffness is very small and is not important here.
+#+end_important
+
+* Generated Number of Charge / Voltage
+:PROPERTIES:
+:header-args:matlab+: :tangle matlab/charge_voltage_estimation.m
+:END:
+<<sec:charge_voltage_estimation>>
+
+** Introduction                                                      :ignore:
+
+
+Two stacks are used as actuator (in parallel) and one stack is used as sensor.
+
+The amplifier gain is 20V/V (Cedrat LA75B).
+
+** Matlab Init                                            :noexport:ignore:
+#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
+  <<matlab-dir>>
+#+end_src
+
+#+begin_src matlab :exports none :results silent :noweb yes
+  <<matlab-init>>
+#+end_src
+
+#+begin_src matlab :tangle no
+  addpath('./matlab/mat/');
+#+end_src
+
+#+begin_src matlab :eval no
+  addpath('./mat/');
+#+end_src
+
+** Steps
+#+begin_src matlab
+  load('force_sensor_steps.mat', 't', 'encoder', 'u', 'v');
+#+end_src
+
+#+begin_src matlab
+  figure;
+  tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
+  nexttile;
+  plot(t, v);
+  xlabel('Time [s]'); ylabel('Measured voltage [V]');
+  nexttile;
+  plot(t, u);
+  xlabel('Time [s]'); ylabel('Actuator Voltage [V]');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/force_sen_steps_time_domain.pdf', 'width', 'wide', 'height', 'tall');
+#+end_src
+
+#+name: fig:force_sen_steps_time_domain
+#+caption: Time domain signal during the 3 actuator voltage steps
+#+RESULTS:
+[[file:figs/force_sen_steps_time_domain.png]]
+
+Three steps are performed at the following time intervals:
+#+begin_src matlab
+  t_s = [ 2.5, 23;
+         23.8, 35;
+         35.8, 50];
+#+end_src
+
+Fit function:
+#+begin_src matlab
+  f = @(b,x) b(1).*exp(b(2).*x) + b(3);
+#+end_src
+
+We are interested by the =b(2)= term, which is the time constant of the exponential.
+#+begin_src matlab
+  tau = zeros(size(t_s, 1),1);
+  V0  = zeros(size(t_s, 1),1);
+#+end_src
+
+#+begin_src matlab
+  for t_i = 1:size(t_s, 1)
+      t_cur = t(t_s(t_i, 1) < t & t < t_s(t_i, 2));
+      t_cur = t_cur - t_cur(1);
+      y_cur = v(t_s(t_i, 1) < t & t < t_s(t_i, 2));
+
+      nrmrsd = @(b) norm(y_cur - f(b,t_cur)); % Residual Norm Cost Function
+      B0 = [0.5, -0.15, 2.2];                        % Choose Appropriate Initial Estimates
+      [B,rnrm] = fminsearch(nrmrsd, B0);     % Estimate Parameters ‘B’
+
+      tau(t_i) = 1/B(2);
+      V0(t_i)  = B(3);
+  end
+#+end_src
+
+#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
+  data2orgtable([abs(tau), V0], {}, {'$tau$ [s]', '$V_0$ [V]'}, ' %.2f ');
+#+end_src
+
+#+RESULTS:
+| $tau$ [s] | $V_0$ [V] |
+|-----------+-----------|
+|      6.47 |      2.26 |
+|      6.76 |      2.26 |
+|      6.49 |      2.25 |
+
+With the capacitance being $C = 4.4 \mu F$, the internal impedance of the Speedgoat ADC can be computed as follows:
+#+begin_src matlab
+  Cp = 4.4e-6; % [F]
+  Rin = abs(mean(tau))/Cp;
+#+end_src
+
+#+begin_src matlab :results value replace :exports results
+  ans = Rin
+#+end_src
+
+#+RESULTS:
+: 1494100.0
+
+The input impedance of the Speedgoat's ADC should then be close to $1.5\,M\Omega$ (specified at $1\,M\Omega$).
+
+#+begin_important
+  How can we explain the voltage offset?
+#+end_important
+
+As shown in Figure [[fig:force_sensor_model_electronics]] (taken from cite:reza06_piezoel_trans_vibrat_contr_dampin), an input voltage offset is due to the input bias current $i_n$.
+
+#+name: fig:force_sensor_model_electronics
+#+caption: Model of a piezoelectric transducer (left) and instrumentation amplifier (right)
+[[file:figs/force_sensor_model_electronics.png]]
+
+The estimated input bias current is then:
+#+begin_src matlab
+  in = mean(V0)/Rin;
+#+end_src
+
+#+begin_src matlab :results value replace :exports results
+   ans = in
+#+end_src
+
+#+RESULTS:
+: 1.5119e-06
+
+An additional resistor in parallel with $R_{in}$ would have two effects:
+- reduce the input voltage offset
+  \[ V_{off} = \frac{R_a R_{in}}{R_a + R_{in}} i_n \]
+- increase the high pass corner frequency $f_c$
+  \[ C_p \frac{R_{in}R_a}{R_{in} + R_a} = \tau_c = \frac{1}{f_c} \]
+  \[ R_a = \frac{R_i}{f_c C_p R_i - 1} \]
+
+
+If we allow the high pass corner frequency to be equals to 3Hz:
+#+begin_src matlab
+  fc = 3;
+  Ra = Rin/(fc*Cp*Rin - 1);
+#+end_src
+
+#+begin_src matlab :results value replace :exports results
+  ans = Ra
+#+end_src
+
+#+RESULTS:
+: 79804
+
+With this parallel resistance value, the voltage offset would be:
+#+begin_src matlab
+  V_offset = Ra*Rin/(Ra + Rin) * in;
+#+end_src
+
+#+begin_src matlab :results value replace :exports results
+  ans = V_offset
+#+end_src
+
+#+RESULTS:
+: 0.11454
+
+Which is much more acceptable.
+
+** Add Parallel Resistor
+A resistor $R_p \approx 100\,k\Omega$ is added in parallel with the force sensor as shown in Figure [[fig:force_sensor_model_electronics_without_R]].
+
+#+name: fig:force_sensor_model_electronics_without_R
+#+caption: Model of a piezoelectric transducer (left) and instrumentation amplifier (right) with added resistor $R_p$
+[[file:figs/force_sensor_model_electronics_without_R.png]]
+
+#+begin_src matlab
+  load('force_sensor_steps_R_82k7.mat', 't', 'encoder', 'u', 'v');
+#+end_src
+
+#+begin_src matlab
+  figure;
+  tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
+  nexttile;
+  plot(t, v);
+  xlabel('Time [s]'); ylabel('Measured voltage [V]');
+  nexttile;
+  plot(t, u);
+  xlabel('Time [s]'); ylabel('Actuator Voltage [V]');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/force_sen_steps_time_domain_par_R.pdf', 'width', 'wide', 'height', 'tall');
+#+end_src
+
+#+name: fig:force_sen_steps_time_domain_par_R
+#+caption: Time domain signal during the actuator voltage steps
+#+RESULTS:
+[[file:figs/force_sen_steps_time_domain_par_R.png]]
+
+Three steps are performed at the following time intervals:
+#+begin_src matlab
+  t_s = [1.9,  6;
+         8.5, 13;
+        15.5, 21;
+        22.6, 26;
+        30.0, 36;
+        37.5, 41;
+        46.2, 49.5]
+#+end_src
+
+Fit function:
+#+begin_src matlab
+  f = @(b,x) b(1).*exp(b(2).*x) + b(3);
+#+end_src
+
+We are interested by the =b(2)= term, which is the time constant of the exponential.
+
+#+begin_src matlab
+  tau = zeros(size(t_s, 1),1);
+  V0  = zeros(size(t_s, 1),1);
+#+end_src
+
+#+begin_src matlab
+  for t_i = 1:size(t_s, 1)
+      t_cur = t(t_s(t_i, 1) < t & t < t_s(t_i, 2));
+      t_cur = t_cur - t_cur(1);
+      y_cur = v(t_s(t_i, 1) < t & t < t_s(t_i, 2));
+
+      nrmrsd = @(b) norm(y_cur - f(b,t_cur)); % Residual Norm Cost Function
+      B0 = [0.5, -0.2, 0.2];                        % Choose Appropriate Initial Estimates
+      [B,rnrm] = fminsearch(nrmrsd, B0);     % Estimate Parameters ‘B’
+
+      tau(t_i) = 1/B(2);
+      V0(t_i)  = B(3);
+  end
+#+end_src
+
+And indeed, we obtain a much smaller offset voltage and a much faster time constant.
+
+#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
+  data2orgtable([abs(tau), V0], {}, {'$tau$ [s]', '$V_0$ [V]'}, ' %.2f ');
+#+end_src
+
+#+RESULTS:
+| $tau$ [s] | $V_0$ [V] |
+|-----------+-----------|
+|      0.43 |      0.15 |
+|      0.45 |      0.16 |
+|      0.43 |      0.15 |
+|      0.43 |      0.15 |
+|      0.45 |      0.15 |
+|      0.46 |      0.16 |
+|      0.48 |      0.16 |
+
+Knowing the capacitance value, we can estimate the value of the added resistor (neglecting the input impedance of $\approx 1\,M\Omega$):
+
+#+begin_src matlab
+  Cp = 4.4e-6; % [F]
+  Rin = abs(mean(tau))/Cp;
+#+end_src
+
+#+begin_src matlab :results value replace :exports results
+  ans = Rin
+#+end_src
+
+#+RESULTS:
+: 101200.0
+
+And we can verify that the bias current estimation stays the same:
+#+begin_src matlab
+  in = mean(V0)/Rin;
+#+end_src
+
+#+begin_src matlab :results value replace :exports results
+   ans = in
+#+end_src
+
+#+RESULTS:
+: 1.5305e-06
+
+This validates the model of the ADC and the effectiveness of the added resistor.
+
+** Sinus
+#+begin_src matlab
+  load('force_sensor_sin.mat', 't', 'encoder', 'u', 'v');
+
+  u       = u(t>25);
+  v       = v(t>25);
+  encoder = encoder(t>25) - mean(encoder(t>25));
+  t       = t(t>25);
+#+end_src
+
+The driving voltage is a sinus at 0.5Hz centered on 3V and with an amplitude of 3V (Figure [[fig:force_sensor_sin_u]]).
+
+#+begin_src matlab :exports none
+  figure;
+  plot(t, u)
+  xlabel('Time [s]'); ylabel('Control Voltage [V]');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/force_sensor_sin_u.pdf', 'width', 'normal', 'height', 'small');
+#+end_src
+
+#+name: fig:force_sensor_sin_u
+#+caption: Driving Voltage
+#+RESULTS:
+[[file:figs/force_sensor_sin_u.png]]
+
+The full stroke as measured by the encoder is:
+#+begin_src matlab :results value replace
+  max(encoder)-min(encoder)
+#+end_src
+
+#+RESULTS:
+: 5.005e-05
+
+Its signal is shown in Figure [[fig:force_sensor_sin_encoder]].
+
+#+begin_src matlab :exports none
+  figure;
+  plot(t, encoder)
+  xlabel('Time [s]'); ylabel('Encoder [m]');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/force_sensor_sin_encoder.pdf', 'width', 'normal', 'height', 'small');
+#+end_src
+
+#+name: fig:force_sensor_sin_encoder
+#+caption: Encoder measurement
+#+RESULTS:
+[[file:figs/force_sensor_sin_encoder.png]]
+
+The generated voltage by the stack is shown in Figure
+
+#+begin_src matlab :exports none
+  figure;
+  plot(t, v)
+  xlabel('Time [s]'); ylabel('Force Sensor Output [V]');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/force_sensor_sin_stack.pdf', 'width', 'normal', 'height', 'small');
+#+end_src
+
+#+name: fig:force_sensor_sin_stack
+#+caption: Voltage measured on the stack used as a sensor
+#+RESULTS:
+[[file:figs/force_sensor_sin_stack.png]]
+
+The capacitance of the stack is
+#+begin_src matlab
+  Cp = 4.4e-6; % [F]
+#+end_src
+
+The corresponding generated charge is then shown in Figure [[fig:force_sensor_sin_charge]].
+#+begin_src matlab :exports none
+  figure;
+  plot(t, 1e6*Cp*(v-mean(v)))
+  xlabel('Time [s]'); ylabel('Generated Charge [$\mu C$]');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/force_sensor_sin_charge.pdf', 'width', 'normal', 'height', 'small');
+#+end_src
+
+#+name: fig:force_sensor_sin_charge
+#+caption: Generated Charge
+#+RESULTS:
+[[file:figs/force_sensor_sin_charge.png]]
+
+
+The relation between the generated voltage and the measured displacement is almost linear as shown in Figure [[fig:force_sensor_linear_relation]].
+
+#+begin_src matlab
+  b1 = encoder\(v-mean(v));
+#+end_src
+
+#+begin_src matlab :exports none
+  figure;
+  hold on;
+  plot(encoder, v-mean(v), 'DisplayName', 'Measured Voltage');
+  plot(encoder, encoder*b1, 'DisplayName', sprintf('Linear Fit: $U_s \\approx %.3f [V/\\mu m] \\cdot d$', 1e-6*abs(b1)));
+  hold off;
+  xlabel('Measured Displacement [m]'); ylabel('Generated Voltage [V]');
+  legend();
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/force_sensor_linear_relation.pdf', 'width', 'normal', 'height', 'small');
+#+end_src
+
+#+name: fig:force_sensor_linear_relation
+#+caption: Almost linear relation between the relative displacement and the generated voltage
+#+RESULTS:
+[[file:figs/force_sensor_linear_relation.png]]
+
+With a 16bits ADC, the resolution will then be equals to (in [nm]):
+#+begin_src matlab :results value replace
+  abs((20/2^16)/(b1/1e9))
+#+end_src
+
+#+RESULTS:
+: 3.9838
--- a/js/bootstrap.min.js
+++ b/js/bootstrap.min.js
--- a/js/jquery.min.js
+++ b/js/jquery.min.js
--- a/js/readtheorg.js
+++ b/js/readtheorg.js
@@ -0,0 +1,87 @@
+$(function() {
+    $('.note').before("<p class='admonition-title note'>Note</p>");
+    $('.seealso').before("<p class='admonition-title seealso'>See also</p>");
+    $('.warning').before("<p class='admonition-title warning'>Warning</p>");
+    $('.caution').before("<p class='admonition-title caution'>Caution</p>");
+    $('.attention').before("<p class='admonition-title attention'>Attention</p>");
+    $('.tip').before("<p class='admonition-title tip'>Tip</p>");
+    $('.important').before("<p class='admonition-title important'>Important</p>");
+    $('.hint').before("<p class='admonition-title hint'>Hint</p>");
+    $('.error').before("<p class='admonition-title error'>Error</p>");
+    $('.danger').before("<p class='admonition-title danger'>Danger</p>");
+    $('.question').before("<p class='admonition-title question'>Question</p>");
+    $('.summary').before("<p class='admonition-title hint'>Summary</p>");
+});
+
+$( document ).ready(function() {
+
+    // Shift nav in mobile when clicking the menu.
+    $(document).on('click', "[data-toggle='wy-nav-top']", function() {
+      $("[data-toggle='wy-nav-shift']").toggleClass("shift");
+      $("[data-toggle='rst-versions']").toggleClass("shift");
+    });
+    // Close menu when you click a link.
+    $(document).on('click', ".wy-menu-vertical .current ul li a", function() {
+      $("[data-toggle='wy-nav-shift']").removeClass("shift");
+      $("[data-toggle='rst-versions']").toggleClass("shift");
+    });
+    $(document).on('click', "[data-toggle='rst-current-version']", function() {
+      $("[data-toggle='rst-versions']").toggleClass("shift-up");
+    });
+    // Make tables responsive
+    $("table.docutils:not(.field-list)").wrap("<div class='wy-table-responsive'></div>");
+});
+
+$( document ).ready(function() {
+    $('#text-table-of-contents ul').first().addClass('nav');
+                                        // ScrollSpy also requires that we use
+                                        // a Bootstrap nav component.
+    $('body').scrollspy({target: '#text-table-of-contents'});
+
+    // add sticky table headers
+    $('table').stickyTableHeaders();
+
+    // set the height of tableOfContents
+    var $postamble = $('#postamble');
+    var $tableOfContents = $('#table-of-contents');
+    $tableOfContents.css({paddingBottom: $postamble.outerHeight()});
+
+    // add TOC button
+    var toggleSidebar = $('<div id="toggle-sidebar"><a href="#table-of-contents"><h2>Table of Contents</h2></a></div>');
+    $('#content').prepend(toggleSidebar);
+
+    // add close button when sidebar showed in mobile screen
+    var closeBtn = $('<a class="close-sidebar" href="#">Close</a>');
+    var tocTitle = $('#table-of-contents').find('h2');
+    tocTitle.append(closeBtn);
+});
+
+window.SphinxRtdTheme = (function (jquery) {
+    var stickyNav = (function () {
+        var navBar,
+            win,
+            stickyNavCssClass = 'stickynav',
+            applyStickNav = function () {
+                if (navBar.height() <= win.height()) {
+                    navBar.addClass(stickyNavCssClass);
+                } else {
+                    navBar.removeClass(stickyNavCssClass);
+                }
+            },
+            enable = function () {
+                applyStickNav();
+                win.on('resize', applyStickNav);
+            },
+            init = function () {
+                navBar = jquery('nav.wy-nav-side:first');
+                win    = jquery(window);
+            };
+        jquery(init);
+        return {
+            enable : enable
+        };
+    }());
+    return {
+        StickyNav : stickyNav
+    };
+}($));
--- a/matlab/charge_voltage_estimation.m
+++ b/matlab/charge_voltage_estimation.m
@@ -0,0 +1,330 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+addpath('./mat/');
+
+% Steps
+
+load('force_sensor_steps.mat', 't', 'encoder', 'u', 'v');
+
+figure;
+tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
+nexttile;
+plot(t, v);
+xlabel('Time [s]'); ylabel('Measured voltage [V]');
+nexttile;
+plot(t, u);
+xlabel('Time [s]'); ylabel('Actuator Voltage [V]');
+
+
+
+% #+name: fig:force_sen_steps_time_domain
+% #+caption: Time domain signal during the 3 actuator voltage steps
+% #+RESULTS:
+% [[file:figs/force_sen_steps_time_domain.png]]
+
+% Three steps are performed at the following time intervals:
+
+t_s = [ 2.5, 23;
+       23.8, 35;
+       35.8, 50];
+
+
+
+% Fit function:
+
+f = @(b,x) b(1).*exp(b(2).*x) + b(3);
+
+
+
+% We are interested by the =b(2)= term, which is the time constant of the exponential.
+
+tau = zeros(size(t_s, 1),1);
+V0  = zeros(size(t_s, 1),1);
+
+for t_i = 1:size(t_s, 1)
+    t_cur = t(t_s(t_i, 1) < t & t < t_s(t_i, 2));
+    t_cur = t_cur - t_cur(1);
+    y_cur = v(t_s(t_i, 1) < t & t < t_s(t_i, 2));
+
+    nrmrsd = @(b) norm(y_cur - f(b,t_cur)); % Residual Norm Cost Function
+    B0 = [0.5, -0.15, 2.2];                        % Choose Appropriate Initial Estimates
+    [B,rnrm] = fminsearch(nrmrsd, B0);     % Estimate Parameters ‘B’
+
+    tau(t_i) = 1/B(2);
+    V0(t_i)  = B(3);
+end
+
+
+
+% #+RESULTS:
+% | $tau$ [s] | $V_0$ [V] |
+% |-----------+-----------|
+% |      6.47 |      2.26 |
+% |      6.76 |      2.26 |
+% |      6.49 |      2.25 |
+
+% With the capacitance being $C = 4.4 \mu F$, the internal impedance of the Speedgoat ADC can be computed as follows:
+
+Cp = 4.4e-6; % [F]
+Rin = abs(mean(tau))/Cp;
+
+ans = Rin
+
+
+
+% #+RESULTS:
+% : 1494100.0
+
+% The input impedance of the Speedgoat's ADC should then be close to $1.5\,M\Omega$ (specified at $1\,M\Omega$).
+
+% #+begin_important
+%   How can we explain the voltage offset?
+% #+end_important
+
+% As shown in Figure [[fig:force_sensor_model_electronics]] (taken from cite:reza06_piezoel_trans_vibrat_contr_dampin), an input voltage offset is due to the input bias current $i_n$.
+
+% #+name: fig:force_sensor_model_electronics
+% #+caption: Model of a piezoelectric transducer (left) and instrumentation amplifier (right)
+% [[file:figs/force_sensor_model_electronics.png]]
+
+% The estimated input bias current is then:
+
+in = mean(V0)/Rin;
+
+ans = in
+
+
+
+% #+RESULTS:
+% : 1.5119e-06
+
+% An additional resistor in parallel with $R_{in}$ would have two effects:
+% - reduce the input voltage offset
+%   \[ V_{off} = \frac{R_a R_{in}}{R_a + R_{in}} i_n \]
+% - increase the high pass corner frequency $f_c$
+%   \[ C_p \frac{R_{in}R_a}{R_{in} + R_a} = \tau_c = \frac{1}{f_c} \]
+%   \[ R_a = \frac{R_i}{f_c C_p R_i - 1} \]
+
+
+% If we allow the high pass corner frequency to be equals to 3Hz:
+
+fc = 3;
+Ra = Rin/(fc*Cp*Rin - 1);
+
+ans = Ra
+
+
+
+% #+RESULTS:
+% : 79804
+
+% With this parallel resistance value, the voltage offset would be:
+
+V_offset = Ra*Rin/(Ra + Rin) * in;
+
+ans = V_offset
+
+% Add Parallel Resistor
+% A resistor $R_p \approx 100\,k\Omega$ is added in parallel with the force sensor as shown in Figure [[fig:force_sensor_model_electronics_without_R]].
+
+% #+name: fig:force_sensor_model_electronics_without_R
+% #+caption: Model of a piezoelectric transducer (left) and instrumentation amplifier (right) with added resistor $R_p$
+% [[file:figs/force_sensor_model_electronics_without_R.png]]
+
+
+load('force_sensor_steps_R_82k7.mat', 't', 'encoder', 'u', 'v');
+
+figure;
+tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
+nexttile;
+plot(t, v);
+xlabel('Time [s]'); ylabel('Measured voltage [V]');
+nexttile;
+plot(t, u);
+xlabel('Time [s]'); ylabel('Actuator Voltage [V]');
+
+
+
+% #+name: fig:force_sen_steps_time_domain_par_R
+% #+caption: Time domain signal during the actuator voltage steps
+% #+RESULTS:
+% [[file:figs/force_sen_steps_time_domain_par_R.png]]
+
+% Three steps are performed at the following time intervals:
+
+t_s = [1.9,  6;
+       8.5, 13;
+      15.5, 21;
+      22.6, 26;
+      30.0, 36;
+      37.5, 41;
+      46.2, 49.5]
+
+
+
+% Fit function:
+
+f = @(b,x) b(1).*exp(b(2).*x) + b(3);
+
+
+
+% We are interested by the =b(2)= term, which is the time constant of the exponential.
+
+
+tau = zeros(size(t_s, 1),1);
+V0  = zeros(size(t_s, 1),1);
+
+for t_i = 1:size(t_s, 1)
+    t_cur = t(t_s(t_i, 1) < t & t < t_s(t_i, 2));
+    t_cur = t_cur - t_cur(1);
+    y_cur = v(t_s(t_i, 1) < t & t < t_s(t_i, 2));
+
+    nrmrsd = @(b) norm(y_cur - f(b,t_cur)); % Residual Norm Cost Function
+    B0 = [0.5, -0.2, 0.2];                        % Choose Appropriate Initial Estimates
+    [B,rnrm] = fminsearch(nrmrsd, B0);     % Estimate Parameters ‘B’
+
+    tau(t_i) = 1/B(2);
+    V0(t_i)  = B(3);
+end
+
+
+
+% #+RESULTS:
+% | $tau$ [s] | $V_0$ [V] |
+% |-----------+-----------|
+% |      0.43 |      0.15 |
+% |      0.45 |      0.16 |
+% |      0.43 |      0.15 |
+% |      0.43 |      0.15 |
+% |      0.45 |      0.15 |
+% |      0.46 |      0.16 |
+% |      0.48 |      0.16 |
+
+% Knowing the capacitance value, we can estimate the value of the added resistor (neglecting the input impedance of $\approx 1\,M\Omega$):
+
+
+Cp = 4.4e-6; % [F]
+Rin = abs(mean(tau))/Cp;
+
+ans = Rin
+
+
+
+% #+RESULTS:
+% : 101200.0
+
+% And we can verify that the bias current estimation stays the same:
+
+in = mean(V0)/Rin;
+
+ans = in
+
+% Sinus
+
+load('force_sensor_sin.mat', 't', 'encoder', 'u', 'v');
+
+u       = u(t>25);
+v       = v(t>25);
+encoder = encoder(t>25) - mean(encoder(t>25));
+t       = t(t>25);
+
+
+
+% The driving voltage is a sinus at 0.5Hz centered on 3V and with an amplitude of 3V (Figure [[fig:force_sensor_sin_u]]).
+
+
+figure;
+plot(t, u)
+xlabel('Time [s]'); ylabel('Control Voltage [V]');
+
+
+
+% #+name: fig:force_sensor_sin_u
+% #+caption: Driving Voltage
+% #+RESULTS:
+% [[file:figs/force_sensor_sin_u.png]]
+
+% The full stroke as measured by the encoder is:
+
+max(encoder)-min(encoder)
+
+
+
+% #+RESULTS:
+% : 5.005e-05
+
+% Its signal is shown in Figure [[fig:force_sensor_sin_encoder]].
+
+
+figure;
+plot(t, encoder)
+xlabel('Time [s]'); ylabel('Encoder [m]');
+
+
+
+% #+name: fig:force_sensor_sin_encoder
+% #+caption: Encoder measurement
+% #+RESULTS:
+% [[file:figs/force_sensor_sin_encoder.png]]
+
+% The generated voltage by the stack is shown in Figure
+
+
+figure;
+plot(t, v)
+xlabel('Time [s]'); ylabel('Force Sensor Output [V]');
+
+
+
+% #+name: fig:force_sensor_sin_stack
+% #+caption: Voltage measured on the stack used as a sensor
+% #+RESULTS:
+% [[file:figs/force_sensor_sin_stack.png]]
+
+% The capacitance of the stack is
+
+Cp = 4.4e-6; % [F]
+
+
+
+% The corresponding generated charge is then shown in Figure [[fig:force_sensor_sin_charge]].
+
+figure;
+plot(t, 1e6*Cp*(v-mean(v)))
+xlabel('Time [s]'); ylabel('Generated Charge [$\mu C$]');
+
+
+
+% #+name: fig:force_sensor_sin_charge
+% #+caption: Generated Charge
+% #+RESULTS:
+% [[file:figs/force_sensor_sin_charge.png]]
+
+
+% The relation between the generated voltage and the measured displacement is almost linear as shown in Figure [[fig:force_sensor_linear_relation]].
+
+
+b1 = encoder\(v-mean(v));
+
+figure;
+hold on;
+plot(encoder, v-mean(v), 'DisplayName', 'Measured Voltage');
+plot(encoder, encoder*b1, 'DisplayName', sprintf('Linear Fit: $U_s \\approx %.3f [V/\\mu m] \\cdot d$', 1e-6*abs(b1)));
+hold off;
+xlabel('Measured Displacement [m]'); ylabel('Generated Voltage [V]');
+legend();
+
+
+
+% #+name: fig:force_sensor_linear_relation
+% #+caption: Almost linear relation between the relative displacement and the generated voltage
+% #+RESULTS:
+% [[file:figs/force_sensor_linear_relation.png]]
+
+% With a 16bits ADC, the resolution will then be equals to (in [nm]):
+
+abs((20/2^16)/(b1/1e9))
--- a/matlab/mat/force_sensor_sin.mat
+++ b/matlab/mat/force_sensor_sin.mat
--- a/matlab/mat/force_sensor_steps.mat
+++ b/matlab/mat/force_sensor_steps.mat
--- a/matlab/mat/force_sensor_steps_R_82k7.mat
+++ b/matlab/mat/force_sensor_steps_R_82k7.mat
--- a/matlab/mat/identification_R_82k7.mat
+++ b/matlab/mat/identification_R_82k7.mat
--- a/matlab/mat/identification_open_circuit.mat
+++ b/matlab/mat/identification_open_circuit.mat
--- a/matlab/mat/identification_short_circuit.mat
+++ b/matlab/mat/identification_short_circuit.mat
--- a/matlab/open_closed_circuit.m
+++ b/matlab/open_closed_circuit.m
@@ -0,0 +1,67 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+addpath('./mat/');
+
+% Load Data
+
+oc = load('identification_open_circuit.mat', 't', 'encoder', 'u');
+sc = load('identification_short_circuit.mat', 't', 'encoder', 'u');
+
+% Transfer Functions
+
+Ts = 1e-4; % Sampling Time [s]
+win = hann(ceil(10/Ts));
+
+[tf_oc_est, f] = tfestimate(oc.u, oc.encoder, win, [], [], 1/Ts);
+[co_oc_est, ~] = mscohere(  oc.u, oc.encoder, win, [], [], 1/Ts);
+
+[tf_sc_est, ~] = tfestimate(sc.u, sc.encoder, win, [], [], 1/Ts);
+[co_sc_est, ~] = mscohere(  sc.u, sc.encoder, win, [], [], 1/Ts);
+
+figure;
+hold on;
+plot(f, co_oc_est, '-')
+plot(f, co_sc_est, '-')
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
+ylabel('Coherence'); xlabel('Frequency [Hz]');
+hold off;
+xlim([0.5, 5e3]);
+
+
+
+% #+name: fig:stiffness_force_sensor_coherence
+% #+caption:
+% #+RESULTS:
+% [[file:figs/stiffness_force_sensor_coherence.png]]
+
+
+
+figure;
+tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile;
+hold on;
+plot(f, abs(tf_oc_est), '-', 'DisplayName', 'Open-Circuit')
+plot(f, abs(tf_sc_est), '-', 'DisplayName', 'Short-Circuit')
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
+ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-7, 3e-4]);
+legend('location', 'southwest');
+
+ax2 = nexttile;
+hold on;
+plot(f, 180/pi*angle(tf_oc_est), '-')
+plot(f, 180/pi*angle(tf_sc_est), '-')
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
+ylabel('Phase'); xlabel('Frequency [Hz]');
+hold off;
+yticks(-360:90:360);
+axis padded 'auto x'
+
+linkaxes([ax1,ax2], 'x');
+xlim([0.5, 5e3]);
--- a/matlab/runtest.m
+++ b/matlab/runtest.m
@@ -0,0 +1,17 @@
+tg = slrt;
+
+%%
+f = SimulinkRealTime.openFTP(tg);
+mget(f, 'int_enc.dat', 'data');
+close(f);
+
+%% Convert the Data
+data = SimulinkRealTime.utils.getFileScopeData('data/int_enc.dat').data;
+
+interferometer = data(:, 1);
+encoder = data(:, 2);
+u = data(:, 3);
+t = data(:, 4);
+
+save('./mat/int_enc_id_noise_bis.mat', 'interferometer', 'encoder', 'u' , 't');
+
--- a/matlab/setup.m
+++ b/matlab/setup.m
@@ -0,0 +1,23 @@
+%%
+s = tf('s');
+Ts = 1e-4; % [s]
+
+%% Pre-Filter
+G_pf = 1/(1 + s/2/pi/20);
+G_pf = c2d(G_pf, Ts, 'tustin');
+
+% %% Force Sensor Filter (HPF)
+% Gf_hpf = s/(s + 2*pi*2);
+% Gf_hpf = tf(1);
+% Gf_hpf = c2d(Gf_hpf, Ts, 'tustin');
+% 
+% %% IFF Controller
+% Kiff = 1/(s + 2*pi*2);
+% Kiff = c2d(Kiff, Ts, 'tustin');
+% 
+% %% Excitation Signal
+Tsim = 100; % Excitation time + Measurement time [s]
+
+t = 0:Ts:Tsim;
+u_exc = timeseries(chirp(t, 10, Tsim, 40, 'logarithmic'), t);
+% u_exc = timeseries(y_v, t);
--- a/matlab/speedgoat_IO318_100k_CI_01585.mat
+++ b/matlab/speedgoat_IO318_100k_CI_01585.mat
--- a/matlab/test_force_sensor.slx
+++ b/matlab/test_force_sensor.slx