Change y to x to match description

figs/test-bench-schematic.svg

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index.html

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-<!-- 2020-12-15 mar. 22:32 -->
-<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
-<title>Flexible Joint - Test Bench</title>
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-<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">Flexible Joint - Test Bench</h1>
-<div id="table-of-contents">
-<h2>Table of Contents</h2>
-<div id="text-table-of-contents">
-<ul>
-<li><a href="#org3ec0fb6">1. Test Bench Description</a>
-<ul>
-<li><a href="#org1c35c31">1.1. Flexible joint Geometry</a></li>
-<li><a href="#org943ed6d">1.2. Required external applied force</a></li>
-<li><a href="#org866642a">1.3. Required actuator stroke and sensors range</a></li>
-<li><a href="#org4789077">1.4. First try with the APA95ML</a></li>
-</ul>
-</li>
-<li><a href="#org75cb5e5">2. Experimental measurement</a></li>
-</ul>
-</div>
-</div>
-
-<div id="outline-container-org3ec0fb6" class="outline-2">
-<h2 id="org3ec0fb6"><span class="section-number-2">1</span> Test Bench Description</h2>
-<div class="outline-text-2" id="text-1">
-<p>
-The main characteristic of the flexible joint that we want to measure is its bending stiffness \(k_{R_x} \approx k_{R_y}\).
-</p>
-
-<p>
-To do so, a test bench is used.
-Specifications of the test bench to precisely measure the bending stiffness are described in this section.
-</p>
-
-<p>
-The basic idea is to measured the angular deflection of the flexible joint as a function of the applied torque.
-</p>
-
-
-<div id="org12e0ba4" class="figure">
-<p><img src="figs/test-bench-schematic.png" alt="test-bench-schematic.png" />
-</p>
-<p><span class="figure-number">Figure 1: </span>Schematic of the test bench to measure the bending stiffness of the flexible joints</p>
-</div>
-</div>
-
-<div id="outline-container-org1c35c31" class="outline-3">
-<h3 id="org1c35c31"><span class="section-number-3">1.1</span> Flexible joint Geometry</h3>
-<div class="outline-text-3" id="text-1-1">
-<p>
-The flexible joint used for the Nano-Hexapod is shown in Figure <a href="#org907b319">2</a>.
-Its bending stiffness is foreseen to be \(k_{R_y}\approx 20\,\frac{Nm}{rad}\) and its stroke \(\theta_{y,\text{max}}\approx 20\,mrad\).
-</p>
-
-
-<div id="org907b319" class="figure">
-<p><img src="figs/flexible_joint_geometry.png" alt="flexible_joint_geometry.png" />
-</p>
-<p><span class="figure-number">Figure 2: </span>Geometry of the flexible joint</p>
-</div>
-
-<p>
-The height between the flexible point (center of the joint) and the point where external forces are applied is \(h = 20\,mm\).
-</p>
-
-<p>
-Let&rsquo;s define the parameters on Matlab.
-</p>
-<div class="org-src-container">
-<pre class="src src-matlab">kRx = 20; <span class="org-comment">% Bending Stiffness [Nm/rad]</span>
-Rxmax = 20e<span class="org-type">-</span>3; <span class="org-comment">% Bending Stroke [rad]</span>
-h = 20e<span class="org-type">-</span>3; <span class="org-comment">% Height [m]</span>
-</pre>
-</div>
-</div>
-</div>
-
-<div id="outline-container-org943ed6d" class="outline-3">
-<h3 id="org943ed6d"><span class="section-number-3">1.2</span> Required external applied force</h3>
-<div class="outline-text-3" id="text-1-2">
-<p>
-The bending \(\theta_y\) of the flexible joint due to the force \(F_x\) is:
-</p>
-\begin{equation}
-  \theta_y = \frac{M_y}{k_{R_y}} = \frac{F_x h}{k_{R_y}}
-\end{equation}
-
-<p>
-Therefore, the applied force to test the full range of the flexible joint is:
-</p>
-\begin{equation}
-  F_{x,\text{max}} = \frac{k_{R_y} \theta_{y,\text{max}}}{h}
-\end{equation}
-
-<div class="org-src-container">
-<pre class="src src-matlab">Fxmax = kRx<span class="org-type">*</span>Rxmax<span class="org-type">/</span>h; <span class="org-comment">% Force to induce maximum stroke [N]</span>
-</pre>
-</div>
-
-<p>
-And we obtain:
-</p>
-\begin{equation} F_{max} = 20.0\, [N] \end{equation}
-
-<p>
-The measurement range of the force sensor should then be higher than \(20\,N\).
-</p>
-</div>
-</div>
-
-<div id="outline-container-org866642a" class="outline-3">
-<h3 id="org866642a"><span class="section-number-3">1.3</span> Required actuator stroke and sensors range</h3>
-<div class="outline-text-3" id="text-1-3">
-<p>
-The flexible joint is designed to allow a bending motion of \(\pm 20\,mrad\).
-The corresponding actuator stroke to impose such motion is:
-</p>
-
-<p>
-\[ d_{x,\text{max}} = h \tan(R_{x,\text{max}}) \]
-</p>
-
-<div class="org-src-container">
-<pre class="src src-matlab">dxmax = h<span class="org-type">*</span>tan(Rxmax);
-</pre>
-</div>
-
-\begin{equation} d_{max} = 0.4\, [mm] \end{equation}
-
-<p>
-In order to test the full range of the flexible joint, the stroke of the actuator should be higher than \(0.4\,mm\).
-The measurement range of the displacement sensor should also be higher than \(0.4\,mm\).
-</p>
-</div>
-</div>
-
-<div id="outline-container-org4789077" class="outline-3">
-<h3 id="org4789077"><span class="section-number-3">1.4</span> First try with the APA95ML</h3>
-<div class="outline-text-3" id="text-1-4">
-<p>
-The APA95ML as a stroke of \(100\,\mu m\) and the encoder in parallel can easily measure the required stroke.
-</p>
-
-<p>
-Suppose the full stroke of the APA can be used to bend the flexible joint (ideal case), the measured force will be:
-</p>
-<div class="org-src-container">
-<pre class="src src-matlab">Fxmax = kRx<span class="org-type">*</span>100e<span class="org-type">-</span>6<span class="org-type">/</span>h<span class="org-type">^</span>2; <span class="org-comment">% Force at maximum stroke [N]</span>
-</pre>
-</div>
-
-\begin{equation} F_{max} = 5.0\, [N] \end{equation}
-
-<p>
-And the tested angular range is:
-</p>
-<div class="org-src-container">
-<pre class="src src-matlab">Rmax = tan(100e<span class="org-type">-</span>6<span class="org-type">/</span>h);
-</pre>
-</div>
-
-\begin{equation} \theta_{max} = 5.0\, [mrad] \end{equation}
-</div>
-</div>
-</div>
-
-<div id="outline-container-org75cb5e5" class="outline-2">
-<h2 id="org75cb5e5"><span class="section-number-2">2</span> Experimental measurement</h2>
-</div>
-</div>
-<div id="postamble" class="status">
-<p class="author">Author: Dehaeze Thomas</p>
-<p class="date">Created: 2020-12-15 mar. 22:32</p>
-</div>
-</body>
-</html>

index.html

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+test-bench-flexible-joints.html

index.org

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-#+TITLE: Flexible Joint - Test Bench
-:DRAWER:
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-#+EMAIL: dehaeze.thomas@gmail.com
-#+AUTHOR: Dehaeze Thomas
-
-#+HTML_LINK_HOME: ../index.html
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-#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
-:END:
-
-* Test Bench Description
-** Introduction                                                      :ignore:
-
-The main characteristic of the flexible joint that we want to measure is its bending stiffness $k_{R_x} \approx k_{R_y}$.
-
-To do so, a test bench is used.
-Specifications of the test bench to precisely measure the bending stiffness are described in this section.
-
-The basic idea is to measured the angular deflection of the flexible joint as a function of the applied torque.
-
-#+name: fig:test-bench-schematic
-#+caption: Schematic of the test bench to measure the bending stiffness of the flexible joints
-[[file:figs/test-bench-schematic.png]]
-
-** 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
-
-** Flexible joint Geometry
-The flexible joint used for the Nano-Hexapod is shown in Figure [[fig:flexible_joint_geometry]].
-Its bending stiffness is foreseen to be $k_{R_y}\approx 20\,\frac{Nm}{rad}$ and its stroke $\theta_{y,\text{max}}\approx 20\,mrad$.
-
-#+name: fig:flexible_joint_geometry
-#+caption: Geometry of the flexible joint
-[[file:figs/flexible_joint_geometry.png]]
-
-The height between the flexible point (center of the joint) and the point where external forces are applied is $h = 20\,mm$.
-
-Let's define the parameters on Matlab.
-#+begin_src matlab
-  kRx = 20; % Bending Stiffness [Nm/rad]
-  Rxmax = 20e-3; % Bending Stroke [rad]
-  h = 20e-3; % Height [m]
-#+end_src
-
-** Required external applied force
-
-The bending $\theta_y$ of the flexible joint due to the force $F_x$ is:
-\begin{equation}
-  \theta_y = \frac{M_y}{k_{R_y}} = \frac{F_x h}{k_{R_y}}
-\end{equation}
-
-Therefore, the applied force to test the full range of the flexible joint is:
-\begin{equation}
-  F_{x,\text{max}} = \frac{k_{R_y} \theta_{y,\text{max}}}{h}
-\end{equation}
-
-#+begin_src matlab
-  Fxmax = kRx*Rxmax/h; % Force to induce maximum stroke [N]
-#+end_src
-
-And we obtain:
-#+begin_src matlab :results value raw replace :exports results
-  sprintf('\\begin{equation} F_{max} = %.1f\\, [N] \\end{equation}', Fxmax)
-#+end_src
-
-#+RESULTS:
-\begin{equation} F_{max} = 20.0\, [N] \end{equation}
-
-The measurement range of the force sensor should then be higher than $20\,N$.
-
-** Required actuator stroke and sensors range
-
-The flexible joint is designed to allow a bending motion of $\pm 20\,mrad$.
-The corresponding actuator stroke to impose such motion is:
-
-\[ d_{x,\text{max}} = h \tan(R_{x,\text{max}}) \]
-
-#+begin_src matlab
-  dxmax = h*tan(Rxmax);
-#+end_src
-
-#+begin_src matlab :results value raw replace :exports results
-  sprintf('\\begin{equation} d_{max} = %.1f\\, [mm] \\end{equation}', 1e3*dxmax)
-#+end_src
-
-#+RESULTS:
-\begin{equation} d_{max} = 0.4\, [mm] \end{equation}
-
-In order to test the full range of the flexible joint, the stroke of the actuator should be higher than $0.4\,mm$.
-The measurement range of the displacement sensor should also be higher than $0.4\,mm$.
-
-** First try with the APA95ML
-
-The APA95ML as a stroke of $100\,\mu m$ and the encoder in parallel can easily measure the required stroke.
-
-Suppose the full stroke of the APA can be used to bend the flexible joint (ideal case), the measured force will be:
-#+begin_src matlab
-  Fxmax = kRx*100e-6/h^2; % Force at maximum stroke [N]
-#+end_src
-
-#+begin_src matlab :results value raw replace :exports results
-  sprintf('\\begin{equation} F_{max} = %.1f\\, [N] \\end{equation}', Fxmax)
-#+end_src
-
-#+RESULTS:
-\begin{equation} F_{max} = 5.0\, [N] \end{equation}
-
-And the tested angular range is:
-#+begin_src matlab
-  Rmax = tan(100e-6/h);
-#+end_src
-
-#+begin_src matlab :results value raw replace :exports results
-  sprintf('\\begin{equation} \\theta_{max} = %.1f\\, [mrad] \\end{equation}', 1e3*Rmax)
-#+end_src
-
-#+RESULTS:
-\begin{equation} \theta_{max} = 5.0\, [mrad] \end{equation}
-
-* Experimental measurement

matlab/bench_dimensioning.m

@@ -0,0 +1,53 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+% Flexible joint Geometry
+% The flexible joint used for the Nano-Hexapod is shown in Figure [[fig:flexible_joint_geometry]].
+% Its bending stiffness is foreseen to be $k_{R_y}\approx 5\,\frac{Nm}{rad}$ and its stroke $\theta_{y,\text{max}}\approx 25\,mrad$.
+
+% #+name: fig:flexible_joint_geometry
+% #+caption: Geometry of the flexible joint
+% [[file:figs/flexible_joint_geometry.png]]
+
+% The height between the flexible point (center of the joint) and the point where external forces are applied is $h = 20\,mm$.
+
+% Let's define the parameters on Matlab.
+
+kRx = 5; % Bending Stiffness [Nm/rad]
+Rxmax = 25e-3; % Bending Stroke [rad]
+h = 20e-3; % Height [m]
+
+% Required external applied force
+
+% The bending $\theta_y$ of the flexible joint due to the force $F_x$ is:
+% \begin{equation}
+%   \theta_y = \frac{M_y}{k_{R_y}} = \frac{F_x h}{k_{R_y}}
+% \end{equation}
+
+% Therefore, the applied force to test the full range of the flexible joint is:
+% \begin{equation}
+%   F_{x,\text{max}} = \frac{k_{R_y} \theta_{y,\text{max}}}{h}
+% \end{equation}
+
+
+Fxmax = kRx*Rxmax/h; % Force to induce maximum stroke [N]
+
+
+
+% And we obtain:
+
+sprintf('\\begin{equation} F_{x,max} = %.1f\\, [N] \\end{equation}', Fxmax)
+
+% Required actuator stroke and sensors range
+
+% The flexible joint is designed to allow a bending motion of $\pm 25\,mrad$.
+% The corresponding stroke at the location of the force sensor is:
+% \[ d_{x,\text{max}} = h \tan(R_{x,\text{max}}) \]
+
+
+dxmax = h*tan(Rxmax);
+
+sprintf('\\begin{equation} d_{max} = %.1f\\, [mm] \\end{equation}', 1e3*dxmax)

matlab/error_budget.m

@@ -0,0 +1,67 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+% Finite Element Model
+% From the Finite Element Model, the stiffness and stroke of the flexible joint have been computed and summarized in Tables [[tab:axial_shear_characteristics]] and [[tab:bending_torsion_characteristics]].
+
+
+%% Stiffness
+ka = 94e6; % Axial Stiffness [N/m]
+ks = 13e6; % Shear Stiffness [N/m]
+kb = 5;    % Bending Stiffness [Nm/rad]
+kt = 260;  % Torsional Stiffness [Nm/rad]
+
+%% Maximum force
+Fa = 469;    % Axial Force before yield [N]
+Fs = 242;    % Shear Force before yield [N]
+Fb = 0.118;  % Bending Force before yield [Nm]
+Ft = 1.508; % Torsional Force before yield [Nm]
+
+%% Compute the corresponding stroke
+Xa = Fa/ka; % Axial Stroke before yield [m]
+Xs = Fs/ks; % Shear Stroke before yield [m]
+Xb = Fb/kb; % Bending Stroke before yield [rad]
+Xt = Ft/kt; % Torsional Stroke before yield [rad]
+
+% Setup
+
+% The setup is schematically represented in Figure [[fig:test_bench_flex_side_bis]].
+
+% The force is applied on top of the flexible joint with a distance $h$ with the joint's center.
+% The displacement of the flexible joint is also measured at the same height.
+
+% The height between the joint's center and the force application point is:
+
+h = 25e-3; % Height [m]
+
+% Estimation error due to force sensor compression
+% The measured displacement is not done directly at the joint's location.
+% The force sensor compression will then induce an error on the joint's stiffness.
+
+% The force sensor stiffness $k_F$ is estimated to be around:
+
+kF = 50/0.05e-3; % [N/m]
+
+sprintf('k_F = %.1e [N/m]', kF)
+
+% Estimation error due to height estimation error
+% Let's consider an error in the estimation of the height from the application of the force to the joint's center:
+% \begin{equation}
+% h_{\text{est}} = h (1 + \epsilon)
+% \end{equation}
+
+% The computed bending stiffness will be:
+% \begin{equation}
+%   k_\text{est} \approx h_{\text{est}}^2 \frac{F_x}{d_x}
+% \end{equation}
+
+% And the stiffness estimation error is:
+% \begin{equation}
+% \frac{k_{\text{est}}}{k_{R_y}} = (1 + \epsilon)^2
+% \end{equation}
+
+
+h_err = 0.2e-3; % Height estimation error [m]