tdehaeze/nass-simscape
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

Update simulation of kinematics

Browse Source
This commit is contained in:
Thomas Dehaeze 2019-12-11 14:47:14 +01:00
parent 6a3481b8ac
commit 7ca627ff91
2 changed files with 75 additions and 43 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

97
kinematics/index.html
View File

@ -3,7 +3,7 @@
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head> <head>
<!-- 2019-12-10 mar. 18:03 --> <!-- 2019-12-11 mer. 14:47 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" /> <meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Kinematics of the station</title> <title>Kinematics of the station</title>
@ -283,16 +283,16 @@ for the JavaScript code in this tag.
<h2>Table of Contents</h2> <h2>Table of Contents</h2>
<div id="text-table-of-contents"> <div id="text-table-of-contents">
<ul> <ul>
<li><a href="#orgc0809a7">1. Micro Hexapod</a> <li><a href="#orgf49d055">1. Micro Hexapod</a>
<ul> <ul>
<li><a href="#org5ed8144">1.1. How the Symetrie Hexapod is controlled on the micro station</a></li> <li><a href="#orgb024fa1">1.1. How the Symetrie Hexapod is controlled on the micro station</a></li>
<li><a href="#orge1f9456">1.2. Control of the Micro-Hexapod using Simscape</a> <li><a href="#org34abe0f">1.2. Control of the Micro-Hexapod using Simscape</a>
<ul> <ul>
<li><a href="#orgaec6c7a">1.2.1. Using Bushing Joint</a></li> <li><a href="#org118cdf5">1.2.1. Using Bushing Joint</a></li>
<li><a href="#org964676d">1.2.2. Using Inverse Kinematics and Leg Actuators</a> <li><a href="#org37b4bdd">1.2.2. Using Inverse Kinematics and Leg Actuators</a>
<ul> <ul>
<li><a href="#org3fcf5f8">1.2.2.1. Theory</a></li> <li><a href="#org9839e83">1.2.2.1. Theory</a></li>
<li><a href="#orgc39da76">1.2.2.2. Matlab Implementation</a></li> <li><a href="#org78fd3cf">1.2.2.2. Matlab Implementation</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
@ -307,12 +307,12 @@ for the JavaScript code in this tag.
In this document, we discuss the way the motion of each stage is defined. In this document, we discuss the way the motion of each stage is defined.
</p> </p>
<div id="outline-container-orgc0809a7" class="outline-2"> <div id="outline-container-orgf49d055" class="outline-2">
<h2 id="orgc0809a7"><span class="section-number-2">1</span> Micro Hexapod</h2> <h2 id="orgf49d055"><span class="section-number-2">1</span> Micro Hexapod</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-1">
</div> </div>
<div id="outline-container-org5ed8144" class="outline-3"> <div id="outline-container-orgb024fa1" class="outline-3">
<h3 id="org5ed8144"><span class="section-number-3">1.1</span> How the Symetrie Hexapod is controlled on the micro station</h3> <h3 id="orgb024fa1"><span class="section-number-3">1.1</span> How the Symetrie Hexapod is controlled on the micro station</h3>
<div class="outline-text-3" id="text-1-1"> <div class="outline-text-3" id="text-1-1">
<p> <p>
For the Micro-Hexapod, the convention for the angles are defined in <code>MAN_A_Software API_4.0.150918_EN.pdf</code> on page 13 (section 2.4 - Rotation Vectors): For the Micro-Hexapod, the convention for the angles are defined in <code>MAN_A_Software API_4.0.150918_EN.pdf</code> on page 13 (section 2.4 - Rotation Vectors):
@ -360,8 +360,8 @@ Thus, it does the translations and then the rotation around the new translated f
</div> </div>
</div> </div>
<div id="outline-container-orge1f9456" class="outline-3"> <div id="outline-container-org34abe0f" class="outline-3">
<h3 id="orge1f9456"><span class="section-number-3">1.2</span> Control of the Micro-Hexapod using Simscape</h3> <h3 id="org34abe0f"><span class="section-number-3">1.2</span> Control of the Micro-Hexapod using Simscape</h3>
<div class="outline-text-3" id="text-1-2"> <div class="outline-text-3" id="text-1-2">
<p> <p>
We can think of two main ways to position the Micro-Hexapod using Simscape. We can think of two main ways to position the Micro-Hexapod using Simscape.
@ -378,15 +378,15 @@ This require a little bit more of mathematical derivations but this is the chose
</p> </p>
</div> </div>
<div id="outline-container-orgaec6c7a" class="outline-4"> <div id="outline-container-org118cdf5" class="outline-4">
<h4 id="orgaec6c7a"><span class="section-number-4">1.2.1</span> Using Bushing Joint</h4> <h4 id="org118cdf5"><span class="section-number-4">1.2.1</span> Using Bushing Joint</h4>
<div class="outline-text-4" id="text-1-2-1"> <div class="outline-text-4" id="text-1-2-1">
<p> <p>
In the documentation of the Bushing Joint (<code>doc "Bushing Joint"</code>) that is used to position the Hexapods, it is mention that the following frame is positioned with respect to the base frame in a way shown in figure <a href="#orgaa7f4a2">1</a>. In the documentation of the Bushing Joint (<code>doc "Bushing Joint"</code>) that is used to position the Hexapods, it is mention that the following frame is positioned with respect to the base frame in a way shown in figure <a href="#orgbf74afe">1</a>.
</p> </p>
<div id="orgaa7f4a2" class="figure"> <div id="orgbf74afe" class="figure">
<p><img src="figs/bushing_joint_transform.png" alt="bushing_joint_transform.png" /> <p><img src="figs/bushing_joint_transform.png" alt="bushing_joint_transform.png" />
</p> </p>
<p><span class="figure-number">Figure 1: </span>Joint Transformation Sequence for the Bushing Joint</p> <p><span class="figure-number">Figure 1: </span>Joint Transformation Sequence for the Bushing Joint</p>
@ -404,8 +404,8 @@ However, the Bushing Joint makes rotations around mobiles axes (X, Y' and then Z
</div> </div>
</div> </div>
<div id="outline-container-org964676d" class="outline-4"> <div id="outline-container-org37b4bdd" class="outline-4">
<h4 id="org964676d"><span class="section-number-4">1.2.2</span> Using Inverse Kinematics and Leg Actuators</h4> <h4 id="org37b4bdd"><span class="section-number-4">1.2.2</span> Using Inverse Kinematics and Leg Actuators</h4>
<div class="outline-text-4" id="text-1-2-2"> <div class="outline-text-4" id="text-1-2-2">
<p> <p>
Here, we can use the Inverse Kinematic of the Hexapod to determine the length of each leg in order to obtain some defined translation and rotation of the mobile platform. Here, we can use the Inverse Kinematic of the Hexapod to determine the length of each leg in order to obtain some defined translation and rotation of the mobile platform.
@ -418,10 +418,21 @@ The advantages are:
<li>we can position the Hexapod as we want by specifying a rotation matrix</li> <li>we can position the Hexapod as we want by specifying a rotation matrix</li>
<li>the hexapod keeps its full flexibility as we don't specify any wanted displacements, only leg's rest position</li> <li>the hexapod keeps its full flexibility as we don't specify any wanted displacements, only leg's rest position</li>
</ul> </ul>
<p>
However:
</p>
<ul class="org-ul">
<li>even though the rest position of each leg (the position where the stiffness force is zero) is set correctly, the hexapod will we deflected due to gravity</li>
</ul>
<p>
Thus, for this simulation, we <b>remove the gravity</b>.
</p>
</div> </div>
<div id="outline-container-org3fcf5f8" class="outline-5"> <div id="outline-container-org9839e83" class="outline-5">
<h5 id="org3fcf5f8"><span class="section-number-5">1.2.2.1</span> Theory</h5> <h5 id="org9839e83"><span class="section-number-5">1.2.2.1</span> Theory</h5>
<div class="outline-text-5" id="text-1-2-2-1"> <div class="outline-text-5" id="text-1-2-2-1">
<p> <p>
For inverse kinematic analysis, it is assumed that the position \({}^A\bm{P}\) and orientation of the moving platform \({}^A\bm{R}_B\) are given and the problem is to obtain the joint variables, namely, \(\bm{L} = [l_1, l_2, \dots, l_6]^T\). For inverse kinematic analysis, it is assumed that the position \({}^A\bm{P}\) and orientation of the moving platform \({}^A\bm{R}_B\) are given and the problem is to obtain the joint variables, namely, \(\bm{L} = [l_1, l_2, \dots, l_6]^T\).
@ -456,20 +467,29 @@ Otherwise, when the limbs' lengths derived yield complex numbers, then the posit
</div> </div>
</div> </div>
<div id="outline-container-orgc39da76" class="outline-5"> <div id="outline-container-org78fd3cf" class="outline-5">
<h5 id="orgc39da76"><span class="section-number-5">1.2.2.2</span> Matlab Implementation</h5> <h5 id="org78fd3cf"><span class="section-number-5">1.2.2.2</span> Matlab Implementation</h5>
<div class="outline-text-5" id="text-1-2-2-2"> <div class="outline-text-5" id="text-1-2-2-2">
<p>
We open the Simulink file.
</p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">open <span class="org-string">'simscape/hexapod_tests.slx'</span> <pre class="src src-matlab">open <span class="org-string">'simscape/hexapod_tests.slx'</span>
</pre> </pre>
</div> </div>
<p>
We load the configuration and set a small <code>StopTime</code>.
</p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'simscape/conf_simscape.mat'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'simscape/conf_simscape.mat'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
<span class="org-matlab-simulink-keyword">set_param</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">conf_simscape</span>, <span class="org-string">'StopTime'</span>, '<span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">5</span><span class="org-type">'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-matlab-simulink-keyword">set_param</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">conf_simscape</span>, <span class="org-string">'StopTime'</span>, '<span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">5</span><span class="org-type">'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre> </pre>
</div> </div>
<p>
We define the wanted position/orientation of the Hexapod under study.
</p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">tx = <span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">1</span>; <span class="org-comment">% [rad]</span> <pre class="src src-matlab">tx = <span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">1</span>; <span class="org-comment">% [rad]</span>
ty = <span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">2</span>; <span class="org-comment">% [rad]</span> ty = <span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">2</span>; <span class="org-comment">% [rad]</span>
@ -494,6 +514,9 @@ hexapod = initializeMicroHexapod<span class="org-rainbow-delimiters-depth-1">(</
</pre> </pre>
</div> </div>
<p>
We run the simulation.
</p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'simscape/hexapod_tests.slx'</span><span class="org-rainbow-delimiters-depth-1">)</span> <pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'simscape/hexapod_tests.slx'</span><span class="org-rainbow-delimiters-depth-1">)</span>
</pre> </pre>
@ -515,15 +538,15 @@ And we verify that we indeed succeed to go to the wanted position.
</colgroup> </colgroup>
<tbody> <tbody>
<tr> <tr>
<td class="org-right">-2.12e-06</td> <td class="org-right">1.611e-10</td>
</tr> </tr>
<tr> <tr>
<td class="org-right">2.9787e-06</td> <td class="org-right">-1.3748e-10</td>
</tr> </tr>
<tr> <tr>
<td class="org-right">-4.4341e-06</td> <td class="org-right">8.4879e-11</td>
</tr> </tr>
</tbody> </tbody>
</table> </table>
@ -545,21 +568,21 @@ And we verify that we indeed succeed to go to the wanted position.
</colgroup> </colgroup>
<tbody> <tbody>
<tr> <tr>
<td class="org-right">-1.5714e-06</td> <td class="org-right">-1.2659e-10</td>
<td class="org-right">1.4513e-06</td> <td class="org-right">6.5603e-11</td>
<td class="org-right">7.8133e-06</td> <td class="org-right">6.2183e-10</td>
</tr> </tr>
<tr> <tr>
<td class="org-right">8.4113e-07</td> <td class="org-right">1.0354e-10</td>
<td class="org-right">-7.1485e-07</td> <td class="org-right">-5.2439e-11</td>
<td class="org-right">-7.4572e-06</td> <td class="org-right">-5.2425e-10</td>
</tr> </tr>
<tr> <tr>
<td class="org-right">-7.5348e-06</td> <td class="org-right">-5.9816e-10</td>
<td class="org-right">7.7112e-06</td> <td class="org-right">5.532e-10</td>
<td class="org-right">-2.3088e-06</td> <td class="org-right">-1.7737e-10</td>
</tr> </tr>
</tbody> </tbody>
</table> </table>
@ -571,7 +594,7 @@ And we verify that we indeed succeed to go to the wanted position.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p> <p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2019-12-10 mar. 18:03</p> <p class="date">Created: 2019-12-11 mer. 14:47</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p> <p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div> </div>
</body> </body>

21
kinematics/index.org
View File

@ -104,6 +104,11 @@ The advantages are:
- we can position the Hexapod as we want by specifying a rotation matrix - we can position the Hexapod as we want by specifying a rotation matrix
- the hexapod keeps its full flexibility as we don't specify any wanted displacements, only leg's rest position - the hexapod keeps its full flexibility as we don't specify any wanted displacements, only leg's rest position
However:
- even though the rest position of each leg (the position where the stiffness force is zero) is set correctly, the hexapod will we deflected due to gravity
Thus, for this simulation, we *remove the gravity*.
**** Theory **** Theory
For inverse kinematic analysis, it is assumed that the position ${}^A\bm{P}$ and orientation of the moving platform ${}^A\bm{R}_B$ are given and the problem is to obtain the joint variables, namely, $\bm{L} = [l_1, l_2, \dots, l_6]^T$. For inverse kinematic analysis, it is assumed that the position ${}^A\bm{P}$ and orientation of the moving platform ${}^A\bm{R}_B$ are given and the problem is to obtain the joint variables, namely, $\bm{L} = [l_1, l_2, \dots, l_6]^T$.
@ -140,15 +145,18 @@ Otherwise, when the limbs' lengths derived yield complex numbers, then the posit
#+end_src #+end_src
**** Matlab Implementation **** Matlab Implementation
We open the Simulink file.
#+begin_src matlab #+begin_src matlab
open 'simscape/hexapod_tests.slx' open 'simscape/hexapod_tests.slx'
#+end_src #+end_src
We load the configuration and set a small =StopTime=.
#+begin_src matlab #+begin_src matlab
load('simscape/conf_simscape.mat'); load('simscape/conf_simscape.mat');
set_param(conf_simscape, 'StopTime', '0.5'); set_param(conf_simscape, 'StopTime', '0.5');
#+end_src #+end_src
We define the wanted position/orientation of the Hexapod under study.
#+begin_src matlab #+begin_src matlab
tx = 0.1; % [rad] tx = 0.1; % [rad]
ty = 0.2; % [rad] ty = 0.2; % [rad]
@ -172,6 +180,7 @@ Otherwise, when the limbs' lengths derived yield complex numbers, then the posit
hexapod = initializeMicroHexapod(struct('AP', AP, 'ARB', ARB)); hexapod = initializeMicroHexapod(struct('AP', AP, 'ARB', ARB));
#+end_src #+end_src
We run the simulation.
#+begin_src matlab #+begin_src matlab
sim('simscape/hexapod_tests.slx') sim('simscape/hexapod_tests.slx')
#+end_src #+end_src
@ -182,15 +191,15 @@ And we verify that we indeed succeed to go to the wanted position.
#+end_src #+end_src
#+RESULTS: #+RESULTS:
| -2.12e-06 | | 1.611e-10 |
| 2.9787e-06 | | -1.3748e-10 |
| -4.4341e-06 | | 8.4879e-11 |
#+begin_src matlab :results table replace #+begin_src matlab :results table replace
simout.R.Data(:, :, end)-ARB simout.R.Data(:, :, end)-ARB
#+end_src #+end_src
#+RESULTS: #+RESULTS:
| -1.5714e-06 | 1.4513e-06 | 7.8133e-06 | | -1.2659e-10 | 6.5603e-11 | 6.2183e-10 |
| 8.4113e-07 | -7.1485e-07 | -7.4572e-06 | | 1.0354e-10 | -5.2439e-11 | -5.2425e-10 |
| -7.5348e-06 | 7.7112e-06 | -2.3088e-06 | | -5.9816e-10 | 5.532e-10 | -1.7737e-10 |