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<title>Kinematics of the station</title>
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<h1 class="title">Kinematics of the station</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org46d4418">1. Micro Hexapod</a>
<li><a href="#org4ed4783">1. Micro Hexapod</a>
<ul>
<li><a href="#org6cc9e73">1.1. How the Symetrie Hexapod is controlled on the micro station</a></li>
<li><a href="#orgfcd44a9">1.2. Control of the Micro-Hexapod using Simscape</a>
<li><a href="#org129001c">1.1. How the Symetrie Hexapod is controlled on the micro station</a></li>
<li><a href="#org310b222">1.2. Control of the Micro-Hexapod using Simscape</a>
<ul>
<li><a href="#org3924132">1.2.1. Using Bushing Joint</a></li>
<li><a href="#org949c942">1.2.2. Using Inverse Kinematics and Leg Actuators</a>
<li><a href="#org8a60dbb">1.2.1. Using Bushing Joint</a></li>
<li><a href="#org207d470">1.2.2. Using Inverse Kinematics and Leg Actuators</a>
<ul>
<li><a href="#orgc9eab88">1.2.2.1. Theory</a></li>
<li><a href="#orge812977">1.2.2.2. Matlab Implementation</a></li>
<li><a href="#org1f70ebc">1.2.2.1. Theory</a></li>
<li><a href="#org8974896">1.2.2.2. Matlab Implementation</a></li>
</ul>
</li>
</ul>
@@ -59,12 +63,12 @@
In this document, we discuss the way the motion of each stage is defined.
</p>
<div id="outline-container-org46d4418" class="outline-2">
<h2 id="org46d4418"><span class="section-number-2">1</span> Micro Hexapod</h2>
<div id="outline-container-org4ed4783" class="outline-2">
<h2 id="org4ed4783"><span class="section-number-2">1</span> Micro Hexapod</h2>
<div class="outline-text-2" id="text-1">
</div>
<div id="outline-container-org6cc9e73" class="outline-3">
<h3 id="org6cc9e73"><span class="section-number-3">1.1</span> How the Symetrie Hexapod is controlled on the micro station</h3>
<div id="outline-container-org129001c" class="outline-3">
<h3 id="org129001c"><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">
<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):
@@ -112,8 +116,8 @@ Thus, it does the translations and then the rotation around the new translated f
</div>
</div>
<div id="outline-container-orgfcd44a9" class="outline-3">
<h3 id="orgfcd44a9"><span class="section-number-3">1.2</span> Control of the Micro-Hexapod using Simscape</h3>
<div id="outline-container-org310b222" class="outline-3">
<h3 id="org310b222"><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">
<p>
We can think of two main ways to position the Micro-Hexapod using Simscape.
@@ -130,15 +134,15 @@ This require a little bit more of mathematical derivations but this is the chose
</p>
</div>
<div id="outline-container-org3924132" class="outline-4">
<h4 id="org3924132"><span class="section-number-4">1.2.1</span> Using Bushing Joint</h4>
<div id="outline-container-org8a60dbb" class="outline-4">
<h4 id="org8a60dbb"><span class="section-number-4">1.2.1</span> Using Bushing Joint</h4>
<div class="outline-text-4" id="text-1-2-1">
<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="#org9af6f4f">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="#org0c4632e">1</a>.
</p>
<div id="org9af6f4f" class="figure">
<div id="org0c4632e" class="figure">
<p><img src="figs/bushing_joint_transform.png" alt="bushing_joint_transform.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Joint Transformation Sequence for the Bushing Joint</p>
@@ -156,8 +160,8 @@ However, the Bushing Joint makes rotations around mobiles axes (X, Y&rsquo; and
</div>
</div>
<div id="outline-container-org949c942" class="outline-4">
<h4 id="org949c942"><span class="section-number-4">1.2.2</span> Using Inverse Kinematics and Leg Actuators</h4>
<div id="outline-container-org207d470" class="outline-4">
<h4 id="org207d470"><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">
<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.
@@ -183,8 +187,8 @@ Thus, for this simulation, we <b>remove the gravity</b>.
</p>
</div>
<div id="outline-container-orgc9eab88" class="outline-5">
<h5 id="orgc9eab88"><span class="section-number-5">1.2.2.1</span> Theory</h5>
<div id="outline-container-org1f70ebc" class="outline-5">
<h5 id="org1f70ebc"><span class="section-number-5">1.2.2.1</span> Theory</h5>
<div class="outline-text-5" id="text-1-2-2-1">
<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, \(\bm{L} = [l_1, l_2, \dots, l_6]^T\).
@@ -219,14 +223,14 @@ Otherwise, when the limbs&rsquo; lengths derived yield complex numbers, then the
</div>
</div>
<div id="outline-container-orge812977" class="outline-5">
<h5 id="orge812977"><span class="section-number-5">1.2.2.2</span> Matlab Implementation</h5>
<div id="outline-container-org8974896" class="outline-5">
<h5 id="org8974896"><span class="section-number-5">1.2.2.2</span> Matlab Implementation</h5>
<div class="outline-text-5" id="text-1-2-2-2">
<p>
We open the Simulink file.
</p>
<div class="org-src-container">
<pre class="src src-matlab">open(<span class="org-string">'nass_model.slx'</span>)
<pre class="src src-matlab"> open(<span class="org-string">'nass_model.slx'</span>)
</pre>
</div>
@@ -234,8 +238,8 @@ We open the Simulink file.
We load the configuration and set a small <code>StopTime</code>.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'mat/conf_simulink.mat'</span>);
<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simulink</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'0.1'</span>);
<pre class="src src-matlab"> load(<span class="org-string">'mat/conf_simulink.mat'</span>);
<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simulink</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'0.1'</span>);
</pre>
</div>
@@ -243,40 +247,40 @@ We load the configuration and set a small <code>StopTime</code>.
We define the wanted position/orientation of the Hexapod under study.
</p>
<div class="org-src-container">
<pre class="src src-matlab">tx = 0.05; <span class="org-comment">% [rad]</span>
ty = 0.1; <span class="org-comment">% [rad]</span>
tz = 0.02; <span class="org-comment">% [rad]</span>
<pre class="src src-matlab"> tx = 0.05; <span class="org-comment">% [rad]</span>
ty = 0.1; <span class="org-comment">% [rad]</span>
tz = 0.02; <span class="org-comment">% [rad]</span>
Rx = [1 0 0;
0 cos(tx) <span class="org-type">-</span>sin(tx);
0 sin(tx) cos(tx)];
Rx = [1 0 0;
0 cos(tx) <span class="org-type">-</span>sin(tx);
0 sin(tx) cos(tx)];
Ry = [ cos(ty) 0 sin(ty);
0 1 0;
<span class="org-type">-</span>sin(ty) 0 cos(ty)];
Ry = [ cos(ty) 0 sin(ty);
0 1 0;
<span class="org-type">-</span>sin(ty) 0 cos(ty)];
Rz = [cos(tz) <span class="org-type">-</span>sin(tz) 0;
sin(tz) cos(tz) 0;
0 0 1];
Rz = [cos(tz) <span class="org-type">-</span>sin(tz) 0;
sin(tz) cos(tz) 0;
0 0 1];
ARB = Rz<span class="org-type">*</span>Ry<span class="org-type">*</span>Rx;
AP = [0.1; 0.005; 0.01]; <span class="org-comment">% [m]</span>
ARB = Rz<span class="org-type">*</span>Ry<span class="org-type">*</span>Rx;
AP = [0.1; 0.005; 0.01]; <span class="org-comment">% [m]</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">false</span>);
initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
initializeGranite(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
initializeTy(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
initializeRy(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
initializeRz(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
initializeMicroHexapod(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>, <span class="org-string">'AP'</span>, AP, <span class="org-string">'ARB'</span>, ARB);
initializeAxisc(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
initializeMirror(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
initializeNanoHexapod(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
initializeSample(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'all'</span>);
<pre class="src src-matlab"> initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">false</span>);
initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
initializeGranite(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
initializeTy(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
initializeRy(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
initializeRz(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
initializeMicroHexapod(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>, <span class="org-string">'AP'</span>, AP, <span class="org-string">'ARB'</span>, ARB);
initializeAxisc(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
initializeMirror(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
initializeNanoHexapod(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
initializeSample(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'all'</span>);
</pre>
</div>
@@ -284,7 +288,7 @@ initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span clas
We run the simulation.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'nass_model'</span>);
<pre class="src src-matlab"> <span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'nass_model'</span>);
</pre>
</div>
@@ -292,7 +296,7 @@ We run the simulation.
And we verify that we indeed succeed to go to the wanted position.
</p>
<div class="org-src-container">
<pre class="src src-matlab">[simout.Dhm.x.Data(end) ; simout.Dhm.y.Data(end) ; simout.Dhm.z.Data(end)] <span class="org-type">-</span> AP
<pre class="src src-matlab"> [simout.Dhm.x.Data(end) ; simout.Dhm.y.Data(end) ; simout.Dhm.z.Data(end)] <span class="org-type">-</span> AP
</pre>
</div>
@@ -318,7 +322,7 @@ And we verify that we indeed succeed to go to the wanted position.
</table>
<div class="org-src-container">
<pre class="src src-matlab">simout.Dhm.R.Data(<span class="org-type">:</span>, <span class="org-type">:</span>, end)<span class="org-type">-</span>ARB
<pre class="src src-matlab"> simout.Dhm.R.Data(<span class="org-type">:</span>, <span class="org-type">:</span>, end)<span class="org-type">-</span>ARB
</pre>
</div>
@@ -360,7 +364,7 @@ And we verify that we indeed succeed to go to the wanted position.
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-04-17 ven. 09:35</p>
<p class="date">Created: 2021-02-20 sam. 23:08</p>
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