652 lines
44 KiB
HTML
652 lines
44 KiB
HTML
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<title>Tomography Experiment</title>
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</head>
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<body>
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<div id="org-div-home-and-up">
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<a accesskey="h" href="../index.html"> UP </a>
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<a accesskey="H" href="../index.html"> HOME </a>
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</div><div id="content">
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<h1 class="title">Tomography Experiment</h1>
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<div id="table-of-contents">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#orgab97c4b">1. Initialize Experiment</a></li>
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<li><a href="#org26c554f">2. Run the Tomography Experiment</a></li>
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<li><a href="#orgc507e1e">3. <span class="todo TODO">TODO</span> Tests on the transformation from reference to wanted position</a>
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<ul>
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<li><a href="#org4537a86">3.1. Wanted Position of the Sample with respect to the Granite</a></li>
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<li><a href="#org6c67ae5">3.2. Measured Position of the Sample with respect to the Granite</a></li>
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<li><a href="#org69e38aa">3.3. Positioning Error with respect to the Granite</a></li>
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<li><a href="#org5246d94">3.4. Position Error Expressed in the Nano-Hexapod Frame</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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<div id="outline-container-orgab97c4b" class="outline-2">
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<h2 id="orgab97c4b"><span class="section-number-2">1</span> Initialize Experiment</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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We first initialize all the stages.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"><code>initializeGround<span class="org-rainbow-delimiters-depth-1">()</span>;</code>
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<code>initializeGranite<span class="org-rainbow-delimiters-depth-1">()</span>;</code>
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<code>initializeTy<span class="org-rainbow-delimiters-depth-1">()</span>;</code>
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<code>initializeRy<span class="org-rainbow-delimiters-depth-1">()</span>;</code>
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<code>initializeRz<span class="org-rainbow-delimiters-depth-1">()</span>;</code>
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<code>initializeMicroHexapod<span class="org-rainbow-delimiters-depth-1">()</span>;</code>
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<code>initializeAxisc<span class="org-rainbow-delimiters-depth-1">()</span>;</code>
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<code>initializeMirror<span class="org-rainbow-delimiters-depth-1">()</span>;</code>
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<code>initializeNanoHexapod<span class="org-rainbow-delimiters-depth-1">(</span>struct<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;</code>
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<code>initializeSample<span class="org-rainbow-delimiters-depth-1">(</span>struct<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'mass'</span>, <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;</code>
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</pre>
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</div>
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<p>
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We initialize the reference path for all the stages.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"><code>initializeReferences<span class="org-rainbow-delimiters-depth-1">(</span>struct<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'Rz_type'</span>, <span class="org-string">'rotating'</span>, <span class="org-string">'Rz_period'</span>, <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;</code>
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</pre>
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</div>
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<p>
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And we initialize the disturbances.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"><code>initDisturbances<span class="org-rainbow-delimiters-depth-1">()</span>;</code>
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</pre>
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</div>
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</div>
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</div>
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<div id="outline-container-org26c554f" class="outline-2">
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<h2 id="org26c554f"><span class="section-number-2">2</span> Run the Tomography Experiment</h2>
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<div class="outline-text-2" id="text-2">
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<p>
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We first load the simulation configuration
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"><code>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>;</code>
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</pre>
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab"><code><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">1</span><span class="org-type">'</span><span class="org-rainbow-delimiters-depth-1">)</span>;</code>
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</pre>
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab"><code><span class="org-matlab-simulink-keyword">set_param</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'sim_nano_station_tomo'</span>, <span class="org-string">'SimulationCommand'</span>, <span class="org-string">'start'</span><span class="org-rainbow-delimiters-depth-1">)</span>;</code>
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</pre>
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</div>
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</div>
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</div>
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<div id="outline-container-orgc507e1e" class="outline-2">
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<h2 id="orgc507e1e"><span class="section-number-2">3</span> <span class="todo TODO">TODO</span> Tests on the transformation from reference to wanted position</h2>
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<div class="outline-text-2" id="text-3">
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<ul class="org-ul">
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<li class="on"><code>[X]</code> Are the rotation matrix commutable? => no</li>
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<li class="on"><code>[X]</code> How to express the measured rotation errors? => screw axis coordinate seems nice (used in Taghirad's book)</li>
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<li class="off"><code>[ ]</code> Should ask Veijo how he specifies the position of the Symetrie Hexapod</li>
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<li class="off"><code>[ ]</code> Create functions for all distinct part and then include that in Simulink</li>
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<li class="off"><code>[ ]</code> How the express the orientation error?</li>
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<li class="off"><code>[ ]</code> If we use screw coordinate, can we add/subtract them?</li>
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<li class="off"><code>[ ]</code> Do some simple tests to verify that the algorithm is working fine</li>
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</ul>
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<blockquote>
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<p>
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Rx = [1 0 0;
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0 cos(t) -sin(t);
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0 sin(t) cos(t)];
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</p>
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<p>
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Ry = [ cos(t) 0 sin(t);
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0 1 0;
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-sin(t) 0 cos(t)];
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</p>
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<p>
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Rz = [cos(t) -sin(t) 0;
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sin(t) cos(t) 0;
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0 0 1];
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</p>
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</blockquote>
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<p>
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Let's define the following frames:
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</p>
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<ul class="org-ul">
|
|
<li>\(\{W\}\) the frame that is <b>fixed to the granite</b> and its origin at the theoretical meeting point between the X-ray and the spindle axis.</li>
|
|
<li>\(\{S\}\) the frame <b>attached to the sample</b> (in reality attached to the top platform of the nano-hexapod) with its origin at 175mm above the top platform of the nano-hexapod.
|
|
Its origin is \(O_S\).</li>
|
|
<li>\(\{T\}\) the theoretical wanted frame that correspond to the wanted pose of the frame \(\{S\}\).
|
|
\(\{T\}\) is computed from the wanted position of each stage. It is thus theoretical and does not correspond to a real position.
|
|
The origin of \(T\) is \(O_T\) and is the wanted position of the sample.</li>
|
|
</ul>
|
|
|
|
<p>
|
|
Thus:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>the <b>measurement</b> of the position of the sample corresponds to \({}^W O_S = \begin{bmatrix} {}^WP_{x,m} & {}^WP_{y,m} & {}^WP_{z,m} \end{bmatrix}^T\) in translation and to \(\theta_m {}^W\bm{s}_m = \theta_m \cdot \begin{bmatrix} {}^Ws_{x,m} & {}^Ws_{y,m} & {}^Ws_{z,m} \end{bmatrix}^T\) in rotations</li>
|
|
<li>the <b>wanted position</b> of the sample expressed w.r.t. the granite is \({}^W O_T = \begin{bmatrix} {}^WP_{x,r} & {}^WP_{y,r} & {}^WP_{z,r} \end{bmatrix}^T\) in translation and to \(\theta_r {}^W\bm{s}_r = \theta_r \cdot \begin{bmatrix} {}^Ws_{x,r} & {}^Ws_{y,r} & {}^Ws_{z,r} \end{bmatrix}^T\) in rotations</li>
|
|
</ul>
|
|
</div>
|
|
|
|
<div id="outline-container-org4537a86" class="outline-3">
|
|
<h3 id="org4537a86"><span class="section-number-3">3.1</span> Wanted Position of the Sample with respect to the Granite</h3>
|
|
<div class="outline-text-3" id="text-3-1">
|
|
<p>
|
|
Let's define the wanted position of each stage.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><code>Ty = <span class="org-highlight-numbers-number">1</span>; <span class="org-comment">% [m]</span></code>
|
|
<code>Ry = <span class="org-highlight-numbers-number">3</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">/</span><span class="org-highlight-numbers-number">180</span>; <span class="org-comment">% [rad]</span></code>
|
|
<code>Rz = <span class="org-highlight-numbers-number">180</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">/</span><span class="org-highlight-numbers-number">180</span>; <span class="org-comment">% [rad]</span></code>
|
|
<code></code>
|
|
<code><span class="org-comment">% Hexapod (first consider only translations)</span></code>
|
|
<code>Thx = <span class="org-highlight-numbers-number">0</span>; <span class="org-comment">% [m]</span></code>
|
|
<code>Thy = <span class="org-highlight-numbers-number">0</span>; <span class="org-comment">% [m]</span></code>
|
|
<code>Thz = <span class="org-highlight-numbers-number">0</span>; <span class="org-comment">% [m]</span></code>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Now, we compute the corresponding wanted translation and rotation of the sample with respect to the granite frame \(\{W\}\).
|
|
This corresponds to \({}^WO_T\) and \(\theta_m {}^Ws_m\).
|
|
</p>
|
|
|
|
<p>
|
|
To do so, we have to define the homogeneous transformation for each stage.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><code><span class="org-comment">% Translation Stage</span></code>
|
|
<code>Rty = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span>;</code>
|
|
<code> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">0</span> Ty;</code>
|
|
<code> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">0</span>;</code>
|
|
<code> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">]</span>;</code>
|
|
<code></code>
|
|
<code><span class="org-comment">% Tilt Stage - Pure rotating aligned with Ob</span></code>
|
|
<code>Rry = <span class="org-rainbow-delimiters-depth-1">[</span> cos<span class="org-rainbow-delimiters-depth-2">(</span>Ry<span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-highlight-numbers-number">0</span> sin<span class="org-rainbow-delimiters-depth-2">(</span>Ry<span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-highlight-numbers-number">0</span>;</code>
|
|
<code> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span>;</code>
|
|
<code> <span class="org-type">-</span>sin<span class="org-rainbow-delimiters-depth-2">(</span>Ry<span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-highlight-numbers-number">0</span> cos<span class="org-rainbow-delimiters-depth-2">(</span>Ry<span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-highlight-numbers-number">0</span>;</code>
|
|
<code> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">]</span>;</code>
|
|
<code></code>
|
|
<code><span class="org-comment">% Spindle - Rotation along the Z axis</span></code>
|
|
<code>Rrz = <span class="org-rainbow-delimiters-depth-1">[</span>cos<span class="org-rainbow-delimiters-depth-2">(</span>Rz<span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-type">-</span>sin<span class="org-rainbow-delimiters-depth-2">(</span>Rz<span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> ;</code>
|
|
<code> sin<span class="org-rainbow-delimiters-depth-2">(</span>Rz<span class="org-rainbow-delimiters-depth-2">)</span> cos<span class="org-rainbow-delimiters-depth-2">(</span>Rz<span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> ;</code>
|
|
<code> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">0</span> ;</code>
|
|
<code> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span> <span class="org-rainbow-delimiters-depth-1">]</span>;</code>
|
|
<code></code>
|
|
<code><span class="org-comment">% Micro-Hexapod (only rotations first)</span></code>
|
|
<code>Rh = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> Thx ;</code>
|
|
<code> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">0</span> Thy ;</code>
|
|
<code> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span> Thz ;</code>
|
|
<code> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span> <span class="org-rainbow-delimiters-depth-1">]</span>;</code>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We combine the individual homogeneous transformations into one homogeneous transformation for all the station.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><code>Ttot = Rty<span class="org-type">*</span>Rry<span class="org-type">*</span>Rrz<span class="org-type">*</span>Rh;</code>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Using this homogeneous transformation, we can compute the wanted position and orientation of the sample with respect to the granite.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><code>WOr = Ttot<span class="org-type">*</span><span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">0</span>;<span class="org-highlight-numbers-number">0</span>;<span class="org-highlight-numbers-number">0</span>;<span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">]</span>;</code>
|
|
<code>WOr = WOr<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">:</span><span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-1">)</span>;</code>
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><code>thetar = acos<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">(</span>trace<span class="org-rainbow-delimiters-depth-3">(</span>Ttot<span class="org-rainbow-delimiters-depth-4">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">:</span><span class="org-highlight-numbers-number">3</span>, <span class="org-highlight-numbers-number">1</span><span class="org-type">:</span><span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">-</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">/</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span></code>
|
|
<code><span class="org-keyword">if</span> thetar <span class="org-type">==</span> <span class="org-highlight-numbers-number">0</span></code>
|
|
<code> WSr = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">0</span>; <span class="org-highlight-numbers-number">0</span>; <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>;</code>
|
|
<code><span class="org-keyword">else</span></code>
|
|
<code> <span class="org-rainbow-delimiters-depth-1">[</span>V, D<span class="org-rainbow-delimiters-depth-1">]</span> = eig<span class="org-rainbow-delimiters-depth-1">(</span>Ttot<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">:</span><span class="org-highlight-numbers-number">3</span>, <span class="org-highlight-numbers-number">1</span><span class="org-type">:</span><span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;</code>
|
|
<code> WSr = thetar<span class="org-type">*</span>V<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-type">:</span>, abs<span class="org-rainbow-delimiters-depth-2">(</span>diag<span class="org-rainbow-delimiters-depth-3">(</span>D<span class="org-rainbow-delimiters-depth-3">)</span> <span class="org-type">-</span> <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-type"><</span> <span class="org-constant">eps</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;</code>
|
|
<code><span class="org-keyword">end</span></code>
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><code>WPr = <span class="org-rainbow-delimiters-depth-1">[</span>WOr ; WSr<span class="org-rainbow-delimiters-depth-1">]</span>;</code>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org6c67ae5" class="outline-3">
|
|
<h3 id="org6c67ae5"><span class="section-number-3">3.2</span> Measured Position of the Sample with respect to the Granite</h3>
|
|
<div class="outline-text-3" id="text-3-2">
|
|
<p>
|
|
The measurement of the position of the sample using the metrology system gives the position and orientation of the sample with respect to the granite.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><code><span class="org-comment">% Measurements: Xm, Ym, Zm, Rx, Ry, Rz</span></code>
|
|
<code>Dxm = <span class="org-highlight-numbers-number">0</span>; <span class="org-comment">% [m]</span></code>
|
|
<code>Dym = <span class="org-highlight-numbers-number">1</span>; <span class="org-comment">% [m]</span></code>
|
|
<code>Dzm = <span class="org-highlight-numbers-number">0</span>; <span class="org-comment">% [m]</span></code>
|
|
<code></code>
|
|
<code>Rxm = <span class="org-highlight-numbers-number">0</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">/</span><span class="org-highlight-numbers-number">180</span>; <span class="org-comment">% [rad]</span></code>
|
|
<code>Rym = <span class="org-highlight-numbers-number">3</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">/</span><span class="org-highlight-numbers-number">180</span>; <span class="org-comment">% [rad]</span></code>
|
|
<code>Rzm = <span class="org-highlight-numbers-number">0</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">/</span><span class="org-highlight-numbers-number">180</span>; <span class="org-comment">% [rad]</span></code>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Let's compute the corresponding orientation using screw axis.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><code>Trxm = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span>;</code>
|
|
<code> <span class="org-highlight-numbers-number">0</span> cos<span class="org-rainbow-delimiters-depth-2">(</span>Rxm<span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-type">-</span>sin<span class="org-rainbow-delimiters-depth-2">(</span>Rxm<span class="org-rainbow-delimiters-depth-2">)</span>;</code>
|
|
<code> <span class="org-highlight-numbers-number">0</span> sin<span class="org-rainbow-delimiters-depth-2">(</span>Rxm<span class="org-rainbow-delimiters-depth-2">)</span> cos<span class="org-rainbow-delimiters-depth-2">(</span>Rxm<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">]</span>;</code>
|
|
<code>Trym = <span class="org-rainbow-delimiters-depth-1">[</span> cos<span class="org-rainbow-delimiters-depth-2">(</span>Rym<span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-highlight-numbers-number">0</span> sin<span class="org-rainbow-delimiters-depth-2">(</span>Rym<span class="org-rainbow-delimiters-depth-2">)</span>;</code>
|
|
<code> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">0</span>;</code>
|
|
<code> <span class="org-type">-</span>sin<span class="org-rainbow-delimiters-depth-2">(</span>Rym<span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-highlight-numbers-number">0</span> cos<span class="org-rainbow-delimiters-depth-2">(</span>Rym<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">]</span>;</code>
|
|
<code>Trzm = <span class="org-rainbow-delimiters-depth-1">[</span>cos<span class="org-rainbow-delimiters-depth-2">(</span>Rzm<span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-type">-</span>sin<span class="org-rainbow-delimiters-depth-2">(</span>Rzm<span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-highlight-numbers-number">0</span>;</code>
|
|
<code> sin<span class="org-rainbow-delimiters-depth-2">(</span>Rzm<span class="org-rainbow-delimiters-depth-2">)</span> cos<span class="org-rainbow-delimiters-depth-2">(</span>Rzm<span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-highlight-numbers-number">0</span>;</code>
|
|
<code> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">]</span>;</code>
|
|
<code></code>
|
|
<code>STw = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-rainbow-delimiters-depth-2">[</span> Trym<span class="org-type">*</span>Trxm<span class="org-type">*</span>Trzm , <span class="org-rainbow-delimiters-depth-3">[</span>Dxm; Dym; Dzm<span class="org-rainbow-delimiters-depth-3">]</span><span class="org-rainbow-delimiters-depth-2">]</span>; <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">]</span>;</code>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We then obtain the orientation measurement in the form of screw coordinate \(\theta_m ({}^Ws_{x,m},\ {}^Ws_{y,m},\ {}^Ws_{z,m})^T\) where:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>\(\theta_m = \cos^{-1} \frac{\text{Tr}(R) - 1}{2}\)</li>
|
|
<li>\({}^W\bm{s}_m\) is the eigen vector of the rotation matrix \(R\) corresponding to the eigen value \(\lambda = 1\)</li>
|
|
</ul>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><code>thetam = acos<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">(</span>trace<span class="org-rainbow-delimiters-depth-3">(</span>STw<span class="org-rainbow-delimiters-depth-4">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">:</span><span class="org-highlight-numbers-number">3</span>, <span class="org-highlight-numbers-number">1</span><span class="org-type">:</span><span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">-</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">/</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% [rad]</span></code>
|
|
<code><span class="org-keyword">if</span> thetam <span class="org-type">==</span> <span class="org-highlight-numbers-number">0</span></code>
|
|
<code> WSm = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">0</span>; <span class="org-highlight-numbers-number">0</span>; <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>;</code>
|
|
<code><span class="org-keyword">else</span></code>
|
|
<code> <span class="org-rainbow-delimiters-depth-1">[</span>V, D<span class="org-rainbow-delimiters-depth-1">]</span> = eig<span class="org-rainbow-delimiters-depth-1">(</span>STw<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">:</span><span class="org-highlight-numbers-number">3</span>, <span class="org-highlight-numbers-number">1</span><span class="org-type">:</span><span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;</code>
|
|
<code> WSm = thetam<span class="org-type">*</span>V<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-type">:</span>, abs<span class="org-rainbow-delimiters-depth-2">(</span>diag<span class="org-rainbow-delimiters-depth-3">(</span>D<span class="org-rainbow-delimiters-depth-3">)</span> <span class="org-type">-</span> <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-type"><</span> <span class="org-constant">eps</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;</code>
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<code><span class="org-keyword">end</span></code>
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</pre>
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab"><code>WPm = <span class="org-rainbow-delimiters-depth-1">[</span>Dxm ; Dym ; Dzm ; WSm<span class="org-rainbow-delimiters-depth-1">]</span>;</code>
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</pre>
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</div>
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</div>
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</div>
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<div id="outline-container-org69e38aa" class="outline-3">
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<h3 id="org69e38aa"><span class="section-number-3">3.3</span> Positioning Error with respect to the Granite</h3>
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<div class="outline-text-3" id="text-3-3">
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<p>
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The wanted position expressed with respect to the granite is \({}^WO_T\) and the measured position with respect to the granite is \({}^WO_S\), thus the <b>position error</b> expressed in \(\{W\}\) is
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\[ {}^W E = {}^W O_T - {}^W O_S \]
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The same is true for rotations:
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\[ \theta_\epsilon {}^W\bm{s}_\epsilon = \theta_r {}^W\bm{s}_r - \theta_m {}^W\bm{s}_m \]
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"><code>WPe = WPr <span class="org-type">-</span> WPm;</code>
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</pre>
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</div>
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|
|
<blockquote>
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<p>
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Now we want to express this error in a frame attached to the <b>base of the nano-hexapod</b> with its origin at the same point where the Jacobian of the nano-hexapod is computed (175mm above the top platform + 90mm of total height of the nano-hexapod).
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</p>
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|
|
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<p>
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Or maybe should we want to express this error with respect to the <b>top platform of the nano-hexapod</b>?
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|
We are measuring the position of the top-platform, and we don't know exactly the position of the bottom platform.
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|
We could compute the position of the bottom platform in two ways:
|
|
</p>
|
|
<ul class="org-ul">
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|
<li>from the encoders of each stage</li>
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|
<li>from the measurement of the nano-hexapod top platform + the internal metrology in the nano-hexapod (capacitive sensors e.g)</li>
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|
</ul>
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|
|
|
<p>
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|
A third option is to say that the maximum stroke of the nano-hexapod is so small that the error should no change to much by the change of base.
|
|
</p>
|
|
</blockquote>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org5246d94" class="outline-3">
|
|
<h3 id="org5246d94"><span class="section-number-3">3.4</span> Position Error Expressed in the Nano-Hexapod Frame</h3>
|
|
<div class="outline-text-3" id="text-3-4">
|
|
<p>
|
|
We now want the position error to be expressed in \(\{S\}\) (the frame attach to the sample):
|
|
\[ {}^S E = {}^S T_W \cdot {}^W E \]
|
|
</p>
|
|
|
|
<p>
|
|
Thus we need to compute the homogeneous transformation \({}^ST_W\).
|
|
Fortunately, this homogeneous transformation can be computed from the measurement of the sample position and orientation with respect to the granite.
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|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><code>Trxm = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span>;</code>
|
|
<code> <span class="org-highlight-numbers-number">0</span> cos<span class="org-rainbow-delimiters-depth-2">(</span>Rxm<span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-type">-</span>sin<span class="org-rainbow-delimiters-depth-2">(</span>Rxm<span class="org-rainbow-delimiters-depth-2">)</span>;</code>
|
|
<code> <span class="org-highlight-numbers-number">0</span> sin<span class="org-rainbow-delimiters-depth-2">(</span>Rxm<span class="org-rainbow-delimiters-depth-2">)</span> cos<span class="org-rainbow-delimiters-depth-2">(</span>Rxm<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">]</span>;</code>
|
|
<code>Trym = <span class="org-rainbow-delimiters-depth-1">[</span> cos<span class="org-rainbow-delimiters-depth-2">(</span>Rym<span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-highlight-numbers-number">0</span> sin<span class="org-rainbow-delimiters-depth-2">(</span>Rym<span class="org-rainbow-delimiters-depth-2">)</span>;</code>
|
|
<code> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">0</span>;</code>
|
|
<code> <span class="org-type">-</span>sin<span class="org-rainbow-delimiters-depth-2">(</span>Rym<span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-highlight-numbers-number">0</span> cos<span class="org-rainbow-delimiters-depth-2">(</span>Rym<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">]</span>;</code>
|
|
<code>Trzm = <span class="org-rainbow-delimiters-depth-1">[</span>cos<span class="org-rainbow-delimiters-depth-2">(</span>Rzm<span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-type">-</span>sin<span class="org-rainbow-delimiters-depth-2">(</span>Rzm<span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-highlight-numbers-number">0</span>;</code>
|
|
<code> sin<span class="org-rainbow-delimiters-depth-2">(</span>Rzm<span class="org-rainbow-delimiters-depth-2">)</span> cos<span class="org-rainbow-delimiters-depth-2">(</span>Rzm<span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-highlight-numbers-number">0</span>;</code>
|
|
<code> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">]</span>;</code>
|
|
<code></code>
|
|
<code>STw = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-rainbow-delimiters-depth-2">[</span> Trym<span class="org-type">*</span>Trxm<span class="org-type">*</span>Trzm , <span class="org-rainbow-delimiters-depth-3">[</span>Dxm; Dym; Dzm<span class="org-rainbow-delimiters-depth-3">]</span><span class="org-rainbow-delimiters-depth-2">]</span>; <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">]</span>;</code>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Translation Error.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><code>SEm = STw <span class="org-type">*</span> <span class="org-rainbow-delimiters-depth-1">[</span>WPe<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">:</span><span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-2">)</span>; <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">]</span>;</code>
|
|
<code>SEm = SEm<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">:</span><span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-1">)</span>;</code>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Rotation Error.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><code>SEr = STw <span class="org-type">*</span> <span class="org-rainbow-delimiters-depth-1">[</span>WPe<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">4</span><span class="org-type">:</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-2">)</span>; <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>;</code>
|
|
<code>SEr = SEr<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">:</span><span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-1">)</span>;</code>
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><code>Etot = <span class="org-rainbow-delimiters-depth-1">[</span>SEm ; SEr<span class="org-rainbow-delimiters-depth-1">]</span></code>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
|
<p class="date">Created: 2019-12-04 mer. 21:55</p>
|
|
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
|
</div>
|
|
</body>
|
|
</html>
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