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Initialize Experiment</a></li> <li><a href="#org26c554f">2. Run the Tomography Experiment</a></li> <li><a href="#orgc507e1e">3. <span class="todo TODO">TODO</span> Tests on the transformation from reference to wanted position</a> <ul> <li><a href="#org4537a86">3.1. Wanted Position of the Sample with respect to the Granite</a></li> <li><a href="#org6c67ae5">3.2. Measured Position of the Sample with respect to the Granite</a></li> <li><a href="#org69e38aa">3.3. Positioning Error with respect to the Granite</a></li> <li><a href="#org5246d94">3.4. Position Error Expressed in the Nano-Hexapod Frame</a></li> </ul> </li> </ul> </div> </div> <div id="outline-container-orgab97c4b" class="outline-2"> <h2 id="orgab97c4b"><span class="section-number-2">1</span> Initialize Experiment</h2> <div class="outline-text-2" id="text-1"> <p> We first initialize all the stages. </p> <div class="org-src-container"> <pre class="src src-matlab"><code>initializeGround<span class="org-rainbow-delimiters-depth-1">()</span>;</code> <code>initializeGranite<span class="org-rainbow-delimiters-depth-1">()</span>;</code> <code>initializeTy<span class="org-rainbow-delimiters-depth-1">()</span>;</code> <code>initializeRy<span class="org-rainbow-delimiters-depth-1">()</span>;</code> <code>initializeRz<span class="org-rainbow-delimiters-depth-1">()</span>;</code> <code>initializeMicroHexapod<span class="org-rainbow-delimiters-depth-1">()</span>;</code> <code>initializeAxisc<span class="org-rainbow-delimiters-depth-1">()</span>;</code> <code>initializeMirror<span class="org-rainbow-delimiters-depth-1">()</span>;</code> <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> <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> </pre> </div> <p> We initialize the reference path for all the stages. </p> <div class="org-src-container"> <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> </pre> </div> <p> And we initialize the disturbances. </p> <div class="org-src-container"> <pre class="src src-matlab"><code>initDisturbances<span class="org-rainbow-delimiters-depth-1">()</span>;</code> </pre> </div> </div> </div> <div id="outline-container-org26c554f" class="outline-2"> <h2 id="org26c554f"><span class="section-number-2">2</span> Run the Tomography Experiment</h2> <div class="outline-text-2" id="text-2"> <p> We first load the simulation configuration </p> <div class="org-src-container"> <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> </pre> </div> <div class="org-src-container"> <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> </pre> </div> <div class="org-src-container"> <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> </pre> </div> </div> </div> <div id="outline-container-orgc507e1e" class="outline-2"> <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> <div class="outline-text-2" id="text-3"> <ul class="org-ul"> <li class="on"><code>[X]</code> Are the rotation matrix commutable? => no</li> <li class="on"><code>[X]</code> How to express the measured rotation errors? => screw axis coordinate seems nice (used in Taghirad's book)</li> <li class="off"><code>[ ]</code> Should ask Veijo how he specifies the position of the Symetrie Hexapod</li> <li class="off"><code>[ ]</code> Create functions for all distinct part and then include that in Simulink</li> <li class="off"><code>[ ]</code> How the express the orientation error?</li> <li class="off"><code>[ ]</code> If we use screw coordinate, can we add/subtract them?</li> <li class="off"><code>[ ]</code> Do some simple tests to verify that the algorithm is working fine</li> </ul> <blockquote> <p> Rx = [1 0 0; 0 cos(t) -sin(t); 0 sin(t) cos(t)]; </p> <p> Ry = [ cos(t) 0 sin(t); 0 1 0; -sin(t) 0 cos(t)]; </p> <p> Rz = [cos(t) -sin(t) 0; sin(t) cos(t) 0; 0 0 1]; </p> </blockquote> <p> Let's define the following frames: </p> <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> <code><span class="org-keyword">end</span></code> </pre> </div> <div class="org-src-container"> <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> </pre> </div> </div> </div> <div id="outline-container-org69e38aa" class="outline-3"> <h3 id="org69e38aa"><span class="section-number-3">3.3</span> Positioning Error with respect to the Granite</h3> <div class="outline-text-3" id="text-3-3"> <p> 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 \[ {}^W E = {}^W O_T - {}^W O_S \] The same is true for rotations: \[ \theta_\epsilon {}^W\bm{s}_\epsilon = \theta_r {}^W\bm{s}_r - \theta_m {}^W\bm{s}_m \] </p> <div class="org-src-container"> <pre class="src src-matlab"><code>WPe = WPr <span class="org-type">-</span> WPm;</code> </pre> </div> <blockquote> <p> 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). </p> <p> Or maybe should we want to express this error with respect to the <b>top platform of the nano-hexapod</b>? We are measuring the position of the top-platform, and we don't know exactly the position of the bottom platform. We could compute the position of the bottom platform in two ways: </p> <ul class="org-ul"> <li>from the encoders of each stage</li> <li>from the measurement of the nano-hexapod top platform + the internal metrology in the nano-hexapod (capacitive sensors e.g)</li> </ul> <p> 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. </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>