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Undamped System</a> <ul> <li><a href="#org919aa7c">1.1. Identification of the plant</a> <ul> <li><a href="#org953f3ee">1.1.1. Initialize the Simulation</a></li> <li><a href="#org4b160cb">1.1.2. Identification</a></li> <li><a href="#orgdb4e9c8">1.1.3. Display TF</a></li> <li><a href="#org7844ee6">1.1.4. Obtained Plants for Active Damping</a></li> </ul> </li> <li><a href="#org3009033">1.2. Tomography Experiment</a> <ul> <li><a href="#org2c866cf">1.2.1. Simulation</a></li> <li><a href="#orga5a7537">1.2.2. Results</a></li> </ul> </li> <li><a href="#org69a746f">1.3. Verification of the transfer function from nano hexapod to metrology</a> <ul> <li><a href="#orgc2ab9cc">1.3.1. Initialize the Simulation</a></li> <li><a href="#org1541bdb">1.3.2. Identification</a></li> <li><a href="#orgd924cdb">1.3.3. Display TF</a></li> <li><a href="#org608ec72">1.3.4. Obtained Plants for Active Damping</a></li> </ul> </li> </ul> </li> </ul> </div> </div> <div id="outline-container-orgbda92b6" class="outline-2"> <h2 id="orgbda92b6"><span class="section-number-2">1</span> Undamped System</h2> <div class="outline-text-2" id="text-1"> <p> <a id="org0dabcfd"></a> </p> </div> <div id="outline-container-org919aa7c" class="outline-3"> <h3 id="org919aa7c"><span class="section-number-3">1.1</span> Identification of the plant</h3> <div class="outline-text-3" id="text-1-1"> </div> <div id="outline-container-org953f3ee" class="outline-4"> <h4 id="org953f3ee"><span class="section-number-4">1.1.1</span> Initialize the Simulation</h4> <div class="outline-text-4" id="text-1-1-1"> <p> We initialize all the stages with the default parameters. </p> <div class="org-src-container"> <pre class="src src-matlab">initializeGround(); initializeGranite(); initializeTy(); initializeRy(); initializeRz(); initializeMicroHexapod(); initializeAxisc(); initializeMirror(); </pre> </div> <p> The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg. </p> <div class="org-src-container"> <pre class="src src-matlab">initializeNanoHexapod(<span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span>); initializeSample(<span class="org-string">'mass'</span>, 50); </pre> </div> <p> No disturbances. </p> <div class="org-src-container"> <pre class="src src-matlab">initializeDisturbances(<span class="org-string">'enable'</span>, <span class="org-constant">false</span>); </pre> </div> <p> We set the references to zero. </p> <div class="org-src-container"> <pre class="src src-matlab">initializeReferences(); </pre> </div> <p> And all the controllers are set to 0. </p> <div class="org-src-container"> <pre class="src src-matlab">K = tf(zeros(6)); save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span>); K_ine = tf(zeros(6)); save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_ine'</span>, <span class="org-string">'-append'</span>); K_iff = tf(zeros(6)); save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span>); K_dvf = tf(zeros(6)); save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span>); </pre> </div> </div> </div> <div id="outline-container-org4b160cb" class="outline-4"> <h4 id="org4b160cb"><span class="section-number-4">1.1.2</span> Identification</h4> <div class="outline-text-4" id="text-1-1-2"> <p> First, we identify the dynamics of the system using the <code>linearize</code> function. </p> <div class="org-src-container"> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span> options = linearizeOptions; options.SampleTime = 0; <span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span> mdl = <span class="org-string">'sim_nass_hac_lac'</span>; <span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span> clear io; io_i = 1; io(io_i) = linio([mdl, <span class="org-string">'/HAC'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; io(io_i) = linio([mdl, <span class="org-string">'/Compute Error in NASS base'</span>], 2, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span> G = linearize(mdl, io, options); G.InputName = {<span class="org-string">'Fnl1'</span>, <span class="org-string">'Fnl2'</span>, <span class="org-string">'Fnl3'</span>, <span class="org-string">'Fnl4'</span>, <span class="org-string">'Fnl5'</span>, <span class="org-string">'Fnl6'</span>}; G.OutputName = {<span class="org-string">'Edx'</span>, <span class="org-string">'Edy'</span>, <span class="org-string">'Edz'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>}; </pre> </div> <div class="org-src-container"> <pre class="src src-matlab">load(<span class="org-string">'mat/stages.mat'</span>, <span class="org-string">'nano_hexapod'</span>); G_cart = minreal(G<span class="org-type">*</span>inv(nano_hexapod.J<span class="org-type">'</span>)); G_cart.InputName = {<span class="org-string">'Fnx'</span>, <span class="org-string">'Fny'</span>, <span class="org-string">'Fnz'</span>, <span class="org-string">'Mnx'</span>, <span class="org-string">'Mny'</span>, <span class="org-string">'Mnz'</span>}; </pre> </div> <div class="org-src-container"> <pre class="src src-matlab">G_legs = minreal(inv(nano_hexapod.J)<span class="org-type">*</span>G); G_legs.OutputName = {<span class="org-string">'e1'</span>, <span class="org-string">'e2'</span>, <span class="org-string">'e3'</span>, <span class="org-string">'e4'</span>, <span class="org-string">'e5'</span>, <span class="org-string">'e6'</span>}; </pre> </div> </div> </div> <div id="outline-container-orgdb4e9c8" class="outline-4"> <h4 id="orgdb4e9c8"><span class="section-number-4">1.1.3</span> Display TF</h4> <div class="outline-text-4" id="text-1-1-3"> <div id="org37b9be3" class="figure"> <p><img src="figs/plant_G_cart.png" alt="plant_G_cart.png" /> </p> <p><span class="figure-number">Figure 1: </span>Transfer Function from forces applied by the nano-hexapod to position error (<a href="./figs/plant_G_cart.png">png</a>, <a href="./figs/plant_G_cart.pdf">pdf</a>)</p> </div> </div> </div> <div id="outline-container-org7844ee6" class="outline-4"> <h4 id="org7844ee6"><span class="section-number-4">1.1.4</span> Obtained Plants for Active Damping</h4> <div class="outline-text-4" id="text-1-1-4"> <div id="org505566d" class="figure"> <p><img src="figs/nass_active_damping_iff_plant.png" alt="nass_active_damping_iff_plant.png" /> </p> <p><span class="figure-number">Figure 2: </span><code>G_iff</code>: IFF Plant (<a href="./figs/nass_active_damping_iff_plant.png">png</a>, <a href="./figs/nass_active_damping_iff_plant.pdf">pdf</a>)</p> </div> <div id="org46c7102" class="figure"> <p><img src="figs/nass_active_damping_ine_plant.png" alt="nass_active_damping_ine_plant.png" /> </p> <p><span class="figure-number">Figure 3: </span><code>G_dvf</code>: Plant for Direct Velocity Feedback (<a href="./figs/nass_active_damping_dvf_plant.png">png</a>, <a href="./figs/nass_active_damping_dvf_plant.pdf">pdf</a>)</p> </div> <div id="org6c61332" class="figure"> <p><img src="figs/nass_active_damping_inertial_plant.png" alt="nass_active_damping_inertial_plant.png" /> </p> <p><span class="figure-number">Figure 4: </span>Inertial Feedback Plant (<a href="./figs/nass_active_damping_inertial_plant.png">png</a>, <a href="./figs/nass_active_damping_inertial_plant.pdf">pdf</a>)</p> </div> </div> </div> </div> <div id="outline-container-org3009033" class="outline-3"> <h3 id="org3009033"><span class="section-number-3">1.2</span> Tomography Experiment</h3> <div class="outline-text-3" id="text-1-2"> </div> <div id="outline-container-org2c866cf" class="outline-4"> <h4 id="org2c866cf"><span class="section-number-4">1.2.1</span> Simulation</h4> <div class="outline-text-4" id="text-1-2-1"> <p> We initialize elements for the tomography experiment. </p> <div class="org-src-container"> <pre class="src src-matlab">prepareTomographyExperiment(); </pre> </div> <p> We change the simulation stop time. </p> <div class="org-src-container"> <pre class="src src-matlab">load(<span class="org-string">'mat/conf_simscape.mat'</span>); <span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simscape</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'3'</span>); </pre> </div> <p> And we simulate the system. </p> <div class="org-src-container"> <pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'sim_nass_active_damping'</span>); </pre> </div> <p> Finally, we save the simulation results for further analysis </p> <div class="org-src-container"> <pre class="src src-matlab">save(<span class="org-string">'./active_damping/mat/tomo_exp.mat'</span>, <span class="org-string">'En'</span>, <span class="org-string">'Eg'</span>, <span class="org-string">'-append'</span>); </pre> </div> </div> </div> <div id="outline-container-orga5a7537" class="outline-4"> <h4 id="orga5a7537"><span class="section-number-4">1.2.2</span> Results</h4> <div class="outline-text-4" id="text-1-2-2"> <p> We load the results of tomography experiments. </p> <div class="org-src-container"> <pre class="src src-matlab">load(<span class="org-string">'./active_damping/mat/tomo_exp.mat'</span>, <span class="org-string">'En'</span>); t = linspace(0, 3, length(En(<span class="org-type">:</span>,1))); </pre> </div> <div id="org43abdf9" class="figure"> <p><img src="figs/nass_act_damp_undamped_sim_tomo_trans.png" alt="nass_act_damp_undamped_sim_tomo_trans.png" /> </p> <p><span class="figure-number">Figure 5: </span>Position Error during tomography experiment - Translations (<a href="./figs/nass_act_damp_undamped_sim_tomo_trans.png">png</a>, <a href="./figs/nass_act_damp_undamped_sim_tomo_trans.pdf">pdf</a>)</p> </div> <div id="orgb984994" class="figure"> <p><img src="figs/nass_act_damp_undamped_sim_tomo_rot.png" alt="nass_act_damp_undamped_sim_tomo_rot.png" /> </p> <p><span class="figure-number">Figure 6: </span>Position Error during tomography experiment - Rotations (<a href="./figs/nass_act_damp_undamped_sim_tomo_rot.png">png</a>, <a href="./figs/nass_act_damp_undamped_sim_tomo_rot.pdf">pdf</a>)</p> </div> </div> </div> </div> <div id="outline-container-org69a746f" class="outline-3"> <h3 id="org69a746f"><span class="section-number-3">1.3</span> Verification of the transfer function from nano hexapod to metrology</h3> <div class="outline-text-3" id="text-1-3"> </div> <div id="outline-container-orgc2ab9cc" class="outline-4"> <h4 id="orgc2ab9cc"><span class="section-number-4">1.3.1</span> Initialize the Simulation</h4> <div class="outline-text-4" id="text-1-3-1"> <p> We initialize all the stages with the default parameters. </p> <div class="org-src-container"> <pre class="src src-matlab">initializeGround(); initializeGranite(); initializeTy(); initializeRy(); initializeRz(); initializeMicroHexapod(); initializeAxisc(); initializeMirror(); </pre> </div> <p> The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg. </p> <div class="org-src-container"> <pre class="src src-matlab">initializeNanoHexapod(<span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span>); initializeSample(<span class="org-string">'mass'</span>, 50); </pre> </div> <p> No disturbances. </p> <div class="org-src-container"> <pre class="src src-matlab">initializeDisturbances(<span class="org-string">'enable'</span>, <span class="org-constant">false</span>); </pre> </div> <p> We set the references to zero. </p> <div class="org-src-container"> <pre class="src src-matlab">initializeReferences(); </pre> </div> <p> And all the controllers are set to 0. </p> <div class="org-src-container"> <pre class="src src-matlab">K = tf(zeros(6)); save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span>); K_ine = tf(zeros(6)); save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_ine'</span>, <span class="org-string">'-append'</span>); K_iff = tf(zeros(6)); save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span>); K_dvf = tf(zeros(6)); save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span>); </pre> </div> </div> </div> <div id="outline-container-org1541bdb" class="outline-4"> <h4 id="org1541bdb"><span class="section-number-4">1.3.2</span> Identification</h4> <div class="outline-text-4" id="text-1-3-2"> <p> First, we identify the dynamics of the system using the <code>linearize</code> function. </p> <div class="org-src-container"> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span> options = linearizeOptions; options.SampleTime = 0; <span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span> mdl = <span class="org-string">'sim_nass_hac_lac'</span>; <span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span> clear io; io_i = 1; io(io_i) = linio([mdl, <span class="org-string">'/HAC'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; io(io_i) = linio([mdl, <span class="org-string">'/Compute Error in NASS base'</span>], 2, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span> G = linearize(mdl, io, options); G.InputName = {<span class="org-string">'Fnl1'</span>, <span class="org-string">'Fnl2'</span>, <span class="org-string">'Fnl3'</span>, <span class="org-string">'Fnl4'</span>, <span class="org-string">'Fnl5'</span>, <span class="org-string">'Fnl6'</span>}; G.OutputName = {<span class="org-string">'Edx'</span>, <span class="org-string">'Edy'</span>, <span class="org-string">'Edz'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>}; </pre> </div> <div class="org-src-container"> <pre class="src src-matlab">load(<span class="org-string">'mat/stages.mat'</span>, <span class="org-string">'nano_hexapod'</span>); G_cart = minreal(G<span class="org-type">*</span>inv(nano_hexapod.J<span class="org-type">'</span>)); G_cart.InputName = {<span class="org-string">'Fnx'</span>, <span class="org-string">'Fny'</span>, <span class="org-string">'Fnz'</span>, <span class="org-string">'Mnx'</span>, <span class="org-string">'Mny'</span>, <span class="org-string">'Mnz'</span>}; </pre> </div> <div class="org-src-container"> <pre class="src src-matlab">G_legs = minreal(inv(nano_hexapod.J)<span class="org-type">*</span>G); G_legs.OutputName = {<span class="org-string">'e1'</span>, <span class="org-string">'e2'</span>, <span class="org-string">'e3'</span>, <span class="org-string">'e4'</span>, <span class="org-string">'e5'</span>, <span class="org-string">'e6'</span>}; </pre> </div> </div> </div> <div id="outline-container-orgd924cdb" class="outline-4"> <h4 id="orgd924cdb"><span class="section-number-4">1.3.3</span> Display TF</h4> <div class="outline-text-4" id="text-1-3-3"> <div id="orgc842d08" class="figure"> <p><img src="figs/plant_G_cart.png" alt="plant_G_cart.png" /> </p> <p><span class="figure-number">Figure 7: </span>Transfer Function from forces applied by the nano-hexapod to position error (<a href="./figs/plant_G_cart.png">png</a>, <a href="./figs/plant_G_cart.pdf">pdf</a>)</p> </div> </div> </div> <div id="outline-container-org608ec72" class="outline-4"> <h4 id="org608ec72"><span class="section-number-4">1.3.4</span> Obtained Plants for Active Damping</h4> <div class="outline-text-4" id="text-1-3-4"> <div id="org253e2e8" class="figure"> <p><img src="figs/nass_active_damping_iff_plant.png" alt="nass_active_damping_iff_plant.png" /> </p> <p><span class="figure-number">Figure 8: </span><code>G_iff</code>: IFF Plant (<a href="./figs/nass_active_damping_iff_plant.png">png</a>, <a href="./figs/nass_active_damping_iff_plant.pdf">pdf</a>)</p> </div> <div id="orgf988b83" class="figure"> <p><img src="figs/nass_active_damping_ine_plant.png" alt="nass_active_damping_ine_plant.png" /> </p> <p><span class="figure-number">Figure 9: </span><code>G_dvf</code>: Plant for Direct Velocity Feedback (<a href="./figs/nass_active_damping_dvf_plant.png">png</a>, <a href="./figs/nass_active_damping_dvf_plant.pdf">pdf</a>)</p> </div> <div id="org54d695a" class="figure"> <p><img src="figs/nass_active_damping_inertial_plant.png" alt="nass_active_damping_inertial_plant.png" /> </p> <p><span class="figure-number">Figure 10: </span>Inertial Feedback Plant (<a href="./figs/nass_active_damping_inertial_plant.png">png</a>, <a href="./figs/nass_active_damping_inertial_plant.pdf">pdf</a>)</p> </div> </div> </div> </div> </div> </div> <div id="postamble" class="status"> <p class="author">Author: Dehaeze Thomas</p> <p class="date">Created: 2020-01-29 mer. 20:35</p> </div> </body> </html>