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< h1 class = "title" > Active Damping applied on the Simscape Model< / h1 >
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< div id = "table-of-contents" >
< h2 > Table of Contents< / h2 >
< div id = "text-table-of-contents" >
< ul >
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< li > < a href = "#org6fe0183" > 1. Undamped System< / a >
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< ul >
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< li > < a href = "#orgc80160b" > 1.1. Identification of the dynamics for Active Damping< / a >
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< ul >
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< li > < a href = "#org636461d" > 1.1.1. Initialize the Simulation< / a > < / li >
< li > < a href = "#orgea36ee5" > 1.1.2. Identification< / a > < / li >
< li > < a href = "#org2dfc68c" > 1.1.3. Obtained Plants for Active Damping< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org954f141" > 1.2. Tomography Experiment< / a >
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< ul >
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< li > < a href = "#org791d4b3" > 1.2.1. Simulation< / a > < / li >
< li > < a href = "#orge0ff76e" > 1.2.2. Results< / a > < / li >
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< / ul >
< / li >
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< / ul >
< / li >
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< li > < a href = "#org00773fe" > 2. Integral Force Feedback< / a >
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< ul >
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< li > < a href = "#orgb873e29" > 2.1. Control Design< / a >
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< ul >
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< li > < a href = "#orgf590470" > 2.1.1. Plant< / a > < / li >
< li > < a href = "#org10ed84e" > 2.1.2. Control Design< / a > < / li >
< li > < a href = "#orgd129b36" > 2.1.3. Diagonal Controller< / a > < / li >
< li > < a href = "#org9b5fe8e" > 2.1.4. IFF with High Pass Filter< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org90345f6" > 2.2. Tomography Experiment< / a >
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< ul >
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< li > < a href = "#org33daed9" > 2.2.1. Simulation with IFF Controller< / a > < / li >
< li > < a href = "#org94161a4" > 2.2.2. Simulation with IFF Controller with added High Pass Filter< / a > < / li >
< li > < a href = "#org9767f66" > 2.2.3. Compare with Undamped system< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgd16ca11" > 2.3. Conclusion< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgfad54f0" > 3. Direct Velocity Feedback< / a >
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< ul >
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< li > < a href = "#orgcf2e837" > 3.1. Control Design< / a >
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< ul >
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< li > < a href = "#org76b1cfe" > 3.1.1. Plant< / a > < / li >
< li > < a href = "#orgd83abae" > 3.1.2. Control Design< / a > < / li >
< li > < a href = "#orge21c84c" > 3.1.3. Diagonal Controller< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgbe85f18" > 3.2. Tomography Experiment< / a >
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< ul >
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< li > < a href = "#orgee67221" > 3.2.1. Initialize the Simulation< / a > < / li >
< li > < a href = "#org76ee3e3" > 3.2.2. Simulation< / a > < / li >
< li > < a href = "#org3c3feec" > 3.2.3. Compare with Undamped system< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org5035edc" > 3.3. Conclusion< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org7454716" > 4. Inertial Control< / a >
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< ul >
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< li > < a href = "#org8e08998" > 4.1. Control Design< / a >
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< ul >
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< li > < a href = "#orgf2db463" > 4.1.1. Plant< / a > < / li >
< li > < a href = "#org93cb5d8" > 4.1.2. Control Design< / a > < / li >
< li > < a href = "#orgd002ae1" > 4.1.3. Diagonal Controller< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgca448ac" > 4.2. Tomography Experiment< / a >
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< ul >
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< li > < a href = "#org632ee5a" > 4.2.1. Initialize the Simulation< / a > < / li >
< li > < a href = "#org10654aa" > 4.2.2. Simulation< / a > < / li >
< li > < a href = "#org1239f7c" > 4.2.3. Compare with Undamped system< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org9674a40" > 4.3. Conclusion< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org251a441" > 5. Variability of the system dynamics for Active Damping< / a >
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< ul >
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< li > < a href = "#orgd178f46" > 5.1. Variation of the Sample Mass< / a > < / li >
< li > < a href = "#org38b8d50" > 5.2. Variation of the Spindle Angle< / a > < / li >
< li > < a href = "#org666ca9f" > 5.3. Variation of the Spindle Rotation Speed< / a > < / li >
< li > < a href = "#org15eb4a9" > 5.4. Variation of the Tilt Angle< / a > < / li >
< li > < a href = "#org8d56329" > 5.5. Conclusion< / a > < / li >
< / ul >
< / li >
< li > < a href = "#org4e51ab2" > 6. Comparison< / a >
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< ul >
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< li > < a href = "#org22beca8" > 6.1. Load the plants< / a > < / li >
< li > < a href = "#orgcaacc66" > 6.2. Sensitivity to Disturbance< / a > < / li >
< li > < a href = "#org1b7fd80" > 6.3. Damped Plant< / a > < / li >
< li > < a href = "#org0f12cf8" > 6.4. Tomography Experiment< / a >
< ul >
< li > < a href = "#orgdb48e43" > 6.4.1. Load the Simulation Data< / a > < / li >
< li > < a href = "#orgfbe1bad" > 6.4.2. Frequency Domain Analysis< / a > < / li >
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< / ul >
< / li >
< / ul >
< / li >
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< li > < a href = "#orgb197610" > 7. Useful Functions< / a >
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< ul >
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< li > < a href = "#org6783ec6" > 7.1. prepareTomographyExperiment< / a >
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< ul >
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< li > < a href = "#org8c75d0b" > Function Description< / a > < / li >
< li > < a href = "#org356aa82" > Optional Parameters< / a > < / li >
< li > < a href = "#org93e527a" > Initialize the Simulation< / a > < / li >
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< / ul >
< / li >
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< / ul >
< / li >
< / ul >
< / div >
< / div >
< p >
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First, in section < a href = "#org4b60bdf" > 1< / a > , we will looked at the undamped system.
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< / p >
< p >
Then, we will compare three active damping techniques:
< / p >
< ul class = "org-ul" >
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< li > In section < a href = "#org0dadbd2" > 2< / a > : the integral force feedback is used< / li >
< li > In section < a href = "#org47344de" > 3< / a > : the direct velocity feedback is used< / li >
< li > In section < a href = "#org6b70b5e" > 4< / a > : inertial control is used< / li >
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< / ul >
< p >
For each of the active damping technique, we will:
< / p >
< ul class = "org-ul" >
< li > Look at the damped plant< / li >
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< li > Simulate tomography experiments< / li >
< li > Compare the sensitivity from disturbances< / li >
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< / ul >
< p >
The disturbances are:
< / p >
< ul class = "org-ul" >
< li > Ground motion< / li >
< li > Motion errors of all the stages< / li >
< / ul >
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< div id = "outline-container-org6fe0183" class = "outline-2" >
< h2 id = "org6fe0183" > < span class = "section-number-2" > 1< / span > Undamped System< / h2 >
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< div class = "outline-text-2" id = "text-1" >
< p >
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< a id = "org4b60bdf" > < / a >
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< / p >
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< div class = "note" >
< p >
All the files (data and Matlab scripts) are accessible < a href = "data/undamped_system.zip" > here< / a > .
< / p >
< / div >
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< p >
We first look at the undamped system.
The performance of this undamped system will be compared with the damped system using various techniques.
< / p >
< / div >
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< div id = "outline-container-orgc80160b" class = "outline-3" >
< h3 id = "orgc80160b" > < span class = "section-number-3" > 1.1< / span > Identification of the dynamics for Active Damping< / h3 >
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< div class = "outline-text-3" id = "text-1-1" >
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< / div >
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< div id = "outline-container-org636461d" class = "outline-4" >
< h4 id = "org636461d" > < span class = "section-number-4" > 1.1.1< / span > Initialize the Simulation< / h4 >
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< div class = "outline-text-4" id = "text-1-1-1" >
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< p >
We initialize all the stages with the default parameters.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
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< / pre >
< / div >
< p >
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The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.
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< / p >
< div class = "org-src-container" >
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< 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);
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< / pre >
< / div >
< p >
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We set the references to zero.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > initializeReferences(< span class = "org-string" > 'Rz_type'< / span > , < span class = "org-string" > 'rotating'< / span > , < span class = "org-string" > 'Ry_period'< / span > , 1);
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< / pre >
< / div >
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< p >
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And all the controllers are set to 0.
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< / p >
< div class = "org-src-container" >
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< 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 > );
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< / pre >
< / div >
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< / div >
< / div >
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< div id = "outline-container-orgea36ee5" class = "outline-4" >
< h4 id = "orgea36ee5" > < span class = "section-number-4" > 1.1.2< / span > Identification< / h4 >
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< div class = "outline-text-4" id = "text-1-1-2" >
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< p >
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First, we identify the dynamics of the system using the < code > linearize< / code > function.
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< / p >
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< 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_active_damping'< / 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" > '/Fnl'< / 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" > '/Micro-Station'< / span > ], 3, < span class = "org-string" > 'openoutput'< / span > , [], < span class = "org-string" > 'Dnlm'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/Micro-Station'< / span > ], 3, < span class = "org-string" > 'openoutput'< / span > , [], < span class = "org-string" > 'Fnlm'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/Micro-Station'< / span > ], 3, < span class = "org-string" > 'openoutput'< / span > , [], < span class = "org-string" > 'Vlm'< / 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 >
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G = linearize(mdl, io, 1, options);
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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" > 'Dnlm1'< / span > , < span class = "org-string" > 'Dnlm2'< / span > , < span class = "org-string" > 'Dnlm3'< / span > , < span class = "org-string" > 'Dnlm4'< / span > , < span class = "org-string" > 'Dnlm5'< / span > , < span class = "org-string" > 'Dnlm6'< / span > , ...
< span class = "org-string" > 'Fnlm1'< / span > , < span class = "org-string" > 'Fnlm2'< / span > , < span class = "org-string" > 'Fnlm3'< / span > , < span class = "org-string" > 'Fnlm4'< / span > , < span class = "org-string" > 'Fnlm5'< / span > , < span class = "org-string" > 'Fnlm6'< / span > , ...
< span class = "org-string" > 'Vnlm1'< / span > , < span class = "org-string" > 'Vnlm2'< / span > , < span class = "org-string" > 'Vnlm3'< / span > , < span class = "org-string" > 'Vnlm4'< / span > , < span class = "org-string" > 'Vnlm5'< / span > , < span class = "org-string" > 'Vnlm6'< / span > };
< / pre >
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< / div >
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< p >
We then create transfer functions corresponding to the active damping plants.
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< / p >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > G_iff = minreal(G({< span class = "org-string" > 'Fnlm1'< / span > , < span class = "org-string" > 'Fnlm2'< / span > , < span class = "org-string" > 'Fnlm3'< / span > , < span class = "org-string" > 'Fnlm4'< / span > , < span class = "org-string" > 'Fnlm5'< / span > , < span class = "org-string" > 'Fnlm6'< / span > }, {< 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_dvf = minreal(G({< span class = "org-string" > 'Dnlm1'< / span > , < span class = "org-string" > 'Dnlm2'< / span > , < span class = "org-string" > 'Dnlm3'< / span > , < span class = "org-string" > 'Dnlm4'< / span > , < span class = "org-string" > 'Dnlm5'< / span > , < span class = "org-string" > 'Dnlm6'< / span > }, {< 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_ine = minreal(G({< span class = "org-string" > 'Vnlm1'< / span > , < span class = "org-string" > 'Vnlm2'< / span > , < span class = "org-string" > 'Vnlm3'< / span > , < span class = "org-string" > 'Vnlm4'< / span > , < span class = "org-string" > 'Vnlm5'< / span > , < span class = "org-string" > 'Vnlm6'< / span > }, {< 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 > }));
< / pre >
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< / div >
< p >
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And we save them for further analysis.
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< / p >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > save(< span class = "org-string" > './active_damping/mat/undamped_plants.mat'< / span > , < span class = "org-string" > 'G_iff'< / span > , < span class = "org-string" > 'G_dvf'< / span > , < span class = "org-string" > 'G_ine'< / span > );
< / pre >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org2dfc68c" class = "outline-4" >
< h4 id = "org2dfc68c" > < span class = "section-number-4" > 1.1.3< / span > Obtained Plants for Active Damping< / h4 >
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< div class = "outline-text-4" id = "text-1-1-3" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > load(< span class = "org-string" > './active_damping/mat/undamped_plants.mat'< / span > , < span class = "org-string" > 'G_iff'< / span > , < span class = "org-string" > 'G_dvf'< / span > , < span class = "org-string" > 'G_ine'< / span > );
< / pre >
< / div >
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< div id = "org9596ae8" class = "figure" >
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< p > < img src = "figs/nass_active_damping_iff_plant.png" alt = "nass_active_damping_iff_plant.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 1: < / 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 >
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< / div >
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< div id = "orgf1a57c2" class = "figure" >
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< p > < img src = "figs/nass_active_damping_ine_plant.png" alt = "nass_active_damping_ine_plant.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 2: < / 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 >
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< / div >
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< div id = "org8b0a38f" class = "figure" >
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< p > < img src = "figs/nass_active_damping_inertial_plant.png" alt = "nass_active_damping_inertial_plant.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 3: < / span > < code > G_ine< / code > : 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 >
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< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org954f141" class = "outline-3" >
< h3 id = "org954f141" > < span class = "section-number-3" > 1.2< / span > Tomography Experiment< / h3 >
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< div class = "outline-text-3" id = "text-1-2" >
< / div >
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< div id = "outline-container-org791d4b3" class = "outline-4" >
< h4 id = "org791d4b3" > < span class = "section-number-4" > 1.2.1< / span > Simulation< / h4 >
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< div class = "outline-text-4" id = "text-1-2-1" >
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< p >
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We initialize elements for the tomography experiment.
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< / p >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > prepareTomographyExperiment();
< / pre >
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< / div >
< p >
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We change the simulation stop time.
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< / p >
< div class = "org-src-container" >
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< 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 > );
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< / pre >
< / div >
< p >
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And we simulate the system.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-matlab-simulink-keyword" > sim< / span > (< span class = "org-string" > 'sim_nass_active_damping'< / span > );
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< / pre >
< / div >
< p >
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Finally, we save the simulation results for further analysis
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< / p >
< div class = "org-src-container" >
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< 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 > );
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< / pre >
< / div >
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< / div >
< / div >
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< div id = "outline-container-orge0ff76e" class = "outline-4" >
< h4 id = "orge0ff76e" > < span class = "section-number-4" > 1.2.2< / span > Results< / h4 >
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< div class = "outline-text-4" id = "text-1-2-2" >
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< p >
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We load the results of tomography experiments.
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< / p >
< div class = "org-src-container" >
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< 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)));
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< / pre >
< / div >
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< div id = "org81f06c2" class = "figure" >
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< p > < img src = "figs/nass_act_damp_undamped_sim_tomo_trans.png" alt = "nass_act_damp_undamped_sim_tomo_trans.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 4: < / 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 >
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< div id = "orgca816f1" class = "figure" >
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< 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 5: < / 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 >
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< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org00773fe" class = "outline-2" >
< h2 id = "org00773fe" > < span class = "section-number-2" > 2< / span > Integral Force Feedback< / h2 >
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< div class = "outline-text-2" id = "text-2" >
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< p >
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< a id = "org0dadbd2" > < / a >
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< / p >
< div class = "note" >
< p >
All the files (data and Matlab scripts) are accessible < a href = "data/iff.zip" > here< / a > .
< / p >
< / div >
< p >
Integral Force Feedback is applied on the simscape model.
< / p >
< / div >
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< div id = "outline-container-orgb873e29" class = "outline-3" >
< h3 id = "orgb873e29" > < span class = "section-number-3" > 2.1< / span > Control Design< / h3 >
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< div class = "outline-text-3" id = "text-2-1" >
< / div >
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< div id = "outline-container-orgf590470" class = "outline-4" >
< h4 id = "orgf590470" > < span class = "section-number-4" > 2.1.1< / span > Plant< / h4 >
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< div class = "outline-text-4" id = "text-2-1-1" >
< p >
Let’ s load the previously indentified undamped plant:
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > load(< span class = "org-string" > './active_damping/mat/undamped_plants.mat'< / span > , < span class = "org-string" > 'G_iff'< / span > );
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< / pre >
< / div >
< p >
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Let’ s look at the transfer function from actuator forces in the nano-hexapod to the force sensor in the nano-hexapod legs for all 6 pairs of actuator/sensor (figure < a href = "#orgdbd61f0" > 6< / a > ).
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< / p >
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< div id = "orgdbd61f0" class = "figure" >
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< p > < img src = "figs/iff_plant.png" alt = "iff_plant.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 6: < / span > Transfer function from forces applied in the legs to force sensor (< a href = "./figs/iff_plant.png" > png< / a > , < a href = "./figs/iff_plant.pdf" > pdf< / a > )< / p >
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< / div >
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< / div >
< / div >
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< div id = "outline-container-org10ed84e" class = "outline-4" >
< h4 id = "org10ed84e" > < span class = "section-number-4" > 2.1.2< / span > Control Design< / h4 >
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< div class = "outline-text-4" id = "text-2-1-2" >
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< p >
The controller for each pair of actuator/sensor is:
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > K_iff = 1000< span class = "org-type" > /< / span > s;
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< / pre >
< / div >
< p >
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The corresponding loop gains are shown in figure < a href = "#org438bb64" > 7< / a > .
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< / p >
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< div id = "org438bb64" class = "figure" >
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< p > < img src = "figs/iff_open_loop.png" alt = "iff_open_loop.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 7: < / span > Loop Gain for the Integral Force Feedback (< a href = "./figs/iff_open_loop.png" > png< / a > , < a href = "./figs/iff_open_loop.pdf" > pdf< / a > )< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-orgd129b36" class = "outline-4" >
< h4 id = "orgd129b36" > < span class = "section-number-4" > 2.1.3< / span > Diagonal Controller< / h4 >
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< div class = "outline-text-4" id = "text-2-1-3" >
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< p >
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We create the diagonal controller and we add a minus sign as we have a positive
feedback architecture.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > K_iff = < span class = "org-type" > -< / span > K_iff< span class = "org-type" > *< / span > eye(6);
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< / pre >
< / div >
< p >
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We save the controller for further analysis.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > save(< span class = "org-string" > './active_damping/mat/K_iff.mat'< / span > , < span class = "org-string" > 'K_iff'< / span > );
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< / pre >
< / div >
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< / div >
< / div >
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< div id = "outline-container-org9b5fe8e" class = "outline-4" >
< h4 id = "org9b5fe8e" > < span class = "section-number-4" > 2.1.4< / span > IFF with High Pass Filter< / h4 >
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< div class = "outline-text-4" id = "text-2-1-4" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > w_hpf = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 10; < span class = "org-comment" > % Cut-off frequency for the high pass filter [rad/s]< / span >
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K_iff = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 200< span class = "org-type" > /< / span > s < span class = "org-type" > *< / span > (s< span class = "org-type" > /< / span > w_hpf)< span class = "org-type" > /< / span > (s< span class = "org-type" > /< / span > w_hpf < span class = "org-type" > +< / span > 1);
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< / pre >
< / div >
< p >
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The corresponding loop gains are shown in figure < a href = "#orgf886eba" > 8< / a > .
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< / p >
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< div id = "orgf886eba" class = "figure" >
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< p > < img src = "figs/iff_hpf_open_loop.png" alt = "iff_hpf_open_loop.png" / >
< / p >
< p > < span class = "figure-number" > Figure 8: < / span > Loop Gain for the Integral Force Feedback with an High pass filter (< a href = "./figs/iff_hpf_open_loop.png" > png< / a > , < a href = "./figs/iff_hpf_open_loop.pdf" > pdf< / a > )< / p >
< / div >
< p >
We create the diagonal controller and we add a minus sign as we have a positive
feedback architecture.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > K_iff = < span class = "org-type" > -< / span > K_iff< span class = "org-type" > *< / span > eye(6);
< / pre >
< / div >
< p >
We save the controller for further analysis.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > save(< span class = "org-string" > './active_damping/mat/K_iff_hpf.mat'< / span > , < span class = "org-string" > 'K_iff'< / span > );
< / pre >
< / div >
< / div >
< / div >
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< / div >
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< div id = "outline-container-org90345f6" class = "outline-3" >
< h3 id = "org90345f6" > < span class = "section-number-3" > 2.2< / span > Tomography Experiment< / h3 >
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< div class = "outline-text-3" id = "text-2-2" >
< / div >
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< div id = "outline-container-org33daed9" class = "outline-4" >
< h4 id = "org33daed9" > < span class = "section-number-4" > 2.2.1< / span > Simulation with IFF Controller< / h4 >
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< div class = "outline-text-4" id = "text-2-2-1" >
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< p >
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We initialize elements for the tomography experiment.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > prepareTomographyExperiment();
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< / pre >
< / div >
< p >
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We set the IFF controller.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > load(< span class = "org-string" > './active_damping/mat/K_iff.mat'< / span > , < span class = "org-string" > 'K_iff'< / span > );
save(< span class = "org-string" > './mat/controllers.mat'< / span > , < span class = "org-string" > 'K_iff'< / span > , < span class = "org-string" > '-append'< / span > );
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< / pre >
< / div >
< p >
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We change the simulation stop time.
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< / p >
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< 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 >
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< / div >
< p >
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And we simulate the system.
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< / p >
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< 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 >
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< / div >
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< p >
Finally, we save the simulation results for further analysis
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< / p >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > En_iff = En;
Eg_iff = Eg;
save(< span class = "org-string" > './active_damping/mat/tomo_exp.mat'< / span > , < span class = "org-string" > 'En_iff'< / span > , < span class = "org-string" > 'Eg_iff'< / span > , < span class = "org-string" > '-append'< / span > );
< / pre >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org94161a4" class = "outline-4" >
< h4 id = "org94161a4" > < span class = "section-number-4" > 2.2.2< / span > Simulation with IFF Controller with added High Pass Filter< / h4 >
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< div class = "outline-text-4" id = "text-2-2-2" >
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< p >
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We initialize elements for the tomography experiment.
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< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > prepareTomographyExperiment();
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< / pre >
< / div >
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< p >
We set the IFF controller with the High Pass Filter.
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > load(< span class = "org-string" > './active_damping/mat/K_iff_hpf.mat'< / span > , < span class = "org-string" > 'K_iff'< / span > );
save(< span class = "org-string" > './mat/controllers.mat'< / span > , < span class = "org-string" > 'K_iff'< / span > , < span class = "org-string" > '-append'< / span > );
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< / pre >
< / div >
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< 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 >
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< 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" > En_iff_hpf = En;
Eg_iff_hpf = Eg;
save(< span class = "org-string" > './active_damping/mat/tomo_exp.mat'< / span > , < span class = "org-string" > 'En_iff_hpf'< / span > , < span class = "org-string" > 'Eg_iff_hpf'< / span > , < span class = "org-string" > '-append'< / span > );
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org9767f66" class = "outline-4" >
< h4 id = "org9767f66" > < span class = "section-number-4" > 2.2.3< / span > Compare with Undamped system< / h4 >
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< div class = "outline-text-4" id = "text-2-2-3" >
< 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 > , < span class = "org-string" > 'En_iff'< / span > , < span class = "org-string" > 'En_iff_hpf'< / span > );
t = linspace(0, 3, length(En(< span class = "org-type" > :< / span > ,1)));
< / pre >
< / div >
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< div id = "org1451aab" class = "figure" >
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< p > < img src = "figs/nass_act_damp_iff_sim_tomo_xy.png" alt = "nass_act_damp_iff_sim_tomo_xy.png" / >
< / p >
< p > < span class = "figure-number" > Figure 9: < / span > Position Error during tomography experiment - XY Motion (< a href = "./figs/nass_act_damp_iff_sim_tomo_xy.png" > png< / a > , < a href = "./figs/nass_act_damp_iff_sim_tomo_xy.pdf" > pdf< / a > )< / p >
< / div >
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< div id = "org89b268b" class = "figure" >
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< p > < img src = "figs/nass_act_damp_iff_sim_tomo_trans.png" alt = "nass_act_damp_iff_sim_tomo_trans.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 10: < / span > Position Error during tomography experiment - Translations (< a href = "./figs/nass_act_damp_iff_sim_tomo_trans.png" > png< / a > , < a href = "./figs/nass_act_damp_iff_sim_tomo_trans.pdf" > pdf< / a > )< / p >
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< / div >
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< div id = "org5d3f340" class = "figure" >
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< p > < img src = "figs/nass_act_damp_iff_sim_tomo_rot.png" alt = "nass_act_damp_iff_sim_tomo_rot.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 11: < / span > Position Error during tomography experiment - Rotations (< a href = "./figs/nass_act_damp_iff_sim_tomo_rot.png" > png< / a > , < a href = "./figs/nass_act_damp_iff_sim_tomo_rot.pdf" > pdf< / a > )< / p >
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< / div >
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< / div >
< / div >
< / div >
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< div id = "outline-container-orgd16ca11" class = "outline-3" >
< h3 id = "orgd16ca11" > < span class = "section-number-3" > 2.3< / span > Conclusion< / h3 >
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< div class = "outline-text-3" id = "text-2-3" >
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< div class = "important" >
< p >
Integral Force Feedback:
< / p >
< ul class = "org-ul" >
< li > Robust (guaranteed stability)< / li >
< li > Acceptable Damping< / li >
< li > Increase the sensitivity to disturbances at low frequencies< / li >
< / ul >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgfad54f0" class = "outline-2" >
< h2 id = "orgfad54f0" > < span class = "section-number-2" > 3< / span > Direct Velocity Feedback< / h2 >
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< div class = "outline-text-2" id = "text-3" >
< p >
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< a id = "org47344de" > < / a >
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< / p >
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< div class = "note" >
< p >
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All the files (data and Matlab scripts) are accessible < a href = "data/dvf.zip" > here< / a > .
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< / p >
< / div >
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< p >
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In the Direct Velocity Feedback (DVF), a derivative feedback is applied between the measured actuator displacement to the actuator force input.
The actuator displacement can be measured with a capacitive sensor for instance.
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< / p >
< / div >
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< div id = "outline-container-orgcf2e837" class = "outline-3" >
< h3 id = "orgcf2e837" > < span class = "section-number-3" > 3.1< / span > Control Design< / h3 >
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< div class = "outline-text-3" id = "text-3-1" >
< / div >
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< div id = "outline-container-org76b1cfe" class = "outline-4" >
< h4 id = "org76b1cfe" > < span class = "section-number-4" > 3.1.1< / span > Plant< / h4 >
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< div class = "outline-text-4" id = "text-3-1-1" >
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< p >
Let’ s load the undamped plant:
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< / p >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > load(< span class = "org-string" > './active_damping/mat/undamped_plants.mat'< / span > , < span class = "org-string" > 'G_dvf'< / span > );
< / pre >
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< / div >
< p >
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Let’ s look at the transfer function from actuator forces in the nano-hexapod to the measured displacement of the actuator for all 6 pairs of actuator/sensor (figure < a href = "#org1a782ba" > 12< / a > ).
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< / p >
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< div id = "org1a782ba" class = "figure" >
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< p > < img src = "figs/dvf_plant.png" alt = "dvf_plant.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 12: < / span > Transfer function from forces applied in the legs to leg displacement sensor (< a href = "./figs/dvf_plant.png" > png< / a > , < a href = "./figs/dvf_plant.pdf" > pdf< / a > )< / p >
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< / div >
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< / div >
< / div >
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< div id = "outline-container-orgd83abae" class = "outline-4" >
< h4 id = "orgd83abae" > < span class = "section-number-4" > 3.1.2< / span > Control Design< / h4 >
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< div class = "outline-text-4" id = "text-3-1-2" >
< p >
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The Direct Velocity Feedback is defined below.
A Low pass Filter is added to make the controller transfer function proper.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > K_dvf = s< span class = "org-type" > *< / span > 20000< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > 2< span class = "org-type" > /< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > /< / span > 10000);
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< / pre >
< / div >
< p >
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The obtained loop gains are shown in figure < a href = "#org6d24b9e" > 13< / a > .
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< / p >
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< div id = "org6d24b9e" class = "figure" >
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< p > < img src = "figs/dvf_open_loop.png" alt = "dvf_open_loop.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 13: < / span > Loop Gain for the Integral Force Feedback (< a href = "./figs/dvf_open_loop.png" > png< / a > , < a href = "./figs/dvf_open_loop.pdf" > pdf< / a > )< / p >
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< / div >
< / div >
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< / div >
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< div id = "outline-container-orge21c84c" class = "outline-4" >
< h4 id = "orge21c84c" > < span class = "section-number-4" > 3.1.3< / span > Diagonal Controller< / h4 >
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< div class = "outline-text-4" id = "text-3-1-3" >
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< p >
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We create the diagonal controller and we add a minus sign as we have a positive feedback architecture.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > K_dvf = < span class = "org-type" > -< / span > K_dvf< span class = "org-type" > *< / span > eye(6);
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< / pre >
< / div >
< p >
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We save the controller for further analysis.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > save(< span class = "org-string" > './active_damping/mat/K_dvf.mat'< / span > , < span class = "org-string" > 'K_dvf'< / span > );
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< / pre >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgbe85f18" class = "outline-3" >
< h3 id = "orgbe85f18" > < span class = "section-number-3" > 3.2< / span > Tomography Experiment< / h3 >
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< div class = "outline-text-3" id = "text-3-2" >
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< / div >
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< div id = "outline-container-orgee67221" class = "outline-4" >
< h4 id = "orgee67221" > < span class = "section-number-4" > 3.2.1< / span > Initialize the Simulation< / h4 >
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< div class = "outline-text-4" id = "text-3-2-1" >
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< p >
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We initialize elements for the tomography experiment.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > prepareTomographyExperiment();
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< / pre >
< / div >
< p >
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We set the DVF controller.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > load(< span class = "org-string" > './active_damping/mat/K_dvf.mat'< / span > , < span class = "org-string" > 'K_dvf'< / span > );
save(< span class = "org-string" > './mat/controllers.mat'< / span > , < span class = "org-string" > 'K_dvf'< / span > , < span class = "org-string" > '-append'< / span > );
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org76ee3e3" class = "outline-4" >
< h4 id = "org76ee3e3" > < span class = "section-number-4" > 3.2.2< / span > Simulation< / h4 >
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< div class = "outline-text-4" id = "text-3-2-2" >
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< p >
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We change the simulation stop time.
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< / p >
< div class = "org-src-container" >
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< 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 > );
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< / pre >
< / div >
< p >
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And we simulate the system.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-matlab-simulink-keyword" > sim< / span > (< span class = "org-string" > 'sim_nass_active_damping'< / span > );
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< / pre >
< / div >
< p >
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Finally, we save the simulation results for further analysis
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > En_dvf = En;
Eg_dvf = Eg;
save(< span class = "org-string" > './active_damping/mat/tomo_exp.mat'< / span > , < span class = "org-string" > 'En_dvf'< / span > , < span class = "org-string" > 'Eg_dvf'< / span > , < span class = "org-string" > '-append'< / span > );
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< / pre >
< / div >
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< / div >
< / div >
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< div id = "outline-container-org3c3feec" class = "outline-4" >
< h4 id = "org3c3feec" > < span class = "section-number-4" > 3.2.3< / span > Compare with Undamped system< / h4 >
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< div class = "outline-text-4" id = "text-3-2-3" >
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< p >
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We load the results of tomography experiments.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > load(< span class = "org-string" > './active_damping/mat/tomo_exp.mat'< / span > , < span class = "org-string" > 'En'< / span > , < span class = "org-string" > 'En_dvf'< / span > );
t = linspace(0, 3, length(En(< span class = "org-type" > :< / span > ,1)));
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< / pre >
< / div >
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< div id = "orgc5a4b37" class = "figure" >
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< p > < img src = "figs/nass_act_damp_dvf_sim_tomo_xy.png" alt = "nass_act_damp_dvf_sim_tomo_xy.png" / >
< / p >
< p > < span class = "figure-number" > Figure 14: < / span > Position Error during tomography experiment - XY Motion (< a href = "./figs/nass_act_damp_dvf_sim_tomo_xy.png" > png< / a > , < a href = "./figs/nass_act_damp_dvf_sim_tomo_xy.pdf" > pdf< / a > )< / p >
< / div >
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< div id = "orgceb56ef" class = "figure" >
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< p > < img src = "figs/nass_act_damp_dvf_sim_tomo_trans.png" alt = "nass_act_damp_dvf_sim_tomo_trans.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 15: < / span > Position Error during tomography experiment - Translations (< a href = "./figs/nass_act_damp_dvf_sim_tomo_trans.png" > png< / a > , < a href = "./figs/nass_act_damp_dvf_sim_tomo_trans.pdf" > pdf< / a > )< / p >
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< / div >
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< div id = "org4f83701" class = "figure" >
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< p > < img src = "figs/nass_act_damp_dvf_sim_tomo_rot.png" alt = "nass_act_damp_dvf_sim_tomo_rot.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 16: < / span > Position Error during tomography experiment - Rotations (< a href = "./figs/nass_act_damp_dvf_sim_tomo_rot.png" > png< / a > , < a href = "./figs/nass_act_damp_dvf_sim_tomo_rot.pdf" > pdf< / a > )< / p >
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< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org5035edc" class = "outline-3" >
< h3 id = "org5035edc" > < span class = "section-number-3" > 3.3< / span > Conclusion< / h3 >
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< div class = "outline-text-3" id = "text-3-3" >
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< div class = "important" >
< p >
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Direct Velocity Feedback:
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< / p >
< ul class = "org-ul" >
< li > < / li >
< / ul >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org7454716" class = "outline-2" >
< h2 id = "org7454716" > < span class = "section-number-2" > 4< / span > Inertial Control< / h2 >
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< div class = "outline-text-2" id = "text-4" >
< p >
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< a id = "org6b70b5e" > < / a >
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< / p >
< div class = "note" >
< p >
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All the files (data and Matlab scripts) are accessible < a href = "data/ine.zip" > here< / a > .
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< / p >
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< / div >
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< p >
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In Inertial Control, a feedback is applied between the measured < b > absolute< / b > motion (velocity or acceleration) of the platform to the actuator force input.
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< / p >
< / div >
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< div id = "outline-container-org8e08998" class = "outline-3" >
< h3 id = "org8e08998" > < span class = "section-number-3" > 4.1< / span > Control Design< / h3 >
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< div class = "outline-text-3" id = "text-4-1" >
< / div >
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< div id = "outline-container-orgf2db463" class = "outline-4" >
< h4 id = "orgf2db463" > < span class = "section-number-4" > 4.1.1< / span > Plant< / h4 >
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< div class = "outline-text-4" id = "text-4-1-1" >
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< p >
Let’ s load the undamped plant:
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< / p >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > load(< span class = "org-string" > './active_damping/mat/undamped_plants.mat'< / span > , < span class = "org-string" > 'G_ine'< / span > );
< / pre >
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< / div >
< p >
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Let’ s look at the transfer function from actuator forces in the nano-hexapod to the measured velocity of the nano-hexapod platform in the direction of the corresponding actuator for all 6 pairs of actuator/sensor (figure < a href = "#org7ac479f" > 17< / a > ).
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< / p >
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< div id = "org7ac479f" class = "figure" >
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< p > < img src = "figs/ine_plant.png" alt = "ine_plant.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 17: < / span > Transfer function from forces applied in the legs to leg velocity sensor (< a href = "./figs/ine_plant.png" > png< / a > , < a href = "./figs/ine_plant.pdf" > pdf< / a > )< / p >
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< / div >
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< / div >
< / div >
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< div id = "outline-container-org93cb5d8" class = "outline-4" >
< h4 id = "org93cb5d8" > < span class = "section-number-4" > 4.1.2< / span > Control Design< / h4 >
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< div class = "outline-text-4" id = "text-4-1-2" >
< p >
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The controller is defined below and the obtained loop gain is shown in figure < a href = "#orgbc888e6" > 18< / a > .
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > K_ine = 1e4< span class = "org-type" > /< / span > (1< span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > (2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 100));
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< / pre >
< / div >
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< div id = "orgbc888e6" class = "figure" >
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< p > < img src = "figs/ine_open_loop_gain.png" alt = "ine_open_loop_gain.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 18: < / span > Loop Gain for Inertial Control (< a href = "./figs/ine_open_loop_gain.png" > png< / a > , < a href = "./figs/ine_open_loop_gain.pdf" > pdf< / a > )< / p >
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< / div >
< / div >
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< / div >
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< div id = "outline-container-orgd002ae1" class = "outline-4" >
< h4 id = "orgd002ae1" > < span class = "section-number-4" > 4.1.3< / span > Diagonal Controller< / h4 >
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< div class = "outline-text-4" id = "text-4-1-3" >
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< p >
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We create the diagonal controller and we add a minus sign as we have a positive feedback architecture.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > K_ine = < span class = "org-type" > -< / span > K_ine< span class = "org-type" > *< / span > eye(6);
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< / pre >
< / div >
< p >
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We save the controller for further analysis.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > save(< span class = "org-string" > './active_damping/mat/K_ine.mat'< / span > , < span class = "org-string" > 'K_ine'< / span > );
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< / pre >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgca448ac" class = "outline-3" >
< h3 id = "orgca448ac" > < span class = "section-number-3" > 4.2< / span > Tomography Experiment< / h3 >
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< div class = "outline-text-3" id = "text-4-2" >
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< / div >
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< div id = "outline-container-org632ee5a" class = "outline-4" >
< h4 id = "org632ee5a" > < span class = "section-number-4" > 4.2.1< / span > Initialize the Simulation< / h4 >
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< div class = "outline-text-4" id = "text-4-2-1" >
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< p >
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We initialize elements for the tomography experiment.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > prepareTomographyExperiment();
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< / pre >
< / div >
< p >
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We set the Inertial controller.
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< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > load(< span class = "org-string" > './active_damping/mat/K_ine.mat'< / span > , < span class = "org-string" > 'K_ine'< / span > );
save(< span class = "org-string" > './mat/controllers.mat'< / span > , < span class = "org-string" > 'K_ine'< / span > , < span class = "org-string" > '-append'< / span > );
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< / pre >
< / div >
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< / div >
< / div >
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< div id = "outline-container-org10654aa" class = "outline-4" >
< h4 id = "org10654aa" > < span class = "section-number-4" > 4.2.2< / span > Simulation< / h4 >
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< div class = "outline-text-4" id = "text-4-2-2" >
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< p >
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We change the simulation stop time.
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< / p >
< div class = "org-src-container" >
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< 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 > );
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< / pre >
< / div >
< p >
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And we simulate the system.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-matlab-simulink-keyword" > sim< / span > (< span class = "org-string" > 'sim_nass_active_damping'< / span > );
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< / pre >
< / div >
< p >
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Finally, we save the simulation results for further analysis
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > En_ine = En;
Eg_ine = Eg;
save(< span class = "org-string" > './active_damping/mat/tomo_exp.mat'< / span > , < span class = "org-string" > 'En_ine'< / span > , < span class = "org-string" > 'Eg_ine'< / span > , < span class = "org-string" > '-append'< / span > );
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< / pre >
< / div >
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< / div >
< / div >
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< div id = "outline-container-org1239f7c" class = "outline-4" >
< h4 id = "org1239f7c" > < span class = "section-number-4" > 4.2.3< / span > Compare with Undamped system< / h4 >
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< div class = "outline-text-4" id = "text-4-2-3" >
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< p >
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We load the results of tomography experiments.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > load(< span class = "org-string" > './active_damping/mat/tomo_exp.mat'< / span > , < span class = "org-string" > 'En'< / span > , < span class = "org-string" > 'En_ine'< / span > );
t = linspace(0, 3, length(En_ine(< span class = "org-type" > :< / span > ,1)));
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< / pre >
< / div >
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< div id = "orga2e86ca" class = "figure" >
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< p > < img src = "figs/nass_act_damp_ine_sim_tomo_xy.png" alt = "nass_act_damp_ine_sim_tomo_xy.png" / >
< / p >
< p > < span class = "figure-number" > Figure 19: < / span > Position Error during tomography experiment - XY Motion (< a href = "./figs/nass_act_damp_ine_sim_tomo_xy.png" > png< / a > , < a href = "./figs/nass_act_damp_ine_sim_tomo_xy.pdf" > pdf< / a > )< / p >
< / div >
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< div id = "orgc2f3c81" class = "figure" >
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< p > < img src = "figs/nass_act_damp_ine_sim_tomo_trans.png" alt = "nass_act_damp_ine_sim_tomo_trans.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 20: < / span > Position Error during tomography experiment - Translations (< a href = "./figs/nass_act_damp_ine_sim_tomo_trans.png" > png< / a > , < a href = "./figs/nass_act_damp_ine_sim_tomo_trans.pdf" > pdf< / a > )< / p >
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< / div >
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< div id = "org063e44f" class = "figure" >
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< p > < img src = "figs/nass_act_damp_ine_sim_tomo_rot.png" alt = "nass_act_damp_ine_sim_tomo_rot.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 21: < / span > Position Error during tomography experiment - Rotations (< a href = "./figs/nass_act_damp_ine_sim_tomo_rot.png" > png< / a > , < a href = "./figs/nass_act_damp_ine_sim_tomo_rot.pdf" > pdf< / a > )< / p >
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< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org9674a40" class = "outline-3" >
< h3 id = "org9674a40" > < span class = "section-number-3" > 4.3< / span > Conclusion< / h3 >
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< div class = "outline-text-3" id = "text-4-3" >
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< div class = "important" >
< p >
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Inertial Control:
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< h2 id = "org251a441" > < span class = "section-number-2" > 5< / span > Variability of the system dynamics for Active Damping< / h2 >
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All the files (data and Matlab scripts) are accessible < a href = "data/act_damp_variability_plant.zip" > here< / a > .
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< h3 id = "orgd178f46" > < span class = "section-number-3" > 5.1< / span > Variation of the Sample Mass< / h3 >
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For all the identifications, the disturbances are disabled and no controller are used.
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We identify the dynamics for the following sample mass.
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< pre class = "src src-matlab" > masses = [1, 10, 50]; < span class = "org-comment" > % [kg]< / span >
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< p > < img src = "figs/act_damp_variability_iff_sample_mass.png" alt = "act_damp_variability_iff_sample_mass.png" / >
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< p > < span class = "figure-number" > Figure 22: < / span > Variability of the IFF plant with the Sample Mass (< a href = "./figs/act_damp_variability_iff_sample_mass.png" > png< / a > , < a href = "./figs/act_damp_variability_iff_sample_mass.pdf" > pdf< / a > )< / p >
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< p > < img src = "figs/act_damp_variability_dvf_sample_mass.png" alt = "act_damp_variability_dvf_sample_mass.png" / >
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< p > < span class = "figure-number" > Figure 23: < / span > Variability of the DVF plant with the Sample Mass (< a href = "./figs/act_damp_variability_dvf_sample_mass.png" > png< / a > , < a href = "./figs/act_damp_variability_dvf_sample_mass.pdf" > pdf< / a > )< / p >
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< p > < img src = "figs/act_damp_variability_ine_sample_mass.png" alt = "act_damp_variability_ine_sample_mass.png" / >
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< p > < span class = "figure-number" > Figure 24: < / span > Variability of the Inertial plant with the Sample Mass (< a href = "./figs/act_damp_variability_ine_sample_mass.png" > png< / a > , < a href = "./figs/act_damp_variability_ine_sample_mass.pdf" > pdf< / a > )< / p >
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< h3 id = "org38b8d50" > < span class = "section-number-3" > 5.2< / span > Variation of the Spindle Angle< / h3 >
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We identify the dynamics for the following Spindle angles.
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< pre class = "src src-matlab" > Rz_amplitudes = [0, < span class = "org-constant" > pi< / span > < span class = "org-type" > /< / span > 4, < span class = "org-constant" > pi< / span > < span class = "org-type" > /< / span > 2, < span class = "org-constant" > pi< / span > ]; < span class = "org-comment" > % [rad]< / span >
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< p > < img src = "figs/act_damp_variability_iff_spindle_angle.png" alt = "act_damp_variability_iff_spindle_angle.png" / >
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< p > < span class = "figure-number" > Figure 25: < / span > Variability of the IFF plant with the Spindle Angle (< a href = "./figs/act_damp_variability_iff_spindle_angle.png" > png< / a > , < a href = "./figs/act_damp_variability_iff_spindle_angle.pdf" > pdf< / a > )< / p >
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< p > < img src = "figs/act_damp_variability_dvf_spindle_angle.png" alt = "act_damp_variability_dvf_spindle_angle.png" / >
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< p > < span class = "figure-number" > Figure 26: < / span > Variability of the DVF plant with the Spindle Angle (< a href = "./figs/act_damp_variability_dvf_spindle_angle.png" > png< / a > , < a href = "./figs/act_damp_variability_dvf_spindle_angle.pdf" > pdf< / a > )< / p >
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< p > < img src = "figs/act_damp_variability_ine_spindle_angle.png" alt = "act_damp_variability_ine_spindle_angle.png" / >
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< p > < span class = "figure-number" > Figure 27: < / span > Variability of the Inertial plant with the Spindle Angle (< a href = "./figs/act_damp_variability_ine_spindle_angle.png" > png< / a > , < a href = "./figs/act_damp_variability_ine_spindle_angle.pdf" > pdf< / a > )< / p >
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< h3 id = "org666ca9f" > < span class = "section-number-3" > 5.3< / span > Variation of the Spindle Rotation Speed< / h3 >
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We identify the dynamics for the following Spindle rotation periods.
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< pre class = "src src-matlab" > Rz_periods = [60, 10, 1]; < span class = "org-comment" > % [s]< / span >
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< p > < img src = "figs/act_damp_variability_iff_spindle_speed.png" alt = "act_damp_variability_iff_spindle_speed.png" / >
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< p > < span class = "figure-number" > Figure 28: < / span > Variability of the IFF plant with the Spindle rotation speed (< a href = "./figs/act_damp_variability_iff_spindle_speed.png" > png< / a > , < a href = "./figs/act_damp_variability_iff_spindle_speed.pdf" > pdf< / a > )< / p >
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< p > < img src = "figs/act_damp_variability_iff_spindle_speed_zoom.png" alt = "act_damp_variability_iff_spindle_speed_zoom.png" / >
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< p > < span class = "figure-number" > Figure 29: < / span > Variability of the IFF plant with the Spindle rotation speed (< a href = "./figs/act_damp_variability_iff_spindle_speed_zoom.png" > png< / a > , < a href = "./figs/act_damp_variability_iff_spindle_speed_zoom.pdf" > pdf< / a > )< / p >
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< div id = "org71d0bf3" class = "figure" >
< p > < img src = "figs/act_damp_variability_dvf_spindle_speed.png" alt = "act_damp_variability_dvf_spindle_speed.png" / >
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< p > < span class = "figure-number" > Figure 30: < / span > Variability of the DVF plant with the Spindle rotation speed (< a href = "./figs/act_damp_variability_dvf_spindle_speed.png" > png< / a > , < a href = "./figs/act_damp_variability_dvf_spindle_speed.pdf" > pdf< / a > )< / p >
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< div id = "orgb10a098" class = "figure" >
< p > < img src = "figs/act_damp_variability_dvf_spindle_speed_zoom.png" alt = "act_damp_variability_dvf_spindle_speed_zoom.png" / >
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< p > < span class = "figure-number" > Figure 31: < / span > Variability of the DVF plant with the Spindle rotation speed (< a href = "./figs/act_damp_variability_dvf_spindle_speed_zoom.png" > png< / a > , < a href = "./figs/act_damp_variability_dvf_spindle_speed_zoom.pdf" > pdf< / a > )< / p >
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< div id = "org5e0bd6d" class = "figure" >
< p > < img src = "figs/act_damp_variability_ine_spindle_speed.png" alt = "act_damp_variability_ine_spindle_speed.png" / >
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< p > < span class = "figure-number" > Figure 32: < / span > Variability of the Inertial plant with the Spindle rotation speed (< a href = "./figs/act_damp_variability_ine_spindle_speed.png" > png< / a > , < a href = "./figs/act_damp_variability_ine_spindle_speed.pdf" > pdf< / a > )< / p >
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< div id = "org563d245" class = "figure" >
< p > < img src = "figs/act_damp_variability_ine_spindle_speed_zoom.png" alt = "act_damp_variability_ine_spindle_speed_zoom.png" / >
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< p > < span class = "figure-number" > Figure 33: < / span > Variability of the Inertial plant with the Spindle rotation speed (< a href = "./figs/act_damp_variability_ine_spindle_speed_zoom.png" > png< / a > , < a href = "./figs/act_damp_variability_ine_spindle_speed_zoom.pdf" > pdf< / a > )< / p >
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< h3 id = "org15eb4a9" > < span class = "section-number-3" > 5.4< / span > Variation of the Tilt Angle< / h3 >
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We identify the dynamics for the following Tilt stage angles.
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< pre class = "src src-matlab" > Ry_amplitudes = [0, 3< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > /< / span > 180]; < span class = "org-comment" > % [rad]< / span >
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< div id = "orge5256c0" class = "figure" >
< p > < img src = "figs/act_damp_variability_iff_tilt_angle.png" alt = "act_damp_variability_iff_tilt_angle.png" / >
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< p > < span class = "figure-number" > Figure 34: < / span > Variability of the IFF plant with the Spindle Angle (< a href = "./figs/act_damp_variability_iff_tilt_angle.png" > png< / a > , < a href = "./figs/act_damp_variability_iff_tilt_angle.pdf" > pdf< / a > )< / p >
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< div id = "orgba17469" class = "figure" >
< p > < img src = "figs/act_damp_variability_dvf_tilt_angle.png" alt = "act_damp_variability_dvf_tilt_angle.png" / >
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< p > < span class = "figure-number" > Figure 35: < / span > Variability of the DVF plant with the Spindle Angle (< a href = "./figs/act_damp_variability_dvf_tilt_angle.png" > png< / a > , < a href = "./figs/act_damp_variability_dvf_tilt_angle.pdf" > pdf< / a > )< / p >
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< div id = "org0980537" class = "figure" >
< p > < img src = "figs/act_damp_variability_ine_tilt_angle.png" alt = "act_damp_variability_ine_tilt_angle.png" / >
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< p > < span class = "figure-number" > Figure 36: < / span > Variability of the Inertial plant with the Spindle Angle (< a href = "./figs/act_damp_variability_ine_tilt_angle.png" > png< / a > , < a href = "./figs/act_damp_variability_ine_tilt_angle.pdf" > pdf< / a > )< / p >
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< h3 id = "org8d56329" > < span class = "section-number-3" > 5.5< / span > Conclusion< / h3 >
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< h2 id = "org4e51ab2" > < span class = "section-number-2" > 6< / span > Comparison< / h2 >
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< h3 id = "org22beca8" > < span class = "section-number-3" > 6.1< / span > Load the plants< / h3 >
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< pre class = "src src-matlab" > load(< span class = "org-string" > './active_damping/mat/plants.mat'< / span > , < span class = "org-string" > 'G'< / span > , < span class = "org-string" > 'G_iff'< / span > , < span class = "org-string" > 'G_ine'< / span > , < span class = "org-string" > 'G_dvf'< / span > );
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< h3 id = "orgcaacc66" > < span class = "section-number-3" > 6.2< / span > Sensitivity to Disturbance< / h3 >
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< div id = "org0c239f2" class = "figure" >
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< p > < img src = "figs/sensitivity_comp_ground_motion_z.png" alt = "sensitivity_comp_ground_motion_z.png" / >
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< p > < span class = "figure-number" > Figure 37: < / span > Sensitivity to ground motion in the Z direction on the Z motion error (< a href = "./figs/sensitivity_comp_ground_motion_z.png" > png< / a > , < a href = "./figs/sensitivity_comp_ground_motion_z.pdf" > pdf< / a > )< / p >
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< p > < img src = "figs/sensitivity_comp_direct_forces_z.png" alt = "sensitivity_comp_direct_forces_z.png" / >
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< p > < span class = "figure-number" > Figure 38: < / span > Compliance in the Z direction: Sensitivity of direct forces applied on the sample in the Z direction on the Z motion error (< a href = "./figs/sensitivity_comp_direct_forces_z.png" > png< / a > , < a href = "./figs/sensitivity_comp_direct_forces_z.pdf" > pdf< / a > )< / p >
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< div id = "orga66cdb4" class = "figure" >
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< p > < img src = "figs/sensitivity_comp_spindle_z.png" alt = "sensitivity_comp_spindle_z.png" / >
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< p > < span class = "figure-number" > Figure 39: < / span > Sensitivity to forces applied in the Z direction by the Spindle on the Z motion error (< a href = "./figs/sensitivity_comp_spindle_z.png" > png< / a > , < a href = "./figs/sensitivity_comp_spindle_z.pdf" > pdf< / a > )< / p >
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< p > < img src = "figs/sensitivity_comp_ty_z.png" alt = "sensitivity_comp_ty_z.png" / >
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< p > < span class = "figure-number" > Figure 40: < / span > Sensitivity to forces applied in the Z direction by the Y translation stage on the Z motion error (< a href = "./figs/sensitivity_comp_ty_z.png" > png< / a > , < a href = "./figs/sensitivity_comp_ty_z.pdf" > pdf< / a > )< / p >
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< p > < img src = "figs/sensitivity_comp_ty_x.png" alt = "sensitivity_comp_ty_x.png" / >
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< p > < span class = "figure-number" > Figure 41: < / span > Sensitivity to forces applied in the X direction by the Y translation stage on the X motion error (< a href = "./figs/sensitivity_comp_ty_x.png" > png< / a > , < a href = "./figs/sensitivity_comp_ty_x.pdf" > pdf< / a > )< / p >
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< h3 id = "org1b7fd80" > < span class = "section-number-3" > 6.3< / span > Damped Plant< / h3 >
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< div id = "org60a4299" class = "figure" >
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< p > < img src = "figs/plant_comp_damping_z.png" alt = "plant_comp_damping_z.png" / >
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< p > < span class = "figure-number" > Figure 42: < / span > Plant for the \(z\) direction for different active damping technique used (< a href = "./figs/plant_comp_damping_z.png" > png< / a > , < a href = "./figs/plant_comp_damping_z.pdf" > pdf< / a > )< / p >
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< p > < img src = "figs/plant_comp_damping_x.png" alt = "plant_comp_damping_x.png" / >
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< p > < span class = "figure-number" > Figure 43: < / span > Plant for the \(x\) direction for different active damping technique used (< a href = "./figs/plant_comp_damping_x.png" > png< / a > , < a href = "./figs/plant_comp_damping_x.pdf" > pdf< / a > )< / p >
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< p > < img src = "figs/plant_comp_damping_coupling.png" alt = "plant_comp_damping_coupling.png" / >
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< p > < span class = "figure-number" > Figure 44: < / span > Comparison of one off-diagonal plant for different damping technique applied (< a href = "./figs/plant_comp_damping_coupling.png" > png< / a > , < a href = "./figs/plant_comp_damping_coupling.pdf" > pdf< / a > )< / p >
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< h3 id = "org0f12cf8" > < span class = "section-number-3" > 6.4< / span > Tomography Experiment< / h3 >
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< h4 id = "orgdb48e43" > < span class = "section-number-4" > 6.4.1< / span > Load the Simulation Data< / h4 >
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< pre class = "src src-matlab" > load(< span class = "org-string" > './active_damping/mat/tomo_exp.mat'< / span > , < span class = "org-string" > 'En'< / span > , < span class = "org-string" > 'En_iff_hpf'< / span > , < span class = "org-string" > 'En_dvf'< / span > , < span class = "org-string" > 'En_ine'< / span > );
En_iff = En_iff_hpf;
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t = linspace(0, 3, length(En(< span class = "org-type" > :< / span > ,1)));
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< h4 id = "orgfbe1bad" > < span class = "section-number-4" > 6.4.2< / span > Frequency Domain Analysis< / h4 >
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< p >
Window used for < code > pwelch< / code > function.
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< pre class = "src src-matlab" > n_av = 8;
han_win = hanning(ceil(length(En(< span class = "org-type" > :< / span > , 1))< span class = "org-type" > /< / span > n_av));
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< p > < img src = "figs/act_damp_tomo_exp_comp_psd_trans.png" alt = "act_damp_tomo_exp_comp_psd_trans.png" / >
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< p > < span class = "figure-number" > Figure 45: < / span > PSD of the translation errors in the X direction for applied Active Damping techniques (< a href = "./figs/act_damp_tomo_exp_comp_psd_trans.png" > png< / a > , < a href = "./figs/act_damp_tomo_exp_comp_psd_trans.pdf" > pdf< / a > )< / p >
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< p > < img src = "figs/act_damp_tomo_exp_comp_psd_rot.png" alt = "act_damp_tomo_exp_comp_psd_rot.png" / >
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< p > < span class = "figure-number" > Figure 46: < / span > PSD of the rotation errors in the X direction for applied Active Damping techniques (< a href = "./figs/act_damp_tomo_exp_comp_psd_rot.png" > png< / a > , < a href = "./figs/act_damp_tomo_exp_comp_psd_rot.pdf" > pdf< / a > )< / p >
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< p > < img src = "figs/act_damp_tomo_exp_comp_cps_trans.png" alt = "act_damp_tomo_exp_comp_cps_trans.png" / >
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< p > < span class = "figure-number" > Figure 47: < / span > CPS of the translation errors in the X direction for applied Active Damping techniques (< a href = "./figs/act_damp_tomo_exp_comp_cps_trans.png" > png< / a > , < a href = "./figs/act_damp_tomo_exp_comp_cps_trans.pdf" > pdf< / a > )< / p >
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< p > < img src = "figs/act_damp_tomo_exp_comp_cps_rot.png" alt = "act_damp_tomo_exp_comp_cps_rot.png" / >
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< p > < span class = "figure-number" > Figure 48: < / span > CPS of the rotation errors in the X direction for applied Active Damping techniques (< a href = "./figs/act_damp_tomo_exp_comp_cps_rot.png" > png< / a > , < a href = "./figs/act_damp_tomo_exp_comp_cps_rot.pdf" > pdf< / a > )< / p >
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< div id = "outline-container-orgb197610" class = "outline-2" >
< h2 id = "orgb197610" > < span class = "section-number-2" > 7< / span > Useful Functions< / h2 >
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< div id = "outline-container-org6783ec6" class = "outline-3" >
< h3 id = "org6783ec6" > < span class = "section-number-3" > 7.1< / span > prepareTomographyExperiment< / h3 >
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< a id = "org6e9cabc" > < / a >
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This Matlab function is accessible < a href = "src/prepareTomographyExperiment.m" > here< / a > .
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< h4 id = "org8c75d0b" > Function Description< / h4 >
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< pre class = "src src-matlab" > < span class = "org-keyword" > function< / span > < span class = "org-variable-name" > []< / span > = < span class = "org-function-name" > prepareTomographyExperiment< / span > (< span class = "org-variable-name" > args< / span > )
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< div id = "outline-container-org356aa82" class = "outline-4" >
< h4 id = "org356aa82" > Optional Parameters< / h4 >
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< pre class = "src src-matlab" > arguments
args.nass_actuator char {mustBeMember(args.nass_actuator,{< span class = "org-string" > 'piezo'< / span > , < span class = "org-string" > 'lorentz'< / span > })} = < span class = "org-string" > 'piezo'< / span >
args.sample_mass (1,1) double {mustBeNumeric, mustBePositive} = 50
args.Ry_period (1,1) double {mustBeNumeric, mustBePositive} = 1
< span class = "org-keyword" > end< / span >
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< h4 id = "org93e527a" > Initialize the Simulation< / h4 >
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< p >
We initialize all the stages with the default parameters.
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< div class = "org-src-container" >
< pre class = "src src-matlab" > initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
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< p >
The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.
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< pre class = "src src-matlab" > initializeNanoHexapod(< span class = "org-string" > 'actuator'< / span > , args.nass_actuator);
initializeSample(< span class = "org-string" > 'mass'< / span > , args.sample_mass);
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We set the references to zero.
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< pre class = "src src-matlab" > initializeReferences(< span class = "org-string" > 'Rz_type'< / span > , < span class = "org-string" > 'rotating'< / span > , < span class = "org-string" > 'Rz_period'< / span > , args.Ry_period);
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< p >
And all the controllers are set to 0.
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< 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 > );
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< div id = "postamble" class = "status" >
< p class = "author" > Author: Dehaeze Thomas< / p >
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< p class = "date" > Created: 2020-01-21 mar. 17:28< / p >
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