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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 = "#org97abc59" > 1. Undamped System< / a >
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< ul >
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< li > < a href = "#org7d459a2" > 1.1. Identification of the dynamics for Active Damping< / a >
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< ul >
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< li > < a href = "#org6a76f18" > 1.1.1. Initialize the Simulation< / a > < / li >
< li > < a href = "#orgceae930" > 1.1.2. Identification< / a > < / li >
< li > < a href = "#org50cac8d" > 1.1.3. Obtained Plants for Active Damping< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgb30c7fd" > 1.2. Tomography Experiment< / a >
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< ul >
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< li > < a href = "#orgf71542f" > 1.2.1. Simulation< / a > < / li >
< li > < a href = "#org15546d3" > 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 = "#org097f664" > 2. Variability of the system dynamics for Active Damping< / a >
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< ul >
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< li > < a href = "#org39af6f9" > 2.1. Variation of the Sample Mass< / a > < / li >
< li > < a href = "#org229c4c5" > 2.2. Variation of the Spindle Angle< / a > < / li >
< li > < a href = "#org352fc0c" > 2.3. Variation of the Spindle Rotation Speed< / a >
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< ul >
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< li > < a href = "#orgc6e2b26" > 2.3.1. Dynamics of the Active Damping plants< / a > < / li >
< li > < a href = "#org8aa8182" > 2.3.2. Variation of the poles and zeros with the Spindle rotation frequency< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org0153a58" > 2.4. Variation of the Tilt Angle< / a > < / li >
< li > < a href = "#org4e27559" > 2.5. Scans of the Translation Stage< / a > < / li >
< li > < a href = "#orgb3ee7e4" > 2.6. Conclusion< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgf08b709" > 3. Integral Force Feedback< / a >
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< ul >
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< li > < a href = "#org87395b8" > 3.1. Control Design< / a >
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< ul >
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< li > < a href = "#org77e4473" > 3.1.1. Plant< / a > < / li >
< li > < a href = "#org1c94029" > 3.1.2. Control Design< / a > < / li >
< li > < a href = "#orga4ebb1c" > 3.1.3. Diagonal Controller< / a > < / li >
< li > < a href = "#orgad90da1" > 3.1.4. IFF with High Pass Filter< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org98b7048" > 3.2. Tomography Experiment< / a >
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< ul >
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< li > < a href = "#org43bf533" > 3.2.1. Simulation with IFF Controller< / a > < / li >
< li > < a href = "#orge0b6480" > 3.2.2. Simulation with IFF Controller with added High Pass Filter< / a > < / li >
< li > < a href = "#org5f65a86" > 3.2.3. Compare with Undamped system< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org22577cc" > 3.3. Conclusion< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org93105e9" > 4. Direct Velocity Feedback< / a >
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< ul >
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< li > < a href = "#org0181bc6" > 4.1. Control Design< / a >
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< ul >
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< li > < a href = "#org0baaad9" > 4.1.1. Plant< / a > < / li >
< li > < a href = "#org9d0d598" > 4.1.2. Control Design< / a > < / li >
< li > < a href = "#orgc664cda" > 4.1.3. Diagonal Controller< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org30a47bd" > 4.2. Tomography Experiment< / a >
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< ul >
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< li > < a href = "#orged4c8ff" > 4.2.1. Initialize the Simulation< / a > < / li >
< li > < a href = "#orgafcf6c3" > 4.2.2. Simulation< / a > < / li >
< li > < a href = "#orgdee965e" > 4.2.3. Compare with Undamped system< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgc1e250f" > 4.3. Conclusion< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orga585807" > 5. Inertial Control< / a >
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< ul >
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< li > < a href = "#orgef3b74a" > 5.1. Control Design< / a >
< ul >
< li > < a href = "#org62bfbc7" > 5.1.1. Plant< / a > < / li >
< li > < a href = "#orgeeec2e5" > 5.1.2. Control Design< / a > < / li >
< li > < a href = "#org240b7fd" > 5.1.3. Diagonal Controller< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgea439d2" > 5.2. Tomography Experiment< / a >
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< ul >
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< li > < a href = "#orgd52597f" > 5.2.1. Initialize the Simulation< / a > < / li >
< li > < a href = "#org2ad0ade" > 5.2.2. Simulation< / a > < / li >
< li > < a href = "#orgfd09099" > 5.2.3. Compare with Undamped system< / a > < / li >
< / ul >
< / li >
< li > < a href = "#org5a63825" > 5.3. Conclusion< / a > < / li >
< / ul >
< / li >
< li > < a href = "#org230c9f4" > 6. Comparison< / a >
< ul >
< li > < a href = "#org4baffd6" > 6.1. Load the plants< / a > < / li >
< li > < a href = "#orga058c69" > 6.2. Sensitivity to Disturbance< / a > < / li >
< li > < a href = "#org7d53c0f" > 6.3. Damped Plant< / a > < / li >
< li > < a href = "#org70d41b8" > 6.4. Tomography Experiment< / a >
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< ul >
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< li > < a href = "#org604ad48" > 6.4.1. Load the Simulation Data< / a > < / li >
< li > < a href = "#orgc2877fd" > 6.4.2. Frequency Domain Analysis< / a > < / li >
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< / ul >
< / li >
< / ul >
< / li >
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< li > < a href = "#orgbe23ba6" > 7. Useful Functions< / a >
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< ul >
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< li > < a href = "#orge26884f" > 7.1. prepareTomographyExperiment< / a >
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< ul >
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< li > < a href = "#orgbd71c69" > Function Description< / a > < / li >
< li > < a href = "#org79eafae" > Optional Parameters< / a > < / li >
< li > < a href = "#orge7b78fc" > 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 = "#orgcd3c1c9" > 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 = "#org81e3ebf" > 3< / a > : the integral force feedback is used< / li >
< li > In section < a href = "#org5349d23" > 4< / a > : the direct velocity feedback is used< / li >
< li > In section < a href = "#org6300794" > 5< / 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-org97abc59" class = "outline-2" >
< h2 id = "org97abc59" > < 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 = "orgcd3c1c9" > < / a >
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< / p >
< 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-org7d459a2" class = "outline-3" >
< h3 id = "org7d459a2" > < 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-org6a76f18" class = "outline-4" >
< h4 id = "org6a76f18" > < 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();
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > initializeDisturbances(< span class = "org-string" > 'enable'< / span > , < span class = "org-constant" > false< / span > );
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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-orgceae930" class = "outline-4" >
< h4 id = "orgceae930" > < 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, 0.5, 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-org50cac8d" class = "outline-4" >
< h4 id = "org50cac8d" > < 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 = "orgae07e00" 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 = "org54ba9ab" 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 = "org8d04315" 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-orgb30c7fd" class = "outline-3" >
< h3 id = "orgb30c7fd" > < 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-orgf71542f" class = "outline-4" >
< h4 id = "orgf71542f" > < 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-org15546d3" class = "outline-4" >
< h4 id = "org15546d3" > < 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 = "orga9eaa7e" 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 = "orge6dfe03" 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-org097f664" class = "outline-2" >
< h2 id = "org097f664" > < span class = "section-number-2" > 2< / span > Variability of the system dynamics for Active Damping< / h2 >
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< div class = "outline-text-2" id = "text-2" >
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< p >
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< a id = "org2ff3e99" > < / a >
< / p >
< p >
The goal of this section is to study how the dynamics of the Active Damping plants are changing with the experimental conditions.
These experimental conditions are:
< / p >
< ul class = "org-ul" >
< li > The mass of the sample (section < a href = "#orgf4c0294" > 2.1< / a > )< / li >
< li > The spindle angle with a null rotating speed (section < a href = "#org103d069" > 2.2< / a > )< / li >
< li > The spindle rotation speed (section < a href = "#org329a267" > 2.3< / a > )< / li >
< li > The tilt angle (section < a href = "#org3f724e4" > 2.4< / a > )< / li >
< li > The scans of the translation stage (section < a href = "#org6499091" > 2.5< / a > )< / li >
< / ul >
< p >
For the identification of the dynamics, the system is simulation for \(\approx 0.5s\) before the linearization is performed.
This is done in order for the transient phase to be over.
< / p >
< / div >
< div id = "outline-container-org39af6f9" class = "outline-3" >
< h3 id = "org39af6f9" > < span class = "section-number-3" > 2.1< / span > Variation of the Sample Mass< / h3 >
< div class = "outline-text-3" id = "text-2-1" >
< p >
< a id = "orgf4c0294" > < / a >
< / p >
< p >
For all the identifications, the disturbances are disabled and no controller are used.
< / p >
< p >
We identify the dynamics for the following sample mass.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > masses = [1, 10, 50]; < span class = "org-comment" > % [kg]< / span >
< / pre >
< / div >
< div id = "org568a9b1" class = "figure" >
< p > < img src = "figs/act_damp_variability_iff_sample_mass.png" alt = "act_damp_variability_iff_sample_mass.png" / >
< / p >
< p > < span class = "figure-number" > Figure 6: < / 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 >
< / div >
< div id = "orgb99e55d" class = "figure" >
< p > < img src = "figs/act_damp_variability_dvf_sample_mass.png" alt = "act_damp_variability_dvf_sample_mass.png" / >
< / p >
< p > < span class = "figure-number" > Figure 7: < / 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 >
< / div >
< div id = "org34d5224" class = "figure" >
< p > < img src = "figs/act_damp_variability_ine_sample_mass.png" alt = "act_damp_variability_ine_sample_mass.png" / >
< / p >
< p > < span class = "figure-number" > Figure 8: < / 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 >
< / div >
< / div >
< / div >
< div id = "outline-container-org229c4c5" class = "outline-3" >
< h3 id = "org229c4c5" > < span class = "section-number-3" > 2.2< / span > Variation of the Spindle Angle< / h3 >
< div class = "outline-text-3" id = "text-2-2" >
< p >
< a id = "org103d069" > < / a >
< / p >
< p >
We identify the dynamics for the following Spindle angles.
< / p >
< div class = "org-src-container" >
< 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 >
< / pre >
< / div >
< div id = "orgd4b2f43" class = "figure" >
< p > < img src = "figs/act_damp_variability_iff_spindle_angle.png" alt = "act_damp_variability_iff_spindle_angle.png" / >
< / p >
< p > < span class = "figure-number" > Figure 9: < / 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 >
< / div >
< div id = "org83b65c2" class = "figure" >
< p > < img src = "figs/act_damp_variability_dvf_spindle_angle.png" alt = "act_damp_variability_dvf_spindle_angle.png" / >
< / p >
< p > < span class = "figure-number" > Figure 10: < / 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 >
< / div >
< div id = "org9d12465" class = "figure" >
< p > < img src = "figs/act_damp_variability_ine_spindle_angle.png" alt = "act_damp_variability_ine_spindle_angle.png" / >
< / p >
< p > < span class = "figure-number" > Figure 11: < / 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 >
< / div >
< / div >
< / div >
< div id = "outline-container-org352fc0c" class = "outline-3" >
< h3 id = "org352fc0c" > < span class = "section-number-3" > 2.3< / span > Variation of the Spindle Rotation Speed< / h3 >
< div class = "outline-text-3" id = "text-2-3" >
< p >
< a id = "org329a267" > < / a >
< / p >
< p >
We identify the dynamics for the following Spindle rotation periods.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Rz_periods = [60, 6, 2, 1]; < span class = "org-comment" > % [s]< / span >
< / pre >
< / div >
< p >
The identification of the dynamics is done at the same Spindle angle position.
< / p >
< / div >
< div id = "outline-container-orgc6e2b26" class = "outline-4" >
< h4 id = "orgc6e2b26" > < span class = "section-number-4" > 2.3.1< / span > Dynamics of the Active Damping plants< / h4 >
< div class = "outline-text-4" id = "text-2-3-1" >
< div id = "org3436e1a" class = "figure" >
< p > < img src = "figs/act_damp_variability_iff_spindle_speed.png" alt = "act_damp_variability_iff_spindle_speed.png" / >
< / p >
< p > < span class = "figure-number" > Figure 12: < / 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 >
< / div >
< div id = "orga1a9583" class = "figure" >
< p > < img src = "figs/act_damp_variability_iff_spindle_speed_zoom.png" alt = "act_damp_variability_iff_spindle_speed_zoom.png" / >
< / p >
< p > < span class = "figure-number" > Figure 13: < / 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 >
< / div >
< div id = "org06821ac" class = "figure" >
< p > < img src = "figs/act_damp_variability_dvf_spindle_speed.png" alt = "act_damp_variability_dvf_spindle_speed.png" / >
< / p >
< p > < span class = "figure-number" > Figure 14: < / 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 >
< / div >
< div id = "org0c492f1" class = "figure" >
< p > < img src = "figs/act_damp_variability_dvf_spindle_speed_zoom.png" alt = "act_damp_variability_dvf_spindle_speed_zoom.png" / >
< / p >
< p > < span class = "figure-number" > Figure 15: < / 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 >
< / div >
< div id = "org61474a1" class = "figure" >
< p > < img src = "figs/act_damp_variability_ine_spindle_speed.png" alt = "act_damp_variability_ine_spindle_speed.png" / >
< / p >
< p > < span class = "figure-number" > Figure 16: < / 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 >
< / div >
< div id = "orgf54e8be" class = "figure" >
< p > < img src = "figs/act_damp_variability_ine_spindle_speed_zoom.png" alt = "act_damp_variability_ine_spindle_speed_zoom.png" / >
< / p >
< p > < span class = "figure-number" > Figure 17: < / 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 >
< / div >
< / div >
< / div >
< div id = "outline-container-org8aa8182" class = "outline-4" >
< h4 id = "org8aa8182" > < span class = "section-number-4" > 2.3.2< / span > Variation of the poles and zeros with the Spindle rotation frequency< / h4 >
< div class = "outline-text-4" id = "text-2-3-2" >
< div id = "org88e0700" class = "figure" >
< p > < img src = "figs/campbell_diagram_spindle_rotation.png" alt = "campbell_diagram_spindle_rotation.png" / >
< / p >
< p > < span class = "figure-number" > Figure 18: < / span > Evolution of the pole with respect to the spindle rotation speed (< a href = "./figs/campbell_diagram_spindle_rotation.png" > png< / a > , < a href = "./figs/campbell_diagram_spindle_rotation.pdf" > pdf< / a > )< / p >
< / div >
< div id = "orgb70444b" class = "figure" >
< p > < img src = "figs/variation_zeros_active_damping_plants.png" alt = "variation_zeros_active_damping_plants.png" / >
< / p >
< p > < span class = "figure-number" > Figure 19: < / span > Evolution of the zero with respect to the spindle rotation speed (< a href = "./figs/variation_zeros_active_damping_plants.png" > png< / a > , < a href = "./figs/variation_zeros_active_damping_plants.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-org0153a58" class = "outline-3" >
< h3 id = "org0153a58" > < span class = "section-number-3" > 2.4< / span > Variation of the Tilt Angle< / h3 >
< div class = "outline-text-3" id = "text-2-4" >
< p >
< a id = "org3f724e4" > < / a >
< / p >
< p >
We identify the dynamics for the following Tilt stage angles.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Ry_amplitudes = [< span class = "org-type" > -< / span > 3< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > /< / span > 180, 3< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > /< / span > 180]; < span class = "org-comment" > % [rad]< / span >
< / pre >
< / div >
< div id = "org3b6edaa" class = "figure" >
< p > < img src = "figs/act_damp_variability_iff_tilt_angle.png" alt = "act_damp_variability_iff_tilt_angle.png" / >
< / p >
< p > < span class = "figure-number" > Figure 20: < / span > Variability of the IFF plant with the Tilt stage 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 >
< / div >
< div id = "org07994f6" class = "figure" >
< p > < img src = "figs/act_damp_variability_dvf_tilt_angle.png" alt = "act_damp_variability_dvf_tilt_angle.png" / >
< / p >
< p > < span class = "figure-number" > Figure 21: < / span > Variability of the DVF plant with the Tilt 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 >
< / div >
< div id = "org8d15388" class = "figure" >
< p > < img src = "figs/act_damp_variability_ine_tilt_angle.png" alt = "act_damp_variability_ine_tilt_angle.png" / >
< / p >
< p > < span class = "figure-number" > Figure 22: < / span > Variability of the Inertial plant with the Tilt 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 >
< / div >
< / div >
< / div >
< div id = "outline-container-org4e27559" class = "outline-3" >
< h3 id = "org4e27559" > < span class = "section-number-3" > 2.5< / span > Scans of the Translation Stage< / h3 >
< div class = "outline-text-3" id = "text-2-5" >
< p >
< a id = "org6499091" > < / a >
< / p >
< p >
We want here to verify if the dynamics used for Active damping is varying when using the translation stage for scans.
< / p >
< p >
We initialize the translation stage reference to be a sinus with an amplitude of 5mm and a period of 1s (Figure < a href = "#orgcd997d7" > 23< / a > ).
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > initializeReferences(< span class = "org-string" > 'Dy_type'< / span > , < span class = "org-string" > 'sinusoidal'< / span > , ...
< span class = "org-string" > 'Dy_amplitude'< / span > , 5e< span class = "org-type" > -< / span > 3, ...< span class = "org-comment" > % [m]< / span >
< span class = "org-string" > 'Dy_period'< / span > , 1); < span class = "org-comment" > % [s]< / span >
< / pre >
< / div >
< div id = "orgcd997d7" class = "figure" >
< p > < img src = "figs/ty_scanning_reference_sinus.png" alt = "ty_scanning_reference_sinus.png" / >
< / p >
< p > < span class = "figure-number" > Figure 23: < / span > Reference path for the translation stage (< a href = "./figs/ty_scanning_reference_sinus.png" > png< / a > , < a href = "./figs/ty_scanning_reference_sinus.pdf" > pdf< / a > )< / p >
< / div >
< p >
We identify the dynamics at different positions (times) when scanning with the Translation stage.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > t_lin = [0.5, 0.75, 1, 1.25];
< / pre >
< / div >
< div id = "orgd5f0d49" class = "figure" >
< p > < img src = "figs/act_damp_variability_iff_ty_scans.png" alt = "act_damp_variability_iff_ty_scans.png" / >
< / p >
< p > < span class = "figure-number" > Figure 24: < / span > Variability of the IFF plant at different Ty scan positions (< a href = "./figs/act_damp_variability_iff_ty_scans.png" > png< / a > , < a href = "./figs/act_damp_variability_iff_ty_scans.pdf" > pdf< / a > )< / p >
< / div >
< div id = "orgf38c00d" class = "figure" >
< p > < img src = "figs/act_damp_variability_dvf_ty_scans.png" alt = "act_damp_variability_dvf_ty_scans.png" / >
< / p >
< p > < span class = "figure-number" > Figure 25: < / span > Variability of the DVF plant at different Ty scan positions (< a href = "./figs/act_damp_variability_dvf_ty_scans.png" > png< / a > , < a href = "./figs/act_damp_variability_dvf_ty_scans.pdf" > pdf< / a > )< / p >
< / div >
< div id = "orga10665d" class = "figure" >
< p > < img src = "figs/act_damp_variability_ine_ty_scans.png" alt = "act_damp_variability_ine_ty_scans.png" / >
< / p >
< p > < span class = "figure-number" > Figure 26: < / span > Variability of the Inertial plant at different Ty scan positions (< a href = "./figs/act_damp_variability_ine_ty_scans.png" > png< / a > , < a href = "./figs/act_damp_variability_ine_ty_scans.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
< div id = "outline-container-orgb3ee7e4" class = "outline-3" >
< h3 id = "orgb3ee7e4" > < span class = "section-number-3" > 2.6< / span > Conclusion< / h3 >
< div class = "outline-text-3" id = "text-2-6" >
< table id = "orgf5b86c1" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< caption class = "t-above" > < span class = "table-number" > Table 1:< / span > Conclusion on the variability of the system dynamics for active damping< / caption >
< colgroup >
< col class = "org-center" / >
< col class = "org-center" / >
< / colgroup >
< thead >
< tr >
< th scope = "col" class = "org-center" >   < / th >
< th scope = "col" class = "org-center" > < b > Change of Dynamics< / b > < / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-center" > < b > Sample Mass< / b > < / td >
< td class = "org-center" > Large< / td >
< / tr >
< tr >
< td class = "org-center" > < b > Spindle Angle< / b > < / td >
< td class = "org-center" > Small< / td >
< / tr >
< tr >
< td class = "org-center" > < b > Spindle Rotation Speed< / b > < / td >
< td class = "org-center" > First resonance is split in two resonances< / td >
< / tr >
< tr >
< td class = "org-center" > < b > Tilt Angle< / b > < / td >
< td class = "org-center" > Negligible< / td >
< / tr >
< tr >
< td class = "org-center" > < b > Translation Stage scans< / b > < / td >
< td class = "org-center" > Negligible< / td >
< / tr >
< / tbody >
< / table >
< p >
Also, for the Inertial Sensor, a RHP zero may appear when the spindle is rotating fast.
< / p >
< div class = "important" >
< p >
When using either a force sensor or a relative motion sensor for active damping, the only “ parameter” that induce a large change for the dynamics to be controlled is the < b > sample mass< / b > .
Thus, the developed damping techniques should be robust to variations of the sample mass.
< / p >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-orgf08b709" class = "outline-2" >
< h2 id = "orgf08b709" > < span class = "section-number-2" > 3< / span > Integral Force Feedback< / h2 >
< div class = "outline-text-2" id = "text-3" >
< p >
< a id = "org81e3ebf" > < / 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-org87395b8" class = "outline-3" >
< h3 id = "org87395b8" > < span class = "section-number-3" > 3.1< / span > Control Design< / h3 >
< div class = "outline-text-3" id = "text-3-1" >
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< / div >
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< div id = "outline-container-org77e4473" class = "outline-4" >
< h4 id = "org77e4473" > < span class = "section-number-4" > 3.1.1< / span > Plant< / h4 >
< div class = "outline-text-4" id = "text-3-1-1" >
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< 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 = "#org3a42d2a" > 27< / a > ).
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< / p >
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< div id = "org3a42d2a" 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 27: < / 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-org1c94029" class = "outline-4" >
< h4 id = "org1c94029" > < span class = "section-number-4" > 3.1.2< / span > Control Design< / h4 >
< div class = "outline-text-4" id = "text-3-1-2" >
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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 = "#org0308d43" > 28< / a > .
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< / p >
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< div id = "org0308d43" 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 28: < / 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-orga4ebb1c" class = "outline-4" >
< h4 id = "orga4ebb1c" > < span class = "section-number-4" > 3.1.3< / span > Diagonal Controller< / h4 >
< 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_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-orgad90da1" class = "outline-4" >
< h4 id = "orgad90da1" > < span class = "section-number-4" > 3.1.4< / span > IFF with High Pass Filter< / h4 >
< div class = "outline-text-4" id = "text-3-1-4" >
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< 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 = "#orgf71e7d4" > 29< / a > .
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< / p >
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< div id = "orgf71e7d4" class = "figure" >
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< p > < img src = "figs/iff_hpf_open_loop.png" alt = "iff_hpf_open_loop.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 29: < / 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 >
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< / 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-org98b7048" class = "outline-3" >
< h3 id = "org98b7048" > < span class = "section-number-3" > 3.2< / span > Tomography Experiment< / h3 >
< div class = "outline-text-3" id = "text-3-2" >
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< / div >
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< div id = "outline-container-org43bf533" class = "outline-4" >
< h4 id = "org43bf533" > < span class = "section-number-4" > 3.2.1< / span > Simulation with IFF Controller< / h4 >
< 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 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-orge0b6480" class = "outline-4" >
< h4 id = "orge0b6480" > < span class = "section-number-4" > 3.2.2< / span > Simulation with IFF Controller with added High Pass Filter< / h4 >
< div class = "outline-text-4" id = "text-3-2-2" >
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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-org5f65a86" class = "outline-4" >
< h4 id = "org5f65a86" > < span class = "section-number-4" > 3.2.3< / span > Compare with Undamped system< / h4 >
< div class = "outline-text-4" id = "text-3-2-3" >
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< 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 = "org623648a" 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 >
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< p > < span class = "figure-number" > Figure 30: < / 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 >
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< / div >
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< div id = "orgfa2a7a6" 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 31: < / 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 = "org016ac9d" 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 32: < / 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-org22577cc" class = "outline-3" >
< h3 id = "org22577cc" > < span class = "section-number-3" > 3.3< / span > Conclusion< / h3 >
< div class = "outline-text-3" id = "text-3-3" >
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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-org93105e9" class = "outline-2" >
< h2 id = "org93105e9" > < span class = "section-number-2" > 4< / span > Direct Velocity Feedback< / h2 >
< div class = "outline-text-2" id = "text-4" >
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< p >
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< a id = "org5349d23" > < / 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-org0181bc6" class = "outline-3" >
< h3 id = "org0181bc6" > < span class = "section-number-3" > 4.1< / span > Control Design< / h3 >
< div class = "outline-text-3" id = "text-4-1" >
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< / div >
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< div id = "outline-container-org0baaad9" class = "outline-4" >
< h4 id = "org0baaad9" > < span class = "section-number-4" > 4.1.1< / span > Plant< / h4 >
< 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_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 = "#orgb6f6883" > 33< / a > ).
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< / p >
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< div id = "orgb6f6883" 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 33: < / 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-org9d0d598" class = "outline-4" >
< h4 id = "org9d0d598" > < span class = "section-number-4" > 4.1.2< / span > Control Design< / h4 >
< div class = "outline-text-4" id = "text-4-1-2" >
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< 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 = "#org684052f" > 34< / a > .
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< / p >
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< div id = "org684052f" 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 34: < / 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-orgc664cda" class = "outline-4" >
< h4 id = "orgc664cda" > < span class = "section-number-4" > 4.1.3< / span > Diagonal Controller< / h4 >
< 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_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-org30a47bd" class = "outline-3" >
< h3 id = "org30a47bd" > < span class = "section-number-3" > 4.2< / span > Tomography Experiment< / h3 >
< div class = "outline-text-3" id = "text-4-2" >
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< / div >
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< div id = "outline-container-orged4c8ff" class = "outline-4" >
< h4 id = "orged4c8ff" > < span class = "section-number-4" > 4.2.1< / span > Initialize the Simulation< / h4 >
< 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 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-orgafcf6c3" class = "outline-4" >
< h4 id = "orgafcf6c3" > < span class = "section-number-4" > 4.2.2< / span > Simulation< / h4 >
< 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_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-orgdee965e" class = "outline-4" >
< h4 id = "orgdee965e" > < span class = "section-number-4" > 4.2.3< / span > Compare with Undamped system< / h4 >
< 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_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 = "org36b30ec" 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 >
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< p > < span class = "figure-number" > Figure 35: < / 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 >
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< / div >
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< div id = "org4df3961" 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 36: < / 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 = "org259181f" 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 37: < / 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-orgc1e250f" class = "outline-3" >
< h3 id = "orgc1e250f" > < span class = "section-number-3" > 4.3< / span > Conclusion< / h3 >
< div class = "outline-text-3" id = "text-4-3" >
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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-orga585807" class = "outline-2" >
< h2 id = "orga585807" > < span class = "section-number-2" > 5< / span > Inertial Control< / h2 >
< div class = "outline-text-2" id = "text-5" >
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< p >
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< a id = "org6300794" > < / 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-orgef3b74a" class = "outline-3" >
< h3 id = "orgef3b74a" > < span class = "section-number-3" > 5.1< / span > Control Design< / h3 >
< div class = "outline-text-3" id = "text-5-1" >
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< / div >
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< div id = "outline-container-org62bfbc7" class = "outline-4" >
< h4 id = "org62bfbc7" > < span class = "section-number-4" > 5.1.1< / span > Plant< / h4 >
< div class = "outline-text-4" id = "text-5-1-1" >
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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 = "#org4952d1e" > 38< / a > ).
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< / p >
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< div id = "org4952d1e" 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 38: < / 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-orgeeec2e5" class = "outline-4" >
< h4 id = "orgeeec2e5" > < span class = "section-number-4" > 5.1.2< / span > Control Design< / h4 >
< div class = "outline-text-4" id = "text-5-1-2" >
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< p >
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The controller is defined below and the obtained loop gain is shown in figure < a href = "#org7db3cd8" > 39< / 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 = "org7db3cd8" 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 39: < / 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-org240b7fd" class = "outline-4" >
< h4 id = "org240b7fd" > < span class = "section-number-4" > 5.1.3< / span > Diagonal Controller< / h4 >
< div class = "outline-text-4" id = "text-5-1-3" >
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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-orgea439d2" class = "outline-3" >
< h3 id = "orgea439d2" > < span class = "section-number-3" > 5.2< / span > Tomography Experiment< / h3 >
< div class = "outline-text-3" id = "text-5-2" >
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< / div >
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< div id = "outline-container-orgd52597f" class = "outline-4" >
< h4 id = "orgd52597f" > < span class = "section-number-4" > 5.2.1< / span > Initialize the Simulation< / h4 >
< div class = "outline-text-4" id = "text-5-2-1" >
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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-org2ad0ade" class = "outline-4" >
< h4 id = "org2ad0ade" > < span class = "section-number-4" > 5.2.2< / span > Simulation< / h4 >
< div class = "outline-text-4" id = "text-5-2-2" >
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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-orgfd09099" class = "outline-4" >
< h4 id = "orgfd09099" > < span class = "section-number-4" > 5.2.3< / span > Compare with Undamped system< / h4 >
< div class = "outline-text-4" id = "text-5-2-3" >
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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 = "org1f18df3" 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 >
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< p > < span class = "figure-number" > Figure 40: < / 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 >
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< / div >
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< div id = "org16410dd" 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 41: < / 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 = "orgb8ccc00" 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 42: < / 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-org5a63825" class = "outline-3" >
< h3 id = "org5a63825" > < span class = "section-number-3" > 5.3< / span > Conclusion< / h3 >
< div class = "outline-text-3" id = "text-5-3" >
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< div class = "important" >
< p >
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Inertial Control:
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< / p >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org230c9f4" class = "outline-2" >
< h2 id = "org230c9f4" > < span class = "section-number-2" > 6< / span > Comparison< / h2 >
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< div class = "outline-text-2" id = "text-6" >
< p >
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< a id = "org36034dc" > < / a >
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< / p >
< / div >
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< div id = "outline-container-org4baffd6" class = "outline-3" >
< h3 id = "org4baffd6" > < span class = "section-number-3" > 6.1< / span > Load the plants< / h3 >
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< div class = "outline-text-3" id = "text-6-1" >
< div class = "org-src-container" >
< 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 > );
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orga058c69" class = "outline-3" >
< h3 id = "orga058c69" > < span class = "section-number-3" > 6.2< / span > Sensitivity to Disturbance< / h3 >
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< div class = "outline-text-3" id = "text-6-2" >
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< div id = "org287ecb6" class = "figure" >
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< p > < img src = "figs/sensitivity_comp_ground_motion_z.png" alt = "sensitivity_comp_ground_motion_z.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 43: < / 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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< / div >
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< div id = "org43ef4d9" class = "figure" >
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< p > < img src = "figs/sensitivity_comp_direct_forces_z.png" alt = "sensitivity_comp_direct_forces_z.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 44: < / 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 >
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< div id = "org694aa4d" class = "figure" >
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< p > < img src = "figs/sensitivity_comp_spindle_z.png" alt = "sensitivity_comp_spindle_z.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 45: < / 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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< div id = "orgefc9b61" class = "figure" >
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< p > < img src = "figs/sensitivity_comp_ty_z.png" alt = "sensitivity_comp_ty_z.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 46: < / 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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< / div >
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< div id = "org09a62ce" class = "figure" >
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< p > < img src = "figs/sensitivity_comp_ty_x.png" alt = "sensitivity_comp_ty_x.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 47: < / 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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< / div >
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< div id = "outline-container-org7d53c0f" class = "outline-3" >
< h3 id = "org7d53c0f" > < span class = "section-number-3" > 6.3< / span > Damped Plant< / h3 >
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< div class = "outline-text-3" id = "text-6-3" >
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< div id = "orgea4578a" class = "figure" >
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< p > < img src = "figs/plant_comp_damping_z.png" alt = "plant_comp_damping_z.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 48: < / 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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< / div >
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< div id = "orgcf7fd3d" class = "figure" >
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< p > < img src = "figs/plant_comp_damping_x.png" alt = "plant_comp_damping_x.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 49: < / 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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< div id = "org0ee741c" class = "figure" >
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< p > < img src = "figs/plant_comp_damping_coupling.png" alt = "plant_comp_damping_coupling.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 50: < / 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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< div id = "outline-container-org70d41b8" class = "outline-3" >
< h3 id = "org70d41b8" > < span class = "section-number-3" > 6.4< / span > Tomography Experiment< / h3 >
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< div class = "outline-text-3" id = "text-6-4" >
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< / div >
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< div id = "outline-container-org604ad48" class = "outline-4" >
< h4 id = "org604ad48" > < span class = "section-number-4" > 6.4.1< / span > Load the Simulation Data< / h4 >
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< div class = "outline-text-4" id = "text-6-4-1" >
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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/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)));
< / pre >
< / div >
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< / div >
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< div id = "outline-container-orgc2877fd" class = "outline-4" >
< h4 id = "orgc2877fd" > < span class = "section-number-4" > 6.4.2< / span > Frequency Domain Analysis< / h4 >
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< div class = "outline-text-4" id = "text-6-4-2" >
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< p >
Window used for < code > pwelch< / code > function.
< / p >
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< div class = "org-src-container" >
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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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< / pre >
< / div >
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< div id = "org50bc9af" class = "figure" >
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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" / >
< / p >
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< p > < span class = "figure-number" > Figure 51: < / 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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< div id = "org333cdfb" class = "figure" >
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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" / >
< / p >
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< p > < span class = "figure-number" > Figure 52: < / 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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< / div >
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< div id = "org29382b7" class = "figure" >
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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" / >
< / p >
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< p > < span class = "figure-number" > Figure 53: < / 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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< div id = "orgcc5d27e" class = "figure" >
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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" / >
< / p >
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< p > < span class = "figure-number" > Figure 54: < / 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-orgbe23ba6" class = "outline-2" >
< h2 id = "orgbe23ba6" > < span class = "section-number-2" > 7< / span > Useful Functions< / h2 >
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< div class = "outline-text-2" id = "text-7" >
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< / div >
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< div id = "outline-container-orge26884f" class = "outline-3" >
< h3 id = "orge26884f" > < span class = "section-number-3" > 7.1< / span > prepareTomographyExperiment< / h3 >
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< div class = "outline-text-3" id = "text-7-1" >
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< p >
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< a id = "org2453c47" > < / a >
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< / p >
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< p >
This Matlab function is accessible < a href = "src/prepareTomographyExperiment.m" > here< / a > .
< / p >
< / div >
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< div id = "outline-container-orgbd71c69" class = "outline-4" >
< h4 id = "orgbd71c69" > Function Description< / h4 >
< div class = "outline-text-4" id = "text-orgbd71c69" >
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< div class = "org-src-container" >
< 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 > )
< / pre >
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< div id = "outline-container-org79eafae" class = "outline-4" >
< h4 id = "org79eafae" > Optional Parameters< / h4 >
< div class = "outline-text-4" id = "text-org79eafae" >
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< div class = "org-src-container" >
< 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 >
< / pre >
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< / div >
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< div id = "outline-container-orge7b78fc" class = "outline-4" >
< h4 id = "orge7b78fc" > Initialize the Simulation< / h4 >
< div class = "outline-text-4" id = "text-orge7b78fc" >
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< p >
We initialize all the stages with the default parameters.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
< / pre >
< / div >
< p >
The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > initializeNanoHexapod(< span class = "org-string" > 'actuator'< / span > , args.nass_actuator);
initializeSample(< span class = "org-string" > 'mass'< / span > , args.sample_mass);
< / pre >
< / div >
< p >
We set the references to zero.
< / p >
< div class = "org-src-container" >
< 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);
< / pre >
< / div >
< p >
And all the controllers are set to 0.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > K = tf(zeros(6));
save(< span class = "org-string" > './mat/controllers.mat'< / span > , < span class = "org-string" > 'K'< / span > , < span class = "org-string" > '-append'< / span > );
K_ine = tf(zeros(6));
save(< span class = "org-string" > './mat/controllers.mat'< / span > , < span class = "org-string" > 'K_ine'< / span > , < span class = "org-string" > '-append'< / span > );
K_iff = tf(zeros(6));
save(< span class = "org-string" > './mat/controllers.mat'< / span > , < span class = "org-string" > 'K_iff'< / span > , < span class = "org-string" > '-append'< / span > );
K_dvf = tf(zeros(6));
save(< span class = "org-string" > './mat/controllers.mat'< / span > , < span class = "org-string" > 'K_dvf'< / span > , < span class = "org-string" > '-append'< / span > );
< / pre >
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< div id = "postamble" class = "status" >
< p class = "author" > Author: Dehaeze Thomas< / p >
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< p class = "date" > Created: 2020-02-04 mar. 16:13< / p >
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