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< title > Active Damping applied on the Simscape Model< / title >
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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 = "#org21b7020" > 1. Undamped System< / a >
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< ul >
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< li > < a href = "#org5851b7a" > 1.1. Identification of the dynamics for Active Damping< / a >
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< ul >
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< li > < a href = "#org74aed57" > 1.1.1. Initialize the Simulation< / a > < / li >
< li > < a href = "#org32dc6b8" > 1.1.2. Identification< / a > < / li >
< li > < a href = "#org467a586" > 1.1.3. Obtained Plants for Active Damping< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgcd8379a" > 1.2. Tomography Experiment< / a >
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< ul >
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< li > < a href = "#org3284883" > 1.2.1. Simulation< / a > < / li >
< li > < a href = "#org700a2b4" > 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 = "#org4f2a70b" > 2. Integral Force Feedback< / a >
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< ul >
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< li > < a href = "#orgf33234d" > 2.1. Control Design< / a >
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< ul >
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< li > < a href = "#orgd9afa23" > 2.1.1. Plant< / a > < / li >
< li > < a href = "#org43379d2" > 2.1.2. Control Design< / a > < / li >
< li > < a href = "#orgbc20ed4" > 2.1.3. Diagonal Controller< / a > < / li >
< li > < a href = "#org90db654" > 2.1.4. IFF with High Pass Filter< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgaaa96ff" > 2.2. Tomography Experiment< / a >
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< ul >
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< li > < a href = "#orgf74a390" > 2.2.1. Simulation with IFF Controller< / a > < / li >
< li > < a href = "#org95de904" > 2.2.2. Simulation with IFF Controller with added High Pass Filter< / a > < / li >
< li > < a href = "#org2b12e17" > 2.2.3. Compare with Undamped system< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org0fd785e" > 2.3. Conclusion< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org9c38d50" > 3. Direct Velocity Feedback< / a >
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< ul >
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< li > < a href = "#org03f9812" > 3.1. Control Design< / a >
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< ul >
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< li > < a href = "#org0426ac1" > 3.1.1. Plant< / a > < / li >
< li > < a href = "#orgbb062d7" > 3.1.2. Control Design< / a > < / li >
< li > < a href = "#org1bdfb4e" > 3.1.3. Diagonal Controller< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org1787430" > 3.2. Tomography Experiment< / a >
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< ul >
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< li > < a href = "#org8c867a4" > 3.2.1. Initialize the Simulation< / a > < / li >
< li > < a href = "#org8ca9d2f" > 3.2.2. Simulation< / a > < / li >
< li > < a href = "#org85c9eb6" > 3.2.3. Compare with Undamped system< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org7ea48ad" > 3.3. Conclusion< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgddee501" > 4. Inertial Control< / a >
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< ul >
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< li > < a href = "#org183d718" > 4.1. Control Design< / a >
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< ul >
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< li > < a href = "#org3b0c93b" > 4.1.1. Plant< / a > < / li >
< li > < a href = "#org0613819" > 4.1.2. Control Design< / a > < / li >
< li > < a href = "#org7eb6d4a" > 4.1.3. Diagonal Controller< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgb837286" > 4.2. Tomography Experiment< / a >
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< ul >
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< li > < a href = "#org088d0cf" > 4.2.1. Initialize the Simulation< / a > < / li >
< li > < a href = "#orge0acdea" > 4.2.2. Simulation< / a > < / li >
< li > < a href = "#orgb9346a9" > 4.2.3. Compare with Undamped system< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org971b7df" > 4.3. Conclusion< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgb30a5ab" > 5. Comparison< / a >
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< ul >
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< li > < a href = "#org056c3e4" > 5.1. Load the plants< / a > < / li >
< li > < a href = "#org75158e7" > 5.2. Sensitivity to Disturbance< / a > < / li >
< li > < a href = "#orgca4880d" > 5.3. Damped Plant< / a > < / li >
< li > < a href = "#org4b14fbd" > 5.4. Tomography Experiment< / a >
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< ul >
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< li > < a href = "#org2213b1c" > 5.4.1. Frequency Domain< / a > < / li >
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< / ul >
< / li >
< / ul >
< / li >
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< li > < a href = "#orge02060f" > 6. Useful Functions< / a >
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< ul >
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< li > < a href = "#org0b9e75d" > 6.1. prepareTomographyExperiment< / a >
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< ul >
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< li > < a href = "#org637f0d7" > Function Description< / a > < / li >
< li > < a href = "#orgac8374d" > Optional Parameters< / a > < / li >
< li > < a href = "#orgb19cd54" > 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 = "#orgcb992ae" > 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 = "#org0e807ab" > 2< / a > : the integral force feedback is used< / li >
< li > In section < a href = "#orgef7f6c4" > 3< / a > : the direct velocity feedback is used< / li >
< li > In section < a href = "#org1930728" > 4< / a > : inertial control is used< / li >
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< / ul >
< p >
For each of the active damping technique, we will:
< / p >
< ul class = "org-ul" >
< li > Look at the damped plant< / li >
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< li > Simulate tomography experiments< / li >
< li > Compare the sensitivity from disturbances< / li >
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< / ul >
< p >
The disturbances are:
< / p >
< ul class = "org-ul" >
< li > Ground motion< / li >
< li > Motion errors of all the stages< / li >
< / ul >
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< div id = "outline-container-org21b7020" class = "outline-2" >
< h2 id = "org21b7020" > < 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 = "orgcb992ae" > < / a >
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< / p >
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< div class = "note" >
< p >
All the files (data and Matlab scripts) are accessible < a href = "data/undamped_system.zip" > here< / a > .
< / p >
< / div >
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< p >
We first look at the undamped system.
The performance of this undamped system will be compared with the damped system using various techniques.
< / p >
< / div >
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< div id = "outline-container-org5851b7a" class = "outline-3" >
< h3 id = "org5851b7a" > < 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-org74aed57" class = "outline-4" >
< h4 id = "org74aed57" > < 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();
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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-org32dc6b8" class = "outline-4" >
< h4 id = "org32dc6b8" > < 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 >
G = linearize(mdl, io, options);
G.InputName = {< span class = "org-string" > 'Fnl1'< / span > , < span class = "org-string" > 'Fnl2'< / span > , < span class = "org-string" > 'Fnl3'< / span > , < span class = "org-string" > 'Fnl4'< / span > , < span class = "org-string" > 'Fnl5'< / span > , < span class = "org-string" > 'Fnl6'< / span > };
G.OutputName = {< span class = "org-string" > '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-org467a586" class = "outline-4" >
< h4 id = "org467a586" > < 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 id = "org9992a9d" 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 = "org5adefaa" 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 = "org071d839" 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 > Inertial Feedback Plant (< a href = "./figs/nass_active_damping_inertial_plant.png" > png< / a > , < a href = "./figs/nass_active_damping_inertial_plant.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgcd8379a" class = "outline-3" >
< h3 id = "orgcd8379a" > < 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-org3284883" class = "outline-4" >
< h4 id = "org3284883" > < 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-org700a2b4" class = "outline-4" >
< h4 id = "org700a2b4" > < 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 = "org9eef789" 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 = "org5e9adc2" 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-org4f2a70b" class = "outline-2" >
< h2 id = "org4f2a70b" > < span class = "section-number-2" > 2< / span > Integral Force Feedback< / h2 >
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< div class = "outline-text-2" id = "text-2" >
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< p >
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< a id = "org0e807ab" > < / 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-orgf33234d" class = "outline-3" >
< h3 id = "orgf33234d" > < span class = "section-number-3" > 2.1< / span > Control Design< / h3 >
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< div class = "outline-text-3" id = "text-2-1" >
< / div >
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< div id = "outline-container-orgd9afa23" class = "outline-4" >
< h4 id = "orgd9afa23" > < span class = "section-number-4" > 2.1.1< / span > Plant< / h4 >
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< div class = "outline-text-4" id = "text-2-1-1" >
< p >
Let’ s load the previously indentified undamped plant:
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > load(< span class = "org-string" > './active_damping/mat/undamped_plants.mat'< / span > , < span class = "org-string" > 'G_iff'< / span > );
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< / pre >
< / div >
< p >
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Let’ s look at the transfer function from actuator forces in the nano-hexapod to the force sensor in the nano-hexapod legs for all 6 pairs of actuator/sensor (figure < a href = "#org2c9189d" > 6< / a > ).
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< / p >
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< div id = "org2c9189d" class = "figure" >
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< p > < img src = "figs/iff_plant.png" alt = "iff_plant.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 6: < / span > Transfer function from forces applied in the legs to force sensor (< a href = "./figs/iff_plant.png" > png< / a > , < a href = "./figs/iff_plant.pdf" > pdf< / a > )< / p >
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< / div >
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< / div >
< / div >
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< div id = "outline-container-org43379d2" class = "outline-4" >
< h4 id = "org43379d2" > < span class = "section-number-4" > 2.1.2< / span > Control Design< / h4 >
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< div class = "outline-text-4" id = "text-2-1-2" >
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< p >
The controller for each pair of actuator/sensor is:
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > K_iff = 1000< span class = "org-type" > /< / span > s;
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< / pre >
< / div >
< p >
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The corresponding loop gains are shown in figure < a href = "#org85fcfd6" > 7< / a > .
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< / p >
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< div id = "org85fcfd6" class = "figure" >
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< p > < img src = "figs/iff_open_loop.png" alt = "iff_open_loop.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 7: < / span > Loop Gain for the Integral Force Feedback (< a href = "./figs/iff_open_loop.png" > png< / a > , < a href = "./figs/iff_open_loop.pdf" > pdf< / a > )< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-orgbc20ed4" class = "outline-4" >
< h4 id = "orgbc20ed4" > < span class = "section-number-4" > 2.1.3< / span > Diagonal Controller< / h4 >
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< div class = "outline-text-4" id = "text-2-1-3" >
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< p >
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We create the diagonal controller and we add a minus sign as we have a positive
feedback architecture.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > K_iff = < span class = "org-type" > -< / span > K_iff< span class = "org-type" > *< / span > eye(6);
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< / pre >
< / div >
< p >
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We save the controller for further analysis.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > save(< span class = "org-string" > './active_damping/mat/K_iff.mat'< / span > , < span class = "org-string" > 'K_iff'< / span > );
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< / pre >
< / div >
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< / div >
< / div >
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< div id = "outline-container-org90db654" class = "outline-4" >
< h4 id = "org90db654" > < span class = "section-number-4" > 2.1.4< / span > IFF with High Pass Filter< / h4 >
< div class = "outline-text-4" id = "text-2-1-4" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > w_hpf = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 10; < span class = "org-comment" > % Cut-off frequency for the high pass filter [rad/s]< / span >
w_lpf = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 200; < span class = "org-comment" > % Cut-off frequency for the low pass filter [rad/s]< / span >
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) < span class = "org-type" > *< / span > 1< span class = "org-type" > /< / span > (s< span class = "org-type" > /< / span > w_lpf < span class = "org-type" > +< / span > 1);
< / pre >
< / div >
< p >
The corresponding loop gains are shown in figure < a href = "#org7f0bc2b" > 8< / a > .
< / p >
< div id = "org7f0bc2b" class = "figure" >
< p > < img src = "figs/iff_hpf_open_loop.png" alt = "iff_hpf_open_loop.png" / >
< / p >
< p > < span class = "figure-number" > Figure 8: < / span > Loop Gain for the Integral Force Feedback with an High pass filter (< a href = "./figs/iff_hpf_open_loop.png" > png< / a > , < a href = "./figs/iff_hpf_open_loop.pdf" > pdf< / a > )< / p >
< / div >
< p >
We create the diagonal controller and we add a minus sign as we have a positive
feedback architecture.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > K_iff = < span class = "org-type" > -< / span > K_iff< span class = "org-type" > *< / span > eye(6);
< / pre >
< / div >
< p >
We save the controller for further analysis.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > save(< span class = "org-string" > './active_damping/mat/K_iff_hpf.mat'< / span > , < span class = "org-string" > 'K_iff'< / span > );
< / pre >
< / div >
< / div >
< / div >
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< / div >
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< div id = "outline-container-orgaaa96ff" class = "outline-3" >
< h3 id = "orgaaa96ff" > < span class = "section-number-3" > 2.2< / span > Tomography Experiment< / h3 >
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< div class = "outline-text-3" id = "text-2-2" >
< / div >
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< div id = "outline-container-orgf74a390" class = "outline-4" >
< h4 id = "orgf74a390" > < span class = "section-number-4" > 2.2.1< / span > Simulation with IFF Controller< / h4 >
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< div class = "outline-text-4" id = "text-2-2-1" >
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< p >
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We initialize elements for the tomography experiment.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > prepareTomographyExperiment();
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< / pre >
< / div >
< p >
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We set the IFF controller.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > load(< span class = "org-string" > './active_damping/mat/K_iff.mat'< / span > , < span class = "org-string" > 'K_iff'< / span > );
save(< span class = "org-string" > './mat/controllers.mat'< / span > , < span class = "org-string" > 'K_iff'< / span > , < span class = "org-string" > '-append'< / span > );
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< / pre >
< / div >
< p >
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We change the simulation stop time.
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< / p >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > load(< span class = "org-string" > 'mat/conf_simscape.mat'< / span > );
< span class = "org-matlab-simulink-keyword" > set_param< / span > (< span class = "org-variable-name" > conf_simscape< / span > , < span class = "org-string" > 'StopTime'< / span > , < span class = "org-string" > '3'< / span > );
< / pre >
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< / div >
< p >
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And we simulate the system.
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< / p >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-simulink-keyword" > sim< / span > (< span class = "org-string" > 'sim_nass_active_damping'< / span > );
< / pre >
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< / div >
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< p >
Finally, we save the simulation results for further analysis
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< / p >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > En_iff = En;
Eg_iff = Eg;
save(< span class = "org-string" > './active_damping/mat/tomo_exp.mat'< / span > , < span class = "org-string" > 'En_iff'< / span > , < span class = "org-string" > 'Eg_iff'< / span > , < span class = "org-string" > '-append'< / span > );
< / pre >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org95de904" class = "outline-4" >
< h4 id = "org95de904" > < span class = "section-number-4" > 2.2.2< / span > Simulation with IFF Controller with added High Pass Filter< / h4 >
< div class = "outline-text-4" id = "text-2-2-2" >
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< p >
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We initialize elements for the tomography experiment.
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< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > prepareTomographyExperiment();
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< / pre >
< / div >
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< p >
We set the IFF controller with the High Pass Filter.
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > load(< span class = "org-string" > './active_damping/mat/K_iff_hpf.mat'< / span > , < span class = "org-string" > 'K_iff'< / span > );
save(< span class = "org-string" > './mat/controllers.mat'< / span > , < span class = "org-string" > 'K_iff'< / span > , < span class = "org-string" > '-append'< / span > );
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< / pre >
< / div >
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< p >
We change the simulation stop time.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > load(< span class = "org-string" > 'mat/conf_simscape.mat'< / span > );
< span class = "org-matlab-simulink-keyword" > set_param< / span > (< span class = "org-variable-name" > conf_simscape< / span > , < span class = "org-string" > 'StopTime'< / span > , < span class = "org-string" > '3'< / span > );
< / pre >
< / div >
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< p >
And we simulate the system.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-simulink-keyword" > sim< / span > (< span class = "org-string" > 'sim_nass_active_damping'< / span > );
< / pre >
< / div >
< p >
Finally, we save the simulation results for further analysis
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > En_iff_hpf = En;
Eg_iff_hpf = Eg;
save(< span class = "org-string" > './active_damping/mat/tomo_exp.mat'< / span > , < span class = "org-string" > 'En_iff_hpf'< / span > , < span class = "org-string" > 'Eg_iff_hpf'< / span > , < span class = "org-string" > '-append'< / span > );
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org2b12e17" class = "outline-4" >
< h4 id = "org2b12e17" > < span class = "section-number-4" > 2.2.3< / span > Compare with Undamped system< / h4 >
< div class = "outline-text-4" id = "text-2-2-3" >
< p >
We load the results of tomography experiments.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > load(< span class = "org-string" > './active_damping/mat/tomo_exp.mat'< / span > , < span class = "org-string" > 'En'< / span > , < span class = "org-string" > 'En_iff'< / span > , < span class = "org-string" > 'En_iff_hpf'< / span > );
t = linspace(0, 3, length(En(< span class = "org-type" > :< / span > ,1)));
< / pre >
< / div >
< div id = "orgb5b8a36" class = "figure" >
< p > < img src = "figs/nass_act_damp_iff_sim_tomo_xy.png" alt = "nass_act_damp_iff_sim_tomo_xy.png" / >
< / p >
< p > < span class = "figure-number" > Figure 9: < / span > Position Error during tomography experiment - XY Motion (< a href = "./figs/nass_act_damp_iff_sim_tomo_xy.png" > png< / a > , < a href = "./figs/nass_act_damp_iff_sim_tomo_xy.pdf" > pdf< / a > )< / p >
< / div >
< div id = "org013fbdc" class = "figure" >
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< p > < img src = "figs/nass_act_damp_iff_sim_tomo_trans.png" alt = "nass_act_damp_iff_sim_tomo_trans.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 10: < / span > Position Error during tomography experiment - Translations (< a href = "./figs/nass_act_damp_iff_sim_tomo_trans.png" > png< / a > , < a href = "./figs/nass_act_damp_iff_sim_tomo_trans.pdf" > pdf< / a > )< / p >
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< / div >
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< div id = "org9a53f55" class = "figure" >
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< p > < img src = "figs/nass_act_damp_iff_sim_tomo_rot.png" alt = "nass_act_damp_iff_sim_tomo_rot.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 11: < / span > Position Error during tomography experiment - Rotations (< a href = "./figs/nass_act_damp_iff_sim_tomo_rot.png" > png< / a > , < a href = "./figs/nass_act_damp_iff_sim_tomo_rot.pdf" > pdf< / a > )< / p >
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< / div >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org0fd785e" class = "outline-3" >
< h3 id = "org0fd785e" > < span class = "section-number-3" > 2.3< / span > Conclusion< / h3 >
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< div class = "outline-text-3" id = "text-2-3" >
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< div class = "important" >
< p >
Integral Force Feedback:
< / p >
< ul class = "org-ul" >
< li > Robust (guaranteed stability)< / li >
< li > Acceptable Damping< / li >
< li > Increase the sensitivity to disturbances at low frequencies< / li >
< / ul >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org9c38d50" class = "outline-2" >
< h2 id = "org9c38d50" > < span class = "section-number-2" > 3< / span > Direct Velocity Feedback< / h2 >
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< div class = "outline-text-2" id = "text-3" >
< p >
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< a id = "orgef7f6c4" > < / 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-org03f9812" class = "outline-3" >
< h3 id = "org03f9812" > < span class = "section-number-3" > 3.1< / span > Control Design< / h3 >
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< div class = "outline-text-3" id = "text-3-1" >
< / div >
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< div id = "outline-container-org0426ac1" class = "outline-4" >
< h4 id = "org0426ac1" > < span class = "section-number-4" > 3.1.1< / span > Plant< / h4 >
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< div class = "outline-text-4" id = "text-3-1-1" >
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< p >
Let’ s load the undamped plant:
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< / p >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > load(< span class = "org-string" > './active_damping/mat/undamped_plants.mat'< / span > , < span class = "org-string" > 'G_dvf'< / span > );
< / pre >
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< / div >
< p >
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Let’ s look at the transfer function from actuator forces in the nano-hexapod to the measured displacement of the actuator for all 6 pairs of actuator/sensor (figure < a href = "#orgce4a669" > 12< / a > ).
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< / p >
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< div id = "orgce4a669" class = "figure" >
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< p > < img src = "figs/dvf_plant.png" alt = "dvf_plant.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 12: < / span > Transfer function from forces applied in the legs to leg displacement sensor (< a href = "./figs/dvf_plant.png" > png< / a > , < a href = "./figs/dvf_plant.pdf" > pdf< / a > )< / p >
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< / div >
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< / div >
< / div >
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< div id = "outline-container-orgbb062d7" class = "outline-4" >
< h4 id = "orgbb062d7" > < span class = "section-number-4" > 3.1.2< / span > Control Design< / h4 >
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< div class = "outline-text-4" id = "text-3-1-2" >
< p >
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The Direct Velocity Feedback is defined below.
A Low pass Filter is added to make the controller transfer function proper.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > K_dvf = s< span class = "org-type" > *< / span > 20000< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > 2< span class = "org-type" > /< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > /< / span > 10000);
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< / pre >
< / div >
< p >
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The obtained loop gains are shown in figure < a href = "#org5e3387d" > 13< / a > .
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< / p >
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< div id = "org5e3387d" class = "figure" >
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< p > < img src = "figs/dvf_open_loop.png" alt = "dvf_open_loop.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 13: < / span > Loop Gain for the Integral Force Feedback (< a href = "./figs/dvf_open_loop.png" > png< / a > , < a href = "./figs/dvf_open_loop.pdf" > pdf< / a > )< / p >
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< / div >
< / div >
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< / div >
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< div id = "outline-container-org1bdfb4e" class = "outline-4" >
< h4 id = "org1bdfb4e" > < span class = "section-number-4" > 3.1.3< / span > Diagonal Controller< / h4 >
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< div class = "outline-text-4" id = "text-3-1-3" >
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< p >
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We create the diagonal controller and we add a minus sign as we have a positive feedback architecture.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > K_dvf = < span class = "org-type" > -< / span > K_dvf< span class = "org-type" > *< / span > eye(6);
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< / pre >
< / div >
< p >
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We save the controller for further analysis.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > save(< span class = "org-string" > './active_damping/mat/K_dvf.mat'< / span > , < span class = "org-string" > 'K_dvf'< / span > );
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< / pre >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org1787430" class = "outline-3" >
< h3 id = "org1787430" > < span class = "section-number-3" > 3.2< / span > Tomography Experiment< / h3 >
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< div class = "outline-text-3" id = "text-3-2" >
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< / div >
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< div id = "outline-container-org8c867a4" class = "outline-4" >
< h4 id = "org8c867a4" > < span class = "section-number-4" > 3.2.1< / span > Initialize the Simulation< / h4 >
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< div class = "outline-text-4" id = "text-3-2-1" >
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< p >
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We initialize elements for the tomography experiment.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > prepareTomographyExperiment();
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< / pre >
< / div >
< p >
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We set the DVF controller.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > load(< span class = "org-string" > './active_damping/mat/K_dvf.mat'< / span > , < span class = "org-string" > 'K_dvf'< / span > );
save(< span class = "org-string" > './mat/controllers.mat'< / span > , < span class = "org-string" > 'K_dvf'< / span > , < span class = "org-string" > '-append'< / span > );
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org8ca9d2f" class = "outline-4" >
< h4 id = "org8ca9d2f" > < span class = "section-number-4" > 3.2.2< / span > Simulation< / h4 >
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< div class = "outline-text-4" id = "text-3-2-2" >
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< p >
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We change the simulation stop time.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > load(< span class = "org-string" > 'mat/conf_simscape.mat'< / span > );
< span class = "org-matlab-simulink-keyword" > set_param< / span > (< span class = "org-variable-name" > conf_simscape< / span > , < span class = "org-string" > 'StopTime'< / span > , < span class = "org-string" > '3'< / span > );
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< / pre >
< / div >
< p >
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And we simulate the system.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-matlab-simulink-keyword" > sim< / span > (< span class = "org-string" > 'sim_nass_active_damping'< / span > );
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< / pre >
< / div >
< p >
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Finally, we save the simulation results for further analysis
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > En_dvf = En;
Eg_dvf = Eg;
save(< span class = "org-string" > './active_damping/mat/tomo_exp.mat'< / span > , < span class = "org-string" > 'En_dvf'< / span > , < span class = "org-string" > 'Eg_dvf'< / span > , < span class = "org-string" > '-append'< / span > );
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< / pre >
< / div >
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< / div >
< / div >
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< div id = "outline-container-org85c9eb6" class = "outline-4" >
< h4 id = "org85c9eb6" > < span class = "section-number-4" > 3.2.3< / span > Compare with Undamped system< / h4 >
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< div class = "outline-text-4" id = "text-3-2-3" >
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< p >
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We load the results of tomography experiments.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > load(< span class = "org-string" > './active_damping/mat/tomo_exp.mat'< / span > , < span class = "org-string" > 'En'< / span > , < span class = "org-string" > 'En_dvf'< / span > );
t = linspace(0, 3, length(En(< span class = "org-type" > :< / span > ,1)));
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< / pre >
< / div >
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< div id = "org8d6a1b3" class = "figure" >
< p > < img src = "figs/nass_act_damp_dvf_sim_tomo_xy.png" alt = "nass_act_damp_dvf_sim_tomo_xy.png" / >
< / p >
< p > < span class = "figure-number" > Figure 14: < / span > Position Error during tomography experiment - XY Motion (< a href = "./figs/nass_act_damp_dvf_sim_tomo_xy.png" > png< / a > , < a href = "./figs/nass_act_damp_dvf_sim_tomo_xy.pdf" > pdf< / a > )< / p >
< / div >
< div id = "org1faad36" class = "figure" >
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< p > < img src = "figs/nass_act_damp_dvf_sim_tomo_trans.png" alt = "nass_act_damp_dvf_sim_tomo_trans.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 15: < / span > Position Error during tomography experiment - Translations (< a href = "./figs/nass_act_damp_dvf_sim_tomo_trans.png" > png< / a > , < a href = "./figs/nass_act_damp_dvf_sim_tomo_trans.pdf" > pdf< / a > )< / p >
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< / div >
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< div id = "org1d30432" class = "figure" >
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< p > < img src = "figs/nass_act_damp_dvf_sim_tomo_rot.png" alt = "nass_act_damp_dvf_sim_tomo_rot.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 16: < / span > Position Error during tomography experiment - Rotations (< a href = "./figs/nass_act_damp_dvf_sim_tomo_rot.png" > png< / a > , < a href = "./figs/nass_act_damp_dvf_sim_tomo_rot.pdf" > pdf< / a > )< / p >
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< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org7ea48ad" class = "outline-3" >
< h3 id = "org7ea48ad" > < span class = "section-number-3" > 3.3< / span > Conclusion< / h3 >
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< div class = "outline-text-3" id = "text-3-3" >
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< div class = "important" >
< p >
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Direct Velocity Feedback:
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< / p >
< ul class = "org-ul" >
< li > < / li >
< / ul >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgddee501" class = "outline-2" >
< h2 id = "orgddee501" > < span class = "section-number-2" > 4< / span > Inertial Control< / h2 >
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< div class = "outline-text-2" id = "text-4" >
< p >
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< a id = "org1930728" > < / 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-org183d718" class = "outline-3" >
< h3 id = "org183d718" > < span class = "section-number-3" > 4.1< / span > Control Design< / h3 >
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< div class = "outline-text-3" id = "text-4-1" >
< / div >
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< div id = "outline-container-org3b0c93b" class = "outline-4" >
< h4 id = "org3b0c93b" > < span class = "section-number-4" > 4.1.1< / span > Plant< / h4 >
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< div class = "outline-text-4" id = "text-4-1-1" >
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< p >
Let’ s load the undamped plant:
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< / p >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > load(< span class = "org-string" > './active_damping/mat/undamped_plants.mat'< / span > , < span class = "org-string" > 'G_ine'< / span > );
< / pre >
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< / div >
< p >
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Let’ s look at the transfer function from actuator forces in the nano-hexapod to the measured velocity of the nano-hexapod platform in the direction of the corresponding actuator for all 6 pairs of actuator/sensor (figure < a href = "#orgd2495ce" > 17< / a > ).
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< / p >
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< div id = "orgd2495ce" class = "figure" >
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< p > < img src = "figs/ine_plant.png" alt = "ine_plant.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 17: < / span > Transfer function from forces applied in the legs to leg velocity sensor (< a href = "./figs/ine_plant.png" > png< / a > , < a href = "./figs/ine_plant.pdf" > pdf< / a > )< / p >
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< / div >
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< / div >
< / div >
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< div id = "outline-container-org0613819" class = "outline-4" >
< h4 id = "org0613819" > < span class = "section-number-4" > 4.1.2< / span > Control Design< / h4 >
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< div class = "outline-text-4" id = "text-4-1-2" >
< p >
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The controller is defined below and the obtained loop gain is shown in figure < a href = "#org766b991" > 18< / a > .
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > K_ine = 1e4< span class = "org-type" > /< / span > (1< span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > (2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 100));
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< / pre >
< / div >
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< div id = "org766b991" class = "figure" >
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< p > < img src = "figs/ine_open_loop_gain.png" alt = "ine_open_loop_gain.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 18: < / span > Loop Gain for Inertial Control (< a href = "./figs/ine_open_loop_gain.png" > png< / a > , < a href = "./figs/ine_open_loop_gain.pdf" > pdf< / a > )< / p >
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< / div >
< / div >
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< / div >
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< div id = "outline-container-org7eb6d4a" class = "outline-4" >
< h4 id = "org7eb6d4a" > < span class = "section-number-4" > 4.1.3< / span > Diagonal Controller< / h4 >
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< div class = "outline-text-4" id = "text-4-1-3" >
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< p >
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We create the diagonal controller and we add a minus sign as we have a positive feedback architecture.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > K_ine = < span class = "org-type" > -< / span > K_ine< span class = "org-type" > *< / span > eye(6);
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< / pre >
< / div >
< p >
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We save the controller for further analysis.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > save(< span class = "org-string" > './active_damping/mat/K_ine.mat'< / span > , < span class = "org-string" > 'K_ine'< / span > );
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< / pre >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgb837286" class = "outline-3" >
< h3 id = "orgb837286" > < span class = "section-number-3" > 4.2< / span > Tomography Experiment< / h3 >
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< div class = "outline-text-3" id = "text-4-2" >
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< / div >
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< div id = "outline-container-org088d0cf" class = "outline-4" >
< h4 id = "org088d0cf" > < span class = "section-number-4" > 4.2.1< / span > Initialize the Simulation< / h4 >
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< div class = "outline-text-4" id = "text-4-2-1" >
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< p >
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We initialize elements for the tomography experiment.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > prepareTomographyExperiment();
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< / pre >
< / div >
< p >
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We set the Inertial controller.
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< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > load(< span class = "org-string" > './active_damping/mat/K_ine.mat'< / span > , < span class = "org-string" > 'K_ine'< / span > );
save(< span class = "org-string" > './mat/controllers.mat'< / span > , < span class = "org-string" > 'K_ine'< / span > , < span class = "org-string" > '-append'< / span > );
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< / pre >
< / div >
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< / div >
< / div >
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< div id = "outline-container-orge0acdea" class = "outline-4" >
< h4 id = "orge0acdea" > < span class = "section-number-4" > 4.2.2< / span > Simulation< / h4 >
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< div class = "outline-text-4" id = "text-4-2-2" >
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< p >
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We change the simulation stop time.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > load(< span class = "org-string" > 'mat/conf_simscape.mat'< / span > );
< span class = "org-matlab-simulink-keyword" > set_param< / span > (< span class = "org-variable-name" > conf_simscape< / span > , < span class = "org-string" > 'StopTime'< / span > , < span class = "org-string" > '3'< / span > );
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< / pre >
< / div >
< p >
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And we simulate the system.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-matlab-simulink-keyword" > sim< / span > (< span class = "org-string" > 'sim_nass_active_damping'< / span > );
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< / pre >
< / div >
< p >
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Finally, we save the simulation results for further analysis
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > En_ine = En;
Eg_ine = Eg;
save(< span class = "org-string" > './active_damping/mat/tomo_exp.mat'< / span > , < span class = "org-string" > 'En_ine'< / span > , < span class = "org-string" > 'Eg_ine'< / span > , < span class = "org-string" > '-append'< / span > );
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< / pre >
< / div >
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< / div >
< / div >
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< div id = "outline-container-orgb9346a9" class = "outline-4" >
< h4 id = "orgb9346a9" > < span class = "section-number-4" > 4.2.3< / span > Compare with Undamped system< / h4 >
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< div class = "outline-text-4" id = "text-4-2-3" >
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< p >
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We load the results of tomography experiments.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > load(< span class = "org-string" > './active_damping/mat/tomo_exp.mat'< / span > , < span class = "org-string" > 'En'< / span > , < span class = "org-string" > 'En_ine'< / span > );
t = linspace(0, 3, length(En_ine(< span class = "org-type" > :< / span > ,1)));
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< / pre >
< / div >
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< div id = "orga44ff4c" class = "figure" >
< p > < img src = "figs/nass_act_damp_ine_sim_tomo_xy.png" alt = "nass_act_damp_ine_sim_tomo_xy.png" / >
< / p >
< p > < span class = "figure-number" > Figure 19: < / span > Position Error during tomography experiment - XY Motion (< a href = "./figs/nass_act_damp_ine_sim_tomo_xy.png" > png< / a > , < a href = "./figs/nass_act_damp_ine_sim_tomo_xy.pdf" > pdf< / a > )< / p >
< / div >
< div id = "orgaf24027" class = "figure" >
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< p > < img src = "figs/nass_act_damp_ine_sim_tomo_trans.png" alt = "nass_act_damp_ine_sim_tomo_trans.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 20: < / span > Position Error during tomography experiment - Translations (< a href = "./figs/nass_act_damp_ine_sim_tomo_trans.png" > png< / a > , < a href = "./figs/nass_act_damp_ine_sim_tomo_trans.pdf" > pdf< / a > )< / p >
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< / div >
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< div id = "org3935c70" class = "figure" >
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< p > < img src = "figs/nass_act_damp_ine_sim_tomo_rot.png" alt = "nass_act_damp_ine_sim_tomo_rot.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 21: < / span > Position Error during tomography experiment - Rotations (< a href = "./figs/nass_act_damp_ine_sim_tomo_rot.png" > png< / a > , < a href = "./figs/nass_act_damp_ine_sim_tomo_rot.pdf" > pdf< / a > )< / p >
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< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org971b7df" class = "outline-3" >
< h3 id = "org971b7df" > < span class = "section-number-3" > 4.3< / span > Conclusion< / h3 >
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< div class = "outline-text-3" id = "text-4-3" >
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< div class = "important" >
< p >
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Inertial Control:
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< / p >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgb30a5ab" class = "outline-2" >
< h2 id = "orgb30a5ab" > < span class = "section-number-2" > 5< / span > Comparison< / h2 >
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< div class = "outline-text-2" id = "text-5" >
< p >
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< a id = "org145a075" > < / a >
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< / p >
< / div >
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< div id = "outline-container-org056c3e4" class = "outline-3" >
< h3 id = "org056c3e4" > < span class = "section-number-3" > 5.1< / span > Load the plants< / h3 >
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< div class = "outline-text-3" id = "text-5-1" >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > load(< span class = "org-string" > './active_damping/mat/plants.mat'< / span > , < span class = "org-string" > 'G'< / span > , < span class = "org-string" > 'G_iff'< / span > , < span class = "org-string" > 'G_ine'< / span > , < span class = "org-string" > 'G_dvf'< / span > );
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org75158e7" class = "outline-3" >
< h3 id = "org75158e7" > < span class = "section-number-3" > 5.2< / span > Sensitivity to Disturbance< / h3 >
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< div class = "outline-text-3" id = "text-5-2" >
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< div id = "org1fd62ca" 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 22: < / span > caption (< 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 = "orgf88bf41" 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 23: < / span > caption (< 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 = "org38bd3f1" 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 24: < / span > caption (< 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 >
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< div id = "org028eb1e" 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 25: < / span > caption (< 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 = "org35bcb61" 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 26: < / span > caption (< 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 >
< / div >
< / div >
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< div id = "outline-container-orgca4880d" class = "outline-3" >
< h3 id = "orgca4880d" > < span class = "section-number-3" > 5.3< / span > Damped Plant< / h3 >
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< div class = "outline-text-3" id = "text-5-3" >
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< div id = "org8b00ca9" 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 27: < / 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 = "org9b6dd84" 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 28: < / 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 >
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< div id = "orga1e06be" 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 29: < / 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 >
< / div >
< / div >
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< div id = "outline-container-org4b14fbd" class = "outline-3" >
< h3 id = "org4b14fbd" > < span class = "section-number-3" > 5.4< / span > Tomography Experiment< / h3 >
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< div class = "outline-text-3" id = "text-5-4" >
< 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_dvf'< / span > , < span class = "org-string" > 'En_ine'< / span > );
t = linspace(0, 3, length(En(< span class = "org-type" > :< / span > ,1)));
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > rms(sqrt(En(< span class = "org-type" > :< / span > , 1)< span class = "org-type" > .^< / span > 2 < span class = "org-type" > +< / span > En(< span class = "org-type" > :< / span > , 2)< span class = "org-type" > .^< / span > 2 < span class = "org-type" > +< / span > En(< span class = "org-type" > :< / span > , 3)< span class = "org-type" > .^< / span > 2))
rms(sqrt(En_ine(< span class = "org-type" > :< / span > , 1)< span class = "org-type" > .^< / span > 2 < span class = "org-type" > +< / span > En_ine(< span class = "org-type" > :< / span > , 2)< span class = "org-type" > .^< / span > 2 < span class = "org-type" > +< / span > En_ine(< span class = "org-type" > :< / span > , 3)< span class = "org-type" > .^< / span > 2))
rms(sqrt(En_dvf(< span class = "org-type" > :< / span > , 1)< span class = "org-type" > .^< / span > 2 < span class = "org-type" > +< / span > En_dvf(< span class = "org-type" > :< / span > , 2)< span class = "org-type" > .^< / span > 2 < span class = "org-type" > +< / span > En_dvf(< span class = "org-type" > :< / span > , 3)< span class = "org-type" > .^< / span > 2))
rms(sqrt(En_iff(< span class = "org-type" > :< / span > , 1)< span class = "org-type" > .^< / span > 2 < span class = "org-type" > +< / span > En_iff(< span class = "org-type" > :< / span > , 2)< span class = "org-type" > .^< / span > 2 < span class = "org-type" > +< / span > En_iff(< span class = "org-type" > :< / span > , 3)< span class = "org-type" > .^< / span > 2))
< / pre >
< / div >
< / div >
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< div id = "outline-container-org2213b1c" class = "outline-4" >
< h4 id = "org2213b1c" > < span class = "section-number-4" > 5.4.1< / span > Frequency Domain< / h4 >
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< div class = "outline-text-4" id = "text-5-4-1" >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > Ts = t(2)< span class = "org-type" > -< / span > t(1); < span class = "org-comment" > % Sample Time for the Data [s]< / span >
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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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[pxx, f] = pwelch(En(< span class = "org-type" > :< / span > , 1), han_win, [], [], 1< span class = "org-type" > /< / span > Ts);
[pxx_ine, < span class = "org-type" > ~< / span > ] = pwelch(En_ine(< span class = "org-type" > :< / span > , 1), han_win, [], [], 1< span class = "org-type" > /< / span > Ts);
[pxx_dvf, < span class = "org-type" > ~< / span > ] = pwelch(En_dvf(< span class = "org-type" > :< / span > , 1), han_win, [], [], 1< span class = "org-type" > /< / span > Ts);
[pxx_iff, < span class = "org-type" > ~< / span > ] = pwelch(En_iff(< span class = "org-type" > :< / span > , 1), han_win, [], [], 1< span class = "org-type" > /< / span > Ts);
[pyy, < span class = "org-type" > ~< / span > ] = pwelch(En(< span class = "org-type" > :< / span > , 2), han_win, [], [], 1< span class = "org-type" > /< / span > Ts);
[pyy_ine, < span class = "org-type" > ~< / span > ] = pwelch(En_ine(< span class = "org-type" > :< / span > , 2), han_win, [], [], 1< span class = "org-type" > /< / span > Ts);
[pyy_dvf, < span class = "org-type" > ~< / span > ] = pwelch(En_dvf(< span class = "org-type" > :< / span > , 2), han_win, [], [], 1< span class = "org-type" > /< / span > Ts);
[pyy_iff, < span class = "org-type" > ~< / span > ] = pwelch(En_iff(< span class = "org-type" > :< / span > , 2), han_win, [], [], 1< span class = "org-type" > /< / span > Ts);
[pzz, < span class = "org-type" > ~< / span > ] = pwelch(En(< span class = "org-type" > :< / span > , 3), han_win, [], [], 1< span class = "org-type" > /< / span > Ts);
[pzz_ine, < span class = "org-type" > ~< / span > ] = pwelch(En_ine(< span class = "org-type" > :< / span > , 3), han_win, [], [], 1< span class = "org-type" > /< / span > Ts);
[pzz_dvf, < span class = "org-type" > ~< / span > ] = pwelch(En_dvf(< span class = "org-type" > :< / span > , 3), han_win, [], [], 1< span class = "org-type" > /< / span > Ts);
[pzz_iff, < span class = "org-type" > ~< / span > ] = pwelch(En_iff(< span class = "org-type" > :< / span > , 3), han_win, [], [], 1< span class = "org-type" > /< / span > Ts);
[prx, < span class = "org-type" > ~< / span > ] = pwelch(En(< span class = "org-type" > :< / span > , 4), han_win, [], [], 1< span class = "org-type" > /< / span > Ts);
[prx_ine, < span class = "org-type" > ~< / span > ] = pwelch(En_ine(< span class = "org-type" > :< / span > , 4), han_win, [], [], 1< span class = "org-type" > /< / span > Ts);
[prx_dvf, < span class = "org-type" > ~< / span > ] = pwelch(En_dvf(< span class = "org-type" > :< / span > , 4), han_win, [], [], 1< span class = "org-type" > /< / span > Ts);
[prx_iff, < span class = "org-type" > ~< / span > ] = pwelch(En_iff(< span class = "org-type" > :< / span > , 4), han_win, [], [], 1< span class = "org-type" > /< / span > Ts);
[pry, < span class = "org-type" > ~< / span > ] = pwelch(En(< span class = "org-type" > :< / span > , 5), han_win, [], [], 1< span class = "org-type" > /< / span > Ts);
[pry_ine, < span class = "org-type" > ~< / span > ] = pwelch(En_ine(< span class = "org-type" > :< / span > , 5), han_win, [], [], 1< span class = "org-type" > /< / span > Ts);
[pry_dvf, < span class = "org-type" > ~< / span > ] = pwelch(En_dvf(< span class = "org-type" > :< / span > , 5), han_win, [], [], 1< span class = "org-type" > /< / span > Ts);
[pry_iff, < span class = "org-type" > ~< / span > ] = pwelch(En_iff(< span class = "org-type" > :< / span > , 5), han_win, [], [], 1< span class = "org-type" > /< / span > Ts);
[prz, < span class = "org-type" > ~< / span > ] = pwelch(En(< span class = "org-type" > :< / span > , 6), han_win, [], [], 1< span class = "org-type" > /< / span > Ts);
[prz_ine, < span class = "org-type" > ~< / span > ] = pwelch(En_ine(< span class = "org-type" > :< / span > , 6), han_win, [], [], 1< span class = "org-type" > /< / span > Ts);
[prz_dvf, < span class = "org-type" > ~< / span > ] = pwelch(En_dvf(< span class = "org-type" > :< / span > , 6), han_win, [], [], 1< span class = "org-type" > /< / span > Ts);
[prz_iff, < span class = "org-type" > ~< / span > ] = pwelch(En_iff(< span class = "org-type" > :< / span > , 6), han_win, [], [], 1< span class = "org-type" > /< / span > Ts);
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
hold on;
plot(f, prx_ine, < span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > 'Inertial'< / span > )
plot(f, prx_dvf, < span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > 'DVF'< / span > )
plot(f, prx_iff, < span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > 'IFF'< / span > )
plot(f, prx, < span class = "org-string" > 'k--'< / span > , < span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > 'Undamped'< / span > )
hold off;
xlabel(< span class = "org-string" > 'Frequency [Hz]'< / span > );
ylabel(< span class = "org-string" > 'Power Spectral Density [$m^2/Hz$]'< / span > );
< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'xscale'< / span > , < span class = "org-string" > 'log'< / span > ); < span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'yscale'< / span > , < span class = "org-string" > 'log'< / span > );
legend(< span class = "org-string" > 'location'< / span > , < span class = "org-string" > 'northeast'< / span > );
xlim([1, 500]);
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
hold on;
plot(f, pxx_ine, < span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > 'Inertial'< / span > )
plot(f, pxx_dvf, < span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > 'DVF'< / span > )
plot(f, pxx_iff, < span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > 'IFF'< / span > )
plot(f, pxx, < span class = "org-string" > 'k--'< / span > , < span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > 'Undamped'< / span > )
hold off;
xlabel(< span class = "org-string" > 'Frequency [Hz]'< / span > );
ylabel(< span class = "org-string" > 'Power Spectral Density [$m^2/Hz$]'< / span > );
< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'xscale'< / span > , < span class = "org-string" > 'log'< / span > ); < span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'yscale'< / span > , < span class = "org-string" > 'log'< / span > );
legend(< span class = "org-string" > 'location'< / span > , < span class = "org-string" > 'northeast'< / span > );
xlim([1, 500]);
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< / pre >
< / div >
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< / div >
< / div >
< / div >
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< / div >
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< div id = "outline-container-orge02060f" class = "outline-2" >
< h2 id = "orge02060f" > < span class = "section-number-2" > 6< / span > Useful Functions< / h2 >
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< div class = "outline-text-2" id = "text-6" >
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< / div >
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< div id = "outline-container-org0b9e75d" class = "outline-3" >
< h3 id = "org0b9e75d" > < span class = "section-number-3" > 6.1< / span > prepareTomographyExperiment< / h3 >
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< div class = "outline-text-3" id = "text-6-1" >
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< p >
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< a id = "orgbc5dfba" > < / 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-org637f0d7" class = "outline-4" >
< h4 id = "org637f0d7" > Function Description< / h4 >
< div class = "outline-text-4" id = "text-org637f0d7" >
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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 >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgac8374d" class = "outline-4" >
< h4 id = "orgac8374d" > Optional Parameters< / h4 >
< div class = "outline-text-4" id = "text-orgac8374d" >
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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 >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgb19cd54" class = "outline-4" >
< h4 id = "orgb19cd54" > Initialize the Simulation< / h4 >
< div class = "outline-text-4" id = "text-orgb19cd54" >
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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 >
< / div >
< / div >
< / div >
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< / div >
< / div >
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< / div >
< div id = "postamble" class = "status" >
< p class = "author" > Author: Dehaeze Thomas< / p >
2020-01-20 17:20:50 +01:00
< p class = "date" > Created: 2020-01-20 lun. 17:18< / p >
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< / div >
< / body >
< / html >
