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< title > Decentralize control to add virtual mass< / title >
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< h1 class = "title" > Decentralize control to add virtual mass< / h1 >
< div id = "table-of-contents" >
< h2 > Table of Contents< / h2 >
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< li > < a href = "#org48b52bd" > 1. Initialization< / a > < / li >
< li > < a href = "#org157dc5d" > 2. Identification< / a > < / li >
< li > < a href = "#orgd072386" > 3. Adding Virtual Mass in the Leg’ s Space< / a >
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< li > < a href = "#org147b003" > 3.1. Plant< / a > < / li >
< li > < a href = "#orgacd4421" > 3.2. Controller Design< / a > < / li >
< li > < a href = "#org4c460cf" > 3.3. Identification of the Primary Plant< / a > < / li >
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< li > < a href = "#org3c74924" > 4. Adding Virtual Mass in the Task Space< / a >
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< ul >
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< li > < a href = "#org3b61568" > 4.1. Plant< / a > < / li >
< li > < a href = "#orgf37b1c0" > 4.2. Controller Design< / a > < / li >
< li > < a href = "#orgcd22c9f" > 4.3. Identification of the Primary Plant< / a > < / li >
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< h2 id = "org48b52bd" > < span class = "section-number-2" > 1< / span > Initialization< / h2 >
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< pre class = "src src-matlab" > initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
initializeSimscapeConfiguration();
initializeDisturbances(< span class = "org-string" > 'enable'< / span > , < span class = "org-constant" > false< / span > );
initializeLoggingConfiguration(< span class = "org-string" > 'log'< / span > , < span class = "org-string" > 'none'< / span > );
initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'hac-dvf'< / span > );
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The nano-hexapod has the following leg’ s stiffness and damping.
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< pre class = "src src-matlab" > initializeNanoHexapod(< span class = "org-string" > 'k'< / span > , 1e5, < span class = "org-string" > 'c'< / span > , 2e2);
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< p >
We set the stiffness of the payload fixation:
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< pre class = "src src-matlab" > Kp = 1e8; < span class = "org-comment" > % [N/m]< / span >
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< h2 id = "org157dc5d" > < span class = "section-number-2" > 2< / span > Identification< / h2 >
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We identify the system for the following payload masses:
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< pre class = "src src-matlab" > Ms = [1, 10, 50];
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Identification of the transfer function from \(\tau\) to \(d\mathcal{L}\).
Identification of the Primary plant without virtual add of mass
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< h2 id = "orgd072386" > < span class = "section-number-2" > 3< / span > Adding Virtual Mass in the Leg’ s Space< / h2 >
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< h3 id = "org147b003" > < span class = "section-number-3" > 3.1< / span > Plant< / h3 >
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< div id = "org74dce28" class = "figure" >
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< p > < img src = "figs/virtual_mass_plant_L.png" alt = "virtual_mass_plant_L.png" / >
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< p > < span class = "figure-number" > Figure 1: < / span > Transfer function from \(\tau_i\) to \(d\mathcal{L}_i\) for three payload masses< / p >
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< h3 id = "orgacd4421" > < span class = "section-number-3" > 3.2< / span > Controller Design< / h3 >
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< pre class = "src src-matlab" > Kdvf = 10< span class = "org-type" > *< / span > s< span class = "org-type" > ^< / span > 2< 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 > 500)< span class = "org-type" > ^< / span > 2< span class = "org-type" > *< / span > eye(6);
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< div id = "orgdf2df59" class = "figure" >
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< p > < img src = "figs/virtual_mass_loop_gain_L.png" alt = "virtual_mass_loop_gain_L.png" / >
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< p > < span class = "figure-number" > Figure 2: < / span > Loop Gain for the addition of virtual mass in the leg’ s space< / p >
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< h3 id = "org4c460cf" > < span class = "section-number-3" > 3.3< / span > Identification of the Primary Plant< / h3 >
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< div class = "outline-text-3" id = "text-3-3" >
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< div id = "org29e9333" class = "figure" >
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< p > < img src = "figs/virtual_mass_L_primary_plant_X.png" alt = "virtual_mass_L_primary_plant_X.png" / >
< / p >
< p > < span class = "figure-number" > Figure 3: < / span > Comparison of the transfer function from \(\mathcal{F}_{x,y,z}\) to \(\mathcal{X}_{x,y,z}\) with and without the virtual addition of mass in the leg’ s space< / p >
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< div id = "orgd96256a" class = "figure" >
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< p > < img src = "figs/virtual_mass_L_primary_plant_L.png" alt = "virtual_mass_L_primary_plant_L.png" / >
< / p >
< p > < span class = "figure-number" > Figure 4: < / span > Comparison of the transfer function from \(\tau_i\) to \(\mathcal{L}_{i}\) with and without the virtual addition of mass in the leg’ s space< / p >
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< h2 id = "org3c74924" > < span class = "section-number-2" > 4< / span > Adding Virtual Mass in the Task Space< / h2 >
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< h3 id = "org3b61568" > < span class = "section-number-3" > 4.1< / span > Plant< / h3 >
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< div class = "outline-text-3" id = "text-4-1" >
< p >
Let’ s look at the transfer function from \(\bm{\mathcal{F}}\) to \(d\bm{\mathcal{X}}\):
\[ \frac{d\bm{\mathcal{L}}}{\bm{\mathcal{F}}} = \bm{J}^{-1} \frac{d\bm{\mathcal{L}}}{\bm{\tau}} \bm{J}^{-T} \]
< / p >
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< div id = "orgb509352" class = "figure" >
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< p > < img src = "figs/virtual_mass_plant_X.png" alt = "virtual_mass_plant_X.png" / >
< / p >
< p > < span class = "figure-number" > Figure 5: < / span > Dynamics from \(\mathcal{F}_{x,y,z}\) to \(\mathcal{X}_{x,y,z}\) used for virtual mass addition in the task space< / p >
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< h3 id = "orgf37b1c0" > < span class = "section-number-3" > 4.2< / span > Controller Design< / h3 >
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< div class = "outline-text-3" id = "text-4-2" >
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< pre class = "src src-matlab" > KmX = (s< span class = "org-type" > ^< / span > 2< span class = "org-type" > *< / span > 1< 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 > 500)< span class = "org-type" > ^< / span > 2< span class = "org-type" > *< / span > diag([1 1 50 0 0 0]));
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< div id = "org18b3b14" class = "figure" >
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< p > < img src = "figs/virtual_mass_loop_gain_X.png" alt = "virtual_mass_loop_gain_X.png" / >
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< p > < span class = "figure-number" > Figure 6: < / span > Loop gain for virtual mass addition in the task space< / p >
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< pre class = "src src-matlab" > Kdvf = inv(nano_hexapod.kinematics.J< span class = "org-type" > '< / span > )< span class = "org-type" > *< / span > KmX< span class = "org-type" > *< / span > inv(nano_hexapod.kinematics.J);
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< div id = "outline-container-orgcd22c9f" class = "outline-3" >
< h3 id = "orgcd22c9f" > < span class = "section-number-3" > 4.3< / span > Identification of the Primary Plant< / h3 >
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< div class = "outline-text-3" id = "text-4-3" >
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< div id = "orgfde1133" class = "figure" >
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< p > < img src = "figs/virtual_mass_X_primary_plant_X.png" alt = "virtual_mass_X_primary_plant_X.png" / >
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< p > < span class = "figure-number" > Figure 7: < / span > Comparison of the transfer function from \(\mathcal{F}_{x,y,z}\) to \(\mathcal{X}_{x,y,z}\) with and without the virtual addition of mass in the task space< / p >
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< div id = "org095b9cd" class = "figure" >
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< p > < img src = "figs/virtual_mass_X_primary_plant_L.png" alt = "virtual_mass_X_primary_plant_L.png" / >
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< p > < span class = "figure-number" > Figure 8: < / span > Comparison of the transfer function from \(\tau_i\) to \(\mathcal{L}_{i}\) with and without the virtual addition of mass in the task space< / p >
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< div id = "postamble" class = "status" >
< p class = "author" > Author: Dehaeze Thomas< / p >
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< p class = "date" > Created: 2021-02-20 sam. 23:09< / p >
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