2020-03-23 10:05:32 +01:00
<?xml version="1.0" encoding="utf-8"?>
<?xml version="1.0" encoding="utf-8"?>
< !DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
< html xmlns = "http://www.w3.org/1999/xhtml" lang = "en" xml:lang = "en" >
< head >
2020-05-05 11:38:52 +02:00
<!-- 2020 - 05 - 05 mar. 10:34 -->
2020-03-23 10:05:32 +01:00
< meta http-equiv = "Content-Type" content = "text/html;charset=utf-8" / >
< title > HAC-LAC applied on the Simscape Model< / title >
< meta name = "generator" content = "Org mode" / >
< meta name = "author" content = "Dehaeze Thomas" / >
< link rel = "stylesheet" type = "text/css" href = "./css/htmlize.css" / >
< link rel = "stylesheet" type = "text/css" href = "./css/readtheorg.css" / >
2020-04-17 10:25:44 +02:00
< script src = "./js/jquery.min.js" > < / script >
< script src = "./js/bootstrap.min.js" > < / script >
< script src = "./js/jquery.stickytableheaders.min.js" > < / script >
< script src = "./js/readtheorg.js" > < / script >
< script > M a t h J a x = {
tex: {
tags: 'ams',
macros: {bm: ["\\boldsymbol{#1}",1],}
}
2020-03-23 10:05:32 +01:00
};
< / script >
2020-04-17 10:25:44 +02:00
< script type = "text/javascript" src = "https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js" > < / script >
2020-03-23 10:05:32 +01:00
< / head >
< body >
< div id = "org-div-home-and-up" >
< a accesskey = "h" href = "./index.html" > UP < / a >
|
< a accesskey = "H" href = "./index.html" > HOME < / a >
< / div > < div id = "content" >
< h1 class = "title" > HAC-LAC applied on the Simscape Model< / h1 >
< div id = "table-of-contents" >
< h2 > Table of Contents< / h2 >
< div id = "text-table-of-contents" >
< ul >
< li > < a href = "#org1fe8594" > 1. Initialization< / a > < / li >
< li > < a href = "#orgfd54447" > 2. Low Authority Control - Direct Velocity Feedback \(\bm{K}_\mathcal{L}\)< / a >
< ul >
< li > < a href = "#orga860160" > 2.1. Identification< / a > < / li >
< li > < a href = "#org7a2c131" > 2.2. Plant< / a > < / li >
< li > < a href = "#org46eb79b" > 2.3. Root Locus< / a > < / li >
< li > < a href = "#orgafbd7d0" > 2.4. Controller and Loop Gain< / a > < / li >
< / ul >
< / li >
2020-03-25 19:23:22 +01:00
< li > < a href = "#org73445c2" > 3. Uncertainty Improvements thanks to the LAC control< / a > < / li >
< li > < a href = "#orged12a17" > 4. High Authority Control - \(\bm{K}_\mathcal{X}\)< / a >
2020-03-23 10:05:32 +01:00
< ul >
2020-03-25 19:23:22 +01:00
< li > < a href = "#orgc22e2f2" > 4.1. Identification of the damped plant< / a > < / li >
< li > < a href = "#org6bca8e2" > 4.2. Controller Design< / a > < / li >
2020-03-23 10:05:32 +01:00
< / ul >
< / li >
2020-03-25 19:23:22 +01:00
< li > < a href = "#orgb7ffa65" > 5. Simulation< / a > < / li >
< li > < a href = "#org448f335" > 6. Results< / a > < / li >
2020-03-23 10:05:32 +01:00
< / ul >
< / div >
< / div >
< p >
The position \(\bm{\mathcal{X}}\) of the Sample with respect to the granite is measured.
< / p >
< p >
It is then compare to the wanted position of the Sample \(\bm{r}_\mathcal{X}\) in order to obtain the position error \(\bm{\epsilon}_\mathcal{X}\) of the Sample with respect to a frame attached to the Stewart top platform.
< / p >
< div id = "orgb987aa6" class = "figure" >
< p > < img src = "figs/hac_lac_control_schematic.png" alt = "hac_lac_control_schematic.png" / >
< / p >
< p > < span class = "figure-number" > Figure 1: < / span > HAC-LAC Control Architecture used for the Control of the NASS< / p >
< / div >
< div id = "outline-container-org1fe8594" class = "outline-2" >
< h2 id = "org1fe8594" > < span class = "section-number-2" > 1< / span > Initialization< / h2 >
< div class = "outline-text-2" id = "text-1" >
< p >
We initialize all the stages with the default parameters.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
< / pre >
< / div >
< p >
The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.
< / p >
< div class = "org-src-container" >
2020-05-05 11:38:52 +02:00
< pre class = "src src-matlab" > initializeNanoHexapod('actuator', 'piezo');
initializeSample('mass', 1);
2020-03-23 10:05:32 +01:00
< / pre >
< / div >
< p >
We set the references that corresponds to a tomography experiment.
< / p >
< div class = "org-src-container" >
2020-05-05 11:38:52 +02:00
< pre class = "src src-matlab" > initializeReferences('Rz_type', 'rotating', 'Rz_period', 1);
2020-03-23 10:05:32 +01:00
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > initializeDisturbances();
< / pre >
< / div >
< p >
Open Loop.
< / p >
< div class = "org-src-container" >
2020-05-05 11:38:52 +02:00
< pre class = "src src-matlab" > initializeController('type', 'open-loop');
2020-03-23 10:05:32 +01:00
< / pre >
< / div >
< p >
And we put some gravity.
< / p >
< div class = "org-src-container" >
2020-05-05 11:38:52 +02:00
< pre class = "src src-matlab" > initializeSimscapeConfiguration('gravity', true);
2020-03-23 10:05:32 +01:00
< / pre >
< / div >
< p >
We log the signals.
< / p >
< div class = "org-src-container" >
2020-05-05 11:38:52 +02:00
< pre class = "src src-matlab" > initializeLoggingConfiguration('log', 'all');
2020-03-23 10:05:32 +01:00
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-orgfd54447" class = "outline-2" >
< h2 id = "orgfd54447" > < span class = "section-number-2" > 2< / span > Low Authority Control - Direct Velocity Feedback \(\bm{K}_\mathcal{L}\)< / h2 >
< div class = "outline-text-2" id = "text-2" >
< p >
The first loop closed corresponds to a direct velocity feedback loop.
< / p >
< p >
The design of the associated decentralized controller is explained in < a href = "control_active_damping.html" > this< / a > file.
< / p >
< / div >
< div id = "outline-container-orga860160" class = "outline-3" >
< h3 id = "orga860160" > < span class = "section-number-3" > 2.1< / span > Identification< / h3 >
< div class = "outline-text-3" id = "text-2-1" >
< div class = "org-src-container" >
2020-05-05 11:38:52 +02:00
< pre class = "src src-matlab" > %% Name of the Simulink File
mdl = 'nass_model';
2020-03-23 10:05:32 +01:00
2020-05-05 11:38:52 +02:00
%% Input/Output definition
2020-03-23 10:05:32 +01:00
clear io; io_i = 1;
2020-05-05 11:38:52 +02:00
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Dnlm'); io_i = io_i + 1; % Relative Motion Outputs
2020-03-23 10:05:32 +01:00
2020-05-05 11:38:52 +02:00
%% Run the linearization
2020-03-23 10:05:32 +01:00
G_dvf = linearize(mdl, io, 0);
2020-05-05 11:38:52 +02:00
G_dvf.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
G_dvf.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'};
2020-03-23 10:05:32 +01:00
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org7a2c131" class = "outline-3" >
< h3 id = "org7a2c131" > < span class = "section-number-3" > 2.2< / span > Plant< / h3 >
< / div >
< div id = "outline-container-org46eb79b" class = "outline-3" >
< h3 id = "org46eb79b" > < span class = "section-number-3" > 2.3< / span > Root Locus< / h3 >
< / div >
< div id = "outline-container-orgafbd7d0" class = "outline-3" >
< h3 id = "orgafbd7d0" > < span class = "section-number-3" > 2.4< / span > Controller and Loop Gain< / h3 >
< div class = "outline-text-3" id = "text-2-4" >
< div class = "org-src-container" >
2020-05-05 11:38:52 +02:00
< pre class = "src src-matlab" > K_dvf = s*15000/(1 + s/2/pi/10000);
2020-03-23 10:05:32 +01:00
< / pre >
< / div >
< div class = "org-src-container" >
2020-05-05 11:38:52 +02:00
< pre class = "src src-matlab" > K_dvf = -K_dvf*eye(6);
2020-03-23 10:05:32 +01:00
< / pre >
< / div >
< / div >
< / div >
< / div >
2020-03-25 19:23:22 +01:00
< div id = "outline-container-org73445c2" class = "outline-2" >
< h2 id = "org73445c2" > < span class = "section-number-2" > 3< / span > Uncertainty Improvements thanks to the LAC control< / h2 >
2020-03-23 10:05:32 +01:00
< div class = "outline-text-2" id = "text-3" >
2020-03-25 19:23:22 +01:00
< div class = "org-src-container" >
< pre class = "src src-matlab" > K_dvf_backup = K_dvf;
2020-05-05 11:38:52 +02:00
initializeController('type', 'hac-dvf');
2020-03-25 19:23:22 +01:00
< / pre >
< / div >
< div class = "org-src-container" >
2020-05-05 11:38:52 +02:00
< pre class = "src src-matlab" > masses = [1, 10, 50]; % [kg]
2020-03-25 19:23:22 +01:00
< / pre >
< / div >
< div class = "org-src-container" >
2020-05-05 11:38:52 +02:00
< pre class = "src src-matlab" > %% Name of the Simulink File
mdl = 'nass_model';
2020-03-25 19:23:22 +01:00
2020-05-05 11:38:52 +02:00
%% Input/Output definition
2020-03-25 19:23:22 +01:00
clear io; io_i = 1;
2020-05-05 11:38:52 +02:00
io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/Tracking Error'], 1, 'output', [], 'En'); io_i = io_i + 1; % Position Errror
2020-03-25 19:23:22 +01:00
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-orged12a17" class = "outline-2" >
< h2 id = "orged12a17" > < span class = "section-number-2" > 4< / span > High Authority Control - \(\bm{K}_\mathcal{X}\)< / h2 >
< div class = "outline-text-2" id = "text-4" >
2020-03-23 10:05:32 +01:00
< / div >
< div id = "outline-container-orgc22e2f2" class = "outline-3" >
2020-03-25 19:23:22 +01:00
< h3 id = "orgc22e2f2" > < span class = "section-number-3" > 4.1< / span > Identification of the damped plant< / h3 >
< div class = "outline-text-3" id = "text-4-1" >
2020-03-23 10:05:32 +01:00
< div class = "org-src-container" >
< pre class = "src src-matlab" > Kx = tf(zeros(6));
< / pre >
< / div >
< div class = "org-src-container" >
2020-05-05 11:38:52 +02:00
< pre class = "src src-matlab" > initializeController('type', 'hac-dvf');
2020-03-23 10:05:32 +01:00
< / pre >
< / div >
< div class = "org-src-container" >
2020-05-05 11:38:52 +02:00
< pre class = "src src-matlab" > %% Name of the Simulink File
mdl = 'nass_model';
2020-03-23 10:05:32 +01:00
2020-05-05 11:38:52 +02:00
%% Input/Output definition
2020-03-23 10:05:32 +01:00
clear io; io_i = 1;
2020-05-05 11:38:52 +02:00
io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/Tracking Error'], 1, 'output', [], 'En'); io_i = io_i + 1; % Position Errror
2020-03-23 10:05:32 +01:00
2020-05-05 11:38:52 +02:00
%% Run the linearization
2020-03-23 10:05:32 +01:00
G = linearize(mdl, io, 0);
2020-05-05 11:38:52 +02:00
G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
2020-03-23 10:05:32 +01:00
< / pre >
< / div >
< p >
The minus sine is put here because there is already a minus sign included due to the computation of the position error.
< / p >
< div class = "org-src-container" >
2020-05-05 11:38:52 +02:00
< pre class = "src src-matlab" > load('mat/stages.mat', 'nano_hexapod');
2020-03-23 10:05:32 +01:00
2020-05-05 11:38:52 +02:00
Gx = -G*inv(nano_hexapod.kinematics.J');
Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
2020-03-23 10:05:32 +01:00
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org6bca8e2" class = "outline-3" >
2020-03-25 19:23:22 +01:00
< h3 id = "org6bca8e2" > < span class = "section-number-3" > 4.2< / span > Controller Design< / h3 >
< div class = "outline-text-3" id = "text-4-2" >
2020-03-23 10:05:32 +01:00
< p >
The controller consists of:
< / p >
< ul class = "org-ul" >
< li > A pure integrator< / li >
< li > A Second integrator up to half the wanted bandwidth< / li >
< li > A Lead around the cross-over frequency< / li >
< li > A low pass filter with a cut-off equal to two times the wanted bandwidth< / li >
< / ul >
< div class = "org-src-container" >
2020-05-05 11:38:52 +02:00
< pre class = "src src-matlab" > wc = 2*pi*15; % Bandwidth Bandwidth [rad/s]
2020-03-23 10:05:32 +01:00
2020-05-05 11:38:52 +02:00
h = 1.5; % Lead parameter
2020-03-23 10:05:32 +01:00
2020-05-05 11:38:52 +02:00
Kx = (1/h) * (1 + s/wc*h)/(1 + s/wc/h) * wc/s * ((s/wc*2 + 1)/(s/wc*2)) * (1/(1 + s/wc/2));
2020-03-23 10:05:32 +01:00
2020-05-05 11:38:52 +02:00
% Normalization of the gain of have a loop gain of 1 at frequency wc
Kx = Kx.*diag(1./diag(abs(freqresp(Gx*Kx, wc))));
2020-03-23 10:05:32 +01:00
< / pre >
< / div >
< div class = "org-src-container" >
2020-05-05 11:38:52 +02:00
< pre class = "src src-matlab" > isstable(feedback(Gx*Kx, eye(6), -1))
2020-03-23 10:05:32 +01:00
< / pre >
< / div >
< div class = "org-src-container" >
2020-05-05 11:38:52 +02:00
< pre class = "src src-matlab" > Kx = inv(nano_hexapod.kinematics.J')*Kx;
2020-03-23 10:05:32 +01:00
< / pre >
< / div >
< div class = "org-src-container" >
2020-05-05 11:38:52 +02:00
< pre class = "src src-matlab" > isstable(feedback(G*Kx, eye(6), 1))
2020-03-23 10:05:32 +01:00
< / pre >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-orgb7ffa65" class = "outline-2" >
2020-03-25 19:23:22 +01:00
< h2 id = "orgb7ffa65" > < span class = "section-number-2" > 5< / span > Simulation< / h2 >
< div class = "outline-text-2" id = "text-5" >
2020-03-23 10:05:32 +01:00
< div class = "org-src-container" >
2020-05-05 11:38:52 +02:00
< pre class = "src src-matlab" > load('mat/conf_simulink.mat');
set_param(conf_simulink, 'StopTime', '2');
2020-03-23 10:05:32 +01:00
< / pre >
< / div >
< p >
And we simulate the system.
< / p >
< div class = "org-src-container" >
2020-05-05 11:38:52 +02:00
< pre class = "src src-matlab" > sim('nass_model');
2020-03-23 10:05:32 +01:00
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > hac_dvf = simout;
2020-05-05 11:38:52 +02:00
save('./mat/tomo_exp_hac_lac.mat', 'hac_dvf');
2020-03-23 10:05:32 +01:00
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org448f335" class = "outline-2" >
2020-03-25 19:23:22 +01:00
< h2 id = "org448f335" > < span class = "section-number-2" > 6< / span > Results< / h2 >
< div class = "outline-text-2" id = "text-6" >
2020-03-23 10:05:32 +01:00
< p >
Let’ s load the simulation when no control is applied.
< / p >
< div class = "org-src-container" >
2020-05-05 11:38:52 +02:00
< pre class = "src src-matlab" > load('./mat/experiment_tomography.mat', 'tomo_align_dist');
load('./mat/tomo_exp_hac_lac.mat', 'hac_dvf');
2020-03-23 10:05:32 +01:00
< / pre >
< / div >
< / div >
< / div >
< / div >
< div id = "postamble" class = "status" >
< p class = "author" > Author: Dehaeze Thomas< / p >
2020-05-05 11:38:52 +02:00
< p class = "date" > Created: 2020-05-05 mar. 10:34< / p >
2020-03-23 10:05:32 +01:00
< / div >
< / body >
< / html >
