commit 8e0831fea5f329ae9989078dcf8b14c54867ad19 Author: Thomas Dehaeze Date: Tue Mar 19 15:50:30 2024 +0100 Initial commit diff --git a/.gitattributes b/.gitattributes new file mode 100644 index 0000000..a06e566 --- /dev/null +++ b/.gitattributes @@ -0,0 +1,3 @@ +*.pdf binary +*.svg binary +*.mat binary diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..ff66960 --- /dev/null +++ b/.gitignore @@ -0,0 +1,64 @@ +sim_data/ + +auto/ +*.tex +*.blg +*-blx.bib +*.bbl +*.aux +*.bcf +*.fdb_latexmk +*.log +*.out +*.pyg +*.toc +*.fls +*.synctex.gz +.auctex-auto/ +_minted* + +# Emacs +auto/ + +# Simulink Real Time +*bio.m +*pt.m +*ref.m +*ri.m +*xcp.m +*.mldatx +*.slxc +*.xml +*_slrt_rtw/ + +# data +data/ + +# Windows default autosave extension +*.asv + +# OSX / *nix default autosave extension +*.m~ + +# Compiled MEX binaries (all platforms) +*.mex* + +# Packaged app and toolbox files +*.mlappinstall +*.mltbx + +# Generated helpsearch folders +helpsearch*/ + +# Simulink code generation folders +slprj/ +sccprj/ + +# Matlab code generation folders +codegen/ + +# Simulink autosave extension +*.autosave + +# Octave session info +octave-workspace diff --git a/.latexmkrc b/.latexmkrc new file mode 100644 index 0000000..bce8659 --- /dev/null +++ b/.latexmkrc @@ -0,0 +1,111 @@ +#!/bin/env perl + +# Shebang is only to get syntax highlighting right across GitLab, GitHub and IDEs. +# This file is not meant to be run, but read by `latexmk`. + +# ====================================================================================== +# Perl `latexmk` configuration file +# ====================================================================================== + +# ====================================================================================== +# PDF Generation/Building/Compilation +# ====================================================================================== + +@default_files=('test-bench-nano-hexapod.tex'); + +# PDF-generating modes are: +# 1: pdflatex, as specified by $pdflatex variable (still largely in use) +# 2: postscript conversion, as specified by the $ps2pdf variable (useless) +# 3: dvi conversion, as specified by the $dvipdf variable (useless) +# 4: lualatex, as specified by the $lualatex variable (best) +# 5: xelatex, as specified by the $xelatex variable (second best) +$pdf_mode = 1; + +# Treat undefined references and citations as well as multiply defined references as +# ERRORS instead of WARNINGS. +# This is only checked in the *last* run, since naturally, there are undefined references +# in initial runs. +# This setting is potentially annoying when debugging/editing, but highly desirable +# in the CI pipeline, where such a warning should result in a failed pipeline, since the +# final document is incomplete/corrupted. +# +# However, I could not eradicate all warnings, so that `latexmk` currently fails with +# this option enabled. +# Specifically, `microtype` fails together with `fontawesome`/`fontawesome5`, see: +# https://tex.stackexchange.com/a/547514/120853 +# The fix in that answer did not help. +# Setting `verbose=silent` to mute `microtype` warnings did not work. +# Switching between `fontawesome` and `fontawesome5` did not help. +$warnings_as_errors = 0; + +# Show used CPU time. Looks like: https://tex.stackexchange.com/a/312224/120853 +$show_time = 1; + +# Default is 5; we seem to need more owed to the complexity of the document. +# Actual documents probably don't need this many since they won't use all features, +# plus won't be compiling from cold each time. +$max_repeat=7; + +# --shell-escape option (execution of code outside of latex) is required for the +#'svg' package. +# It converts raw SVG files to the PDF+PDF_TEX combo using InkScape. +# +# SyncTeX allows to jump between source (code) and output (PDF) in IDEs with support +# (many have it). A value of `1` is enabled (gzipped), `-1` is enabled but uncompressed, +# `0` is off. +# Testing in VSCode w/ LaTeX Workshop only worked for the compressed version. +# Adjust this as needed. Of course, only relevant for local use, no effect on a remote +# CI pipeline (except for slower compilation, probably). +# +# %O and %S will forward Options and the Source file, respectively, given to latexmk. +# +# `set_tex_cmds` applies to all *latex commands (latex, xelatex, lualatex, ...), so +# no need to specify these each. This allows to simply change `$pdf_mode` to get a +# different engine. Check if this works with `latexmk --commands`. +set_tex_cmds("--shell-escape -interaction=nonstopmode --synctex=1 %O %S"); + +# Use default pdf viewer +$pdf_previewer = 'zathura'; + +# option 2 is same as 1 (run biber when necessary), but also deletes the +# regeneratable bbl-file in a clenaup (`latexmk -c`). Do not use if original +# bib file is not available! +$bibtex_use = 2; # default: 1 + +# Change default `biber` call, help catch errors faster/clearer. See +# https://web.archive.org/web/20200526101657/https://www.semipol.de/2018/06/12/latex-best-practices.html#database-entries +$biber = "biber --validate-datamodel %O %S"; + +# Glossaries +add_cus_dep('glo', 'gls', 0, 'run_makeglossaries'); +add_cus_dep('acn', 'acr', 0, 'run_makeglossaries'); + +sub run_makeglossaries { + if ( $silent ) { + system "makeglossaries -q -s '$_[0].ist' '$_[0]'"; + } + else { + system "makeglossaries -s '$_[0].ist' '$_[0]'"; + }; +} + +# ====================================================================================== +# Auxiliary Files +# ====================================================================================== + +# Let latexmk know about generated files, so they can be used to detect if a +# rerun is required, or be deleted in a cleanup. +# loe: List of Examples (KOMAScript) +# lol: List of Listings (`listings` and `minted` packages) +# run.xml: biber runs +# glg: glossaries log +# glstex: generated from glossaries-extra +push @generated_exts, 'loe', 'lol', 'run.xml', 'glstex', 'glo', 'gls', 'glg', 'acn', 'acr', 'alg'; + +# Also delete the *.glstex files from package glossaries-extra. Problem is, +# that that package generates files of the form "basename-digit.glstex" if +# multiple glossaries are present. Latexmk looks for "basename.glstex" and so +# does not find those. 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Lafarga, V and + Hellegouarch, Sylvain and Rondineau, Alan and Rodrigues, + Gon{\c{c}}alo and Collette, Christophe}, + title = {A Concept of Active Mount for Space Applications}, + journal = {CEAS Space Journal}, + volume = 10, + number = 2, + pages = {157--165}, + year = 2018, + publisher = {Springer}, +} + +@phdthesis{poel10_explor_activ_hard_mount_vibrat, + author = {van der Poel, Gerrit Wijnand}, + doi = {10.3990/1.9789036530163}, + isbn = {978-90-365-3016-3}, + keywords = {parallel robot}, + school = {University of Twente}, + title = {An Exploration of Active Hard Mount Vibration Isolation for + Precision Equipment}, + year = 2010, +} + +@book{indri20_mechat_robot, + author = {Indri, Marina and Oboe, Roberto}, + title = {Mechatronics and Robotics: New Trends and Challenges}, + year = 2020, + publisher = {CRC Press}, +} + +@book{skogestad07_multiv_feedb_contr, + author = {Skogestad, Sigurd and Postlethwaite, Ian}, + title = {Multivariable Feedback Control: Analysis and Design - + Second Edition}, + year = 2007, + publisher = {John Wiley}, + isbn = {978-0470011683}, + keywords = {favorite}, +} + +@article{oomen15_ident_robus_contr_compl_system, + author = {Oomen, Tom and Steinbuch, Maarten}, + title = {Identification for Robust Control of Complex Systems: + Algorithm and Motion Application}, + journal = {Control-oriented modelling and identification: theory and + applications, IET}, + year = 2015, +} diff --git a/test-bench-nano-hexapod.html b/test-bench-nano-hexapod.html new file mode 100644 index 0000000..f982462 --- /dev/null +++ b/test-bench-nano-hexapod.html @@ -0,0 +1,5087 @@ + + + + + + +Nano-Hexapod - Test Bench + + + + + + + + +
+ UP + | + HOME +
+

Nano-Hexapod - Test Bench

+
+

Table of Contents

+
+ +
+
+
+

This report is also available as a pdf.

+
+ +

+This document is dedicated to the experimental study of the nano-hexapod shown in Figure 1. +

+ + +
+

IMG_20210608_152917.jpg +

+

Figure 1: Nano-Hexapod

+
+ +
+

+Here are the documentation of the equipment used for this test bench (lots of them are shwon in Figure 2): +

+
    +
  • Voltage Amplifier: PiezoDrive PD200
    • +
  • Amplified Piezoelectric Actuator: Cedrat APA300ML
    • +
  • DAC/ADC: Speedgoat IO313
    • +
  • Encoder: Renishaw Vionic and used Ruler
    • +
  • Interferometers: Attocube
    • +
+ +
+ + +
+

IMG_20210608_154722.jpg +

+

Figure 2: Nano-Hexapod and the control electronics

+
+ +

+In Figure 3 is shown a block diagram of the experimental setup. +When possible, the notations are consistent with this diagram and summarized in Table 1. +

+ + +
+

nano_hexapod_signals.png +

+

Figure 3: Block diagram of the system with named signals

+
+ + + + +++ ++ ++ ++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Table 1: List of signals
 UnitMatlabVectorElements
Control Input (wanted DAC voltage)[V]u\(\bm{u}\)\(u_i\)
DAC Output Voltage[V]u\(\tilde{\bm{u}}\)\(\tilde{u}_i\)
PD200 Output Voltage[V]ua\(\bm{u}_a\)\(u_{a,i}\)
Actuator applied force[N]tau\(\bm{\tau}\)\(\tau_i\)
Strut motion[m]dL\(d\bm{\mathcal{L}}\)\(d\mathcal{L}_i\)
Encoder measured displacement[m]dLm\(d\bm{\mathcal{L}}_m\)\(d\mathcal{L}_{m,i}\)
Force Sensor strain[m]epsilon\(\bm{\epsilon}\)\(\epsilon_i\)
Force Sensor Generated Voltage[V]taum\(\tilde{\bm{\tau}}_m\)\(\tilde{\tau}_{m,i}\)
Measured Generated Voltage[V]taum\(\bm{\tau}_m\)\(\tau_{m,i}\)
Motion of the top platform[m,rad]dX\(d\bm{\mathcal{X}}\)\(d\mathcal{X}_i\)
Metrology measured displacement[m,rad]dXm\(d\bm{\mathcal{X}}_m\)\(d\mathcal{X}_{m,i}\)
+ +

+This document is divided in the following sections: +

+ + +
+

1. Encoders fixed to the Struts - Dynamics

+
+

+ +

+

+In this section, the encoders are fixed to the struts. +

+ +

+It is divided in the following sections: +

+
    +
  • Section 1.1: the transfer function matrix from the actuators to the force sensors and to the encoders is experimentally identified.
    • +
  • Section 1.2: the obtained FRF matrix is compared with the dynamics of the simscape model
    • +
  • Section 1.3: decentralized Integral Force Feedback (IFF) is applied and its performances are evaluated.
    • +
  • Section 1.4: a modal analysis of the nano-hexapod is performed
    • +
+
+ +
+

1.1. Identification of the dynamics

+
+

+ +

+
+
+

1.1.1. Load Measurement Data

+
+
+
%% Load Identification Data
+meas_data_lf = {};
+
+for i = 1:6
+    meas_data_lf(i) = {load(sprintf('mat/frf_data_exc_strut_%i_noise_lf.mat', i), 't', 'Va', 'Vs', 'de')};
+    meas_data_hf(i) = {load(sprintf('mat/frf_data_exc_strut_%i_noise_hf.mat', i), 't', 'Va', 'Vs', 'de')};
+end
+
+
+
+
+ +
+

1.1.2. Spectral Analysis - Setup

+
+
+
%% Setup useful variables
+% Sampling Time [s]
+Ts = (meas_data_lf{1}.t(end) - (meas_data_lf{1}.t(1)))/(length(meas_data_lf{1}.t)-1);
+
+% Sampling Frequency [Hz]
+Fs = 1/Ts;
+
+% Hannning Windows
+win = hanning(ceil(1*Fs));
+
+% And we get the frequency vector
+[~, f] = tfestimate(meas_data_lf{1}.Va, meas_data_lf{1}.de, win, [], [], 1/Ts);
+
+i_lf = f < 250; % Points for low frequency excitation
+i_hf = f > 250; % Points for high frequency excitation
+
+
+
+
+ +
+

1.1.3. Transfer function from Actuator to Encoder

+
+

+First, let’s compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure 4). +

+ +
+
%% Coherence
+coh_dvf = zeros(length(f), 6, 6);
+
+for i = 1:6
+    coh_dvf_lf = mscohere(meas_data_lf{i}.Va, meas_data_lf{i}.de, win, [], [], 1/Ts);
+    coh_dvf_hf = mscohere(meas_data_hf{i}.Va, meas_data_hf{i}.de, win, [], [], 1/Ts);
+    coh_dvf(:,:,i) = [coh_dvf_lf(i_lf, :); coh_dvf_hf(i_hf, :)];
+end
+
+
+ + +
+

enc_struts_dvf_coh.png +

+

Figure 4: Obtained coherence for the DVF plant

+
+ +

+Then the 6x6 transfer function matrix is estimated (Figure 5). +

+
+
%% DVF Plant (transfer function from u to dLm)
+G_dvf = zeros(length(f), 6, 6);
+
+for i = 1:6
+    G_dvf_lf = tfestimate(meas_data_lf{i}.Va, meas_data_lf{i}.de, win, [], [], 1/Ts);
+    G_dvf_hf = tfestimate(meas_data_hf{i}.Va, meas_data_hf{i}.de, win, [], [], 1/Ts);
+    G_dvf(:,:,i) = [G_dvf_lf(i_lf, :); G_dvf_hf(i_hf, :)];
+end
+
+
+ + +
+

enc_struts_dvf_frf.png +

+

Figure 5: Measured FRF for the DVF plant

+
+
+
+ +
+

1.1.4. Transfer function from Actuator to Force Sensor

+
+

+First, let’s compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure 6). +

+ +
+
%% Coherence for the IFF plant
+coh_iff = zeros(length(f), 6, 6);
+
+for i = 1:6
+    coh_iff_lf = mscohere(meas_data_lf{i}.Va, meas_data_lf{i}.Vs, win, [], [], 1/Ts);
+    coh_iff_hf = mscohere(meas_data_hf{i}.Va, meas_data_hf{i}.Vs, win, [], [], 1/Ts);
+    coh_iff(:,:,i) = [coh_iff_lf(i_lf, :); coh_iff_hf(i_hf, :)];
+end
+
+
+ + +
+

enc_struts_iff_coh.png +

+

Figure 6: Obtained coherence for the IFF plant

+
+ +

+Then the 6x6 transfer function matrix is estimated (Figure 7). +

+
+
%% IFF Plant
+G_iff = zeros(length(f), 6, 6);
+
+for i = 1:6
+    G_iff_lf = tfestimate(meas_data_lf{i}.Va, meas_data_lf{i}.Vs, win, [], [], 1/Ts);
+    G_iff_hf = tfestimate(meas_data_hf{i}.Va, meas_data_hf{i}.Vs, win, [], [], 1/Ts);
+    G_iff(:,:,i) = [G_iff_lf(i_lf, :); G_iff_hf(i_hf, :)];
+end
+
+
+ + +
+

enc_struts_iff_frf.png +

+

Figure 7: Measured FRF for the IFF plant

+
+
+
+ +
+

1.1.5. Save Identified Plants

+
+
+
save('matlab/mat/identified_plants_enc_struts.mat', 'f', 'Ts', 'G_iff', 'G_dvf')
+
+
+
+
+
+ +
+

1.2. Comparison with the Simscape Model

+
+

+ +

+

+In this section, the measured dynamics is compared with the dynamics estimated from the Simscape model. +

+
+
+

1.2.1. Load measured FRF

+
+
+
%% Load data
+load('identified_plants_enc_struts.mat', 'f', 'Ts', 'G_iff', 'G_dvf')
+
+
+
+
+ +
+

1.2.2. Dynamics from Actuator to Force Sensors

+
+
+
%% Initialize Nano-Hexapod
+n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
+                                       'flex_top_type', '4dof', ...
+                                       'motion_sensor_type', 'struts', ...
+                                       'actuator_type', '2dof');
+
+
+ +
+
%% Identify the IFF Plant (transfer function from u to taum)
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/du'],  1, 'openinput');   io_i = io_i + 1; % Actuator Inputs
+io(io_i) = linio([mdl, '/dum'],  1, 'openoutput'); io_i = io_i + 1; % Force Sensors
+
+Giff = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
+
+
+ + +
+

enc_struts_iff_comp_simscape.png +

+

Figure 8: Diagonal elements of the IFF Plant

+
+ + +
+

enc_struts_iff_comp_offdiag_simscape.png +

+

Figure 9: Off diagonal elements of the IFF Plant

+
+
+
+ +
+

1.2.3. Dynamics from Actuator to Encoder

+
+
+
%% Initialization of the Nano-Hexapod
+n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
+                                       'flex_top_type', '4dof', ...
+                                       'motion_sensor_type', 'struts', ...
+                                       'actuator_type', 'flexible');
+
+
+ +
+
%% Identify the DVF Plant (transfer function from u to dLm)
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/du'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
+io(io_i) = linio([mdl, '/D'],  1, 'openoutput'); io_i = io_i + 1; % Encoders
+
+Gdvf = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
+
+
+ + +
+

enc_struts_dvf_comp_simscape.png +

+

Figure 10: Diagonal elements of the DVF Plant

+
+ + +
+

enc_struts_dvf_comp_offdiag_simscape.png +

+

Figure 11: Off diagonal elements of the DVF Plant

+
+
+
+ +
+

1.2.4. Effect of a change in bending damping of the joints

+
+
+
%% Tested bending dampings [Nm/(rad/s)]
+cRs = [1e-3, 5e-3, 1e-2, 5e-2, 1e-1];
+
+
+ +
+
%% Identify the DVF Plant (transfer function from u to dLm)
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/du'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
+io(io_i) = linio([mdl, '/D'],  1, 'openoutput'); io_i = io_i + 1; % Encoders
+
+
+ +

+Then the identification is performed for all the values of the bending damping. +

+
+
%% Idenfity the transfer function from actuator to encoder for all bending dampins
+Gs = {zeros(length(cRs), 1)};
+
+for i = 1:length(cRs)
+    n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
+                                           'flex_top_type', '4dof', ...
+                                           'motion_sensor_type', 'struts', ...
+                                           'actuator_type', 'flexible', ...
+                                           'flex_bot_cRx', cRs(i), ...
+                                           'flex_bot_cRy', cRs(i), ...
+                                           'flex_top_cRx', cRs(i), ...
+                                           'flex_top_cRy', cRs(i));
+
+    G = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
+    G.InputName  = {'Va1', 'Va2', 'Va3', 'Va4', 'Va5', 'Va6'};
+    G.OutputName = {'dL1', 'dL2', 'dL3', 'dL4', 'dL5', 'dL6'};
+
+    Gs(i) = {G};
+end
+
+
+ +
    +
  • Could be nice
    • +
  • Actual damping is very small
    • +
+
+
+ +
+

1.2.5. Effect of a change in damping factor of the APA

+
+
+
%% Tested bending dampings [Nm/(rad/s)]
+xis = [1e-3, 5e-3, 1e-2, 5e-2, 1e-1];
+
+
+ +
+
%% Identify the DVF Plant (transfer function from u to dLm)
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/du'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
+io(io_i) = linio([mdl, '/D'],  1, 'openoutput'); io_i = io_i + 1; % Encoders
+
+
+ +
+
%% Idenfity the transfer function from actuator to encoder for all bending dampins
+Gs = {zeros(length(xis), 1)};
+
+for i = 1:length(xis)
+    n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
+                                           'flex_top_type', '4dof', ...
+                                           'motion_sensor_type', 'struts', ...
+                                           'actuator_type', 'flexible', ...
+                                           'actuator_xi', xis(i));
+
+    G = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
+    G.InputName  = {'Va1', 'Va2', 'Va3', 'Va4', 'Va5', 'Va6'};
+    G.OutputName = {'dL1', 'dL2', 'dL3', 'dL4', 'dL5', 'dL6'};
+
+    Gs(i) = {G};
+end
+
+
+ + +
+

bode_Va_dL_effect_xi_damp.png +

+

Figure 12: Effect of the APA damping factor \(\xi\) on the dynamics from \(u\) to \(d\mathcal{L}\)

+
+ +
+

+Damping factor \(\xi\) has a large impact on the damping of the “spurious resonances” at 200Hz and 300Hz. +

+ +
+ +
+

+Why is the damping factor does not change the damping of the first peak? +

+ +
+
+
+ +
+

1.2.6. Effect of a change in stiffness damping coef of the APA

+
+
+
m_coef = 1e1;
+
+
+ +
+
%% Tested bending dampings [Nm/(rad/s)]
+k_coefs = [1e-6, 5e-6, 1e-5, 5e-5, 1e-4];
+
+
+ +
+
%% Identify the DVF Plant (transfer function from u to dLm)
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/du'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
+io(io_i) = linio([mdl, '/D'],  1, 'openoutput'); io_i = io_i + 1; % Encoders
+
+
+ +
+
%% Idenfity the transfer function from actuator to encoder for all bending dampins
+Gs = {zeros(length(k_coefs), 1)};
+n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
+                                       'flex_top_type', '4dof', ...
+                                       'motion_sensor_type', 'struts', ...
+                                       'actuator_type', 'flexible');
+
+for i = 1:length(k_coefs)
+    k_coef = k_coefs(i);
+
+    G = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
+    G.InputName  = {'Va1', 'Va2', 'Va3', 'Va4', 'Va5', 'Va6'};
+    G.OutputName = {'dL1', 'dL2', 'dL3', 'dL4', 'dL5', 'dL6'};
+
+    Gs(i) = {G};
+end
+
+
+ + +
+

bode_Va_dL_effect_k_coef.png +

+

Figure 13: Effect of a change of the damping “stiffness coeficient” on the transfer function from \(u\) to \(d\mathcal{L}\)

+
+
+
+ +
+

1.2.7. Effect of a change in mass damping coef of the APA

+
+
+
k_coef = 1e-6;
+
+
+ +
+
%% Tested bending dampings [Nm/(rad/s)]
+m_coefs = [1e1, 5e1, 1e2, 5e2, 1e3];
+
+
+ +
+
%% Identify the DVF Plant (transfer function from u to dLm)
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/du'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
+io(io_i) = linio([mdl, '/D'],  1, 'openoutput'); io_i = io_i + 1; % Encoders
+
+
+ +
+
%% Idenfity the transfer function from actuator to encoder for all bending dampins
+Gs = {zeros(length(m_coefs), 1)};
+n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
+                                       'flex_top_type', '4dof', ...
+                                       'motion_sensor_type', 'struts', ...
+                                       'actuator_type', 'flexible');
+
+for i = 1:length(m_coefs)
+    m_coef = m_coefs(i);
+
+    G = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
+    G.InputName  = {'Va1', 'Va2', 'Va3', 'Va4', 'Va5', 'Va6'};
+    G.OutputName = {'dL1', 'dL2', 'dL3', 'dL4', 'dL5', 'dL6'};
+
+    Gs(i) = {G};
+end
+
+
+ + +
+

bode_Va_dL_effect_m_coef.png +

+

Figure 14: Effect of a change of the damping “mass coeficient” on the transfer function from \(u\) to \(d\mathcal{L}\)

+
+
+
+ +
+

1.2.8. Using Flexible model

+
+
+
d_aligns = [[-0.05,  -0.3,  0];
+            [ 0,      0.5,  0];
+            [-0.1,   -0.3,  0];
+            [ 0,      0.3,  0];
+            [-0.05,   0.05, 0];
+            [0,       0,    0]]*1e-3;
+
+
+ +
+
d_aligns = zeros(6,3);
+% d_aligns(1,:) = [-0.05,  -0.3,  0]*1e-3;
+d_aligns(2,:) = [ 0,      0.3,  0]*1e-3;
+
+
+ +
+
%% Initialize Nano-Hexapod
+n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
+                                       'flex_top_type', '4dof', ...
+                                       'motion_sensor_type', 'struts', ...
+                                       'actuator_type', 'flexible', ...
+                                       'actuator_d_align', d_aligns);
+
+
+ +
+

+Why do we have smaller resonances when using flexible APA? +On the test bench we have the same resonance as the 2DoF model. +Could it be due to the compliance in other dof of the flexible model? +

+ +
+ +
+
%% Identify the DVF Plant (transfer function from u to dLm)
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/du'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
+io(io_i) = linio([mdl, '/D'],  1, 'openoutput'); io_i = io_i + 1; % Encoders
+
+Gdvf = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
+
+
+ +
+
%% Identify the IFF Plant (transfer function from u to taum)
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/du'],  1, 'openinput');   io_i = io_i + 1; % Actuator Inputs
+io(io_i) = linio([mdl, '/dum'],  1, 'openoutput'); io_i = io_i + 1; % Force Sensors
+
+Giff = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
+
+
+
+
+ +
+

1.2.9. Flexible model + encoders fixed to the plates

+
+
+
%% Identify the IFF Plant (transfer function from u to taum)
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/du'],  1, 'openinput');   io_i = io_i + 1; % Actuator Inputs
+io(io_i) = linio([mdl, '/D'],  1, 'openoutput'); io_i = io_i + 1; % Force Sensors
+
+
+ +
+
d_aligns = [[-0.05,  -0.3,  0];
+            [ 0,      0.5,  0];
+            [-0.1,   -0.3,  0];
+            [ 0,      0.3,  0];
+            [-0.05,   0.05, 0];
+            [0,       0,    0]]*1e-3;
+
+
+ +
+
%% Initialize Nano-Hexapod
+n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
+                                       'flex_top_type', '4dof', ...
+                                       'motion_sensor_type', 'struts', ...
+                                       'actuator_type', 'flexible', ...
+                                       'actuator_d_align', d_aligns);
+
+
+ +
+
Gdvf_struts = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
+
+
+ +
+
%% Initialize Nano-Hexapod
+n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
+                                       'flex_top_type', '4dof', ...
+                                       'motion_sensor_type', 'plates', ...
+                                       'actuator_type', 'flexible', ...
+                                       'actuator_d_align', d_aligns);
+
+
+ +
+
Gdvf_plates = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
+
+
+ + +
+

dvf_plant_comp_struts_plates.png +

+

Figure 15: Comparison of the dynamics from \(V_a\) to \(d_L\) when the encoders are fixed to the struts (blue) and to the plates (red). APA are modeled as a flexible element.

+
+
+
+
+ +
+

1.3. Integral Force Feedback

+
+

+ +

+

+In this section, the Integral Force Feedback (IFF) control strategy is applied to the nano-hexapod. +The main goal of this to add damping to the nano-hexapod’s modes. +

+ +

+The control architecture is shown in Figure 16 where \(\bm{K}_\text{IFF}\) is a diagonal \(6 \times 6\) controller. +

+ +

+The system as then a new input \(\bm{u}^\prime\), and the transfer function from \(\bm{u}^\prime\) to \(d\bm{\mathcal{L}}_m\) should be easier to control than the initial transfer function from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\). +

+ + +
+

control_architecture_iff_struts.png +

+

Figure 16: Integral Force Feedback Strategy

+
+ +

+This section is structured as follow: +

+
    +
  • Section 1.3.1: Using the Simscape model (APA taken as 2DoF model), the transfer function from \(\bm{u}\) to \(\bm{\tau}_m\) is identified. Based on the obtained dynamics, the control law is developed and the optimal gain is estimated using the Root Locus.
    • +
  • Section 1.3.2: Still using the Simscape model, the effect of the IFF gain on the the transfer function from \(\bm{u}^\prime\) to \(d\bm{\mathcal{L}}_m\) is studied.
    • +
  • Section 1.3.3: The same is performed experimentally: several IFF gains are used and the damped plant is identified each time.
    • +
  • Section 1.3.4: The damped model and the identified damped system are compared for the optimal IFF gain. It is found that IFF indeed adds a lot of damping into the system. However it is not efficient in damping the spurious struts modes.
    • +
  • Section 1.3.5: Finally, a “flexible” model of the APA is used in the Simscape model and the optimally damped model is compared with the measurements.
    • +
+
+
+

1.3.1. IFF Control Law and Optimal Gain

+
+

+ +

+ +

+Let’s use a model of the Nano-Hexapod with the encoders fixed to the struts and the APA taken as 2DoF model. +

+
+
%% Initialize Nano-Hexapod
+n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
+                                       'flex_top_type', '4dof', ...
+                                       'motion_sensor_type', 'struts', ...
+                                       'actuator_type', '2dof');
+
+
+ +

+The transfer function from \(\bm{u}\) to \(\bm{\tau}_m\) is identified. +

+
+
%% Identify the IFF Plant (transfer function from u to taum)
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/du'],  1, 'openinput');   io_i = io_i + 1; % Actuator Inputs
+io(io_i) = linio([mdl, '/dum'],  1, 'openoutput'); io_i = io_i + 1; % Force Sensors
+
+Giff = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
+
+
+ +

+The IFF controller is defined as shown below: +

+
+
%% IFF Controller
+Kiff_g1 = -(1/(s + 2*pi*40))*...    % LPF: provides integral action above 40Hz
+           (s/(s + 2*pi*30))*...    % HPF: limit low frequency gain
+           (1/(1 + s/2/pi/500))*... % LPF: more robust to high frequency resonances
+           eye(6);                  % Diagonal 6x6 controller
+
+
+ +

+Then, the poles of the system are shown in the complex plane as a function of the controller gain (i.e. Root Locus plot) in Figure 17. +A gain of \(400\) is chosen as the “optimal” gain as it visually seems to be the gain that adds the maximum damping to all the suspension modes simultaneously. +

+ + +
+

enc_struts_iff_root_locus.png +

+

Figure 17: Root Locus for the IFF control strategy

+
+ +

+Then the “optimal” IFF controller is: +

+
+
%% IFF controller with Optimal gain
+Kiff = 400*Kiff_g1;
+
+
+ +

+And it is saved for further use. +

+
+
save('mat/Kiff.mat', 'Kiff')
+
+
+ +

+The bode plots of the “diagonal” elements of the loop gain are shown in Figure 18. +It is shown that the phase and gain margins are quite high and the loop gain is large arround the resonances. +

+ +
+

enc_struts_iff_opt_loop_gain.png +

+

Figure 18: Bode plot of the “decentralized loop gain” \(G_\text{iff}(i,i) \times K_\text{iff}(i,i)\)

+
+
+
+ +
+

1.3.2. Effect of IFF on the plant - Simulations

+
+

+ +

+ +

+Still using the Simscape model with encoders fixed to the struts and 2DoF APA, the IFF strategy is tested. +

+
+
%% Initialize the Simscape model in closed loop
+n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
+                                       'flex_top_type', '4dof', ...
+                                       'motion_sensor_type', 'struts', ...
+                                       'actuator_type', '2dof', ...
+                                       'controller_type', 'iff');
+
+
+ +

+The following IFF gains are tried: +

+
+
%% Tested IFF gains
+iff_gains = [4, 10, 20, 40, 100, 200, 400];
+
+
+ +

+And the transfer functions from \(\bm{u}^\prime\) to \(d\bm{\mathcal{L}}_m\) are identified for all the IFF gains. +

+
+
%% Identify the (damped) transfer function from u to dLm for different values of the IFF gain
+Gd_iff = {zeros(1, length(iff_gains))};
+
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/du'], 1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
+io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Strut Displacement (encoder)
+
+for i = 1:length(iff_gains)
+    Kiff = iff_gains(i)*Kiff_g1*eye(6); % IFF Controller
+    Gd_iff(i) = {exp(-s*Ts)*linearize(mdl, io, 0.0, options)};
+
+    isstable(Gd_iff{i})
+end
+
+
+ +

+The obtained dynamics are shown in Figure 19. +

+ +
+

enc_struts_iff_gains_effect_dvf_plant.png +

+

Figure 19: Effect of the IFF gain \(g\) on the transfer function from \(\bm{\tau}\) to \(d\bm{\mathcal{L}}_m\)

+
+
+
+ +
+

1.3.3. Effect of IFF on the plant - Experimental Results

+
+

+ +

+

+The IFF strategy is applied experimentally and the transfer function from \(\bm{u}^\prime\) to \(d\bm{\mathcal{L}}_m\) is identified for all the defined values of the gain. +

+
+ +
+
1.3.3.1. Load Data
+
+

+First load the identification data. +

+
+
%% Load Identification Data
+meas_iff_gains = {};
+
+for i = 1:length(iff_gains)
+    meas_iff_gains(i) = {load(sprintf('mat/iff_strut_1_noise_g_%i.mat', iff_gains(i)), 't', 'Vexc', 'Vs', 'de', 'u')};
+end
+
+
+
+
+ +
+
1.3.3.2. Spectral Analysis - Setup
+
+

+And define the useful variables that will be used for the identification using the tfestimate function. +

+
+
%% Setup useful variables
+% Sampling Time [s]
+Ts = (meas_iff_gains{1}.t(end) - (meas_iff_gains{1}.t(1)))/(length(meas_iff_gains{1}.t)-1);
+
+% Sampling Frequency [Hz]
+Fs = 1/Ts;
+
+% Hannning Windows
+win = hanning(ceil(1*Fs));
+
+% And we get the frequency vector
+[~, f] = tfestimate(meas_iff_gains{1}.Vexc, meas_iff_gains{1}.de, win, [], [], 1/Ts);
+
+
+
+
+ +
+
1.3.3.3. DVF Plant
+
+

+The transfer functions are estimated for all the values of the gain. +

+
+
%% DVF Plant (transfer function from u to dLm)
+G_iff_gains = {};
+
+for i = 1:length(iff_gains)
+    G_iff_gains{i} = tfestimate(meas_iff_gains{i}.Vexc, meas_iff_gains{i}.de(:,1), win, [], [], 1/Ts);
+end
+
+
+ +

+The obtained dynamics as shown in the bode plot in Figure 20. +The dashed curves are the results obtained using the model, and the solid curves the results from the experimental identification. +

+ +
+

comp_iff_gains_dvf_plant.png +

+

Figure 20: Transfer function from \(u\) to \(d\mathcal{L}_m\) for multiple values of the IFF gain

+
+ +

+The bode plot is then zoomed on the suspension modes of the nano-hexapod in Figure 21. +

+ +
+

comp_iff_gains_dvf_plant_zoom.png +

+

Figure 21: Transfer function from \(u\) to \(d\mathcal{L}_m\) for multiple values of the IFF gain (Zoom)

+
+ +
+

+The IFF control strategy is very effective for the damping of the suspension modes. +It however does not damp the modes at 200Hz, 300Hz and 400Hz (flexible modes of the APA). +

+ +

+Also, the experimental results and the models obtained from the Simscape model are in agreement concerning the damped system (up to the flexible modes). +

+ +
+
+
+ +
+
1.3.3.4. Experimental Results - Comparison of the un-damped and fully damped system
+
+

+The un-damped and damped experimental plants are compared in Figure 22 (diagonal terms). +

+ +

+It is very clear that all the suspension modes are very well damped thanks to IFF. +However, there is little to no effect on the flexible modes of the struts and of the plate. +

+ + +
+

comp_undamped_opt_iff_gain_diagonal.png +

+

Figure 22: Comparison of the diagonal elements of the tranfer function from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\) without active damping and with optimal IFF gain

+
+
+
+
+ +
+

1.3.4. Experimental Results - Damped Plant with Optimal gain

+
+

+ +

+

+Let’s now look at the \(6 \times 6\) damped plant with the optimal gain \(g = 400\). +

+
+
+
1.3.4.1. Load Data
+
+

+The experimental data are loaded. +

+
+
%% Load Identification Data
+meas_iff_struts = {};
+
+for i = 1:6
+    meas_iff_struts(i) = {load(sprintf('mat/iff_strut_%i_noise_g_400.mat', i), 't', 'Vexc', 'Vs', 'de', 'u')};
+end
+
+
+
+
+ +
+
1.3.4.2. Spectral Analysis - Setup
+
+

+And the parameters useful for the spectral analysis are defined. +

+
+
%% Setup useful variables
+% Sampling Time [s]
+Ts = (meas_iff_struts{1}.t(end) - (meas_iff_struts{1}.t(1)))/(length(meas_iff_struts{1}.t)-1);
+
+% Sampling Frequency [Hz]
+Fs = 1/Ts;
+
+% Hannning Windows
+win = hanning(ceil(1*Fs));
+
+% And we get the frequency vector
+[~, f] = tfestimate(meas_iff_struts{1}.Vexc, meas_iff_struts{1}.de, win, [], [], 1/Ts);
+
+
+
+
+ +
+
1.3.4.3. DVF Plant
+
+

+Finally, the \(6 \times 6\) plant is identified using the tfestimate function. +

+
+
%% DVF Plant (transfer function from u to dLm)
+G_iff_opt = {};
+
+for i = 1:6
+    G_iff_opt{i} = tfestimate(meas_iff_struts{i}.Vexc, meas_iff_struts{i}.de, win, [], [], 1/Ts);
+end
+
+
+ +

+The obtained diagonal elements are compared with the model in Figure 23. +

+ +
+

damped_iff_plant_comp_diagonal.png +

+

Figure 23: Comparison of the diagonal elements of the transfer functions from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\) with active damping (IFF) applied with an optimal gain \(g = 400\)

+
+ +

+And all the off-diagonal elements are compared with the model in Figure 24. +

+ +
+

damped_iff_plant_comp_off_diagonal.png +

+

Figure 24: Comparison of the off-diagonal elements of the transfer functions from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\) with active damping (IFF) applied with an optimal gain \(g = 400\)

+
+ +
+

+With the IFF control strategy applied and the optimal gain used, the suspension modes are very well damped. +Remains the un-damped flexible modes of the APA (200Hz, 300Hz, 400Hz), and the modes of the plates (700Hz). +

+ +

+The Simscape model and the experimental results are in very good agreement. +

+ +
+
+
+
+ +
+

1.3.5. Comparison with the Flexible model

+
+

+ +

+ +

+When using the 2-DoF model for the APA, the flexible modes of the struts were not modelled, and it was the main limitation of the model. +Now, let’s use a flexible model for the APA, and see if the obtained damped plant using the model is similar to the measured dynamics. +

+ +

+First, the nano-hexapod is initialized. +

+
+
%% Estimated misalignement of the struts
+d_aligns = [[-0.05,  -0.3,  0];
+            [ 0,      0.5,  0];
+            [-0.1,   -0.3,  0];
+            [ 0,      0.3,  0];
+            [-0.05,   0.05, 0];
+            [0,       0,    0]]*1e-3;
+
+
+%% Initialize Nano-Hexapod
+n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
+                                       'flex_top_type', '4dof', ...
+                                       'motion_sensor_type', 'struts', ...
+                                       'actuator_type', 'flexible', ...
+                                       'actuator_d_align', d_aligns, ...
+                                       'controller_type', 'iff');
+
+
+ +

+And the “optimal” controller is loaded. +

+
+
%% Optimal IFF controller
+load('Kiff.mat', 'Kiff');
+
+
+ +

+The transfer function from \(\bm{u}^\prime\) to \(d\bm{\mathcal{L}}_m\) is identified using the Simscape model. +

+
+
%% Linearized inputs/outputs
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/du'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
+io(io_i) = linio([mdl, '/dL'],  1, 'openoutput'); io_i = io_i + 1; % Strut Displacement (encoder)
+
+%% Identification of the plant
+Gd_iff = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
+
+
+ +

+The obtained diagonal elements are shown in Figure 25 while the off-diagonal elements are shown in Figure 26. +

+ +
+

enc_struts_iff_opt_damp_comp_flex_model_diag.png +

+

Figure 25: Diagonal elements of the transfer function from \(\bm{u}^\prime\) to \(d\bm{\mathcal{L}}_m\) - comparison of the measured FRF and the identified dynamics using the flexible model

+
+ + + +
+

enc_struts_iff_opt_damp_comp_flex_model_off_diag.png +

+

Figure 26: Off-diagonal elements of the transfer function from \(\bm{u}^\prime\) to \(d\bm{\mathcal{L}}_m\) - comparison of the measured FRF and the identified dynamics using the flexible model

+
+ +
+

+Using flexible models for the APA, the agreement between the Simscape model of the nano-hexapod and the measured FRF is very good. +

+ +

+Only the flexible mode of the top-plate is not appearing in the model which is very logical as the top plate is taken as a solid body. +

+ +
+
+
+ +
+

1.3.6. Conclusion

+
+
+

+The decentralized Integral Force Feedback strategy applied on the nano-hexapod is very effective in damping all the suspension modes. +

+ +

+The Simscape model (especially when using a flexible model for the APA) is shown to be very accurate, even when IFF is applied. +

+ +
+
+
+
+ +
+

1.4. Modal Analysis

+
+

+ +

+

+Several 3-axis accelerometers are fixed on the top platform of the nano-hexapod as shown in Figure 31. +

+ + +
+

accelerometers_nano_hexapod.jpg +

+

Figure 27: Location of the accelerometers on top of the nano-hexapod

+
+ +

+The top platform is then excited using an instrumented hammer as shown in Figure 28. +

+ + +
+

hammer_excitation_compliance_meas.jpg +

+

Figure 28: Example of an excitation using an instrumented hammer

+
+ +

+From this experiment, the resonance frequencies and the associated mode shapes can be computed (Section 1.4.1). +Then, in Section 1.4.2, the vertical compliance of the nano-hexapod is experimentally estimated. +Finally, in Section 1.4.3, the measured compliance is compare with the estimated one from the Simscape model. +

+
+ +
+

1.4.1. Obtained Mode Shapes

+
+

+ +

+ +

+We can observe the mode shapes of the first 6 modes that are the suspension modes (the plate is behaving as a solid body) in Figure 29. +

+ + +
+

mode_shapes_annotated.gif +

+

Figure 29: Measured mode shapes for the first six modes

+
+ +

+Then, there is a mode at 692Hz which corresponds to a flexible mode of the top plate (Figure 30). +

+ + +
+

ModeShapeFlex1_crop.gif +

+

Figure 30: First flexible mode at 692Hz

+
+ +

+The obtained modes are summarized in Table 2. +

+ + + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Table 2: Description of the identified modes
ModeFreq. [Hz]Description
1105Suspension Mode: Y-translation
2107Suspension Mode: X-translation
3131Suspension Mode: Z-translation
4161Suspension Mode: Y-tilt
5162Suspension Mode: X-tilt
6180Suspension Mode: Z-rotation
7692(flexible) Membrane mode of the top platform
+
+
+ +
+

1.4.2. Nano-Hexapod Compliance - Effect of IFF

+
+

+ +

+ +

+In this section, we wish to estimated the effectiveness of the IFF strategy concerning the compliance. +

+ +

+The top plate is excited vertically using the instrumented hammer two times: +

+
    +
  1. no control loop is used
    2. +
  2. decentralized IFF is used
    3. +
+ +

+The data is loaded. +

+
+
frf_ol  = load('Measurement_Z_axis.mat'); % Open-Loop
+frf_iff = load('Measurement_Z_axis_damped.mat'); % IFF
+
+
+ +

+The mean vertical motion of the top platform is computed by averaging all 5 accelerometers. +

+
+
%% Multiply by 10 (gain in m/s^2/V) and divide by 5 (number of accelerometers)
+d_frf_ol = 10/5*(frf_ol.FFT1_H1_4_1_RMS_Y_Mod + frf_ol.FFT1_H1_7_1_RMS_Y_Mod + frf_ol.FFT1_H1_10_1_RMS_Y_Mod + frf_ol.FFT1_H1_13_1_RMS_Y_Mod + frf_ol.FFT1_H1_16_1_RMS_Y_Mod)./(2*pi*frf_ol.FFT1_H1_16_1_RMS_X_Val).^2;
+d_frf_iff = 10/5*(frf_iff.FFT1_H1_4_1_RMS_Y_Mod + frf_iff.FFT1_H1_7_1_RMS_Y_Mod + frf_iff.FFT1_H1_10_1_RMS_Y_Mod + frf_iff.FFT1_H1_13_1_RMS_Y_Mod + frf_iff.FFT1_H1_16_1_RMS_Y_Mod)./(2*pi*frf_iff.FFT1_H1_16_1_RMS_X_Val).^2;
+
+
+ +

+The vertical compliance (magnitude of the transfer function from a vertical force applied on the top plate to the vertical motion of the top plate) is shown in Figure 31. +

+ +
+

compliance_vertical_comp_iff.png +

+

Figure 31: Measured vertical compliance with and without IFF

+
+ +
+

+From Figure 31, it is clear that the IFF control strategy is very effective in damping the suspensions modes of the nano-hexapod. +It also has the effect of (slightly) degrading the vertical compliance at low frequency. +

+ +

+It also seems some damping can be added to the modes at around 205Hz which are flexible modes of the struts. +

+ +
+
+
+ +
+

1.4.3. Comparison with the Simscape Model

+
+

+ +

+ +

+Let’s now compare the measured vertical compliance with the vertical compliance as estimated from the Simscape model. +

+ +

+The transfer function from a vertical external force to the absolute motion of the top platform is identified (with and without IFF) using the Simscape model. +The comparison is done in Figure 32. +Again, the model is quite accurate! +

+ +
+

compliance_vertical_comp_model_iff.png +

+

Figure 32: Measured vertical compliance with and without IFF

+
+
+
+
+ +
+

1.5. Conclusion

+
+
+

+From the previous analysis, several conclusions can be drawn: +

+
    +
  • Decentralized IFF is very effective in damping the “suspension” modes of the nano-hexapod (Figure 22)
    • +
  • Decentralized IFF does not damp the “spurious” modes of the struts nor the flexible modes of the top plate (Figure 22)
    • +
  • Even though the Simscape model and the experimentally measured FRF are in good agreement (Figures 25 and 26), the obtain dynamics from the control inputs \(\bm{u}\) and the encoders \(d\bm{\mathcal{L}}_m\) is very difficult to control
    • +
+ +

+Therefore, in the following sections, the encoders will be fixed to the plates. +The goal is to be less sensitive to the flexible modes of the struts. +

+ +
+
+
+
+ +
+

2. Encoders fixed to the plates - Dynamics

+
+

+ +

+

+In this section, the encoders are fixed to the plates rather than to the struts as shown in Figure 33. +

+ + +
+

IMG_20210625_083801.jpg +

+

Figure 33: Nano-Hexapod with encoders fixed to the struts

+
+ +

+It is structured as follow: +

+
    +
  • Section 2.1: The dynamics of the nano-hexapod is identified.
    • +
  • Section 2.2: The identified dynamics is compared with the Simscape model.
    • +
  • Section 2.3: The Integral Force Feedback (IFF) control strategy is applied and the dynamics of the damped nano-hexapod is identified and compare with the Simscape model.
    • +
+
+ +
+

2.1. Identification of the dynamics

+
+

+ +

+

+In this section, the dynamics of the nano-hexapod with the encoders fixed to the plates is identified. +

+ +

+First, the measurement data are loaded in Section 2.1.1, then the transfer function matrix from the actuators to the encoders are estimated in Section 2.1.2. +Finally, the transfer function matrix from the actuators to the force sensors is estimated in Section 2.1.3. +

+
+
+

2.1.1. Data Loading and Spectral Analysis Setup

+
+

+ +

+ +

+The actuators are excited one by one using a low pass filtered white noise. +For each excitation, the 6 force sensors and 6 encoders are measured and saved. +

+
+
%% Load Identification Data
+meas_data_lf = {};
+
+for i = 1:6
+    meas_data_lf(i) = {load(sprintf('mat/frf_exc_strut_%i_enc_plates_noise.mat', i), 't', 'Va', 'Vs', 'de')};
+end
+
+
+
+
+ +
+

2.1.2. Transfer function from Actuator to Encoder

+
+

+ +

+ +

+Let’s compute the coherence from the excitation voltage \(\bm{u}\) and the displacement \(d\bm{\mathcal{L}}_m\) as measured by the encoders. +

+
+
%% Coherence
+coh_dvf = zeros(length(f), 6, 6);
+
+for i = 1:6
+    coh_dvf(:, :, i) = mscohere(meas_data_lf{i}.Va, meas_data_lf{i}.de, win, [], [], 1/Ts);
+end
+
+
+ +

+The obtained coherence shown in Figure 34 is quite good up to 400Hz. +

+ + +
+

enc_plates_dvf_coh.png +

+

Figure 34: Obtained coherence for the DVF plant

+
+ +

+Then the 6x6 transfer function matrix is estimated. +

+
+
%% DVF Plant (transfer function from u to dLm)
+G_dvf = zeros(length(f), 6, 6);
+
+for i = 1:6
+    G_dvf(:,:,i) = tfestimate(meas_data_lf{i}.Va, meas_data_lf{i}.de, win, [], [], 1/Ts);
+end
+
+
+ +

+The diagonal and off-diagonal terms of this transfer function matrix are shown in Figure 35. +

+ +
+

enc_plates_dvf_frf.png +

+

Figure 35: Measured FRF for the DVF plant

+
+ +
+

+From Figure 35, we can draw few conclusions on the transfer functions from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\) when the encoders are fixed to the plates: +

+
    +
  • the decoupling is rather good at low frequency (below the first suspension mode). +The low frequency gain is constant for the off diagonal terms, whereas when the encoders where fixed to the struts, the low frequency gain of the off-diagonal terms were going to zero (Figure 5).
    • +
  • the flexible modes of the struts at 226Hz and 337Hz are indeed shown in the transfer functions, but their amplitudes are rather low.
    • +
  • the diagonal terms have alternating poles and zeros up to at least 600Hz: the flexible modes of the struts are not affecting the alternating pole/zero pattern. This what not the case when the encoders were fixed to the struts (Figure 5).
    • +
+ +
+
+
+ +
+

2.1.3. Transfer function from Actuator to Force Sensor

+
+

+ +

+ +

+Let’s now compute the coherence from the excitation voltage \(\bm{u}\) and the voltage \(\bm{\tau}_m\) generated by the Force senors. +

+
+
%% Coherence for the IFF plant
+coh_iff = zeros(length(f), 6, 6);
+
+for i = 1:6
+    coh_iff(:,:,i) = mscohere(meas_data_lf{i}.Va, meas_data_lf{i}.Vs, win, [], [], 1/Ts);
+end
+
+
+ +

+The coherence is shown in Figure 36, and is very good for from 10Hz up to 2kHz. +

+ +
+

enc_plates_iff_coh.png +

+

Figure 36: Obtained coherence for the IFF plant

+
+ +

+Then the 6x6 transfer function matrix is estimated. +

+
+
%% IFF Plant
+G_iff = zeros(length(f), 6, 6);
+
+for i = 1:6
+    G_iff(:,:,i) = tfestimate(meas_data_lf{i}.Va, meas_data_lf{i}.Vs, win, [], [], 1/Ts);
+end
+
+
+ +

+The bode plot of the diagonal and off-diagonal terms are shown in Figure 37. +

+ +
+

enc_plates_iff_frf.png +

+

Figure 37: Measured FRF for the IFF plant

+
+ +
+

+It is shown in Figure 38 that: +

+
    +
  • The IFF plant has alternating poles and zeros
    • +
  • The first flexible mode of the struts as 235Hz is appearing, and therefore is should be possible to add some damping to this mode using IFF
    • +
  • The decoupling is quite good at low frequency (below the first model) as well as high frequency (above the last suspension mode, except near the flexible modes of the top plate)
    • +
+ +
+
+
+ +
+

2.1.4. Save Identified Plants

+
+

+The identified dynamics is saved for further use. +

+
+
save('mat/identified_plants_enc_plates.mat', 'f', 'Ts', 'G_iff', 'G_dvf')
+
+
+
+
+
+ +
+

2.2. Comparison with the Simscape Model

+
+

+ +

+

+In this section, the measured dynamics done in Section 2.1 is compared with the dynamics estimated from the Simscape model. +

+
+
+

2.2.1. Identification Setup

+
+

+The nano-hexapod is initialized with the APA taken as flexible models. +

+
+
%% Initialize Nano-Hexapod
+n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
+                                       'flex_top_type', '4dof', ...
+                                       'motion_sensor_type', 'plates', ...
+                                       'actuator_type', 'flexible');
+
+
+
+
+ +
+

2.2.2. Dynamics from Actuator to Force Sensors

+
+

+Then the transfer function from \(\bm{u}\) to \(\bm{\tau}_m\) is identified using the Simscape model. +

+
+
%% Identify the IFF Plant (transfer function from u to taum)
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/du'], 1, 'openinput');   io_i = io_i + 1; % Actuator Inputs
+io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensors
+
+Giff = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
+
+
+ +

+The identified dynamics is compared with the measured FRF: +

+
    +
  • Figure 38: the individual transfer function from \(u_1\) (the DAC voltage for the first actuator) to the force sensors of all 6 struts are compared
    • +
  • Figure 39: all the diagonal elements are compared
    • +
  • Figure 40: all the off-diagonal elements are compared
    • +
+ + +
+

enc_plates_iff_comp_simscape_all.png +

+

Figure 38: IFF Plant for the first actuator input and all the force senosrs

+
+ + +
+

enc_plates_iff_comp_simscape.png +

+

Figure 39: Diagonal elements of the IFF Plant

+
+ + +
+

enc_plates_iff_comp_offdiag_simscape.png +

+

Figure 40: Off diagonal elements of the IFF Plant

+
+
+
+ +
+

2.2.3. Dynamics from Actuator to Encoder

+
+

+Now, the dynamics from the DAC voltage \(\bm{u}\) to the encoders \(d\bm{\mathcal{L}}_m\) is estimated using the Simscape model. +

+
+
%% Identify the DVF Plant (transfer function from u to dLm)
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/du'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
+io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Encoders
+
+Gdvf = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
+
+
+ +

+The identified dynamics is compared with the measured FRF: +

+
    +
  • Figure 41: the individual transfer function from \(u_3\) (the DAC voltage for the actuator number 3) to the six encoders
    • +
  • Figure 42: all the diagonal elements are compared
    • +
  • Figure 43: all the off-diagonal elements are compared
    • +
+ + +
+

enc_plates_dvf_comp_simscape_all.png +

+

Figure 41: DVF Plant for the first actuator input and all the encoders

+
+ + +
+

enc_plates_dvf_comp_simscape.png +

+

Figure 42: Diagonal elements of the DVF Plant

+
+ + +
+

enc_plates_dvf_comp_offdiag_simscape.png +

+

Figure 43: Off diagonal elements of the DVF Plant

+
+
+
+ +
+

2.2.4. Flexible Top Plate

+
+
+
%% Initialize Nano-Hexapod
+n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '2dof', ...
+                                       'flex_top_type', '3dof', ...
+                                       'motion_sensor_type', 'struts', ...
+                                       'actuator_type', '2dof', ...
+                                       'top_plate_type', 'rigid');
+
+
+ +
+
%% Identify the DVF Plant (transfer function from u to dLm)
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/du'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
+io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Encoders
+
+Gdvf = linearize(mdl, io, 0.0, options);
+
+
+ +
+
size(Gdvf)
+isstable(Gdvf)
+
+
+ +
+
[sys,g] = balreal(Gdvf);  % Compute balanced realization
+elim = (g<1e-4);         % Small entries of g are negligible states
+rsys = modred(sys,elim); % Remove negligible states
+size(rsys)
+
+
+ +
+
%% Identify the IFF Plant (transfer function from u to taum)
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/du'], 1, 'openinput');   io_i = io_i + 1; % Actuator Inputs
+io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensors
+
+Giff = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
+
+
+
+
+ +
+

2.2.5. Conclusion

+
+
+

+The Simscape model is quite accurate for the transfer function matrices from \(\bm{u}\) to \(\bm{\tau}_m\) and from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\) except at frequencies of the flexible modes of the top-plate. +The Simscape model can therefore be used to develop the control strategies. +

+ +
+
+
+
+ +
+

2.3. Integral Force Feedback

+
+

+ +

+

+In this section, the Integral Force Feedback (IFF) control strategy is applied to the nano-hexapod in order to add damping to the suspension modes. +

+ +

+The control architecture is shown in Figure 44: +

+
    +
  • \(\bm{\tau}_m\) is the measured voltage of the 6 force sensors
    • +
  • \(\bm{K}_{\text{IFF}}\) is the \(6 \times 6\) diagonal controller
    • +
  • \(\bm{u}\) is the plant input (voltage generated by the 6 DACs)
    • +
  • \(\bm{u}^\prime\) is the new plant inputs with added damping
    • +
+ + +
+

control_architecture_iff.png +

+

Figure 44: Integral Force Feedback Strategy

+
+ + +
+
+

2.3.1. Effect of IFF on the plant - Simscape Model

+
+

+ +

+ +

+The nano-hexapod is initialized with flexible APA and the encoders fixed to the struts. +

+
+
%% Initialize the Simscape model in closed loop
+n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
+                                       'flex_top_type', '4dof', ...
+                                       'motion_sensor_type', 'plates', ...
+                                       'actuator_type', 'flexible');
+
+
+ +

+The same controller as the one developed when the encoder were fixed to the struts is used. +

+
+
%% Optimal IFF controller
+load('Kiff.mat', 'Kiff')
+
+
+ +

+The transfer function from \(\bm{u}^\prime\) to \(d\bm{\mathcal{L}}_m\) is identified. +

+
+
%% Identify the (damped) transfer function from u to dLm for different values of the IFF gain
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/du'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
+io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Plate Displacement (encoder)
+
+
+ +

+First in Open-Loop: +

+
+
%% Transfer function from u to dL (open-loop)
+Gd_ol = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
+
+
+ +

+And then with the IFF controller: +

+
+
%% Initialize the Simscape model in closed loop
+n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
+                                       'flex_top_type', '4dof', ...
+                                       'motion_sensor_type', 'plates', ...
+                                       'actuator_type', 'flexible', ...
+                                       'controller_type', 'iff');
+
+%% Transfer function from u to dL (IFF)
+Gd_iff = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
+
+
+ +

+It is first verified that the system is stable: +

+
+
isstable(Gd_iff)
+
+
+ +
+1
+
+ + +

+The diagonal and off-diagonal terms of the \(6 \times 6\) transfer function matrices identified are compared in Figure 45. +It is shown, as was the case when the encoders were fixed to the struts, that the IFF control strategy is very effective in damping the suspension modes of the nano-hexapod. +

+ +
+

enc_plates_iff_gains_effect_dvf_plant.png +

+

Figure 45: Effect of the IFF control strategy on the transfer function from \(\bm{\tau}\) to \(d\bm{\mathcal{L}}_m\)

+
+
+
+ +
+

2.3.2. Effect of IFF on the plant - FRF

+
+

+The IFF control strategy is experimentally implemented. +The (damped) transfer function from \(\bm{u}^\prime\) to \(d\bm{\mathcal{L}}_m\) is experimentally identified. +

+ +

+The identification data are loaded: +

+
+
%% Load Identification Data
+meas_iff_plates = {};
+
+for i = 1:6
+    meas_iff_plates(i) = {load(sprintf('mat/frf_exc_iff_strut_%i_enc_plates_noise.mat', i), 't', 'Va', 'Vs', 'de', 'u')};
+end
+
+
+ +

+And the parameters used for the transfer function estimation are defined below. +

+
+
% Sampling Time [s]
+Ts = (meas_iff_plates{1}.t(end) - (meas_iff_plates{1}.t(1)))/(length(meas_iff_plates{1}.t)-1);
+
+% Hannning Windows
+win = hanning(ceil(1/Ts));
+
+% And we get the frequency vector
+[~, f] = tfestimate(meas_iff_plates{1}.Va, meas_iff_plates{1}.de, win, [], [], 1/Ts);
+
+
+ +

+The estimation is performed using the tfestimate command. +

+
+
%% Estimation of the transfer function matrix from u to dL when IFF is applied
+G_enc_iff_opt = zeros(length(f), 6, 6);
+
+for i = 1:6
+    G_enc_iff_opt(:,:,i) = tfestimate(meas_iff_plates{i}.Va, meas_iff_plates{i}.de, win, [], [], 1/Ts);
+end
+
+
+ +

+The obtained diagonal and off-diagonal elements of the transfer function from \(\bm{u}^\prime\) to \(d\bm{\mathcal{L}}_m\) are shown in Figure 46 both without and with IFF. +

+ +
+

enc_plant_plates_effect_iff.png +

+

Figure 46: Effect of the IFF control strategy on the transfer function from \(\bm{\tau}\) to \(d\bm{\mathcal{L}}_m\)

+
+ +
+

+As was predicted with the Simscape model, the IFF control strategy is very effective in damping the suspension modes of the nano-hexapod. +Little damping is also applied on the first flexible mode of the strut at 235Hz. +However, no damping is applied on other modes, such as the flexible modes of the top plate. +

+ +
+
+
+ +
+

2.3.3. Comparison of the measured FRF and the Simscape model

+
+

+Let’s now compare the obtained damped plants obtained experimentally with the one extracted from Simscape: +

+
    +
  • Figure 47: the individual transfer function from \(u_1^\prime\) to the six encoders are comapred
    • +
  • Figure 48: all the diagonal elements are compared
    • +
  • Figure 49: all the off-diagonal elements are compared
    • +
+ + +
+

enc_plates_opt_iff_comp_simscape_all.png +

+

Figure 47: FRF from one actuator to all the encoders when the plant is damped using IFF

+
+ + +
+

damped_iff_plates_plant_comp_diagonal.png +

+

Figure 48: Comparison of the diagonal elements of the transfer functions from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\) with active damping (IFF) applied with an optimal gain \(g = 400\)

+
+ + +
+

damped_iff_plates_plant_comp_off_diagonal.png +

+

Figure 49: Comparison of the off-diagonal elements of the transfer functions from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\) with active damping (IFF) applied with an optimal gain \(g = 400\)

+
+ +
+

+From Figures 48 and 49, it is clear that the Simscape model very well represents the dynamics of the nano-hexapod. +This is true to around 400Hz, then the dynamics depends on the flexible modes of the top plate which are not modelled. +

+ +
+
+
+ +
+

2.3.4. Save Damped Plant

+
+

+The experimentally identified plant is saved for further use. +

+
+
save('matlab/mat/damped_plant_enc_plates.mat', 'f', 'Ts', 'G_enc_iff_opt')
+
+
+ +
+
save('mat/damped_plant_enc_plates.mat', 'f', 'Ts', 'G_enc_iff_opt')
+
+
+
+
+
+ +
+

2.4. Effect of Payload mass - Robust IFF

+
+

+ +

+

+In this section, the encoders are fixed to the plates, and we identify the dynamics for several payloads. +The added payload are half cylinders, and three layers can be added for a total of around 40kg (Figure 50). +

+ + +
+

picture_added_3_masses.jpg +

+

Figure 50: Picture of the nano-hexapod with added mass

+
+ +

+First the dynamics from \(\bm{u}\) to \(d\mathcal{L}_m\) and \(\bm{\tau}_m\) is identified. +Then, the Integral Force Feedback controller is developed and applied as shown in Figure 51. +Finally, the dynamics from \(\bm{u}^\prime\) to \(d\mathcal{L}_m\) is identified and the added damping can be estimated. +

+ + +
+

nano_hexapod_signals_iff.png +

+

Figure 51: Block Diagram of the experimental setup and model

+
+
+
+

2.4.1. Measured Frequency Response Functions

+
+
+
+
2.4.1.1. Compute FRF in open-loop
+
+

+The identification is performed without added mass, and with one, two and three layers of added cylinders. +

+
+
i_masses = 0:3;
+
+
+ +

+The following data are loaded: +

+
    +
  • Va: the excitation voltage (corresponding to \(u_i\))
    • +
  • Vs: the generated voltage by the 6 force sensors (corresponding to \(\bm{\tau}_m\))
    • +
  • de: the measured motion by the 6 encoders (corresponding to \(d\bm{\mathcal{L}}_m\))
    • +
+
+
%% Load Identification Data
+meas_added_mass = {};
+
+for i_mass = i_masses
+    for i_strut = 1:6
+        meas_added_mass(i_strut, i_mass+1) = {load(sprintf('frf_data_exc_strut_%i_realigned_vib_table_%im.mat', i_strut, i_mass), 't', 'Va', 'Vs', 'de')};
+    end
+end
+
+
+ +

+The window win and the frequency vector f are defined. +

+
+
% Sampling Time [s]
+Ts = (meas_added_mass{1,1}.t(end) - (meas_added_mass{1,1}.t(1)))/(length(meas_added_mass{1,1}.t)-1);
+
+% Hannning Windows
+win = hanning(ceil(1/Ts));
+
+% And we get the frequency vector
+[~, f] = tfestimate(meas_added_mass{1,1}.Va, meas_added_mass{1,1}.de, win, [], [], 1/Ts);
+
+
+ +

+Finally the \(6 \times 6\) transfer function matrices from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\) and from \(\bm{u}\) to \(\bm{\tau}_m\) are identified: +

+
+
%% DVF Plant (transfer function from u to dLm)
+G_dL = {};
+
+for i_mass = i_masses
+    G_dL(i_mass+1) = {zeros(length(f), 6, 6)};
+    for i_strut = 1:6
+        G_dL{i_mass+1}(:,:,i_strut) = tfestimate(meas_added_mass{i_strut, i_mass+1}.Va, meas_added_mass{i_strut, i_mass+1}.de, win, [], [], 1/Ts);
+    end
+end
+
+%% IFF Plant (transfer function from u to taum)
+G_tau = {};
+
+for i_mass = i_masses
+    G_tau(i_mass+1) = {zeros(length(f), 6, 6)};
+    for i_strut = 1:6
+        G_tau{i_mass+1}(:,:,i_strut) = tfestimate(meas_added_mass{i_strut, i_mass+1}.Va, meas_added_mass{i_strut, i_mass+1}.Vs, win, [], [], 1/Ts);
+    end
+end
+
+
+ +

+The identified dynamics are then saved for further use. +

+
+
save('mat/frf_vib_table_m.mat', 'f', 'Ts', 'G_tau', 'G_dL')
+
+
+
+
+
+ +
+

2.4.2. Transfer function from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\)

+
+

+The transfer functions from \(u_i\) to \(d\mathcal{L}_{m,i}\) are shown in Figure 52. +

+ + +
+

comp_plant_payloads_dvf.png +

+

Figure 52: Measured Frequency Response Functions from \(u_i\) to \(d\mathcal{L}_{m,i}\) for all 4 payload conditions

+
+ + +
+

+From Figure 52, we can observe few things: +

+
    +
  • The obtained dynamics is changing a lot between the case without mass and when there is at least one added mass.
    • +
  • Between 1, 2 and 3 added masses, the dynamics is not much different, and it would be easier to design a controller only for these cases.
    • +
  • The flexible modes of the top plate is first decreased a lot when the first mass is added (from 700Hz to 400Hz). +This is due to the fact that the added mass is composed of two half cylinders which are not fixed together. +Therefore is adds a lot of mass to the top plate without adding a lot of rigidity in one direction. +When more than 1 mass layer is added, the half cylinders are added with some angles such that rigidity are added in all directions (see Figure 50). +In that case, the frequency of these flexible modes are increased. +In practice, the payload should be one solid body, and we should not see a massive decrease of the frequency of this flexible mode.
    • +
  • Flexible modes of the top plate are becoming less problematic as masses are added.
    • +
  • First flexible mode of the strut at 230Hz is not much decreased when mass is added. +However, its apparent amplitude is much decreased.
    • +
+ +
+
+
+ +
+

2.4.3. Transfer function from \(\bm{u}\) to \(\bm{\tau}_m\)

+
+

+The transfer functions from \(u_i\) to \(\tau_{m,i}\) are shown in Figure 53. +

+ + +
+

comp_plant_payloads_iff.png +

+

Figure 53: Measured Frequency Response Functions from \(u_i\) to \(\tau_{m,i}\) for all 4 payload conditions

+
+ +
+

+From Figure 53, we can see that for all added payloads, the transfer function from \(u_i\) to \(\tau_{m,i}\) always has alternating poles and zeros. +

+ +
+
+
+
+ +
+

2.5. Comparison with the Simscape model

+
+
+
+

2.5.1. System Identification

+
+

+Let’s initialize the simscape model with the nano-hexapod fixed on top of the vibration table. +

+
+
support.type = 1; % On top of vibration table
+
+
+ +

+The model of the nano-hexapod is defined as shown bellow: +

+
+
%% Initialize Nano-Hexapod
+n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '2dof', ...
+                                       'flex_top_type', '3dof', ...
+                                       'motion_sensor_type', 'plates', ...
+                                       'actuator_type', '2dof');
+
+
+ +

+And finally, we add the same payloads as during the experiments: +

+
+
payload.type = 1; % Payload / 1 "mass layer"
+
+
+ +

+First perform the identification for the transfer functions from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\): +

+
+
%% Identify the DVF Plant (transfer function from u to dLm)
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/du'], 1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
+io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Encoders
+
+%% Identification for all the added payloads
+G_dL = {};
+
+for i = i_masses
+    fprintf('i = %i\n', i)
+    payload.type = i;
+    G_dL(i+1) = {exp(-s*frf_ol.Ts)*linearize(mdl, io, 0.0, options)};
+end
+
+
+ +
+
%% Identify the IFF Plant (transfer function from u to taum)
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/du'], 1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
+io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensors
+
+%% Identification for all the added payloads
+G_tau = {};
+
+for i = 0:3
+    fprintf('i = %i\n', i)
+    payload.type = i;
+    G_tau(i+1) = {exp(-s*frf_ol.Ts)*linearize(mdl, io, 0.0, options)};
+end
+
+
+ +

+The identified dynamics are then saved for further use. +

+
+
save('mat/sim_vib_table_m.mat', 'G_tau', 'G_dL')
+
+
+
+
+ +
+

2.5.2. Transfer function from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\)

+
+

+The measured FRF and the identified dynamics from \(u_i\) to \(d\mathcal{L}_{m,i}\) are compared in Figure 54. +A zoom near the “suspension” modes is shown in Figure 55. +

+ + +
+

comp_masses_model_exp_dvf.png +

+

Figure 54: Comparison of the transfer functions from \(u_i\) to \(d\mathcal{L}_{m,i}\) - measured FRF and identification from the Simscape model

+
+ + +
+

comp_masses_model_exp_dvf_zoom.png +

+

Figure 55: Comparison of the transfer functions from \(u_i\) to \(d\mathcal{L}_{m,i}\) - measured FRF and identification from the Simscape model (Zoom)

+
+ +
+

+The Simscape model is very accurately representing the measured dynamics up. +Only the flexible modes of the struts and of the top plate are not represented here as these elements are modelled as rigid bodies. +

+ +
+
+
+ +
+

2.5.3. Transfer function from \(\bm{u}\) to \(\bm{\tau}_m\)

+
+

+The measured FRF and the identified dynamics from \(u_i\) to \(\tau_{m,i}\) are compared in Figure 56. +A zoom near the “suspension” modes is shown in Figure 57. +

+ + +
+

comp_masses_model_exp_iff.png +

+

Figure 56: Comparison of the transfer functions from \(u_i\) to \(\tau_{m,i}\) - measured FRF and identification from the Simscape model

+
+ + +
+

comp_masses_model_exp_iff_zoom.png +

+

Figure 57: Comparison of the transfer functions from \(u_i\) to \(\tau_{m,i}\) - measured FRF and identification from the Simscape model (Zoom)

+
+
+
+
+ +
+

2.6. Integral Force Feedback Controller

+
+
+
+

2.6.1. Robust IFF Controller

+
+

+Based on the measured FRF from \(u_i\) to \(\tau_{m,i}\), the following IFF controller is developed: +

+
+
%% IFF Controller
+Kiff_g1 = (1/(s + 2*pi*20))*... % LPF: provides integral action above 20[Hz]
+          (s/(s + 2*pi*20))*... % HPF: limit low frequency gain
+          (1/(1 + s/2/pi/400)); % LPF: more robust to high frequency resonances
+
+
+ +

+Then, the Root Locus plot of Figure 58 is used to estimate the optimal gain. +This Root Locus plot is computed from the Simscape model. +

+ +
+

iff_root_locus_masses.png +

+

Figure 58: Root Locus for the IFF control strategy (for all payload conditions).

+
+ +

+The found optimal IFF controller is: +

+
+
%% Optimal controller
+g_opt = -2e2;
+Kiff = g_opt*Kiff_g1*eye(6);
+
+
+ +

+It is saved for further use. +

+
+
save('mat/Kiff_opt.mat', 'Kiff')
+
+
+ +

+The corresponding experimental loop gains are shown in Figure 59. +

+ +
+

iff_loop_gain_masses.png +

+

Figure 59: Loop gain for the Integral Force Feedback controller

+
+ +
+

+Based on the above analysis: +

+
    +
  • The same IFF controller can be used to damp the suspension modes for all payload conditions
    • +
  • The IFF controller should be robust
    • +
+ +
+
+
+ +
+

2.6.2. Estimated Damped Plant from the Simscape model

+
+

+Let’s initialize the simscape model with the nano-hexapod fixed on top of the vibration table. +

+
+
support.type = 1; % On top of vibration table
+
+
+ +

+The model of the nano-hexapod is defined as shown bellow: +

+
+
%% Initialize the Simscape model in closed loop
+n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '2dof', ...
+                                       'flex_top_type', '3dof', ...
+                                       'motion_sensor_type', 'plates', ...
+                                       'actuator_type', '2dof', ...
+                                       'controller_type', 'iff');
+
+
+ +

+And finally, we add the same payloads as during the experiments: +

+
+
payload.type = 1; % Payload / 1 "mass layer"
+
+
+ +
+
%% Identify the (damped) transfer function from u to dLm
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/du'], 1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
+io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Plate Displacement (encoder)
+
+%% Identify for all add masses
+G_dL = {};
+
+for i = i_masses
+    payload.type = i;
+    G_dL(i+1) = {exp(-s*frf_ol.Ts)*linearize(mdl, io, 0.0, options)};
+end
+
+
+ +

+The identified dynamics are then saved for further use. +

+
+
save('mat/sim_iff_vib_table_m.mat', 'G_dL');
+
+
+ +
+
sim_iff = load('sim_iff_vib_table_m.mat', 'G_dL');
+
+
+ + +
+

damped_plant_model_masses.png +

+

Figure 60: Transfer function from \(u_i\) to \(d\mathcal{L}_{m,i}\) (without active damping) and from \(u^\prime_i\) to \(d\mathcal{L}_{m,i}\) (with IFF)

+
+
+
+ +
+

2.6.3. Compute the identified FRF with IFF

+
+

+The identification is performed without added mass, and with one, two and three layers of added cylinders. +

+
+
i_masses = 0:3;
+
+
+ +

+The following data are loaded: +

+
    +
  • Va: the excitation voltage for the damped plant (corresponding to \(u^\prime_i\))
    • +
  • de: the measured motion by the 6 encoders (corresponding to \(d\bm{\mathcal{L}}_m\))
    • +
+
+
%% Load Identification Data
+meas_added_mass = {};
+
+for i_mass = i_masses
+    for i_strut = 1:6
+        meas_iff_mass(i_strut, i_mass+1) = {load(sprintf('frf_data_exc_strut_%i_iff_vib_table_%im.mat', i_strut, i_mass), 't', 'Va', 'de')};
+    end
+end
+
+
+ +

+The window win and the frequency vector f are defined. +

+
+
% Sampling Time [s]
+Ts = (meas_iff_mass{1,1}.t(end) - (meas_iff_mass{1,1}.t(1)))/(length(meas_iff_mass{1,1}.t)-1);
+
+% Hannning Windows
+win = hanning(ceil(1/Ts));
+
+% And we get the frequency vector
+[~, f] = tfestimate(meas_iff_mass{1,1}.Va, meas_iff_mass{1,1}.de, win, [], [], 1/Ts);
+
+
+ +

+Finally the \(6 \times 6\) transfer function matrix from \(\bm{u}^\prime\) to \(d\bm{\mathcal{L}}_m\) is estimated: +

+
+
%% DVF Plant (transfer function from u to dLm)
+G_dL = {};
+
+for i_mass = i_masses
+    G_dL(i_mass+1) = {zeros(length(f), 6, 6)};
+    for i_strut = 1:6
+        G_dL{i_mass+1}(:,:,i_strut) = tfestimate(meas_iff_mass{i_strut, i_mass+1}.Va, meas_iff_mass{i_strut, i_mass+1}.de, win, [], [], 1/Ts);
+    end
+end
+
+
+ +

+The identified dynamics are then saved for further use. +

+
+
save('mat/frf_iff_vib_table_m.mat', 'f', 'Ts', 'G_dL');
+
+
+
+
+ +
+

2.6.4. Comparison of the measured FRF and the Simscape model

+
+

+The following figures are computed: +

+
    +
  • Figure 61: the measured damped FRF are displayed
    • +
  • Figure 62: the open-loop and damped FRF are compared (diagonal elements)
    • +
  • Figure 63: the obtained damped FRF is compared with the identified damped from using the Simscape model
    • +
+ + +
+

damped_iff_plant_meas_frf.png +

+

Figure 61: Diagonal and off-diagonal of the measured FRF matrix for the damped plant

+
+ + +
+

comp_undamped_damped_plant_meas_frf.png +

+

Figure 62: Damped and Undamped measured FRF (diagonal elements)

+
+ + +
+

comp_iff_plant_frf_sim.png +

+

Figure 63: Comparison of the measured FRF and the identified dynamics from the Simscape model

+
+ +
+

+The IFF control strategy effectively damps all the suspensions modes of the nano-hexapod whatever the payload is. +The obtained plant is easier to control (provided the flexible modes of the top platform are well damped). +

+ +
+
+
+ +
+

2.6.5. Change of coupling with IFF

+
+

+The added damping using IFF reduces the coupling in the system near the suspensions modes that are damped. +It can be estimated by taking the ratio of the diagonal-term and the off-diagonal term. +

+ +

+This is shown in Figure 64. +

+ + +
+

reduced_coupling_iff_masses.png +

+

Figure 64: Comparison of the coupling with and without IFF

+
+
+
+
+ +
+

2.7. Un-Balanced mass

+
+
+
+

2.7.1. Introduction

+
+ +
+

picture_unbalanced_payload.jpg +

+

Figure 65: Nano-Hexapod with unbalanced payload

+
+
+
+ +
+

2.7.2. Compute the identified FRF with IFF

+
+

+The following data are loaded: +

+
    +
  • Va: the excitation voltage for the damped plant (corresponding to \(u^\prime_i\))
    • +
  • de: the measured motion by the 6 encoders (corresponding to \(d\bm{\mathcal{L}}_m\))
    • +
+
+
%% Load Identification Data
+meas_added_mass = {zeros(6,1)};
+
+for i_strut = 1:6
+    meas_iff_mass(i_strut) = {load(sprintf('frf_data_exc_strut_%i_iff_vib_table_1m_unbalanced.mat', i_strut), 't', 'Va', 'de')};
+end
+
+
+ +

+The window win and the frequency vector f are defined. +

+
+
% Sampling Time [s]
+Ts = (meas_iff_mass{1}.t(end) - (meas_iff_mass{1}.t(1)))/(length(meas_iff_mass{1}.t)-1);
+
+% Hannning Windows
+win = hanning(ceil(1/Ts));
+
+% And we get the frequency vector
+[~, f] = tfestimate(meas_iff_mass{1}.Va, meas_iff_mass{1}.de, win, [], [], 1/Ts);
+
+
+ +

+Finally the \(6 \times 6\) transfer function matrix from \(\bm{u}^\prime\) to \(d\bm{\mathcal{L}}_m\) is estimated: +

+
+
%% DVF Plant (transfer function from u to dLm)
+G_dL = zeros(length(f), 6, 6);
+for i_strut = 1:6
+    G_dL(:,:,i_strut) = tfestimate(meas_iff_mass{i_strut}.Va, meas_iff_mass{i_strut}.de, win, [], [], 1/Ts);
+end
+
+
+ +

+The identified dynamics are then saved for further use. +

+
+
save('mat/frf_iff_unbalanced_vib_table_m.mat', 'f', 'Ts', 'G_dL');
+
+
+
+
+ +
+

2.7.3. Effect of an unbalanced payload

+
+

+The transfer functions from \(u_i\) to \(d\mathcal{L}_i\) are shown in Figure 66. +Due to the unbalanced payload, the system is not symmetrical anymore, and therefore each of the diagonal elements are not equal. +This is due to the fact that each strut is not affected by the same inertia. +

+ + +
+

frf_damp_unbalanced_mass.png +

+

Figure 66: Transfer function from \(u_i\) to \(d\mathcal{L}_i\) for the nano-hexapod with an unbalanced payload

+
+
+
+
+ + + +
+

2.8. Conclusion

+
+
+

+In this section, the dynamics of the nano-hexapod with the encoders fixed to the plates is studied. +

+ +

+It has been found that: +

+
    +
  • The measured dynamics is in agreement with the dynamics of the simscape model, up to the flexible modes of the top plate. +See figures 39 and 40 for the transfer function to the force sensors and Figures 42 and 43for the transfer functions to the encoders
    • +
  • The Integral Force Feedback strategy is very effective in damping the suspension modes of the nano-hexapod (Figure 46).
    • +
  • The transfer function from \(\bm{u}^\prime\) to \(d\bm{\mathcal{L}}_m\) show nice dynamical properties and is a much better candidate for the high-authority-control than when the encoders were fixed to the struts. +At least up to the flexible modes of the top plate, the diagonal elements of the transfer function matrix have alternating poles and zeros, and the phase is moving smoothly. +Only the flexible modes of the top plates seems to be problematic for control.
    • +
+ +
+
+
+
+ +
+

3. Decentralized High Authority Control with Integral Force Feedback

+
+

+ +

+

+In this section is studied the HAC-IFF architecture for the Nano-Hexapod. +More precisely: +

+
    +
  • The LAC control is a decentralized integral force feedback as studied in Section 2.3
    • +
  • The HAC control is a decentralized controller working in the frame of the struts
    • +
+ +

+The corresponding control architecture is shown in Figure 67 with: +

+
    +
  • \(\bm{r}_{\mathcal{X}_n}\): the \(6 \times 1\) reference signal in the cartesian frame
    • +
  • \(\bm{r}_{d\mathcal{L}}\): the \(6 \times 1\) reference signal transformed in the frame of the struts thanks to the inverse kinematic
    • +
  • \(\bm{\epsilon}_{d\mathcal{L}}\): the \(6 \times 1\) length error of the 6 struts
    • +
  • \(\bm{u}^\prime\): input of the damped plant
    • +
  • \(\bm{u}\): generated DAC voltages
    • +
  • \(\bm{\tau}_m\): measured force sensors
    • +
  • \(d\bm{\mathcal{L}}_m\): measured displacement of the struts by the encoders
    • +
+ + +
+

control_architecture_hac_iff_struts.png +

+

Figure 67: HAC-LAC: IFF + Control in the frame of the legs

+
+ +

+This part is structured as follow: +

+
    +
  • Section 3.1: some reference tracking tests are performed
    • +
  • Section 3.2: the decentralized high authority controller is tuned using the Simscape model and is implemented and tested experimentally
    • +
  • Section 3.3: an interaction analysis is performed, from which the best decoupling strategy can be determined
    • +
  • Section 3.4: Robust High Authority Controller are designed
    • +
+
+ +
+

3.1. Reference Tracking - Trajectories

+
+

+ +

+

+In this section, several trajectories representing the wanted pose (position and orientation) of the top platform with respect to the bottom platform are defined. +

+ +

+These trajectories will be used to test the HAC-LAC architecture. +

+ +

+In order to transform the wanted pose to the wanted displacement of the 6 struts, the inverse kinematic is required. +As a first approximation, the Jacobian matrix \(\bm{J}\) can be used instead of using the full inverse kinematic equations. +

+ +

+Therefore, the control architecture with the input trajectory \(\bm{r}_{\mathcal{X}_n}\) is shown in Figure 68. +

+ + +
+

control_architecture_hac_iff_struts_L.png +

+

Figure 68: HAC-LAC: IFF + Control in the frame of the legs

+
+ +

+In the following sections, several reference trajectories are defined: +

+
    +
  • Section 3.1.1: simple scans in the Y-Z plane
    • +
  • Section 3.1.2: scans in tilt are performed
    • +
  • Section 3.1.3: scans with X-Y-Z translations in order to draw the word “NASS”
    • +
+
+
+

3.1.1. Y-Z Scans

+
+

+ +A function generateYZScanTrajectory has been developed (accessible here) in order to easily generate scans in the Y-Z plane. +

+ +

+For instance, the following generated trajectory is represented in Figure 69. +

+
+
%% Generate the Y-Z trajectory scan
+Rx_yz = generateYZScanTrajectory(...
+    'y_tot', 4e-6, ... % Length of Y scans [m]
+    'z_tot', 4e-6, ... % Total Z distance [m]
+    'n', 5, ...     % Number of Y scans
+    'Ts', 1e-3, ... % Sampling Time [s]
+    'ti', 1, ...    % Time to go to initial position [s]
+    'tw', 0, ...    % Waiting time between each points [s]
+    'ty', 0.6, ...  % Time for a scan in Y [s]
+    'tz', 0.2);     % Time for a scan in Z [s]
+
+
+ + +
+

yz_scan_example_trajectory_yz_plane.png +

+

Figure 69: Generated scan in the Y-Z plane

+
+ +

+The Y and Z positions as a function of time are shown in Figure 70. +

+ + +
+

yz_scan_example_trajectory.png +

+

Figure 70: Y and Z trajectories as a function of time

+
+ +

+Using the Jacobian matrix, it is possible to compute the wanted struts lengths as a function of time: +

+\begin{equation} + \bm{r}_{d\mathcal{L}} = \bm{J} \bm{r}_{\mathcal{X}_n} +\end{equation} + +
+
%% Compute the reference in the frame of the legs
+dL_ref = [J*Rx_yz(:, 2:7)']';
+
+
+ +

+The reference signal for the strut length is shown in Figure 71. +

+ +
+

yz_scan_example_trajectory_struts.png +

+

Figure 71: Trajectories for the 6 individual struts

+
+
+
+ +
+

3.1.2. Tilt Scans

+
+

+ +

+ +

+A function generalSpiralAngleTrajectory has been developed in order to easily generate \(R_x,R_y\) tilt scans. +

+ +

+For instance, the following generated trajectory is represented in Figure 72. +

+
+
%% Generate the "tilt-spiral" trajectory scan
+R_tilt = generateSpiralAngleTrajectory(...
+    'R_tot',  20e-6, ... % Total Tilt [ad]
+    'n_turn', 5, ...     % Number of scans
+    'Ts',     1e-3, ...  % Sampling Time [s]
+    't_turn', 1, ...     % Turn time [s]
+    't_end',  1);        % End time to go back to zero [s]
+
+
+ + +
+

tilt_scan_example_trajectory.png +

+

Figure 72: Generated “spiral” scan

+
+ +

+The reference signal for the strut length is shown in Figure 73. +

+ +
+

tilt_scan_example_trajectory_struts.png +

+

Figure 73: Trajectories for the 6 individual struts - Tilt scan

+
+
+
+ +
+

3.1.3. “NASS” reference path

+
+

+ +In this section, a reference path that “draws” the work “NASS” is developed. +

+ +

+First, a series of points representing each letter are defined. +Between each letter, a negative Z motion is performed. +

+
+
%% List of points that draws "NASS"
+ref_path = [ ...
+    0, 0,0; % Initial Position
+    0,0,1; 0,4,1; 3,0,1; 3,4,1; % N
+    3,4,0; 4,0,0; % Transition
+    4,0,1; 4,3,1; 5,4,1; 6,4,1; 7,3,1; 7,2,1; 4,2,1; 4,3,1; 5,4,1; 6,4,1; 7,3,1; 7,0,1; % A
+    7,0,0; 8,0,0; % Transition
+    8,0,1; 11,0,1; 11,2,1; 8,2,1; 8,4,1; 11,4,1; % S
+    11,4,0; 12,0,0; % Transition
+    12,0,1; 15,0,1; 15,2,1; 12,2,1; 12,4,1; 15,4,1; % S
+    15,4,0;
+           ];
+
+%% Center the trajectory arround zero
+ref_path = ref_path - (max(ref_path) - min(ref_path))/2;
+
+%% Define the X-Y-Z cuboid dimensions containing the trajectory
+X_max = 10e-6;
+Y_max =  4e-6;
+Z_max =  2e-6;
+
+ref_path = ([X_max, Y_max, Z_max]./max(ref_path)).*ref_path; % [m]
+
+
+ +

+Then, using the generateXYZTrajectory function, the \(6 \times 1\) trajectory signal is computed. +

+
+
%% Generating the trajectory
+Rx_nass = generateXYZTrajectory('points', ref_path);
+
+
+ +

+The trajectory in the X-Y plane is shown in Figure 74 (the transitions between the letters are removed). +

+ +
+

ref_track_test_nass.png +

+

Figure 74: Reference path corresponding to the “NASS” acronym

+
+ +

+It can also be better viewed in a 3D representation as in Figure 75. +

+ + +
+

ref_track_test_nass_3d.png +

+

Figure 75: Reference path that draws “NASS” - 3D view

+
+
+
+
+ +
+

3.2. First Basic High Authority Controller

+
+

+ +

+

+In this section, a simple decentralized high authority controller \(\bm{K}_{\mathcal{L}}\) is developed to work without any payload. +

+ +

+The diagonal controller is tuned using classical Loop Shaping in Section 3.2.1. +The stability is verified in Section 3.2.2 using the Simscape model. +

+
+
+

3.2.1. HAC Controller

+
+

+ +

+ +

+Let’s first try to design a first decentralized controller with: +

+
    +
  • a bandwidth of 100Hz
    • +
  • sufficient phase margin
    • +
  • simple and understandable components
    • +
+ +

+After some very basic and manual loop shaping, A diagonal controller is developed. +Each diagonal terms are identical and are composed of: +

+
    +
  • A lead around 100Hz
    • +
  • A first order low pass filter starting at 200Hz to add some robustness to high frequency modes
    • +
  • A notch at 700Hz to cancel the flexible modes of the top plate
    • +
  • A pure integrator
    • +
+ +
+
%% Lead to increase phase margin
+a  = 2;  % Amount of phase lead / width of the phase lead / high frequency gain
+wc = 2*pi*100; % Frequency with the maximum phase lead [rad/s]
+
+H_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)));
+
+%% Low Pass filter to increase robustness
+H_lpf = 1/(1 + s/2/pi/200);
+
+%% Notch at the top-plate resonance
+gm = 0.02;
+xi = 0.3;
+wn = 2*pi*700;
+
+H_notch = (s^2 + 2*gm*xi*wn*s + wn^2)/(s^2 + 2*xi*wn*s + wn^2);
+
+%% Decentralized HAC
+Khac_iff_struts = -(1/(2.87e-5)) * ... % Gain
+                  H_lead * ...       % Lead
+                  H_notch * ...      % Notch
+                  (2*pi*100/s) * ... % Integrator
+                  eye(6);            % 6x6 Diagonal
+
+
+ +

+This controller is saved for further use. +

+
+
save('mat/Khac_iff_struts.mat', 'Khac_iff_struts')
+
+
+ +

+The experimental loop gain is computed and shown in Figure 76. +

+
+
L_hac_iff_struts = pagemtimes(permute(frf_iff.G_dL{1}, [2 3 1]), squeeze(freqresp(Khac_iff_struts, frf_iff.f, 'Hz')));
+
+
+ + +
+

loop_gain_hac_iff_struts.png +

+

Figure 76: Diagonal and off-diagonal elements of the Loop gain for “HAC-IFF-Struts”

+
+
+
+ +
+

3.2.2. Verification of the Stability using the Simscape model

+
+

+ +

+ +

+The HAC-IFF control strategy is implemented using Simscape. +

+
+
%% Initialize the Simscape model in closed loop
+n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
+                                       'flex_top_type', '4dof', ...
+                                       'motion_sensor_type', 'plates', ...
+                                       'actuator_type', 'flexible', ...
+                                       'controller_type', 'hac-iff-struts');
+
+
+ +
+
%% Identify the (damped) transfer function from u to dLm
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/du'], 1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
+io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Plate Displacement (encoder)
+
+
+ +

+We identify the closed-loop system. +

+
+
%% Identification
+Gd_iff_hac_opt = linearize(mdl, io, 0.0, options);
+
+
+ +

+And verify that it is indeed stable. +

+
+
%% Verify the stability
+isstable(Gd_iff_hac_opt)
+
+
+ +
+1
+
+
+
+ +
+

3.2.3. Experimental Validation

+
+

+Both the Integral Force Feedback controller (developed in Section 2.3) and the high authority controller working in the frame of the struts (developed in Section 3.2) are implemented experimentally. +

+ +

+Two reference tracking experiments are performed to evaluate the stability and performances of the implemented control. +

+ +
+
%% Load the experimental data
+load('hac_iff_struts_yz_scans.mat', 't', 'de')
+
+
+ +

+The position of the top-platform is estimated using the Jacobian matrix: +

+
+
%% Pose of the top platform from the encoder values
+load('jacobian.mat', 'J');
+Xe = [inv(J)*de']';
+
+
+ +
+
%% Generate the Y-Z trajectory scan
+Rx_yz = generateYZScanTrajectory(...
+    'y_tot', 4e-6, ... % Length of Y scans [m]
+    'z_tot', 8e-6, ... % Total Z distance [m]
+    'n', 5, ...     % Number of Y scans
+    'Ts', 1e-3, ... % Sampling Time [s]
+    'ti', 1, ...    % Time to go to initial position [s]
+    'tw', 0, ...    % Waiting time between each points [s]
+    'ty', 0.6, ...  % Time for a scan in Y [s]
+    'tz', 0.2);     % Time for a scan in Z [s]
+
+
+ +

+The reference path as well as the measured position are partially shown in the Y-Z plane in Figure 77. +

+ +
+

yz_scans_exp_results_first_K.png +

+

Figure 77: Measured position \(\bm{\mathcal{X}}_n\) and reference signal \(\bm{r}_{\mathcal{X}_n}\) in the Y-Z plane - Zoom on a change of direction

+
+ +
+

+It is clear from Figure 77 that the position of the nano-hexapod effectively tracks to reference signal. +However, oscillations with amplitudes as large as 50nm can be observe. +

+ +

+It turns out that the frequency of these oscillations is 100Hz which is corresponding to the crossover frequency of the High Authority Control loop. +This clearly indicates poor stability margins. +In the next section, the controller is re-designed to improve the stability margins. +

+ +
+
+
+ +
+

3.2.4. Controller with increased stability margins

+
+

+The High Authority Controller is re-designed in order to improve the stability margins. +

+
+
%% Lead
+a  = 5;  % Amount of phase lead / width of the phase lead / high frequency gain
+wc = 2*pi*110; % Frequency with the maximum phase lead [rad/s]
+
+H_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)));
+
+%% Low Pass Filter
+H_lpf = 1/(1 + s/2/pi/300);
+
+%% Notch
+gm = 0.02;
+xi = 0.5;
+wn = 2*pi*700;
+
+H_notch = (s^2 + 2*gm*xi*wn*s + wn^2)/(s^2 + 2*xi*wn*s + wn^2);
+
+%% HAC Controller
+Khac_iff_struts = -2.2e4 * ... % Gain
+            H_lead * ...       % Lead
+            H_lpf * ...        % Lead
+            H_notch * ...      % Notch
+            (2*pi*100/s) * ... % Integrator
+            eye(6);            % 6x6 Diagonal
+
+
+ +

+The bode plot of the new loop gain is shown in Figure 78. +

+ +
+

hac_iff_plates_exp_loop_gain_redesigned_K.png +

+

Figure 78: Loop Gain for the updated decentralized HAC controller

+
+ +

+This new controller is implemented experimentally and several tracking tests are performed. +

+
+
%% Load Measurements
+load('hac_iff_more_lead_nass_scan.mat', 't', 'de')
+
+
+ +

+The pose of the top platform is estimated from the encoder position using the Jacobian matrix. +

+
+
%% Compute the pose of the top platform
+load('jacobian.mat', 'J');
+Xe = [inv(J)*de']';
+
+
+ +

+The measured motion as well as the trajectory are shown in Figure 79. +

+ +
+

nass_scans_first_test_exp.png +

+

Figure 79: Measured position \(\bm{\mathcal{X}}_n\) and reference signal \(\bm{r}_{\mathcal{X}_n}\) for the “NASS” trajectory

+
+ +

+The trajectory and measured motion are also shown in the X-Y plane in Figure 80. +

+ +
+

ref_track_nass_exp_hac_iff_struts.png +

+

Figure 80: Reference path and measured motion in the X-Y plane

+
+ +

+The orientation errors during all the scans are shown in Figure 81. +

+ +
+

nass_ref_rx_ry.png +

+

Figure 81: Orientation errors during the scan

+
+ +
+

+Using the updated High Authority Controller, the nano-hexapod can follow trajectories with high accuracy (the position errors are in the order of 50nm peak to peak, and the orientation errors 300nrad peak to peak). +

+ +
+
+
+
+ +
+

3.3. Interaction Analysis and Decoupling

+
+

+ +

+

+In this section, the interaction in the identified plant is estimated using the Relative Gain Array (RGA) (Skogestad and Postlethwaite 2007, 3.4). +

+ +

+Then, several decoupling strategies are compared for the nano-hexapod. +

+ +

+The RGA Matrix is defined as follow: +

+\begin{equation} + \text{RGA}(G(f)) = G(f) \times (G(f)^{-1})^T +\end{equation} + +

+Then, the RGA number is defined: +

+\begin{equation} +\text{RGA-num}(f) = \| \text{I - RGA(G(f))} \|_{\text{sum}} +\end{equation} + + +

+In this section, the plant with 2 added mass is studied. +

+
+
+

3.3.1. Parameters

+
+
+
wc = 100; % Wanted crossover frequency [Hz]
+[~, i_wc] = min(abs(frf_iff.f - wc)); % Indice corresponding to wc
+
+
+ +
+
%% Plant to be decoupled
+frf_coupled = frf_iff.G_dL{2};
+G_coupled = sim_iff.G_dL{2};
+
+
+
+
+ +
+

3.3.2. No Decoupling (Decentralized)

+
+

+ +

+ + +
+

decoupling_arch_decentralized.png +

+

Figure 82: Block diagram representing the plant.

+
+ + +
+

interaction_decentralized_plant.png +

+

Figure 83: Bode Plot of the decentralized plant (diagonal and off-diagonal terms)

+
+ + +
+

interaction_rga_decentralized.png +

+

Figure 84: RGA number for the decentralized plant

+
+
+
+ +
+

3.3.3. Static Decoupling

+
+

+ +

+ + +
+

decoupling_arch_static.png +

+

Figure 85: Decoupling using the inverse of the DC gain of the plant

+
+ +

+The DC gain is evaluated from the model as be have bad low frequency identification. +

+ + + + +++ ++ ++ ++ ++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
-62011.53910.64299.3660.7-4016.5-4373.6
3914.4-61991.2-4356.8-4019.2640.24281.6
-4020.0-4370.5-62004.53914.64295.8653.8
660.94292.43903.3-62012.2-4366.5-4008.9
4302.8655.6-4025.8-4377.8-62006.03919.7
-4377.9-4013.2668.64303.73906.8-62019.3
+ + +
+

interaction_static_dec_plant.png +

+

Figure 86: Bode Plot of the static decoupled plant

+
+ + +
+

interaction_rga_static_dec.png +

+

Figure 87: RGA number for the statically decoupled plant

+
+
+
+ +
+

3.3.4. Decoupling at the Crossover

+
+

+ +

+ + +
+

decoupling_arch_crossover.png +

+

Figure 88: Decoupling using the inverse of a dynamical model \(\bm{\hat{G}}\) of the plant dynamics \(\bm{G}\)

+
+ + + + +++ ++ ++ ++ ++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
67229.83769.3-13704.6-23084.8-6318.223378.7
3486.267708.923220.0-6314.5-22699.8-14060.6
-5731.722471.766701.43070.2-13205.6-21944.6
-23305.5-14542.62743.270097.624846.8-5295.0
-14882.9-22957.8-5344.425786.270484.62979.9
24353.3-5195.2-22449.0-14459.22203.669484.2
+ + +
+

interaction_wc_plant.png +

+

Figure 89: Bode Plot of the plant decoupled at the crossover

+
+ +
+
%% Compute RGA Matrix
+RGA_wc = zeros(size(frf_coupled));
+for i = 1:length(frf_iff.f)
+    RGA_wc(i,:,:) = squeeze(G_dL_wc(i,:,:)).*inv(squeeze(G_dL_wc(i,:,:))).';
+end
+
+%% Compute RGA-number
+RGA_wc_sum = zeros(size(RGA_wc, 1), 1);
+for i = 1:size(RGA_wc, 1)
+    RGA_wc_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_wc(i,:,:)))));
+end
+
+
+ + +
+

interaction_rga_wc.png +

+

Figure 90: RGA number for the plant decoupled at the crossover

+
+
+
+ +
+

3.3.5. SVD Decoupling

+
+

+ +

+ + +
+

decoupling_arch_svd.png +

+

Figure 91: Decoupling using the Singular Value Decomposition

+
+ + +
+

interaction_svd_plant.png +

+

Figure 92: Bode Plot of the plant decoupled using the Singular Value Decomposition

+
+ +
+
%% Compute the RGA matrix for the SVD decoupled plant
+RGA_svd = zeros(size(frf_coupled));
+for i = 1:length(frf_iff.f)
+    RGA_svd(i,:,:) = squeeze(G_dL_svd(i,:,:)).*inv(squeeze(G_dL_svd(i,:,:))).';
+end
+
+%% Compute the RGA-number
+RGA_svd_sum = zeros(size(RGA_svd, 1), 1);
+for i = 1:length(frf_iff.f)
+    RGA_svd_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_svd(i,:,:)))));
+end
+
+
+ +
+
%% RGA Number for the SVD decoupled plant
+figure;
+plot(frf_iff.f, RGA_svd_sum, 'k-');
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); ylabel('RGA Number');
+xlim([10, 1e3]); ylim([1e-2, 1e2]);
+
+
+ + +
+

interaction_rga_svd.png +

+

Figure 93: RGA number for the plant decoupled using the SVD

+
+
+
+ +
+

3.3.6. Dynamic decoupling

+
+

+ +

+ + +
+

decoupling_arch_dynamic.png +

+

Figure 94: Decoupling using the inverse of a dynamical model \(\bm{\hat{G}}\) of the plant dynamics \(\bm{G}\)

+
+ + +
+

interaction_dynamic_dec_plant.png +

+

Figure 95: Bode Plot of the dynamically decoupled plant

+
+ + +
+

interaction_rga_dynamic_dec.png +

+

Figure 96: RGA number for the dynamically decoupled plant

+
+
+
+ +
+

3.3.7. Jacobian Decoupling - Center of Stiffness

+
+

+ +

+ + +
+

decoupling_arch_jacobian_cok.png +

+

Figure 97: Decoupling using Jacobian matrices evaluated at the Center of Stiffness

+
+ + +
+

interaction_J_cok_plant.png +

+

Figure 98: Bode Plot of the plant decoupled using the Jacobian evaluated at the “center of stiffness”

+
+ + +
+

interaction_rga_J_cok.png +

+

Figure 99: RGA number for the plant decoupled using the Jacobian evaluted at the Center of Stiffness

+
+
+
+ +
+

3.3.8. Jacobian Decoupling - Center of Mass

+
+

+ +

+ + +
+

decoupling_arch_jacobian_com.png +

+

Figure 100: Decoupling using Jacobian matrices evaluated at the Center of Mass

+
+ + +
+

interaction_J_com_plant.png +

+

Figure 101: Bode Plot of the plant decoupled using the Jacobian evaluated at the Center of Mass

+
+ + +
+

interaction_rga_J_com.png +

+

Figure 102: RGA number for the plant decoupled using the Jacobian evaluted at the Center of Mass

+
+
+
+ +
+

3.3.9. Decoupling Comparison

+
+

+ +

+ +

+Let’s now compare all of the decoupling methods (Figure 103). +

+ +
+

+From Figure 103, the following remarks are made: +

+
    +
  • Decentralized plant: well decoupled below suspension modes
    • +
  • Static inversion: similar to the decentralized plant as the decentralized plant has already a good decoupling at low frequency
    • +
  • Crossover inversion: the decoupling is improved around the crossover frequency as compared to the decentralized plant. However, the decoupling is increased at lower frequency.
    • +
  • SVD decoupling: Very good decoupling up to 235Hz. Especially between 100Hz and 200Hz.
    • +
  • Dynamic Inversion: the plant is very well decoupled at frequencies where the model is accurate (below 235Hz where flexible modes are not modelled).
    • +
  • Jacobian - Stiffness: good decoupling at low frequency. The decoupling increases at the frequency of the suspension modes, but is acceptable up to the strut flexible modes (235Hz).
    • +
  • Jacobian - Mass: bad decoupling at low frequency. Better decoupling above the frequency of the suspension modes, and acceptable decoupling up to the strut flexible modes (235Hz).
    • +
+ +
+ + +
+

interaction_compare_rga_numbers.png +

+

Figure 103: Comparison of the obtained RGA-numbers for all the decoupling methods

+
+
+
+ +
+

3.3.10. Decoupling Robustness

+
+

+ +

+ +

+Let’s now see how the decoupling is changing when changing the payload’s mass. +

+
+
frf_new = frf_iff.G_dL{3};
+
+
+ +

+The obtained RGA-numbers are shown in Figure 104. +

+ +
+

+From Figure 104: +

+
    +
  • The decoupling using the Jacobian evaluated at the “center of stiffness” seems to give the most robust results.
    • +
+ +
+ + +
+

interaction_compare_rga_numbers_rob.png +

+

Figure 104: Change of the RGA-number with a change of the payload. Indication of the robustness of the inversion method.

+
+
+
+ +
+

3.3.11. Conclusion

+
+
+

+Several decoupling methods can be used: +

+
    +
  • SVD
    • +
  • Inverse
    • +
  • Jacobian (CoK)
    • +
+ +
+ + + + +++ ++ ++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Table 3: Summary of the interaction analysis and different decoupling strategies
MethodRGADiag PlantRobustness
Decentralized--Equal++
Static dec.--Equal++
Crossover dec.-Equal0
SVD++Diff+
Dynamic dec.++Unity, equal-
Jacobian - CoK+Diff++
Jacobian - CoM0Diff+
+
+
+
+ +
+

3.4. Robust High Authority Controller

+
+

+ +

+

+In this section we wish to develop a robust High Authority Controller (HAC) that is working for all payloads. +

+ +

+(Indri and Oboe 2020) +

+
+
+

3.4.1. Using Jacobian evaluated at the center of stiffness

+
+
+
+
3.4.1.1. Decoupled Plant
+
+
+
G_nom = frf_iff.G_dL{2}; % Nominal Plant
+
+
+ + +
+

bode_plot_hac_iff_plant_jacobian_cok.png +

+

Figure 105: Bode plot of the decoupled plant using the Jacobian evaluated at the Center of Stiffness

+
+
+
+ +
+
3.4.1.2. SISO Controller Design
+
+

+As the diagonal elements of the plant are not equal, several SISO controllers are designed and then combined to form a diagonal controller. +All the diagonal terms of the controller consists of: +

+
    +
  • A double integrator to have high gain at low frequency
    • +
  • A lead around the crossover frequency to increase stability margins
    • +
  • Two second order low pass filters above the crossover frequency to increase the robustness to high frequency modes
    • +
+
+
+ +
+
3.4.1.3. Obtained Loop Gain
+
+ +
+

bode_plot_hac_iff_loop_gain_jacobian_cok.png +

+

Figure 106: Bode plot of the Loop Gain when using the Jacobian evaluated at the Center of Stiffness to decouple the system

+
+ +
+
%% Controller to be implemented
+Kd = inv(J_cok')*input_normalize*ss(Kd_diag)*inv(Js_cok);
+
+
+
+
+ +
+
3.4.1.4. Verification of the Stability
+
+

+Now the stability of the feedback loop is verified using the generalized Nyquist criteria. +

+ + +
+

loci_hac_iff_loop_gain_jacobian_cok.png +

+

Figure 107: Loci of \(L(j\omega)\) in the complex plane.

+
+
+
+ +
+
3.4.1.5. Save for further analysis
+
+
+
save('mat/Khac_iff_struts_jacobian_cok.mat', 'Kd')
+
+
+
+
+
+ +
+

3.4.2. Using Singular Value Decomposition

+
+
+
+
3.4.2.1. Decoupled Plant
+
+
+
G_nom = frf_iff.G_dL{2}; % Nominal Plant
+
+
+ + +
+

bode_plot_hac_iff_plant_svd.png +

+

Figure 108: Bode plot of the decoupled plant using the SVD

+
+
+
+ +
+
3.4.2.2. Controller Design
+
+
+
3.4.2.3. Loop Gain
+
+ +
+

bode_plot_hac_iff_loop_gain_svd.png +

+

Figure 109: Bode plot of Loop Gain when using the SVD

+
+
+
+ +
+
3.4.2.4. Stability Verification
+
+
+
%% Compute the Eigenvalues of the loop gain
+Ldet = zeros(3, 6, length(frf_iff.f));
+
+for i = 1:3
+    Lmimo = pagemtimes(permute(frf_iff.G_dL{i}, [2,3,1]),squeeze(freqresp(Kd, frf_iff.f, 'Hz')));
+    for i_f = 2:length(frf_iff.f)
+        Ldet(i,:, i_f) = eig(squeeze(Lmimo(:,:,i_f)));
+    end
+end
+
+
+ + +
+

loci_hac_iff_loop_gain_svd.png +

+

Figure 110: Locis of \(L(j\omega)\) in the complex plane.

+
+
+
+ +
+
3.4.2.5. Save for further analysis
+
+
+
save('mat/Khac_iff_struts_svd.mat', 'Kd')
+
+
+
+
+
+
+
+ +
+

4. Functions

+
+
+
+

4.1. generateXYZTrajectory

+
+

+ +

+
+ +
+

Function description

+
+
+
function [ref] = generateXYZTrajectory(args)
+% generateXYZTrajectory -
+%
+% Syntax: [ref] = generateXYZTrajectory(args)
+%
+% Inputs:
+%    - args
+%
+% Outputs:
+%    - ref - Reference Signal
+
+
+
+
+ +
+

Optional Parameters

+
+
+
arguments
+    args.points double {mustBeNumeric} = zeros(2, 3) % [m]
+
+    args.ti    (1,1) double {mustBeNumeric, mustBeNonnegative} = 1 % Time to go to first point and after last point [s]
+    args.tw    (1,1) double {mustBeNumeric, mustBeNonnegative} = 0.5 % Time wait between each point [s]
+    args.tm    (1,1) double {mustBeNumeric, mustBeNonnegative} = 1 % Motion time between points [s]
+
+    args.Ts    (1,1) double {mustBeNumeric, mustBePositive} = 1e-3 % Sampling Time [s]
+end
+
+
+
+
+ +
+

Initialize Time Vectors

+
+
+
time_i = 0:args.Ts:args.ti;
+time_w = 0:args.Ts:args.tw;
+time_m = 0:args.Ts:args.tm;
+
+
+
+
+ +
+

XYZ Trajectory

+
+
+
% Go to initial position
+xyz = (args.points(1,:))'*(time_i/args.ti);
+
+% Wait
+xyz = [xyz, xyz(:,end).*ones(size(time_w))];
+
+% Scans
+for i = 2:size(args.points, 1)
+    % Go to next point
+    xyz = [xyz, xyz(:,end) + (args.points(i,:)' - xyz(:,end))*(time_m/args.tm)];
+    % Wait a litle bit
+    xyz = [xyz, xyz(:,end).*ones(size(time_w))];
+end
+
+% End motion
+xyz = [xyz, xyz(:,end) - xyz(:,end)*(time_i/args.ti)];
+
+
+
+
+ +
+

Reference Signal

+
+
+
t = 0:args.Ts:args.Ts*(length(xyz) - 1);
+
+
+ +
+
ref = zeros(length(xyz), 7);
+
+ref(:, 1) = t;
+ref(:, 2:4) = xyz';
+
+
+
+
+
+ +
+

4.2. generateYZScanTrajectory

+
+

+ +

+
+ +
+

Function description

+
+
+
function [ref] = generateYZScanTrajectory(args)
+% generateYZScanTrajectory -
+%
+% Syntax: [ref] = generateYZScanTrajectory(args)
+%
+% Inputs:
+%    - args
+%
+% Outputs:
+%    - ref - Reference Signal
+
+
+
+
+ +
+

Optional Parameters

+
+
+
arguments
+    args.y_tot (1,1) double {mustBeNumeric, mustBePositive} = 10e-6 % [m]
+    args.z_tot (1,1) double {mustBeNumeric, mustBePositive} = 10e-6 % [m]
+
+    args.n     (1,1) double {mustBeInteger, mustBePositive} = 10 % [-]
+
+    args.Ts    (1,1) double {mustBeNumeric, mustBePositive} = 1e-4 % [s]
+
+    args.ti    (1,1) double {mustBeNumeric, mustBeNonnegative} = 1 % [s]
+    args.tw    (1,1) double {mustBeNumeric, mustBeNonnegative} = 1 % [s]
+    args.ty    (1,1) double {mustBeNumeric, mustBeNonnegative} = 1 % [s]
+    args.tz    (1,1) double {mustBeNumeric, mustBeNonnegative} = 1 % [s]
+end
+
+
+
+
+ +
+

Initialize Time Vectors

+
+
+
time_i = 0:args.Ts:args.ti;
+time_w = 0:args.Ts:args.tw;
+time_y = 0:args.Ts:args.ty;
+time_z = 0:args.Ts:args.tz;
+
+
+
+
+ +
+

Y and Z vectors

+
+
+
% Go to initial position
+y = (time_i/args.ti)*(args.y_tot/2);
+
+% Wait
+y = [y, y(end)*ones(size(time_w))];
+
+% Scans
+for i = 1:args.n
+    if mod(i,2) == 0
+        y = [y, -(args.y_tot/2) + (time_y/args.ty)*args.y_tot];
+    else
+        y = [y,  (args.y_tot/2) - (time_y/args.ty)*args.y_tot];
+    end
+
+    if i < args.n
+        y = [y, y(end)*ones(size(time_z))];
+    end
+end
+
+% Wait a litle bit
+y = [y, y(end)*ones(size(time_w))];
+
+% End motion
+y = [y, y(end) - y(end)*time_i/args.ti];
+
+
+ +
+
% Go to initial position
+z = (time_i/args.ti)*(args.z_tot/2);
+
+% Wait
+z = [z, z(end)*ones(size(time_w))];
+
+% Scans
+for i = 1:args.n
+    z = [z, z(end)*ones(size(time_y))];
+
+    if i < args.n
+        z = [z, z(end) - (time_z/args.tz)*args.z_tot/(args.n-1)];
+    end
+end
+
+% Wait a litle bit
+z = [z, z(end)*ones(size(time_w))];
+
+% End motion
+z = [z, z(end) - z(end)*time_i/args.ti];
+
+
+
+
+ +
+

Reference Signal

+
+
+
t = 0:args.Ts:args.Ts*(length(y) - 1);
+
+
+ +
+
ref = zeros(length(y), 7);
+
+ref(:, 1) = t;
+ref(:, 3) = y;
+ref(:, 4) = z;
+
+
+
+
+
+ +
+

4.3. generateSpiralAngleTrajectory

+
+

+ +

+
+ +
+

Function description

+
+
+
function [ref] = generateSpiralAngleTrajectory(args)
+% generateSpiralAngleTrajectory -
+%
+% Syntax: [ref] = generateSpiralAngleTrajectory(args)
+%
+% Inputs:
+%    - args
+%
+% Outputs:
+%    - ref - Reference Signal
+
+
+
+
+ +
+

Optional Parameters

+
+
+
arguments
+    args.R_tot  (1,1) double {mustBeNumeric, mustBePositive} = 10e-6 % [rad]
+    args.n_turn (1,1) double {mustBeInteger, mustBePositive} = 5 % [-]
+    args.Ts     (1,1) double {mustBeNumeric, mustBePositive} = 1e-3 % [s]
+    args.t_turn (1,1) double {mustBeNumeric, mustBePositive} = 1 % [s]
+    args.t_end  (1,1) double {mustBeNumeric, mustBePositive} = 1 % [s]
+end
+
+
+
+
+ +
+

Initialize Time Vectors

+
+
+
time_s = 0:args.Ts:args.n_turn*args.t_turn;
+time_e = 0:args.Ts:args.t_end;
+
+
+
+
+ +
+

Rx and Ry vectors

+
+
+
Rx = sin(2*pi*time_s/args.t_turn).*(args.R_tot*time_s/(args.n_turn*args.t_turn));
+Ry = cos(2*pi*time_s/args.t_turn).*(args.R_tot*time_s/(args.n_turn*args.t_turn));
+
+
+ +
+
Rx = [Rx, 0*time_e];
+Ry = [Ry, Ry(end) - Ry(end)*time_e/args.t_end];
+
+
+
+
+ +
+

Reference Signal

+
+
+
t = 0:args.Ts:args.Ts*(length(Rx) - 1);
+
+
+ +
+
ref = zeros(length(Rx), 7);
+
+ref(:, 1) = t;
+ref(:, 5) = Rx;
+ref(:, 6) = Ry;
+
+
+
+
+
+ +
+

4.4. getTransformationMatrixAcc

+
+

+ +

+
+ +
+

Function description

+
+
+
function [M] = getTransformationMatrixAcc(Opm, Osm)
+% getTransformationMatrixAcc -
+%
+% Syntax: [M] = getTransformationMatrixAcc(Opm, Osm)
+%
+% Inputs:
+%    - Opm - Nx3 (N = number of accelerometer measurements) X,Y,Z position of accelerometers
+%    - Opm - Nx3 (N = number of accelerometer measurements) Unit vectors representing the accelerometer orientation
+%
+% Outputs:
+%    - M - Transformation Matrix
+
+
+
+
+ +
+

Transformation matrix from motion of the solid body to accelerometer measurements

+
+

+Let’s try to estimate the x-y-z acceleration of any point of the solid body from the acceleration/angular acceleration of the solid body expressed in \(\{O\}\). +For any point \(p_i\) of the solid body (corresponding to an accelerometer), we can write: +

+\begin{equation} +\begin{bmatrix} +a_{i,x} \\ a_{i,y} \\ a_{i,z} +\end{bmatrix} = \begin{bmatrix} +\dot{v}_x \\ \dot{v}_y \\ \dot{v}_z +\end{bmatrix} + p_i \times \begin{bmatrix} +\dot{\omega}_x \\ \dot{\omega}_y \\ \dot{\omega}_z +\end{bmatrix} +\end{equation} + +

+We can write the cross product as a matrix product using the skew-symmetric transformation: +

+\begin{equation} +\begin{bmatrix} +a_{i,x} \\ a_{i,y} \\ a_{i,z} +\end{bmatrix} = \begin{bmatrix} +\dot{v}_x \\ \dot{v}_y \\ \dot{v}_z +\end{bmatrix} + \underbrace{\begin{bmatrix} + 0 & p_{i,z} & -p_{i,y} \\ + -p_{i,z} & 0 & p_{i,x} \\ + p_{i,y} & -p_{i,x} & 0 +\end{bmatrix}}_{P_{i,[\times]}} \cdot \begin{bmatrix} +\dot{\omega}_x \\ \dot{\omega}_y \\ \dot{\omega}_z +\end{bmatrix} +\end{equation} + +

+If we now want to know the (scalar) acceleration \(a_i\) of the point \(p_i\) in the direction of the accelerometer direction \(\hat{s}_i\), we can just project the 3d acceleration on \(\hat{s}_i\): +

+\begin{equation} +a_i = \hat{s}_i^T \cdot \begin{bmatrix} +a_{i,x} \\ a_{i,y} \\ a_{i,z} +\end{bmatrix} = \hat{s}_i^T \cdot \begin{bmatrix} +\dot{v}_x \\ \dot{v}_y \\ \dot{v}_z +\end{bmatrix} + \left( \hat{s}_i^T \cdot P_{i,[\times]} \right) \cdot \begin{bmatrix} +\dot{\omega}_x \\ \dot{\omega}_y \\ \dot{\omega}_z +\end{bmatrix} +\end{equation} + +

+Which is equivalent as a simple vector multiplication: +

+\begin{equation} +a_i = \begin{bmatrix} +\hat{s}_i^T & \hat{s}_i^T \cdot P_{i,[\times]} +\end{bmatrix} +\begin{bmatrix} +\dot{v}_x \\ \dot{v}_y \\ \dot{v}_z \\ \dot{\omega}_x \\ \dot{\omega}_y \\ \dot{\omega}_z +\end{bmatrix} = \begin{bmatrix} +\hat{s}_i^T & \hat{s}_i^T \cdot P_{i,[\times]} +\end{bmatrix} {}^O\vec{x} +\end{equation} + +

+And finally we can combine the 6 (line) vectors for the 6 accelerometers to write that in a matrix form. +We obtain Eq. \eqref{eq:M_matrix}. +

+
+

+The transformation from solid body acceleration \({}^O\vec{x}\) from sensor measured acceleration \(\vec{a}\) is: +

+\begin{equation} \label{eq:M_matrix} +\vec{a} = \underbrace{\begin{bmatrix} +\hat{s}_1^T & \hat{s}_1^T \cdot P_{1,[\times]} \\ +\vdots & \vdots \\ +\hat{s}_6^T & \hat{s}_6^T \cdot P_{6,[\times]} +\end{bmatrix}}_{M} {}^O\vec{x} +\end{equation} + +

+with \(\hat{s}_i\) the unit vector representing the measured direction of the i’th accelerometer expressed in frame \(\{O\}\) and \(P_{i,[\times]}\) the skew-symmetric matrix representing the cross product of the position of the i’th accelerometer expressed in frame \(\{O\}\). +

+ +
+ +

+Let’s define such matrix using matlab: +

+
+
M = zeros(length(Opm), 6);
+
+for i = 1:length(Opm)
+    Ri = [0,         Opm(3,i), -Opm(2,i);
+         -Opm(3,i),  0,         Opm(1,i);
+          Opm(2,i), -Opm(1,i),  0];
+    M(i, 1:3) = Osm(:,i)';
+    M(i, 4:6) = Osm(:,i)'*Ri;
+end
+
+
+ +
+
end
+
+
+
+
+
+ + +
+

4.5. getJacobianNanoHexapod

+
+

+ +

+
+ +
+

Function description

+
+
+
function [J] = getJacobianNanoHexapod(Hbm)
+% getJacobianNanoHexapod -
+%
+% Syntax: [J] = getJacobianNanoHexapod(Hbm)
+%
+% Inputs:
+%    - Hbm - Height of {B} w.r.t. {M} [m]
+%
+% Outputs:
+%    - J - Jacobian Matrix
+
+
+
+
+ +
+

Transformation matrix from motion of the solid body to accelerometer measurements

+
+
+
Fa = [[-86.05,  -74.78, 22.49],
+      [ 86.05,  -74.78, 22.49],
+      [ 107.79, -37.13, 22.49],
+      [ 21.74,  111.91, 22.49],
+      [-21.74,  111.91, 22.49],
+      [-107.79, -37.13, 22.49]]'*1e-3; % Ai w.r.t. {F} [m]
+
+Mb = [[-28.47, -106.25, -22.50],
+      [ 28.47, -106.25, -22.50],
+      [ 106.25, 28.47,  -22.50],
+      [ 77.78,  77.78,  -22.50],
+      [-77.78,  77.78,  -22.50],
+      [-106.25, 28.47,  -22.50]]'*1e-3; % Bi w.r.t. {M} [m]
+
+H = 95e-3; % Stewart platform height [m]
+Fb = Mb + [0; 0; H]; % Bi w.r.t. {F} [m]
+
+si = Fb - Fa;
+si = si./vecnorm(si); % Normalize
+
+Bb = Mb - [0; 0; Hbm];
+
+J = [si', cross(Bb, si)'];
+
+
+
+
+
+
+ +

Bibliography

+
+
Indri, Marina, and Roberto Oboe. 2020. Mechatronics and Robotics: New Trends and Challenges. CRC Press.
+
Skogestad, Sigurd, and Ian Postlethwaite. 2007. Multivariable Feedback Control: Analysis and Design - Second Edition. John Wiley.
+
+
+
+

Author: Dehaeze Thomas

+

Created: 2021-08-11 mer. 00:07

+
+ + diff --git a/test-bench-nano-hexapod.org b/test-bench-nano-hexapod.org new file mode 100644 index 0000000..03fd5a1 --- /dev/null +++ b/test-bench-nano-hexapod.org @@ -0,0 +1,8898 @@ +#+TITLE: Nano-Hexapod - Test Bench +:DRAWER: +#+LANGUAGE: en +#+EMAIL: dehaeze.thomas@gmail.com +#+AUTHOR: Dehaeze Thomas + +#+HTML_LINK_HOME: ../index.html +#+HTML_LINK_UP: ../index.html + +#+HTML_HEAD: +#+HTML_HEAD: + +#+BIND: org-latex-image-default-option "scale=1" +#+BIND: org-latex-image-default-width "" + +#+LaTeX_CLASS: scrreprt +#+LaTeX_CLASS_OPTIONS: [a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc] +#+LaTeX_HEADER_EXTRA: \input{preamble.tex} +#+LATEX_HEADER_EXTRA: \bibliography{test-bench-nano-hexapod.bib} + +#+BIND: org-latex-bib-compiler "biber" + +#+PROPERTY: header-args:matlab :session *MATLAB* +#+PROPERTY: header-args:matlab+ :comments org +#+PROPERTY: header-args:matlab+ :exports none +#+PROPERTY: header-args:matlab+ :results none +#+PROPERTY: header-args:matlab+ :eval no-export +#+PROPERTY: header-args:matlab+ :noweb yes +#+PROPERTY: header-args:matlab+ :mkdirp yes +#+PROPERTY: header-args:matlab+ :output-dir figs +#+PROPERTY: header-args:matlab+ :tangle no + +#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/tikz/org/}{config.tex}") +#+PROPERTY: header-args:latex+ :imagemagick t :fit yes +#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150 +#+PROPERTY: header-args:latex+ :imoutoptions -quality 100 +#+PROPERTY: header-args:latex+ :results file raw replace +#+PROPERTY: header-args:latex+ :buffer no +#+PROPERTY: header-args:latex+ :tangle no +#+PROPERTY: header-args:latex+ :eval no-export +#+PROPERTY: header-args:latex+ :exports results +#+PROPERTY: header-args:latex+ :mkdirp yes +#+PROPERTY: header-args:latex+ :output-dir figs +#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png") +:END: + +#+begin_export html +
+

This report is also available as a pdf.

+
+#+end_export + +#+latex: \clearpage + +* Build :noexport: +#+NAME: startblock +#+BEGIN_SRC emacs-lisp :results none :tangle no +(add-to-list 'org-latex-classes + '("scrreprt" + "\\documentclass{scrreprt}" + ("\\chapter{%s}" . "\\chapter*{%s}") + ("\\section{%s}" . "\\section*{%s}") + ("\\subsection{%s}" . "\\subsection*{%s}") + ("\\paragraph{%s}" . "\\paragraph*{%s}") + )) + + +;; Remove automatic org heading labels +(defun my-latex-filter-removeOrgAutoLabels (text backend info) + "Org-mode automatically generates labels for headings despite explicit use of `#+LABEL`. This filter forcibly removes all automatically generated org-labels in headings." + (when (org-export-derived-backend-p backend 'latex) + (replace-regexp-in-string "\\\\label{sec:org[a-f0-9]+}\n" "" text))) +(add-to-list 'org-export-filter-headline-functions + 'my-latex-filter-removeOrgAutoLabels) + +;; Remove all org comments in the output LaTeX file +(defun delete-org-comments (backend) + (loop for comment in (reverse (org-element-map (org-element-parse-buffer) + 'comment 'identity)) + do + (setf (buffer-substring (org-element-property :begin comment) + (org-element-property :end comment)) + ""))) +(add-hook 'org-export-before-processing-hook 'delete-org-comments) + +;; Use no package by default +(setq org-latex-packages-alist nil) +(setq org-latex-default-packages-alist nil) + +;; Do not include the subtitle inside the title +(setq org-latex-subtitle-separate t) +(setq org-latex-subtitle-format "\\subtitle{%s}") + +(setq org-export-before-parsing-hook '(org-ref-glossary-before-parsing + org-ref-acronyms-before-parsing)) +#+END_SRC + +* Notes :noexport: + +Add these documents: +- [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/nano_hexapod.org][nano_hexapod]] +- [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-nano-hexapod-assembly/nass-nano-hexapod-assembly.org][nass-nano-hexapod-assembly]] +- [[file:~/Cloud/work-projects/ID31-NASS/matlab/test-bench-vibration-table/vibration-table.org][test-bench-vibration-table]] + +* Introduction :ignore: +This document is dedicated to the experimental study of the nano-hexapod shown in Figure ref:fig:picture_bench_granite_nano_hexapod. + +#+name: fig:picture_bench_granite_nano_hexapod +#+caption: Nano-Hexapod +#+attr_latex: :width \linewidth +[[file:figs/IMG_20210608_152917.jpg]] + +#+begin_note +Here are the documentation of the equipment used for this test bench (lots of them are shwon in Figure ref:fig:picture_bench_granite_overview): +- Voltage Amplifier: PiezoDrive [[file:doc/PD200-V7-R1.pdf][PD200]] +- Amplified Piezoelectric Actuator: Cedrat [[file:doc/APA300ML.pdf][APA300ML]] +- DAC/ADC: Speedgoat [[file:doc/IO131-OEM-Datasheet.pdf][IO313]] +- Encoder: Renishaw [[file:doc/L-9517-9678-05-A_Data_sheet_VIONiC_series_en.pdf][Vionic]] and used [[file:doc/L-9517-9862-01-C_Data_sheet_RKLC_EN.pdf][Ruler]] +- Interferometers: Attocube +#+end_note + +#+name: fig:picture_bench_granite_overview +#+caption: Nano-Hexapod and the control electronics +#+attr_latex: :width \linewidth +[[file:figs/IMG_20210608_154722.jpg]] + +In Figure ref:fig:nano_hexapod_signals is shown a block diagram of the experimental setup. +When possible, the notations are consistent with this diagram and summarized in Table ref:tab:list_signals. + +#+begin_src latex :file nano_hexapod_signals.pdf +\definecolor{instrumentation}{rgb}{0, 0.447, 0.741} +\definecolor{mechanics}{rgb}{0.8500, 0.325, 0.098} + +\begin{tikzpicture} + % Blocs + \node[block={4.0cm}{3.0cm}, fill=mechanics!20!white] (nano_hexapod) {Mechanics}; + \coordinate[] (inputF) at (nano_hexapod.west); + \coordinate[] (outputL) at ($(nano_hexapod.south east)!0.8!(nano_hexapod.north east)$); + \coordinate[] (outputF) at ($(nano_hexapod.south east)!0.2!(nano_hexapod.north east)$); + + \node[block, left= 0.8 of inputF, fill=instrumentation!20!white, align=center] (F_stack) {\tiny Actuator \\ \tiny stacks}; + \node[block, left= 0.8 of F_stack, fill=instrumentation!20!white] (PD200) {PD200}; + \node[DAC, left= 0.8 of PD200, fill=instrumentation!20!white] (F_DAC) {DAC}; + \node[block, right=0.8 of outputF, fill=instrumentation!20!white, align=center] (Fm_stack){\tiny Sensor \\ \tiny stack}; + \node[ADC, right=0.8 of Fm_stack,fill=instrumentation!20!white] (Fm_ADC) {ADC}; + \node[block, right=0.8 of outputL, fill=instrumentation!20!white] (encoder) {\tiny Encoder}; + + % Connections and labels + \draw[->] ($(F_DAC.west)+(-0.8,0)$) node[above right]{$\bm{u}$} node[below right]{$[V]$} -- node[sloped]{$/$} (F_DAC.west); + \draw[->] (F_DAC.east) -- node[midway, above]{$\tilde{\bm{u}}$}node[midway, below]{$[V]$} (PD200.west); + \draw[->] (PD200.east) -- node[midway, above]{$\bm{u}_a$}node[midway, below]{$[V]$} (F_stack.west); + \draw[->] (F_stack.east) -- (inputF) node[above left]{$\bm{\tau}$}node[below left]{$[N]$}; + + \draw[->] (outputF) -- (Fm_stack.west) node[above left]{$\bm{\epsilon}$} node[below left]{$[m]$}; + \draw[->] (Fm_stack.east) -- node[midway, above]{$\tilde{\bm{\tau}}_m$}node[midway, below]{$[V]$} (Fm_ADC.west); + \draw[->] (Fm_ADC.east) -- node[sloped]{$/$} ++(0.8, 0)coordinate(end) node[above left]{$\bm{\tau}_m$}node[below left]{$[V]$}; + + \draw[->] (outputL) -- (encoder.west) node[above left]{$d\bm{\mathcal{L}}$} node[below left]{$[m]$}; + \draw[->] (encoder.east) -- node[sloped]{$/$} (encoder-|end) node[above left]{$d\bm{\mathcal{L}}_m$}node[below left]{$[m]$}; + + % Nano-Hexapod + \begin{scope}[on background layer] + \node[fit={(F_stack.west|-nano_hexapod.south) (Fm_stack.east|-nano_hexapod.north)}, fill=black!20!white, draw, inner sep=2pt] (system) {}; + \node[above] at (system.north) {Nano-Hexapod}; + \end{scope} +\end{tikzpicture} +#+end_src + +#+name: fig:nano_hexapod_signals +#+caption: Block diagram of the system with named signals +#+attr_latex: :scale 1 +[[file:figs/nano_hexapod_signals.png]] + +#+name: tab:list_signals +#+caption: List of signals +#+attr_latex: :environment tabularx :width \linewidth :align Xllll +#+attr_latex: :center t :booktabs t +| | *Unit* | *Matlab* | *Vector* | *Elements* | +|------------------------------------+-----------+-----------+-----------------------+----------------------| +| Control Input (wanted DAC voltage) | =[V]= | =u= | $\bm{u}$ | $u_i$ | +| DAC Output Voltage | =[V]= | =u= | $\tilde{\bm{u}}$ | $\tilde{u}_i$ | +| PD200 Output Voltage | =[V]= | =ua= | $\bm{u}_a$ | $u_{a,i}$ | +| Actuator applied force | =[N]= | =tau= | $\bm{\tau}$ | $\tau_i$ | +|------------------------------------+-----------+-----------+-----------------------+----------------------| +| Strut motion | =[m]= | =dL= | $d\bm{\mathcal{L}}$ | $d\mathcal{L}_i$ | +| Encoder measured displacement | =[m]= | =dLm= | $d\bm{\mathcal{L}}_m$ | $d\mathcal{L}_{m,i}$ | +|------------------------------------+-----------+-----------+-----------------------+----------------------| +| Force Sensor strain | =[m]= | =epsilon= | $\bm{\epsilon}$ | $\epsilon_i$ | +| Force Sensor Generated Voltage | =[V]= | =taum= | $\tilde{\bm{\tau}}_m$ | $\tilde{\tau}_{m,i}$ | +| Measured Generated Voltage | =[V]= | =taum= | $\bm{\tau}_m$ | $\tau_{m,i}$ | +|------------------------------------+-----------+-----------+-----------------------+----------------------| +| Motion of the top platform | =[m,rad]= | =dX= | $d\bm{\mathcal{X}}$ | $d\mathcal{X}_i$ | +| Metrology measured displacement | =[m,rad]= | =dXm= | $d\bm{\mathcal{X}}_m$ | $d\mathcal{X}_{m,i}$ | + +This document is divided in the following sections: +- Section ref:sec:encoders_struts: the dynamics of the nano-hexapod when the encoders are fixed to the struts is studied. +- Section ref:sec:encoders_plates: the same is done when the encoders are fixed to the plates. +- Section ref:sec:decentralized_hac_iff: a decentralized HAC-LAC strategy is studied and implemented. + +* Encoders fixed to the struts +<> + +** Introduction :ignore: +In this section, the encoders are fixed to the struts. + +It is divided in the following sections: +- Section ref:sec:enc_struts_plant_id: the transfer function matrix from the actuators to the force sensors and to the encoders is experimentally identified. +- Section ref:sec:enc_struts_comp_simscape: the obtained FRF matrix is compared with the dynamics of the simscape model +- Section ref:sec:enc_struts_iff: decentralized Integral Force Feedback (IFF) is applied and its performances are evaluated. +- Section ref:sec:enc_struts_modal_analysis: a modal analysis of the nano-hexapod is performed + +** Identification of the dynamics +:PROPERTIES: +:header-args:matlab+: :tangle matlab/scripts/id_frf_enc_struts.m +:END: +<> +*** Introduction :ignore: + +*** Matlab Init :noexport:ignore: +#+begin_src matlab +%% id_frf_enc_struts.m +% Identification of the nano-hexapod dynamics from u to dL and to taum +% Encoders are fixed to the struts +#+end_src + +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) +<> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes +<> +#+end_src + +#+begin_src matlab :tangle no :noweb yes +<> +#+end_src + +#+begin_src matlab :eval no :noweb yes +<> +#+end_src + +#+begin_src matlab :noweb yes +<> +#+end_src + +*** Load Measurement Data +Two identifications are performed, one for low frequencies and one for higher frequencies. + +#+begin_src matlab +%% Load Identification Data +meas_data_lf = {}; + +for i = 1:6 + meas_data_lf(i) = {load(sprintf('frf_data_exc_strut_%i_noise_lf.mat', i), 't', 'Va', 'Vs', 'de')}; + meas_data_hf(i) = {load(sprintf('frf_data_exc_strut_%i_noise_hf.mat', i), 't', 'Va', 'Vs', 'de')}; +end +#+end_src + +*** Spectral Analysis - Setup +The window used for the spectral analysis is defined as well as the frequency vector =f=. + +#+begin_src matlab +%% Setup useful variables +% Sampling Time [s] +Ts = (meas_data_lf{1}.t(end) - (meas_data_lf{1}.t(1)))/(length(meas_data_lf{1}.t)-1); + +% Sampling Frequency [Hz] +Fs = 1/Ts; + +% Hannning Windows +win = hanning(ceil(1*Fs)); + +% And we get the frequency vector +[~, f] = tfestimate(meas_data_lf{1}.Va, meas_data_lf{1}.de, win, [], [], 1/Ts); + +i_lf = f < 250; % Points for low frequency excitation +i_hf = f > 250; % Points for high frequency excitation +#+end_src + +*** Transfer function from Actuator to Encoder +The 6x6 transfer function matrix from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ is estimated and shown in Figure ref:fig:enc_struts_dvf_frf. +#+begin_src matlab +%% Transfer function from u to dLm +G_dL = zeros(length(f), 6, 6); + +for i = 1:6 + G_dL_lf = tfestimate(meas_data_lf{i}.Va, meas_data_lf{i}.de, win, [], [], 1/Ts); + G_dL_hf = tfestimate(meas_data_hf{i}.Va, meas_data_hf{i}.de, win, [], [], 1/Ts); + G_dL(:,:,i) = [G_dL_lf(i_lf, :); G_dL_hf(i_hf, :)]; +end +#+end_src + +#+begin_src matlab :exports none +%% Bode plot for the transfer function from u to dLm +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(f, abs(G_dL(:, i, j)), 'color', [0, 0, 0, 0.2], ... + 'HandleVisibility', 'off'); + end +end +for i =1:6 + set(gca,'ColorOrderIndex',i) + plot(f, abs(G_dL(:,i, i)), ... + 'DisplayName', sprintf('$d\\mathcal{L}_{m,%i}/u_{%i}$', i, i)); + set(gca,'ColorOrderIndex',i) + plot(f, abs(G_dL(:,i, i)), ... + 'HandleVisibility', 'off'); +end +plot(f, abs(G_dL(:, 1, 2)), 'color', [0, 0, 0, 0.2], ... + 'DisplayName', '$d\mathcal{L}_{m,i}/u_{j}$'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); +ylim([1e-9, 1e-3]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); + +ax2 = nexttile; +hold on; +for i =1:6 + set(gca,'ColorOrderIndex',i) + plot(f, 180/pi*angle(G_dL(:,i, i))); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); + +linkaxes([ax1,ax2],'x'); +xlim([20, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_struts_dvf_frf.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:enc_struts_dvf_frf +#+caption: Measured FRF for the transfer function from $\bm{u}$ to $d\bm{\mathcal{L}_m}$ +#+RESULTS: +[[file:figs/enc_struts_dvf_frf.png]] + +#+begin_important +The low frequency gain of the transfer function from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ is close to diagonal. +Then the coupling is quite large starting from the frequency of the first suspension mode (110Hz to 180Hz). +Additional modes at 220Hz, 300Hz and 380Hz which are the modes of the struts are adding a lot of complexity in the plant and phase drops. +Then there is a mode at 700Hz which correspond to a flexible mode of the top plate. +#+end_important + +*** Transfer function from Actuator to Force Sensor +The 6x6 transfer function matrix from $\bm{u}$ to $\bm{\tau}_m$ is estimated and shown in Figure ref:fig:enc_struts_iff_frf. +#+begin_src matlab +%% Transfer function from u to taum +G_tau = zeros(length(f), 6, 6); + +for i = 1:6 + G_tau_lf = tfestimate(meas_data_lf{i}.Va, meas_data_lf{i}.Vs, win, [], [], 1/Ts); + G_tau_hf = tfestimate(meas_data_hf{i}.Va, meas_data_hf{i}.Vs, win, [], [], 1/Ts); + G_tau(:,:,i) = [G_tau_lf(i_lf, :); G_tau_hf(i_hf, :)]; +end +#+end_src + +#+begin_src matlab :exports none +%% Bode plot of the transfer function from u to taum +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(f, abs(G_tau(:, i, j)), 'color', [0, 0, 0, 0.2], ... + 'HandleVisibility', 'off'); + end +end +for i =1:6 + set(gca,'ColorOrderIndex',i) + plot(f, abs(G_tau(:,i , i)), ... + 'DisplayName', sprintf('$\\tau_{m,%i}/u_{%i}$', i, i)); +end +plot(f, abs(G_tau(:, 1, 2)), 'color', [0, 0, 0, 0.2], ... + 'DisplayName', '$\tau_{m,i}/u_{j}$'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); +ylim([1e-3, 1e2]); + +ax2 = nexttile; +hold on; +for i =1:6 + set(gca,'ColorOrderIndex',i) + plot(f, 180/pi*angle(G_tau(:,i, i))); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); + +linkaxes([ax1,ax2],'x'); +xlim([20, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_struts_iff_frf.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:enc_struts_iff_frf +#+caption: Measured FRF for the transfer function from $\bm{u}$ to $\bm{\tau}_m$ +#+RESULTS: +[[file:figs/enc_struts_iff_frf.png]] + +#+begin_important +The transfer function matrix from $\bm{u}$ to $\bm{\tau}_m$ in Figure ref:fig:enc_struts_iff_frf has alternating poles and zeros as expected. +The 4 suspensions modes have quite separated poles and zeros which indicates that is should be possible to add a good amount of damping to these modes using Integral Force Feedback. +#+end_important + +*** Save Identified Plants +The identified plants are now saved for further analysis. +#+begin_src matlab :exports none :tangle no +save('matlab/data_frf/identified_plants_enc_struts.mat', 'f', 'Ts', 'G_tau', 'G_dL') +#+end_src + +#+begin_src matlab :eval no +save('data_frf/identified_plants_enc_struts.mat', 'f', 'Ts', 'G_tau', 'G_dL') +#+end_src + +** Jacobian matrices and centralized plants +:PROPERTIES: +:header-args:matlab+: :tangle matlab/scripts/frf_centralized_struts.m +:END: +*** Introduction :ignore: +The Jacobian is used to transform the excitation force in the cartesian frame as well as the displacements. + +Consider the plant shown in Figure ref:fig:schematic_jacobian_in_out with: +- $\bm{u}$ the 6 input voltages (going to the PD200 amplifier and then to the APA) +- $d\bm{\mathcal{L}}$ the relative motion sensor outputs (encoders) +- $\bm{\tau}_m$ the generated voltage of the force sensor stacks +- $J_a$ and $J_s$ the Jacobians for the actuators and sensors + +#+begin_src latex :file schematic_jacobian_in_out.pdf +\begin{tikzpicture} + % Blocs + \node[block={2.0cm}{2.0cm}] (P) {Plant}; + \coordinate[] (inputF) at (P.west); + \coordinate[] (outputL) at ($(P.south east)!0.8!(P.north east)$); + \coordinate[] (outputF) at ($(P.south east)!0.2!(P.north east)$); + + \node[block, left= of inputF] (Ja) {$\bm{J}^{-T}_a$}; + \node[block, right= of outputL] (Js) {$\bm{J}^{-1}_s$}; + \node[block, right= of outputF] (Jf) {$\bm{J}^{T}_a$}; + + % Connections and labels + \draw[->] ($(Ja.west)+(-1,0)$) -- (Ja.west) node[above left]{$\bm{\mathcal{F}}$}; + \draw[->] (Ja.east) -- (inputF) node[above left]{$\bm{u}$}; + \draw[->] (outputL) -- (Js.west) node[above left]{$d\bm{\mathcal{L}}$}; + \draw[->] (Js.east) -- ++(1, 0) node[above left]{$d\bm{\mathcal{X}}$}; + \draw[->] (outputF) -- (Jf.west) node[above left]{$\bm{\tau}_m$}; + \draw[->] (Jf.east) -- ++(1, 0) node[above left]{$\bm{\mathcal{F}}_m$}; +\end{tikzpicture} +#+end_src + +#+name: fig:schematic_jacobian_in_out +#+caption: Plant in the cartesian Frame +#+RESULTS: +[[file:figs/schematic_jacobian_in_out.png]] + +*** Matlab Init :noexport:ignore: +#+begin_src matlab +%% id_frf_enc_struts.m +% Compute centralized plant using the Jacobian matrices +#+end_src + +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) +<> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes +<> +#+end_src + +#+begin_src matlab :tangle no :noweb yes +<> +#+end_src + +#+begin_src matlab :eval no :noweb yes +<> +#+end_src + +#+begin_src matlab :noweb yes +<> +#+end_src + +#+begin_src matlab +%% Load identified FRF +load('identified_plants_enc_struts.mat', 'f', 'Ts', 'G_tau', 'G_dL') +#+end_src + +*** Jacobian Matrices +The Jacobian matrices (for the sensors and for the actuators) are estimated in two cases: +- when the point of interest is at 150mm above the top platform +- when the point of interest is at the "center of stiffness" (42mm below the top platform) + +#+begin_src matlab +%% Initialize the Hexapod with the Interest point 150mm above the top platform +n_hexapod = initializeNanoHexapodFinal('MO_B', 150e-3, ... + 'motion_sensor_type', 'struts'); + +% Jacobian matrices +Ja_coi = n_hexapod.geometry.J; +Js_coi = n_hexapod.geometry.Js; + +%% Initialize the Hexapod with Interest point at the Center of Stiffness +n_hexapod = initializeNanoHexapodFinal('MO_B', -42e-3, ... + 'motion_sensor_type', 'struts'); + +% Jacobian matrices +Ja_cok = n_hexapod.geometry.J; +Js_cok = n_hexapod.geometry.Js; +#+end_src + +*** Obtained Centralized plants +The transfer function matrices from $\bm{\mathcal{F}}$ to $d\bm{\mathcal{X}}$ are computed for both Jacobians and shown in Figures ref:fig:enc_struts_cok_dvf_cart_frf (Center of Stiffness) and ref:fig:enc_struts_coi_dvf_cart_frf (Point of Interest). + +#+begin_src matlab +%% "Centralized" plant at the "center of stiffness" +G_dL_J_cok = permute(pagemtimes(inv(Js_cok), pagemtimes(permute(G_dL, [2 3 1]), inv(Ja_cok'))), [3 1 2]); + +%% "Centralized" plant at the "point of interest" +G_dL_J_coi = permute(pagemtimes(inv(Js_coi), pagemtimes(permute(G_dL, [2 3 1]), inv(Ja_coi'))), [3 1 2]); +#+end_src + +#+begin_src matlab :exports none +%% Bode plot of the transfer function from F to dX using the Jacobians (CoK) +labels = {'$D_x/F_{x}$', '$D_y/F_{y}$', '$D_z/F_{z}$', '$R_{x}/M_{x}$', '$R_{y}/M_{y}$', '$R_{R}/M_{z}$'}; + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(f, abs(G_dL_J_cok(:, i, j)), 'color', [0, 0, 0, 0.2], ... + 'HandleVisibility', 'off'); + end +end +for i =1:6 + set(gca,'ColorOrderIndex',i) + plot(f, abs(G_dL_J_cok(:,i , i)), ... + 'DisplayName', labels{i}); +end +plot(f, abs(G_dL_J_cok(:, 1, 2)), 'color', [0, 0, 0, 0.2], ... + 'DisplayName', '$D_i/F_j$'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude'); set(gca, 'XTickLabel',[]); +ylim([1e-7, 2e-1]); +legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 3); + +ax2 = nexttile; +hold on; +for i =1:6 + set(gca,'ColorOrderIndex',i) + plot(f, 180/pi*angle(G_dL_J_cok(:,i , i))); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); + +linkaxes([ax1,ax2],'x'); +xlim([20, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_struts_cok_dvf_cart_frf.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:enc_struts_cok_dvf_cart_frf +#+caption: Bode plot of the transfer function from $\bm{\mathcal{F}}$ to $d\bm{\mathcal{X}}$ which is computed using the Jacobian matrices evaluted at the Center of Stiffness +#+RESULTS: +[[file:figs/enc_struts_cok_dvf_cart_frf.png]] + +#+begin_src matlab :exports none +%% Bode plot of the transfer function from F to dX using the Jacobians (CoI) +labels = {'$D_x/F_{x}$', '$D_y/F_{y}$', '$D_z/F_{z}$', '$R_{x}/M_{x}$', '$R_{y}/M_{y}$', '$R_{R}/M_{z}$'}; + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(f, abs(G_dL_J_coi(:, i, j)), 'color', [0, 0, 0, 0.2], ... + 'HandleVisibility', 'off'); + end +end +for i =1:6 + set(gca,'ColorOrderIndex',i) + plot(f, abs(G_dL_J_coi(:,i , i)), ... + 'DisplayName', labels{i}); +end +plot(f, abs(G_dL_J_coi(:, 1, 2)), 'color', [0, 0, 0, 0.2], ... + 'DisplayName', '$D_i/F_j$'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude'); set(gca, 'XTickLabel',[]); +ylim([1e-7, 2e-1]); +legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 3); + +ax2 = nexttile; +hold on; +for i =1:6 + set(gca,'ColorOrderIndex',i) + plot(f, 180/pi*angle(G_dL_J_coi(:,i , i))); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); + +linkaxes([ax1,ax2],'x'); +xlim([20, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_struts_coi_dvf_cart_frf.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:enc_struts_coi_dvf_cart_frf +#+caption: Bode plot of the transfer function from $\bm{\mathcal{F}}$ to $d\bm{\mathcal{X}}$ which is computed using the Jacobian matrices evaluated at the Point of Interest +#+RESULTS: +[[file:figs/enc_struts_coi_dvf_cart_frf.png]] + +#+begin_important +The centralized plant evaluated at the Center of Stiffness (Figure ref:fig:enc_struts_cok_dvf_cart_frf) is presenting a good decoupling at low frequency which is not the case for the Centralized plant at the point of interest in Figure ref:fig:enc_struts_coi_dvf_cart_frf. +#+end_important + +*** Centralized plant using the Force sensors +The transfer function from $\bm{\mathcal{F}}$ to $\bm{\mathcal{F}}_m$ is computed using the Jacobian evaluated at the Center of Stiffness and shown in Figure ref:fig:enc_struts_iff_cart_frf. + +#+begin_src matlab +%% Computed Centralized plant: transfer function from F to Fm +G_tau_J = permute(pagemtimes(inv(Js_cok), pagemtimes(permute(G_tau, [2 3 1]), inv(Ja_cok'))), [3 1 2]); +#+end_src + +From Figure ref:fig:enc_struts_iff_cart_frf, we can well estimate the directions of the suspension modes: +- modes at 110Hz and 160Hz are horizontal translation/rotation modes +- mode at 135Hz is a pure vertical mode +- mode at 185Hz is a pure rotation mode around the vertical axis + +#+begin_src matlab :exports none +%% Bode plot of the centralized plant +labels = {'$F_{m,x}/F_{x}$', '$F_{m,y}/F_{y}$', '$F_{m,z}/F_{z}$', '$M_{m,x}/M_{x}$', '$M_{m,y}/M_{y}$', '$M_{m,z}/M_{z}$'}; + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(f, abs(G_tau_J(:, i, j)), 'color', [0, 0, 0, 0.2], ... + 'HandleVisibility', 'off'); + end +end +for i =1:6 + set(gca,'ColorOrderIndex',i) + plot(f, abs(G_tau_J(:,i, i)), ... + 'DisplayName', labels{i}); +end +plot(f, abs(G_tau_J(:, 1, 2)), 'color', [0, 0, 0, 0.2], ... + 'DisplayName', '$D_i/F_j$'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]); +ylim([1e-3, 1e4]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); + +ax2 = nexttile; +hold on; +for i =1:6 + set(gca,'ColorOrderIndex',i) + plot(f, 180/pi*angle(G_tau_J(:,i, i))); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); + +linkaxes([ax1,ax2],'x'); +xlim([20, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_struts_iff_cart_frf.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:enc_struts_iff_cart_frf +#+caption: Measured FRF for the IFF plant in the cartesian frame +#+RESULTS: +[[file:figs/enc_struts_iff_cart_frf.png]] + +** Comparison with the Simscape Model +:PROPERTIES: +:header-args:matlab+: :tangle matlab/scripts/frf_struts_comp_simscape.m +:END: +<> +*** Introduction :ignore: +In this section, the measured dynamics is compared with the dynamics estimated from the Simscape model. + +*** Matlab Init :noexport:ignore: +#+begin_src matlab +%% frf_struts_comp_simscape.m +% Compared the measured FRF with the Simscape Models +% Encoders are fixed to the struts +#+end_src + +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) +<> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes +<> +#+end_src + +#+begin_src matlab :tangle no :noweb yes +<> +#+end_src + +#+begin_src matlab :eval no :noweb yes +<> +#+end_src + +#+begin_src matlab :noweb yes +<> +#+end_src + +#+begin_src matlab :noweb yes +<> +#+end_src + +#+begin_src matlab +%% Load data +frf_ol = load('identified_plants_enc_struts.mat', 'f', 'Ts', 'G_tau', 'G_dL'); +#+end_src + +*** Simscape Model - Identification +Let's initialize the nano-hexapod using 2 DoF models for the APA. + +#+begin_src matlab +%% Initialize Nano-Hexapod +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof'); +#+end_src + +The Nano-hexapod is fixed on a rigid granite and no payload is fixed on top of the hexapod. +#+begin_src matlab +%% Nano-Hexapod is fixed on a rigid granite +support.type = 0; + +%% No Payload on top of the Nano-Hexapod +payload.type = 0; +#+end_src + +The transfer function matrix from $\bm{u}$ to $\bm{\tau}_m$ using the Simscape model is now extracted. + +#+begin_src matlab +%% Identify the transfer function from u to taum +clear io; io_i = 1; +io(io_i) = linio([mdl, '/du'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensors + +%% Perform the model extraction +G_tau = exp(-s*frf_ol.Ts)*linearize(mdl, io, 0.0, options); +#+end_src + +And the plant from $\bm{u}$ to $d\bm{\mathcal{L}}_m$. + +#+begin_src matlab +%% Identify the DVF Plant (transfer function from u to dLm) +clear io; io_i = 1; +io(io_i) = linio([mdl, '/du'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Encoders + +%% Perform the identification +G_dL = exp(-s*frf_ol.Ts)*linearize(mdl, io, 0.0, options); +#+end_src + +Both transfer functions are saved for further use. +#+begin_src matlab :exports none :tangle no +save('matlab/data_frf/simscape_plants_enc_struts.mat', 'G_tau', 'G_dL') +#+end_src + +#+begin_src matlab :eval no +save('data_frf/simscape_plants_enc_struts.mat', 'G_tau', 'G_dL') +#+end_src + +#+begin_src matlab +%% Load the Previsouly saved plants from the Simscape model +sim_ol = load('simscape_plants_enc_struts.mat', 'G_tau', 'G_dL'); +#+end_src + +*** Dynamics from Actuator to Force Sensors +The comparison between the extracted model and the measurement is done in Figures ref:fig:enc_struts_iff_comp_simscape (diagonal elements) and ref:fig:enc_struts_iff_comp_offdiag_simscape (off-diagonal elements). +#+begin_src matlab :exports none +%% Bode plot of the identified Plant (Simscape) and measured FRF data +freqs = 2*logspace(1, 3, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(frf_ol.f, abs(frf_ol.G_tau(:,1, 1)), 'color', [colors(1,:),0.2], ... + 'DisplayName', '$\tau_{m,i}/u_i$ - FRF') +for i = 2:6 + set(gca,'ColorOrderIndex',2) + plot(frf_ol.f, abs(frf_ol.G_tau(:,i, i)), 'color', [colors(1,:),0.2], ... + 'HandleVisibility', 'off'); +end +plot(freqs, abs(squeeze(freqresp(sim_ol.G_tau(1,1), freqs, 'Hz'))), 'color', [colors(2,:),0.2], ... + 'DisplayName', '$\tau_{m,i}/u_i$ - Model') +for i = 2:6 + plot(freqs, abs(squeeze(freqresp(sim_ol.G_tau(i,i), freqs, 'Hz'))), 'color', [colors(2,:),0.2], ... + 'HandleVisibility', 'off'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]); +legend('location', 'southeast'); + +ax2 = nexttile; +hold on; +for i = 1:6 + plot(frf_ol.f, 180/pi*angle(frf_ol.G_tau(:,i, i)), 'color', [colors(1,:),0.2]); + plot(freqs, 180/pi*angle(squeeze(freqresp(sim_ol.G_tau(i,i), freqs, 'Hz'))), 'color', [colors(2,:),0.2]); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); +ylim([-180, 180]); +yticks([-180, -90, 0, 90, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_struts_iff_comp_simscape.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:enc_struts_iff_comp_simscape +#+caption: Comparison of the diagonal elements of the transfer function matrix from $\bm{u}$ to $\bm{\tau}_m$ - Simscape Model and measured FRF +#+RESULTS: +[[file:figs/enc_struts_iff_comp_simscape.png]] + +#+begin_src matlab :exports none +%% Bode plot of the identified IFF Plant (Simscape) and measured FRF data (off-diagonal elements) +freqs = 2*logspace(1, 3, 1000); + +figure; +hold on; +% Off diagonal terms +plot(frf_ol.f, abs(frf_ol.G_tau(:, 1, 2)), 'color', [colors(1,:),0.2], ... + 'DisplayName', '$\tau_{m,i}/u_j$ - FRF') +for i = 1:5 + for j = i+1:6 + plot(frf_ol.f, abs(frf_ol.G_tau(:, i, j)), 'color', [colors(1,:),0.2], ... + 'HandleVisibility', 'off'); + end +end +plot(freqs, abs(squeeze(freqresp(sim_ol.G_tau(1, 2), freqs, 'Hz'))), 'color', [colors(2,:),0.2], ... + 'DisplayName', '$\tau_{m,i}/u_j$ - Model') +for i = 1:5 + for j = i+1:6 + plot(freqs, abs(squeeze(freqresp(sim_ol.G_tau(i, j), freqs, 'Hz'))), 'color', [colors(2,:),0.2], ... + 'HandleVisibility', 'off'); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Amplitude [V/V]'); +xlim([freqs(1), freqs(end)]); ylim([1e-3, 1e2]); +legend('location', 'northeast'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_struts_iff_comp_offdiag_simscape.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:enc_struts_iff_comp_offdiag_simscape +#+caption: Comparison of the off-diagonal elements of the transfer function matrix from $\bm{u}$ to $\bm{\tau}_m$ - Simscape Model and measured FRF +#+RESULTS: +[[file:figs/enc_struts_iff_comp_offdiag_simscape.png]] + +*** Dynamics from Actuator to Encoder +The comparison between the model is done in Figures ref:fig:enc_struts_dvf_comp_simscape and ref:fig:enc_struts_dvf_comp_offdiag_simscape. + +#+begin_src matlab :exports none +%% Diagonal elements of the DVF plant +freqs = 2*logspace(1, 3, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(frf_ol.f, abs(frf_ol.G_dL(:,1, 1)), 'color', [colors(1,:),0.2], ... + 'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - FRF') +for i = 2:6 + plot(frf_ol.f, abs(frf_ol.G_dL(:,i, i)), 'color', [colors(1,:),0.2], ... + 'HandleVisibility', 'off'); +end +plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(1,1), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ... + 'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - Model') +for i = 2:6 + plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ... + 'HandleVisibility', 'off'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); +ylim([1e-8, 1e-3]); +legend('location', 'northeast'); + +ax2 = nexttile; +hold on; +for i = 1:6 + plot(frf_ol.f, 180/pi*angle(frf_ol.G_dL(:,i, i)), 'color', [colors(1,:),0.2]); + plot(freqs, 180/pi*angle(squeeze(freqresp(sim_ol.G_dL(i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.2]); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); +ylim([-180, 180]); +yticks([-180, -90, 0, 90, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_struts_dvf_comp_simscape.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:enc_struts_dvf_comp_simscape +#+caption: Diagonal elements of the DVF Plant +#+RESULTS: +[[file:figs/enc_struts_dvf_comp_simscape.png]] + +#+begin_src matlab :exports none +%% Off-diagonal elements of the DVF plant +freqs = 2*logspace(1, 3, 1000); + +figure; +hold on; +% Off diagonal terms +plot(frf_ol.f, abs(frf_ol.G_dL(:, 1, 2)), 'color', [colors(1,:),0.2], ... + 'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - FRF') +for i = 1:5 + for j = i+1:6 + plot(frf_ol.f, abs(frf_ol.G_dL(:, i, j)), 'color', [colors(1,:),0.2], ... + 'HandleVisibility', 'off'); + end +end +plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(1, 2), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ... + 'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - Model') +for i = 1:5 + for j = i+1:6 + plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(i, j), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ... + 'HandleVisibility', 'off'); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]'); +xlim([freqs(1), freqs(end)]); ylim([1e-8, 1e-3]); +legend('location', 'northeast'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_struts_dvf_comp_offdiag_simscape.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:enc_struts_dvf_comp_offdiag_simscape +#+caption: Off diagonal elements of the DVF Plant +#+RESULTS: +[[file:figs/enc_struts_dvf_comp_offdiag_simscape.png]] + +#+begin_important +The Simscape model is quite accurately representing the system dynamics. +Using 2dof models for the APA, we don't model the flexible modes of the APA. +Similarly, the flexible modes of the top plate are not modelled. +But up to 200Hz, the model is accurate. +#+end_important + +*** Using Flexible model +In order to model flexible modes of the APA, flexible elements are used and some misalignment between the APA and the flexible joints is introduced. + +These misalignments are estimated from measurements performed on each of the struts. +#+begin_src matlab +%% Misalignement between the APA and +d_aligns = [[-0.05, -0.3, 0]; + [ 0, 0.5, 0]; + [-0.1, -0.3, 0]; + [ 0, 0.3, 0]; + [-0.05, 0.05, 0]; + [0, 0, 0]]*1e-3; +#+end_src + +The dynamics from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ is estimated. +#+begin_src matlab +%% Initialize Nano-Hexapod +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', 'flexible', ... + 'actuator_d_align', d_aligns); + +%% Identify the DVF Plant (transfer function from u to dLm) +clear io; io_i = 1; +io(io_i) = linio([mdl, '/du'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Encoders + +sim_flex.G_dL = exp(-s*frf_ol.Ts)*linearize(mdl, io, 0.0, options); +#+end_src + +The obtained diagonal elements are compared with the measurements in Figure ref:fig:comp_frf_sim_enc_struts_modes. +Modeling the APA as flexible elements can indeed yields to more accurate models. +However this also adds a lot of states to the system, and time domain simulations can become realy long to perform. + +#+begin_src matlab :exports none +%% Comparison of the plants (encoder output) when tuning the misalignment +freqs = 2*logspace(0, 3, 1000); + +figure; +tiledlayout(2, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); +ax1 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_dL(:, 1, 1))); +plot(freqs, abs(squeeze(freqresp(sim_flex.G_dL(1,1), freqs, 'Hz')))); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/V]'); + +ax2 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_dL(:, 2, 2))); +plot(freqs, abs(squeeze(freqresp(sim_flex.G_dL(2,2), freqs, 'Hz')))); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); + +ax3 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_dL(:, 3, 3)), 'DisplayName', 'Meas.'); +plot(freqs, abs(squeeze(freqresp(sim_flex.G_dL(3,3), freqs, 'Hz'))), ... + 'DisplayName', 'Model'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); +legend('location', 'southwest', 'FontSize', 8); + +ax4 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_dL(:, 4, 4))); +plot(freqs, abs(squeeze(freqresp(sim_flex.G_dL(4,4), freqs, 'Hz')))); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]'); + +ax5 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_dL(:, 5, 5))); +plot(freqs, abs(squeeze(freqresp(sim_flex.G_dL(5,5), freqs, 'Hz')))); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); + +ax6 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_dL(:, 6, 6))); +plot(freqs, abs(squeeze(freqresp(sim_flex.G_dL(6,6), freqs, 'Hz')))); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); + +linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'xy'); +xlim([50, 5e2]); ylim([1e-6, 1e-3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/comp_frf_sim_enc_struts_modes.pdf', 'width', 'full', 'height', 'tall'); +#+end_src + +#+name: fig:comp_frf_sim_enc_struts_modes +#+caption: Comparison of the measured transfer functions from $u_i$ to $d\mathcal{L}_{m,i}$ - Simscape model and measured FRF +#+RESULTS: +[[file:figs/comp_frf_sim_enc_struts_modes.png]] + +** Integral Force Feedback +:PROPERTIES: +:header-args:matlab+: :tangle matlab/scripts/iff_enc_struts.m +:END: +<> +*** Introduction :ignore: + +In this section, the Integral Force Feedback (IFF) control strategy is applied to the nano-hexapod. +The main goal of this to add damping to the nano-hexapod's modes. + +The control architecture is shown in Figure ref:fig:control_architecture_iff_struts where $\bm{K}_\text{IFF}$ is a diagonal $6 \times 6$ controller. + +The system as then a new input $\bm{u}^\prime$, and the transfer function from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ should be easier to control than the initial transfer function from $\bm{u}$ to $d\bm{\mathcal{L}}_m$. + +#+begin_src latex :file control_architecture_iff_struts.pdf +\begin{tikzpicture} + % Blocs + \node[block={3.0cm}{2.0cm}] (P) {Plant}; + \coordinate[] (inputF) at ($(P.south west)!0.5!(P.north west)$); + \coordinate[] (outputF) at ($(P.south east)!0.7!(P.north east)$); + \coordinate[] (outputL) at ($(P.south east)!0.3!(P.north east)$); + + \node[block, above=0.4 of P] (Kiff) {$\bm{K}_\text{IFF}$}; + \node[addb, left= of inputF] (addF) {}; + + % Connections and labels + \draw[->] (outputF) -- ++(1, 0) node[below left]{$\bm{\tau}_m$}; + \draw[->] (outputL) -- ++(1, 0) node[below left]{$d\bm{\mathcal{L}}_m$}; + + \draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east); + \draw[->] (Kiff.west) -| (addF.north); + \draw[->] (addF.east) -- (inputF) node[above left]{$\bm{u}$}; + \draw[<-] (addF.west) -- ++(-1, 0) node[above right]{$\bm{u}^\prime$}; +\end{tikzpicture} +#+end_src + +#+name: fig:control_architecture_iff_struts +#+caption: Integral Force Feedback Strategy +#+RESULTS: +[[file:figs/control_architecture_iff_struts.png]] + +This section is structured as follow: +- Section ref:sec:iff_struts_plant_id: Using the Simscape model (APA taken as 2DoF model), the transfer function from $\bm{u}$ to $\bm{\tau}_m$ is identified. Based on the obtained dynamics, the control law is developed and the optimal gain is estimated using the Root Locus. +- Section ref:sec:iff_struts_effect_plant: Still using the Simscape model, the effect of the IFF gain on the the transfer function from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ is studied. +- Section ref:sec:iff_struts_effect_plant_exp: The same is performed experimentally: several IFF gains are used and the damped plant is identified each time. +- Section ref:sec:iff_struts_opt_gain: The damped model and the identified damped system are compared for the optimal IFF gain. It is found that IFF indeed adds a lot of damping into the system. However it is not efficient in damping the spurious struts modes. +- Section ref:sec:iff_struts_comp_flex_model: Finally, a "flexible" model of the APA is used in the Simscape model and the optimally damped model is compared with the measurements. + +*** Matlab Init :noexport:ignore: +#+begin_src matlab +%% iff_enc_struts.m +% Apply Integral Force Feedback for the nano-hexapod with encoders +% fixed to the struts +#+end_src + +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) +<> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes +<> +#+end_src + +#+begin_src matlab :tangle no :noweb yes +<> +#+end_src + +#+begin_src matlab :eval no :noweb yes +<> +#+end_src + +#+begin_src matlab :noweb yes +<> +#+end_src + +#+begin_src matlab :noweb yes +<> +#+end_src + +#+begin_src matlab +%% Load Plants +frf_ol = load('identified_plants_enc_struts.mat', 'f', 'Ts', 'G_tau', 'G_dL'); +sim_ol = load('simscape_plants_enc_struts.mat', 'G_tau', 'G_dL'); +#+end_src + +*** IFF Control Law and Optimal Gain +<> + +The IFF controller is defined as shown below: +#+begin_src matlab +%% IFF Controller +Kiff_g1 = -(1/(s + 2*pi*40))*... % LPF: provides integral action above 40Hz + (s/(s + 2*pi*30))*... % HPF: limit low frequency gain + (1/(1 + s/2/pi/500))*... % LPF: more robust to high frequency resonances + eye(6); % Diagonal 6x6 controller +#+end_src + +Then, the poles of the system are shown in the complex plane as a function of the controller gain (i.e. Root Locus plot) in Figure ref:fig:enc_struts_iff_root_locus. +A gain of $400$ is chosen as the "optimal" gain as it visually seems to be the gain that adds the maximum damping to all the suspension modes simultaneously. + +#+begin_src matlab :exports none +%% Root Locus for IFF +gains = logspace(1, 4, 100); + +figure; + +hold on; +plot(real(pole(sim_ol.G_tau)), imag(pole(sim_ol.G_tau)), 'x', 'color', colors(1,:), ... + 'DisplayName', '$g = 0$'); +plot(real(tzero(sim_ol.G_tau)), imag(tzero(sim_ol.G_tau)), 'o', 'color', colors(1,:), ... + 'HandleVisibility', 'off'); + +for g = gains + clpoles = pole(feedback(sim_ol.G_tau, g*Kiff_g1*eye(6), +1)); + plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ... + 'HandleVisibility', 'off'); +end + +% Optimal gain +g = 4e2; +clpoles = pole(feedback(sim_ol.G_tau, g*Kiff_g1*eye(6), +1)); +plot(real(clpoles), imag(clpoles), 'x', 'color', colors(2,:), ... + 'DisplayName', sprintf('$g=%.0f$', g)); +hold off; +axis square; +xlim([-1250, 0]); ylim([0, 1250]); +xlabel('Real Part'); ylabel('Imaginary Part'); +legend('location', 'northwest'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_struts_iff_root_locus.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:enc_struts_iff_root_locus +#+caption: Root Locus for the IFF control strategy +#+RESULTS: +[[file:figs/enc_struts_iff_root_locus.png]] + +Then the "optimal" IFF controller is defined: +#+begin_src matlab +%% IFF controller with Optimal gain +Kiff = 400*Kiff_g1; +#+end_src + +And it is saved for further use. +#+begin_src matlab :exports none :tangle no +save('matlab/data_sim/Kiff_struts_no_payload.mat', 'Kiff') +#+end_src + +#+begin_src matlab :eval no +save('data_sim/Kiff_struts_no_payload.mat', 'Kiff') +#+end_src + +The bode plots of the "diagonal" elements of the loop gain are shown in Figure ref:fig:enc_struts_iff_opt_loop_gain. +It is shown that the phase and gain margins are quite high and the loop gain is large around the resonances. +#+begin_src matlab :exports none +%% Bode plot of the "decentralized loop gain" +freqs = 2*logspace(1, 3, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:6 + plot(freqs, abs(squeeze(freqresp(Kiff(1,1)*sim_ol.G_tau(i,i), freqs, 'Hz'))), 'color', [colors(1,:), 0.2]); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Loop Gain'); set(gca, 'XTickLabel',[]); + +ax2 = nexttile; +hold on; +for i = 1:6 + plot(freqs, 180/pi*angle(squeeze(freqresp(Kiff(1,1)*sim_ol.G_tau(i,i), freqs, 'Hz'))), 'color', [colors(1,:), 0.2]); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); +ylim([-180, 180]); +yticks([-180, -90, 0, 90, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_struts_iff_opt_loop_gain.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:enc_struts_iff_opt_loop_gain +#+caption: Bode plot of the "decentralized loop gain" $G_\text{iff}(i,i) \times K_\text{iff}(i,i)$ +#+RESULTS: +[[file:figs/enc_struts_iff_opt_loop_gain.png]] + +*** Effect of IFF on the transfer function from actuator to encoder - Simulations +<> + +Still using the Simscape model with encoders fixed to the struts and 2DoF APA, the IFF strategy is tested. +#+begin_src matlab +%% Initialize the Simscape model in closed loop +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof', ... + 'controller_type', 'iff'); +#+end_src + +The Nano-hexapod is fixed on a rigid granite and no payload is fixed on top of the hexapod. +#+begin_src matlab +%% Nano-Hexapod is fixed on a rigid granite +support.type = 0; + +%% No Payload on top of the Nano-Hexapod +payload.type = 0; +#+end_src + +The following IFF gains are tested: +#+begin_src matlab +%% Tested IFF gains +iff_gains = [4, 10, 20, 40, 100, 200, 400]; +#+end_src + +And the transfer functions from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ are identified for all the IFF gains. +#+begin_src matlab +%% Identify the (damped) transfer function from u to dLm for different values of the IFF gain +Gd_iff = {zeros(1, length(iff_gains))}; + +clear io; io_i = 1; +io(io_i) = linio([mdl, '/du'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Strut Displacement (encoder) + +for i = 1:length(iff_gains) + Kiff = iff_gains(i)*Kiff_g1*eye(6); % IFF Controller + Gd_iff(i) = {exp(-s*frf_ol.Ts)*linearize(mdl, io, 0.0, options)}; + + isstable(Gd_iff{i}) +end +#+end_src + +The obtained dynamics are shown in Figure ref:fig:enc_struts_iff_gains_effect_dvf_plant. +#+begin_src matlab :exports none +%% Bode plot of the transfer function from u to dLm for tested values of the IFF gain +freqs = 2*logspace(1, 3, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:length(iff_gains) + plot(freqs, abs(squeeze(freqresp(Gd_iff{i}(1,1), freqs, 'Hz'))), '-', ... + 'DisplayName', sprintf('$g = %.0f$', iff_gains(i))); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); +legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); + +ax2 = nexttile; +hold on; +for i = 1:length(iff_gains) + plot(freqs, 180/pi*angle(squeeze(freqresp(Gd_iff{i}(1,1), freqs, 'Hz'))), '-'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); +ylim([-180, 180]); +yticks([-180, -90, 0, 90, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_struts_iff_gains_effect_dvf_plant.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:enc_struts_iff_gains_effect_dvf_plant +#+caption: Effect of the IFF gain $g$ on the transfer function from $\bm{\tau}$ to $d\bm{\mathcal{L}}_m$ +#+RESULTS: +[[file:figs/enc_struts_iff_gains_effect_dvf_plant.png]] + +*** Effect of IFF on the plant - Experimental Results +<> + +**** Introduction :ignore: +The IFF strategy is applied experimentally and the transfer function from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ is identified for all the defined values of the gain. + +**** Load Data +First load the identification data. +#+begin_src matlab +%% Load Identification Data +meas_iff_gains = {}; + +for i = 1:length(iff_gains) + meas_iff_gains(i) = {load(sprintf('iff_strut_1_noise_g_%i.mat', iff_gains(i)), 't', 'Vexc', 'de')}; +end +#+end_src + +**** Spectral Analysis - Setup +And define the useful variables that will be used for the identification using the =tfestimate= function. +#+begin_src matlab +%% Setup useful variables +% Sampling Time [s] +Ts = (meas_iff_gains{1}.t(end) - (meas_iff_gains{1}.t(1)))/(length(meas_iff_gains{1}.t)-1); + +% Sampling Frequency [Hz] +Fs = 1/Ts; + +% Hannning Windows +win = hanning(ceil(1*Fs)); + +% And we get the frequency vector +[~, f] = tfestimate(meas_iff_gains{1}.Vexc, meas_iff_gains{1}.de, win, [], [], 1/Ts); +#+end_src + +**** DVF Plant +The transfer functions are estimated for all the values of the gain. +#+begin_src matlab +%% Transfer function from u to dLm +G_iff_gains = {}; + +for i = 1:length(iff_gains) + G_iff_gains{i} = tfestimate(meas_iff_gains{i}.Vexc, meas_iff_gains{i}.de(:,1), win, [], [], 1/Ts); +end +#+end_src + +The obtained dynamics as shown in the bode plot in Figure ref:fig:comp_iff_gains_dvf_plant. +The dashed curves are the results obtained using the model, and the solid curves the results from the experimental identification. +#+begin_src matlab :exports none +%% Bode plot of the transfer function from u to dLm for tested values of the IFF gain +freqs = 2*logspace(1, 3, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:length(iff_gains) + plot(f, abs(G_iff_gains{i}), '-', ... + 'DisplayName', sprintf('$g = %.0f$', iff_gains(i))); +end +set(gca,'ColorOrderIndex',1) +for i = 1:length(iff_gains) + plot(freqs, abs(squeeze(freqresp(Gd_iff{i}(1,1), freqs, 'Hz'))), '--', ... + 'HandleVisibility', 'off'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); +legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); + +ax2 = nexttile; +hold on; +for i =1:length(iff_gains) + plot(f, 180/pi*angle(G_iff_gains{i}), '-'); +end +set(gca,'ColorOrderIndex',1) +for i = 1:length(iff_gains) + plot(freqs, 180/pi*angle(squeeze(freqresp(Gd_iff{i}(1,1), freqs, 'Hz'))), '--'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); +ylim([-180, 180]); +yticks([-180, -90, 0, 90, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/comp_iff_gains_dvf_plant.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:comp_iff_gains_dvf_plant +#+caption: Transfer function from $u$ to $d\mathcal{L}_m$ for multiple values of the IFF gain +#+RESULTS: +[[file:figs/comp_iff_gains_dvf_plant.png]] + +The bode plot is then zoomed on the suspension modes of the nano-hexapod in Figure ref:fig:comp_iff_gains_dvf_plant_zoom. +#+begin_src matlab :exports none +xlim([20, 200]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/comp_iff_gains_dvf_plant_zoom.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:comp_iff_gains_dvf_plant_zoom +#+caption: Transfer function from $u$ to $d\mathcal{L}_m$ for multiple values of the IFF gain (Zoom) +#+RESULTS: +[[file:figs/comp_iff_gains_dvf_plant_zoom.png]] + +#+begin_important +The IFF control strategy is very effective for the damping of the suspension modes. +It however does not damp the modes at 200Hz, 300Hz and 400Hz (flexible modes of the APA). + +Also, the experimental results and the models obtained from the Simscape model are in agreement concerning the damped system (up to the flexible modes). +#+end_important + +**** Experimental Results - Comparison of the un-damped and fully damped system +The un-damped and damped experimental plants are compared in Figure ref:fig:comp_undamped_opt_iff_gain_diagonal (diagonal terms). + +It is very clear that all the suspension modes are very well damped thanks to IFF. +However, there is little to no effect on the flexible modes of the struts and of the plate. + +#+begin_src matlab :exports none +%% Bode plot for the transfer function from u to dLm +freqs = 2*logspace(1, 3, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +% Un Damped measurement +set(gca,'ColorOrderIndex',1) +plot(f, abs(G_dvf(:, 1, 1)), ... + 'DisplayName', 'Un-Damped') +for i = 2:6 + set(gca,'ColorOrderIndex',1) + plot(f, abs(G_dvf(:,i , i)), ... + 'HandleVisibility', 'off'); +end + +% IFF Damped measurement +set(gca,'ColorOrderIndex',2) +plot(f, abs(G_iff_opt{1}(:,1)), ... + 'DisplayName', 'Optimal gain') +for i = 2:6 + set(gca,'ColorOrderIndex',2) + plot(f, abs(G_iff_opt{i}(:,i)), ... + 'HandleVisibility', 'off'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d_e/V_{exc}$ [m/V]'); set(gca, 'XTickLabel',[]); +ylim([1e-9, 1e-3]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); + +ax2 = nexttile; +hold on; +for i =1:6 + set(gca,'ColorOrderIndex',1) + plot(f, 180/pi*angle(G_dvf(i,i, i))); + set(gca,'ColorOrderIndex',2) + plot(f, 180/pi*angle(G_iff_opt{i}(:,i))); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); + +linkaxes([ax1,ax2],'x'); +xlim([20, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/comp_undamped_opt_iff_gain_diagonal.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:comp_undamped_opt_iff_gain_diagonal +#+caption: Comparison of the diagonal elements of the tranfer function from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ without active damping and with optimal IFF gain +#+RESULTS: +[[file:figs/comp_undamped_opt_iff_gain_diagonal.png]] + +*** Experimental Results - Damped Plant with Optimal gain +<> +**** Introduction :ignore: +Let's now look at the $6 \times 6$ damped plant with the optimal gain $g = 400$. + +**** Load Data +The experimental data are loaded. +#+begin_src matlab +%% Load Identification Data +meas_iff_struts = {}; + +for i = 1:6 + meas_iff_struts(i) = {load(sprintf('iff_strut_%i_noise_g_400.mat', i), 't', 'Vexc', 'Vs', 'de', 'u')}; +end +#+end_src + +**** Spectral Analysis - Setup +And the parameters useful for the spectral analysis are defined. +#+begin_src matlab +%% Setup useful variables +% Sampling Time [s] +Ts = (meas_iff_struts{1}.t(end) - (meas_iff_struts{1}.t(1)))/(length(meas_iff_struts{1}.t)-1); + +% Sampling Frequency [Hz] +Fs = 1/Ts; + +% Hannning Windows +win = hanning(ceil(1*Fs)); + +% And we get the frequency vector +[~, f] = tfestimate(meas_iff_struts{1}.Vexc, meas_iff_struts{1}.de, win, [], [], 1/Ts); +#+end_src + +**** DVF Plant +Finally, the $6 \times 6$ plant is identified using the =tfestimate= function. +#+begin_src matlab +%% DVF Plant (transfer function from u to dLm) +G_iff_opt = {}; + +for i = 1:6 + G_iff_opt{i} = tfestimate(meas_iff_struts{i}.Vexc, meas_iff_struts{i}.de, win, [], [], 1/Ts); +end +#+end_src + +The obtained diagonal elements are compared with the model in Figure ref:fig:damped_iff_plant_comp_diagonal. +#+begin_src matlab :exports none +%% Bode plot for the transfer function from u to dLm +freqs = 2*logspace(1, 3, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +% Diagonal Elements FRF +plot(f, abs(G_iff_opt{1}(:,1)), 'color', [0,0,0,0.2], ... + 'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - FRF') +for i = 2:6 + plot(f, abs(G_iff_opt{i}(:,i)), 'color', [0,0,0,0.2], ... + 'HandleVisibility', 'off'); +end + +% Diagonal Elements Model +set(gca,'ColorOrderIndex',2) +plot(freqs, abs(squeeze(freqresp(Gd_iff{end}(1,1), freqs, 'Hz'))), '-', ... + 'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - Model') +for i = 2:6 + set(gca,'ColorOrderIndex',2) + plot(freqs, abs(squeeze(freqresp(Gd_iff{end}(i,i), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); +end + +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d_e/V_{exc}$ [m/V]'); set(gca, 'XTickLabel',[]); +ylim([1e-9, 1e-3]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); + +ax2 = nexttile; +hold on; +for i =1:6 + plot(f, 180/pi*angle(G_iff_opt{i}(:,i)), 'color', [0,0,0,0.2]); + set(gca,'ColorOrderIndex',2) + plot(freqs, 180/pi*angle(squeeze(freqresp(Gd_iff{end}(i,i), freqs, 'Hz')))); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); + +linkaxes([ax1,ax2],'x'); +xlim([20, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/damped_iff_plant_comp_diagonal.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:damped_iff_plant_comp_diagonal +#+caption: Comparison of the diagonal elements of the transfer functions from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ with active damping (IFF) applied with an optimal gain $g = 400$ +#+RESULTS: +[[file:figs/damped_iff_plant_comp_diagonal.png]] + +And all the off-diagonal elements are compared with the model in Figure ref:fig:damped_iff_plant_comp_off_diagonal. +#+begin_src matlab :exports none +%% Bode plot for the transfer function from u to dLm +freqs = 2*logspace(1, 3, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +% Off diagonal FRF +plot(f, abs(G_iff_opt{1}(:,2)), 'color', [0,0,0,0.2], ... + 'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - FRF') +for i = 1:5 + for j = i+1:6 + plot(f, abs(G_iff_opt{i}(:,j)), 'color', [0, 0, 0, 0.2], ... + 'HandleVisibility', 'off'); + end +end + +% Off diagonal Model +set(gca,'ColorOrderIndex',2) +plot(freqs, abs(squeeze(freqresp(Gd_iff{end}(1,2), freqs, 'Hz'))), '-', ... + 'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - Model') +for i = 1:5 + for j = i+1:6 + set(gca,'ColorOrderIndex',2) + plot(freqs, abs(squeeze(freqresp(Gd_iff{end}(i,j), freqs, 'Hz'))), ... + 'HandleVisibility', 'off'); + end +end + +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d_e/V_{exc}$ [m/V]'); set(gca, 'XTickLabel',[]); +ylim([1e-9, 1e-3]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); + +ax2 = nexttile; +hold on; +% Off diagonal FRF +for i = 1:5 + for j = i+1:6 + plot(f, 180/pi*angle(G_iff_opt{i}(:,j)), 'color', [0, 0, 0, 0.2]); + end +end + +% Off diagonal Model +for i = 1:5 + for j = i+1:6 + set(gca,'ColorOrderIndex',2) + plot(freqs, 180/pi*angle(squeeze(freqresp(Gd_iff{end}(i,j), freqs, 'Hz')))); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); + +linkaxes([ax1,ax2],'x'); +xlim([20, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/damped_iff_plant_comp_off_diagonal.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:damped_iff_plant_comp_off_diagonal +#+caption: Comparison of the off-diagonal elements of the transfer functions from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ with active damping (IFF) applied with an optimal gain $g = 400$ +#+RESULTS: +[[file:figs/damped_iff_plant_comp_off_diagonal.png]] + +#+begin_important +With the IFF control strategy applied and the optimal gain used, the suspension modes are very well damped. +Remains the un-damped flexible modes of the APA (200Hz, 300Hz, 400Hz), and the modes of the plates (700Hz). + +The Simscape model and the experimental results are in very good agreement. +#+end_important + +*** Comparison with the Flexible model +<> + +When using the 2-DoF model for the APA, the flexible modes of the struts were not modelled, and it was the main limitation of the model. +Now, let's use a flexible model for the APA, and see if the obtained damped plant using the model is similar to the measured dynamics. + +First, the nano-hexapod is initialized. +#+begin_src matlab +%% Estimated misalignement of the struts +d_aligns = [[-0.05, -0.3, 0]; + [ 0, 0.5, 0]; + [-0.1, -0.3, 0]; + [ 0, 0.3, 0]; + [-0.05, 0.05, 0]; + [0, 0, 0]]*1e-3; + + +%% Initialize Nano-Hexapod +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', 'flexible', ... + 'actuator_d_align', d_aligns, ... + 'controller_type', 'iff'); +#+end_src + +And the "optimal" controller is loaded. +#+begin_src matlab +%% Optimal IFF controller +load('Kiff_struts_no_payload.mat', 'Kiff'); +#+end_src + +The transfer function from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ is identified using the Simscape model. +#+begin_src matlab +%% Linearized inputs/outputs +clear io; io_i = 1; +io(io_i) = linio([mdl, '/du'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Strut Displacement (encoder) + +%% Identification of the plant +Gd_iff = exp(-s*frf_ol.Ts)*linearize(mdl, io, 0.0, options); +#+end_src + +The obtained diagonal elements are shown in Figure ref:fig:enc_struts_iff_opt_damp_comp_flex_model_diag while the off-diagonal elements are shown in Figure ref:fig:enc_struts_iff_opt_damp_comp_flex_model_off_diag. +#+begin_src matlab :exports none +%% Bode plot for the transfer function from u' to dLm - Diagonal elements - Simscape and FRF +freqs = 2*logspace(1, 3, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +% Diagonal Elements FRF +plot(f, abs(G_iff_opt{1}(:,1)), 'color', [0,0,0,0.2], ... + 'DisplayName', '$d\mathcal{L}_{m,i}/u^\prime_i$ - FRF') +for i = 2:6 + plot(f, abs(G_iff_opt{i}(:,i)), 'color', [0,0,0,0.2], ... + 'HandleVisibility', 'off'); +end + +% Diagonal Elements Model +set(gca,'ColorOrderIndex',2) +plot(freqs, abs(squeeze(freqresp(Gd_iff(1,1), freqs, 'Hz'))), '-', ... + 'DisplayName', '$d\mathcal{L}_{m,i}/u^\prime_i$ - Model') +for i = 2:6 + set(gca,'ColorOrderIndex',2) + plot(freqs, abs(squeeze(freqresp(Gd_iff(i,i), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); +end + +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d\mathcal{L}_m/u^\prime$ [m/V]'); set(gca, 'XTickLabel',[]); +ylim([1e-9, 1e-3]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); + +ax2 = nexttile; +hold on; +for i =1:6 + plot(f, 180/pi*angle(G_iff_opt{i}(:,i)), 'color', [0,0,0,0.2]); + set(gca,'ColorOrderIndex',2) + plot(freqs, 180/pi*angle(squeeze(freqresp(Gd_iff(i,i), freqs, 'Hz')))); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); + +linkaxes([ax1,ax2],'x'); +xlim([20, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_struts_iff_opt_damp_comp_flex_model_diag.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:enc_struts_iff_opt_damp_comp_flex_model_diag +#+caption: Diagonal elements of the transfer function from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ - comparison of the measured FRF and the identified dynamics using the flexible model +#+RESULTS: +[[file:figs/enc_struts_iff_opt_damp_comp_flex_model_diag.png]] + +#+begin_src matlab :exports none +%% Bode plot for the transfer function from u' to dLm - Off diagonal elements - Simscape and FRF +freqs = 2*logspace(1, 3, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +% Off diagonal FRF +plot(f, abs(G_iff_opt{1}(:,2)), 'color', [0,0,0,0.2], ... + 'DisplayName', '$d\mathcal{L}_{m,i}/u^\prime_j$ - FRF') +for i = 1:5 + for j = i+1:6 + plot(f, abs(G_iff_opt{i}(:,j)), 'color', [0, 0, 0, 0.2], ... + 'HandleVisibility', 'off'); + end +end + +% Off diagonal Model +set(gca,'ColorOrderIndex',2) +plot(freqs, abs(squeeze(freqresp(Gd_iff(1,2), freqs, 'Hz'))), '-', ... + 'DisplayName', '$d\mathcal{L}_{m,i}/u^\prime_j$ - Model') +for i = 1:5 + for j = i+1:6 + set(gca,'ColorOrderIndex',2) + plot(freqs, abs(squeeze(freqresp(Gd_iff(i,j), freqs, 'Hz'))), ... + 'HandleVisibility', 'off'); + end +end + +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d\mathcal{L}_m/u^\prime$ [m/V]'); set(gca, 'XTickLabel',[]); +ylim([1e-9, 1e-3]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); + +ax2 = nexttile; +hold on; +% Off diagonal FRF +for i = 1:5 + for j = i+1:6 + plot(f, 180/pi*angle(G_iff_opt{i}(:,j)), 'color', [0, 0, 0, 0.2]); + end +end + +% Off diagonal Model +for i = 1:5 + for j = i+1:6 + set(gca,'ColorOrderIndex',2) + plot(freqs, 180/pi*angle(squeeze(freqresp(Gd_iff(i,j), freqs, 'Hz')))); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); + +linkaxes([ax1,ax2],'x'); +xlim([20, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_struts_iff_opt_damp_comp_flex_model_off_diag.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:enc_struts_iff_opt_damp_comp_flex_model_off_diag +#+caption: Off-diagonal elements of the transfer function from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ - comparison of the measured FRF and the identified dynamics using the flexible model +#+RESULTS: +[[file:figs/enc_struts_iff_opt_damp_comp_flex_model_off_diag.png]] + +#+begin_important +Using flexible models for the APA, the agreement between the Simscape model of the nano-hexapod and the measured FRF is very good. + +Only the flexible mode of the top-plate is not appearing in the model which is very logical as the top plate is taken as a solid body. +#+end_important + +*** Conclusion +#+begin_important +The decentralized Integral Force Feedback strategy applied on the nano-hexapod is very effective in damping all the suspension modes. + +The Simscape model (especially when using a flexible model for the APA) is shown to be very accurate, even when IFF is applied. +#+end_important + +** Modal Analysis +:PROPERTIES: +:header-args:matlab+: :tangle matlab/scripts/enc_struts_compliance_iff.m +:END: +<> +*** Introduction :ignore: +Several 3-axis accelerometers are fixed on the top platform of the nano-hexapod as shown in Figure ref:fig:compliance_vertical_comp_iff. + +#+name: fig:accelerometers_nano_hexapod +#+caption: Location of the accelerometers on top of the nano-hexapod +#+attr_latex: :width \linewidth +[[file:figs/accelerometers_nano_hexapod.jpg]] + +The top platform is then excited using an instrumented hammer as shown in Figure ref:fig:hammer_excitation_compliance_meas. + +#+name: fig:hammer_excitation_compliance_meas +#+caption: Example of an excitation using an instrumented hammer +#+attr_latex: :width \linewidth +[[file:figs/hammer_excitation_compliance_meas.jpg]] + +From this experiment, the resonance frequencies and the associated mode shapes can be computed (Section ref:sec:modal_analysis_mode_shapes). +Then, in Section ref:sec:compliance_effect_iff, the vertical compliance of the nano-hexapod is experimentally estimated. +Finally, in Section ref:sec:compliance_effect_iff_comp_model, the measured compliance is compare with the estimated one from the Simscape model. + +*** Matlab Init :noexport:ignore: +#+begin_src matlab +%% enc_struts_compliance_iff.m +% Compare measured compliance and estimated compliance from the Simscape model +#+end_src + +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) +<> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes +<> +#+end_src + +#+begin_src matlab :tangle no :noweb yes +<> +#+end_src + +#+begin_src matlab :eval no :noweb yes +<> +#+end_src + +#+begin_src matlab :noweb yes +<> +#+end_src + +#+begin_src matlab :noweb yes +<> +#+end_src + +*** Obtained Mode Shapes +<> + +We can observe the mode shapes of the first 6 modes that are the suspension modes (the plate is behaving as a solid body) in Figure ref:fig:mode_shapes_annotated. + +#+name: fig:mode_shapes_annotated +#+caption: Measured mode shapes for the first six modes +#+attr_latex: :width \linewidth +[[file:figs/mode_shapes_annotated.gif]] + +Then, there is a mode at 692Hz which corresponds to a flexible mode of the top plate (Figure ref:fig:mode_shapes_flexible_annotated). + +#+name: fig:mode_shapes_flexible_annotated +#+caption: First flexible mode at 692Hz +#+attr_latex: :width 0.3\linewidth +[[file:figs/ModeShapeFlex1_crop.gif]] + +The obtained modes are summarized in Table ref:tab:description_modes. + +#+name: tab:description_modes +#+caption: Description of the identified modes +#+attr_latex: :environment tabularx :width 0.7\linewidth :align ccX +#+attr_latex: :center t :booktabs t +| *Mode* | *Freq. [Hz]* | *Description* | +|--------+--------------+----------------------------------------------| +| 1 | 105 | Suspension Mode: Y-translation | +| 2 | 107 | Suspension Mode: X-translation | +| 3 | 131 | Suspension Mode: Z-translation | +| 4 | 161 | Suspension Mode: Y-tilt | +| 5 | 162 | Suspension Mode: X-tilt | +| 6 | 180 | Suspension Mode: Z-rotation | +| 7 | 692 | (flexible) Membrane mode of the top platform | + +*** Nano-Hexapod Compliance - Effect of IFF +<> + +In this section, we wish to estimate the effectiveness of the IFF strategy regarding the compliance. + +The top plate is excited vertically using the instrumented hammer two times: +1. no control loop is used +2. decentralized IFF is used + +The data are loaded. +#+begin_src matlab +frf_ol = load('Measurement_Z_axis.mat'); % Open-Loop +frf_iff = load('Measurement_Z_axis_damped.mat'); % IFF +#+end_src + +The mean vertical motion of the top platform is computed by averaging all 5 vertical accelerometers. +#+begin_src matlab +%% Multiply by 10 (gain in m/s^2/V) and divide by 5 (number of accelerometers) +d_frf_ol = 10/5*(frf_ol.FFT1_H1_4_1_RMS_Y_Mod + frf_ol.FFT1_H1_7_1_RMS_Y_Mod + frf_ol.FFT1_H1_10_1_RMS_Y_Mod + frf_ol.FFT1_H1_13_1_RMS_Y_Mod + frf_ol.FFT1_H1_16_1_RMS_Y_Mod)./(2*pi*frf_ol.FFT1_H1_16_1_RMS_X_Val).^2; +d_frf_iff = 10/5*(frf_iff.FFT1_H1_4_1_RMS_Y_Mod + frf_iff.FFT1_H1_7_1_RMS_Y_Mod + frf_iff.FFT1_H1_10_1_RMS_Y_Mod + frf_iff.FFT1_H1_13_1_RMS_Y_Mod + frf_iff.FFT1_H1_16_1_RMS_Y_Mod)./(2*pi*frf_iff.FFT1_H1_16_1_RMS_X_Val).^2; +#+end_src + +The vertical compliance (magnitude of the transfer function from a vertical force applied on the top plate to the vertical motion of the top plate) is shown in Figure ref:fig:compliance_vertical_comp_iff. +#+begin_src matlab :exports none +%% Comparison of the vertical compliances (OL and IFF) +figure; +hold on; +plot(frf_ol.FFT1_H1_16_1_RMS_X_Val, d_frf_ol, 'DisplayName', 'OL'); +plot(frf_iff.FFT1_H1_16_1_RMS_X_Val, d_frf_iff, 'DisplayName', 'IFF'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Vertical Compliance [$m/N$]'); +xlim([20, 2e3]); ylim([2e-9, 2e-5]); +legend('location', 'northeast'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/compliance_vertical_comp_iff.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:compliance_vertical_comp_iff +#+caption: Measured vertical compliance with and without IFF +#+RESULTS: +[[file:figs/compliance_vertical_comp_iff.png]] + +#+begin_important +From Figure ref:fig:compliance_vertical_comp_iff, it is clear that the IFF control strategy is very effective in damping the suspensions modes of the nano-hexapod. +It also has the effect of (slightly) degrading the vertical compliance at low frequency. + +It also seems some damping can be added to the modes at around 205Hz which are flexible modes of the struts. +#+end_important + +*** Comparison with the Simscape Model +<> + +Let's initialize the Simscape model such that it corresponds to the experiment. +#+begin_src matlab +%% Nano-Hexapod is fixed on a rigid granite +support.type = 0; + +%% No Payload on top of the Nano-Hexapod +payload.type = 0; + +%% Initialize Nano-Hexapod in Open Loop +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof'); + +#+end_src + +And let's compare the measured vertical compliance with the vertical compliance as estimated from the Simscape model. + +The transfer function from a vertical external force to the absolute motion of the top platform is identified (with and without IFF) using the Simscape model. +#+begin_src matlab :exports none +%% Identify the IFF Plant (transfer function from u to taum) +clear io; io_i = 1; +io(io_i) = linio([mdl, '/Fz_ext'], 1, 'openinput'); io_i = io_i + 1; % External - Vertical force +io(io_i) = linio([mdl, '/Z_top_plat'], 1, 'openoutput'); io_i = io_i + 1; % Absolute vertical motion of top platform +#+end_src + +#+begin_src matlab :exports none +%% Perform the identifications +G_compl_z_ol = linearize(mdl, io, 0.0, options); +#+end_src + +#+begin_src matlab :exports none +%% Initialize Nano-Hexapod with IFF +Kiff = 400*(1/(s + 2*pi*40))*... % Low pass filter (provides integral action above 40Hz) + (s/(s + 2*pi*30))*... % High pass filter to limit low frequency gain + (1/(1 + s/2/pi/500))*... % Low pass filter to be more robust to high frequency resonances + eye(6); % Diagonal 6x6 controller + +%% Initialize the Nano-Hexapod with IFF +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof', ... + 'controller_type', 'iff'); + +%% Perform the identification +G_compl_z_iff = linearize(mdl, io, 0.0, options); +#+end_src + +The comparison is done in Figure ref:fig:compliance_vertical_comp_model_iff. +Again, the model is quite accurate in predicting the (closed-loop) behavior of the system. + +#+begin_src matlab :exports none +%% Comparison of the measured compliance and the one obtained from the model +freqs = 2*logspace(1,3,1000); + +figure; +hold on; +plot(frf_ol.FFT1_H1_16_1_RMS_X_Val, d_frf_ol, '-', 'DisplayName', 'OL - Meas.'); +plot(frf_iff.FFT1_H1_16_1_RMS_X_Val, d_frf_iff, '-', 'DisplayName', 'IFF - Meas.'); +set(gca,'ColorOrderIndex',1) +plot(freqs, abs(squeeze(freqresp(G_compl_z_ol, freqs, 'Hz'))), '--', 'DisplayName', 'OL - Model') +plot(freqs, abs(squeeze(freqresp(G_compl_z_iff, freqs, 'Hz'))), '--', 'DisplayName', 'IFF - Model') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Vertical Compliance [$m/N$]'); +xlim([20, 2e3]); ylim([2e-9, 2e-5]); +legend('location', 'northeast', 'FontSize', 8); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/compliance_vertical_comp_model_iff.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:compliance_vertical_comp_model_iff +#+caption: Measured vertical compliance with and without IFF +#+RESULTS: +[[file:figs/compliance_vertical_comp_model_iff.png]] + +** Conclusion +#+begin_important +From the previous analysis, several conclusions can be drawn: +- Decentralized IFF is very effective in damping the "suspension" modes of the nano-hexapod (Figure ref:fig:comp_undamped_opt_iff_gain_diagonal) +- Decentralized IFF does not damp the "spurious" modes of the struts nor the flexible modes of the top plate (Figure ref:fig:comp_undamped_opt_iff_gain_diagonal) +- Even though the Simscape model and the experimentally measured FRF are in good agreement (Figures ref:fig:enc_struts_iff_opt_damp_comp_flex_model_diag and ref:fig:enc_struts_iff_opt_damp_comp_flex_model_off_diag), the obtain dynamics from the control inputs $\bm{u}$ and the encoders $d\bm{\mathcal{L}}_m$ is very difficult to control + +Therefore, in the following sections, the encoders will be fixed to the plates. +The goal is to be less sensitive to the flexible modes of the struts. +#+end_important + +* Encoders fixed to the plates +<> + +** Introduction :ignore: +In this section, the encoders are fixed to the plates rather than to the struts as shown in Figure ref:fig:enc_fixed_to_struts. + +#+name: fig:enc_fixed_to_struts +#+caption: Nano-Hexapod with encoders fixed to the struts +#+attr_latex: :width \linewidth +[[file:figs/IMG_20210625_083801.jpg]] + +It is structured as follow: +- Section ref:sec:enc_plates_plant_id: The dynamics of the nano-hexapod is identified. +- Section ref:sec:enc_plates_comp_simscape: The identified dynamics is compared with the Simscape model. +- Section ref:sec:enc_plates_iff: The Integral Force Feedback (IFF) control strategy is applied and the dynamics of the damped nano-hexapod is identified and compare with the Simscape model. + +** Identification of the dynamics +:PROPERTIES: +:header-args:matlab+: :tangle matlab/scripts/id_frf_enc_plates.m +:END: +<> +*** Introduction :ignore: +In this section, the dynamics of the nano-hexapod with the encoders fixed to the plates is identified. + +First, the measurement data are loaded in Section ref:sec:enc_plates_plant_id_setup, then the transfer function matrix from the actuators to the encoders are estimated in Section ref:sec:enc_plates_plant_id_dvf. +Finally, the transfer function matrix from the actuators to the force sensors is estimated in Section ref:sec:enc_plates_plant_id_iff. + +*** Matlab Init :noexport:ignore: +#+begin_src matlab +%% id_frf_enc_plates.m +% Identification of the nano-hexapod dynamics from u to dL and to taum +% Encoders are fixed to the plates +#+end_src + +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) +<> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes +<> +#+end_src + +#+begin_src matlab :tangle no :noweb yes +<> +#+end_src + +#+begin_src matlab :eval no :noweb yes +<> +#+end_src + +#+begin_src matlab :noweb yes +<> +#+end_src + +*** Data Loading and Spectral Analysis Setup +<> + +The actuators are excited one by one using a low pass filtered white noise. +For each excitation, the 6 force sensors and 6 encoders are measured and saved. +#+begin_src matlab +%% Load Identification Data +meas_data = {}; + +for i = 1:6 + meas_data(i) = {load(sprintf('frf_exc_strut_%i_enc_plates_noise.mat', i), 't', 'Va', 'Vs', 'de')}; +end +#+end_src + +#+begin_src matlab :exports none +%% Setup useful variables +% Sampling Time [s] +Ts = (meas_data{1}.t(end) - (meas_data{1}.t(1)))/(length(meas_data{1}.t)-1); + +% Sampling Frequency [Hz] +Fs = 1/Ts; + +% Hannning Windows +win = hanning(ceil(1*Fs)); + +% And we get the frequency vector +[~, f] = tfestimate(meas_data{1}.Va, meas_data{1}.de, win, [], [], 1/Ts); +#+end_src + +*** Transfer function from Actuator to Encoder +<> + +The 6x6 transfer function matrix from the excitation voltage $\bm{u}$ and the displacement $d\bm{\mathcal{L}}_m$ as measured by the encoders is estimated. + +#+begin_src matlab +%% Transfer function from u to dLm +G_dL = zeros(length(f), 6, 6); + +for i = 1:6 + G_dL(:,:,i) = tfestimate(meas_data{i}.Va, meas_data{i}.de, win, [], [], 1/Ts); +end +#+end_src + +The diagonal and off-diagonal terms of this transfer function matrix are shown in Figure ref:fig:enc_plates_dvf_frf. +#+begin_src matlab :exports none +%% Bode plot for the transfer function from u to dLm +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(f, abs(G_dL(:, i, j)), 'color', [0, 0, 0, 0.2], ... + 'HandleVisibility', 'off'); + end +end +for i =1:6 + set(gca,'ColorOrderIndex',i) + plot(f, abs(G_dL(:,i, i)), ... + 'DisplayName', sprintf('$d\\mathcal{L}_%i/u_%i$', i, i)); +end +plot(f, abs(G_dL(:, 1, 2)), 'color', [0, 0, 0, 0.2], ... + 'DisplayName', '$d\mathcal{L}_i/u_j$'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); +ylim([1e-9, 1e-3]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); + +ax2 = nexttile; +hold on; +for i =1:6 + set(gca,'ColorOrderIndex',i) + plot(f, 180/pi*angle(G_dL(:,i, i))); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); + +linkaxes([ax1,ax2],'x'); +xlim([20, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_plates_dvf_frf.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:enc_plates_dvf_frf +#+caption: Measured FRF for the transfer function from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ +#+RESULTS: +[[file:figs/enc_plates_dvf_frf.png]] + +#+begin_important +From Figure ref:fig:enc_plates_dvf_frf, we can draw few conclusions on the transfer functions from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ when the encoders are fixed to the plates: +- the decoupling is rather good at low frequency (below the first suspension mode). + The low frequency gain is constant for the off diagonal terms, whereas when the encoders where fixed to the struts, the low frequency gain of the off-diagonal terms were going to zero (Figure ref:fig:enc_struts_dvf_frf). +- the flexible modes of the struts at 226Hz and 337Hz are indeed shown in the transfer functions, but their amplitudes are rather low. +- the diagonal terms have alternating poles and zeros up to at least 600Hz: the flexible modes of the struts are not affecting the alternating pole/zero pattern. This what not the case when the encoders were fixed to the struts (Figure ref:fig:enc_struts_dvf_frf). +#+end_important + +*** Transfer function from Actuator to Force Sensor +<> +Then the 6x6 transfer function matrix from the excitation voltage $\bm{u}$ and the voltage $\bm{\tau}_m$ generated by the Force senors is estimated. +#+begin_src matlab +%% IFF Plant +G_tau = zeros(length(f), 6, 6); + +for i = 1:6 + G_tau(:,:,i) = tfestimate(meas_data{i}.Va, meas_data{i}.Vs, win, [], [], 1/Ts); +end +#+end_src + +The bode plot of the diagonal and off-diagonal terms are shown in Figure ref:fig:enc_plates_iff_frf. +#+begin_src matlab :exports none +%% Bode plot of the IFF Plant (transfer function from u to taum) +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(f, abs(G_tau(:, i, j)), 'color', [0, 0, 0, 0.2], ... + 'HandleVisibility', 'off'); + end +end +for i =1:6 + set(gca,'ColorOrderIndex',i) + plot(f, abs(G_tau(:,i , i)), ... + 'DisplayName', sprintf('$\\tau_{m,%i}/u_%i$', i, i)); +end +plot(f, abs(G_tau(:, 1, 2)), 'color', [0, 0, 0, 0.2], ... + 'DisplayName', '$\tau_{m,i}/u_j$'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); +ylim([1e-3, 1e2]); + +ax2 = nexttile; +hold on; +for i =1:6 + set(gca,'ColorOrderIndex',i) + plot(f, 180/pi*angle(G_tau(:,i, i))); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); + +linkaxes([ax1,ax2],'x'); +xlim([20, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_plates_iff_frf.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:enc_plates_iff_frf +#+caption: Measured FRF for the IFF plant +#+RESULTS: +[[file:figs/enc_plates_iff_frf.png]] + +#+begin_important +It is shown in Figure ref:fig:enc_plates_iff_comp_simscape_all that: +- The IFF plant has alternating poles and zeros +- The first flexible mode of the struts as 235Hz is appearing, and therefore is should be possible to add some damping to this mode using IFF +- The decoupling is quite good at low frequency (below the first model) as well as high frequency (above the last suspension mode, except near the flexible modes of the top plate) +#+end_important + +*** Save Identified Plants +The identified dynamics is saved for further use. +#+begin_src matlab :exports none :tangle no +save('matlab/data_frf/identified_plants_enc_plates.mat', 'f', 'Ts', 'G_tau', 'G_dL') +#+end_src + +#+begin_src matlab :eval no +save('data_frf/identified_plants_enc_plates.mat', 'f', 'Ts', 'G_tau', 'G_dL') +#+end_src + +** Comparison with the Simscape Model +:PROPERTIES: +:header-args:matlab+: :tangle matlab/scripts/frf_enc_plates_comp_simscape.m +:END: +<> +*** Introduction :ignore: +In this section, the measured dynamics done in Section ref:sec:enc_plates_plant_id is compared with the dynamics estimated from the Simscape model. + +*** Matlab Init :noexport:ignore: +#+begin_src matlab +%% frf_enc_plates_comp_simscape.m +% Compare the measured dynamics from u to dL and to taum with the Simscape model +% Encoders are fixed to the plates +#+end_src + +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) +<> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes +<> +#+end_src + +#+begin_src matlab :tangle no :noweb yes +<> +#+end_src + +#+begin_src matlab :eval no :noweb yes +<> +#+end_src + +#+begin_src matlab :noweb yes +<> +#+end_src + +#+begin_src matlab :noweb yes +<> +#+end_src + +#+begin_src matlab +%% Load identification data +frf_ol = load('identified_plants_enc_plates.mat', 'f', 'Ts', 'G_tau', 'G_dL'); +#+end_src + +*** Identification with the Simscape Model +The nano-hexapod is initialized with the APA taken as 2dof models. +#+begin_src matlab +%% Initialize Nano-Hexapod +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'plates', ... + 'actuator_type', '2dof'); + +support.type = 1; % On top of vibration table +payload.type = 0; % No Payload +#+end_src + +Then the transfer function from $\bm{u}$ to $\bm{\tau}_m$ is identified using the Simscape model. +#+begin_src matlab +%% Identify the transfer function from u to taum +clear io; io_i = 1; +io(io_i) = linio([mdl, '/du'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensors + +G_tau = exp(-s*frf_ol.Ts)*linearize(mdl, io, 0.0, options); +#+end_src + +Now, the dynamics from the DAC voltage $\bm{u}$ to the encoders $d\bm{\mathcal{L}}_m$ is estimated using the Simscape model. +#+begin_src matlab +%% Identify the DVtransfer function from u to dLm +clear io; io_i = 1; +io(io_i) = linio([mdl, '/du'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Encoders + +G_dL = exp(-s*frf_ol.Ts)*linearize(mdl, io, 0.0, options); +#+end_src + +The identified dynamics is saved for further use. +#+begin_src matlab :exports none :tangle no +%% Save Identified Plants +save('matlab/data_frf/simscape_plants_enc_plates.mat', 'G_tau', 'G_dL'); +#+end_src + +#+begin_src matlab :eval no +save('data_frf/simscape_plants_enc_plates.mat', 'G_tau', 'G_dL'); +#+end_src + +#+begin_src matlab :exports none +%% Load the Simscape model +sim_ol = load('simscape_plants_enc_plates.mat', 'G_tau', 'G_dL'); +#+end_src + +*** Dynamics from Actuator to Force Sensors +The identified dynamics is compared with the measured FRF: +- Figure ref:fig:enc_plates_iff_comp_simscape_all: the individual transfer function from $u_1$ (the DAC voltage for the first actuator) to the force sensors of all 6 struts are compared +- Figure ref:fig:enc_plates_iff_comp_simscape: all the diagonal elements are compared +- Figure ref:fig:enc_plates_iff_comp_offdiag_simscape: all the off-diagonal elements are compared + +#+begin_src matlab :exports none +%% Comparison of the plants (encoder output) when tuning the misalignment +freqs = 2*logspace(1, 3, 1000); + +i_input = 1; + +figure; +tiledlayout(2, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_tau(:, 1, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_tau(1, i_input), freqs, 'Hz')))); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +set(gca, 'XTickLabel',[]); ylabel('Amplitude [V/V]'); +title(sprintf('$d\\tau_{m1}/u_{%i}$', i_input)); + +ax2 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_tau(:, 2, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_tau(2, i_input), freqs, 'Hz')))); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); +title(sprintf('$d\\tau_{m2}/u_{%i}$', i_input)); + +ax3 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_tau(:, 3, i_input)), ... + 'DisplayName', 'Meas.'); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_tau(3, i_input), freqs, 'Hz'))), ... + 'DisplayName', 'Model'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); +legend('location', 'southeast', 'FontSize', 8); +title(sprintf('$d\\tau_{m3}/u_{%i}$', i_input)); + +ax4 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_tau(:, 4, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_tau(4, i_input), freqs, 'Hz')))); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Amplitude [V/V]'); +title(sprintf('$d\\tau_{m4}/u_{%i}$', i_input)); + +ax5 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_tau(:, 5, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_tau(5, i_input), freqs, 'Hz')))); +hold off; +xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +title(sprintf('$d\\tau_{m5}/u_{%i}$', i_input)); + +ax6 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_tau(:, 6, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_tau(6, i_input), freqs, 'Hz')))); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); +title(sprintf('$d\\tau_{m6}/u_{%i}$', i_input)); + +linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'xy'); +xlim([20, 2e3]); ylim([1e-2, 1e2]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_plates_iff_comp_simscape_all.pdf', 'width', 'full', 'height', 'tall'); +#+end_src + +#+name: fig:enc_plates_iff_comp_simscape_all +#+caption: IFF Plant for the first actuator input and all the force senosrs +#+RESULTS: +[[file:figs/enc_plates_iff_comp_simscape_all.png]] + +#+begin_src matlab :exports none +%% Bode plot of the identified IFF Plant (Simscape) and measured FRF data +freqs = 2*logspace(1, 3, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(frf_ol.f, abs(frf_ol.G_tau(:,1, 1)), 'color', [colors(1,:),0.2], ... + 'DisplayName', '$\tau_{m,i}/u_i$ - FRF') +for i = 2:6 + plot(frf_ol.f, abs(frf_ol.G_tau(:,i, i)), 'color', [colors(1,:),0.2], ... + 'HandleVisibility', 'off'); +end +plot(freqs, abs(squeeze(freqresp(sim_ol.G_tau(1,1), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ... + 'DisplayName', '$\tau_{m,i}/u_i$ - Model') +for i = 2:6 + plot(freqs, abs(squeeze(freqresp(sim_ol.G_tau(i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ... + 'HandleVisibility', 'off'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]); +legend('location', 'southeast'); + +ax2 = nexttile; +hold on; +for i = 1:6 + plot(frf_ol.f, 180/pi*angle(frf_ol.G_tau(:,i, i)), 'color', [colors(1,:),0.2]); + plot(freqs, 180/pi*angle(squeeze(freqresp(sim_ol.G_tau(i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.2]); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); +ylim([-180, 180]); +yticks([-180, -90, 0, 90, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_plates_iff_comp_simscape.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:enc_plates_iff_comp_simscape +#+caption: Diagonal elements of the IFF Plant +#+RESULTS: +[[file:figs/enc_plates_iff_comp_simscape.png]] + +#+begin_src matlab :exports none +%% Bode plot of the identified IFF Plant (Simscape) and measured FRF data (off-diagonal elements) +freqs = 2*logspace(1, 3, 1000); + +figure; +hold on; +% Off diagonal terms +plot(frf_ol.f, abs(frf_ol.G_tau(:, 1, 2)), 'color', [colors(1,:),0.2], ... + 'DisplayName', '$\tau_{m,i}/u_j$ - FRF') +for i = 1:5 + for j = i+1:6 + plot(frf_ol.f, abs(frf_ol.G_tau(:, i, j)), 'color', [colors(1,:),0.2], ... + 'HandleVisibility', 'off'); + end +end +set(gca,'ColorOrderIndex',2); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_tau(1, 2), freqs, 'Hz'))), 'color', [colors(2,:),0.2], ... + 'DisplayName', '$\tau_{m,i}/u_j$ - Model') +for i = 1:5 + for j = i+1:6 + set(gca,'ColorOrderIndex',2); + plot(freqs, abs(squeeze(freqresp(sim_ol.G_tau(i, j), freqs, 'Hz'))), 'color', [colors(2,:),0.2], ... + 'HandleVisibility', 'off'); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Amplitude [V/V]'); +xlim([freqs(1), freqs(end)]); ylim([1e-3, 1e2]); +legend('location', 'northeast'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_plates_iff_comp_offdiag_simscape.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:enc_plates_iff_comp_offdiag_simscape +#+caption: Off diagonal elements of the IFF Plant +#+RESULTS: +[[file:figs/enc_plates_iff_comp_offdiag_simscape.png]] + +*** Dynamics from Actuator to Encoder +The identified dynamics is compared with the measured FRF: +- Figure ref:fig:enc_plates_dvf_comp_simscape_all: the individual transfer function from $u_3$ (the DAC voltage for the actuator number 3) to the six encoders +- Figure ref:fig:enc_plates_dvf_comp_simscape: all the diagonal elements are compared +- Figure ref:fig:enc_plates_dvf_comp_offdiag_simscape: all the off-diagonal elements are compared + +#+begin_src matlab :exports none +%% Comparison of the plants (encoder output) when tuning the misalignment +freqs = 2*logspace(1, 3, 1000); + +i_input = 3; + +figure; +tiledlayout(2, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_dL(:, 1, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(1, i_input), freqs, 'Hz')))); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/V]'); +title(sprintf('$d\\mathcal{L}_{m1}/u_{%i}$', i_input)); + +ax2 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_dL(:, 2, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(2, i_input), freqs, 'Hz')))); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); +title(sprintf('$d\\mathcal{L}_{m2}/u_{%i}$', i_input)); + +ax3 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_dL(:, 3, i_input)), ... + 'DisplayName', 'Meas.'); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(3, i_input), freqs, 'Hz'))), ... + 'DisplayName', 'Model'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); +legend('location', 'southeast', 'FontSize', 8); +title(sprintf('$d\\mathcal{L}_{m3}/u_{%i}$', i_input)); + +ax4 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_dL(:, 4, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(4, i_input), freqs, 'Hz')))); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]'); +title(sprintf('$d\\mathcal{L}_{m4}/u_{%i}$', i_input)); + +ax5 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_dL(:, 5, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(5, i_input), freqs, 'Hz')))); +hold off; +xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +title(sprintf('$d\\mathcal{L}_{m5}/u_{%i}$', i_input)); + +ax6 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_dL(:, 6, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(6, i_input), freqs, 'Hz')))); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); +title(sprintf('$d\\mathcal{L}_{m6}/u_{%i}$', i_input)); + +linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'xy'); +xlim([40, 4e2]); ylim([1e-8, 1e-2]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_plates_dvf_comp_simscape_all.pdf', 'width', 'full', 'height', 'tall'); +#+end_src + +#+name: fig:enc_plates_dvf_comp_simscape_all +#+caption: DVF Plant for the first actuator input and all the encoders +#+RESULTS: +[[file:figs/enc_plates_dvf_comp_simscape_all.png]] + +#+begin_src matlab :exports none +%% Diagonal elements of the DVF plant +freqs = 2*logspace(1, 3, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(frf_ol.f, abs(frf_ol.G_dL(:,1, 1)), 'color', [colors(1,:),0.2], ... + 'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - FRF') +for i = 2:6 + plot(frf_ol.f, abs(frf_ol.G_dL(:,i, i)), 'color', [colors(1,:),0.2], ... + 'HandleVisibility', 'off'); +end +plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(1,1), freqs, 'Hz'))), 'color', [colors(2,:),0.2], ... + 'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - Model') +for i = 2:6 + plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(i,i), freqs, 'Hz'))), 'color', [colors(2,:),0.2], ... + 'HandleVisibility', 'off'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); +ylim([1e-8, 1e-3]); +legend('location', 'northeast'); + +ax2 = nexttile; +hold on; +for i = 1:6 + plot(frf_ol.f, 180/pi*angle(frf_ol.G_dL(:,i, i)), 'color', [colors(1,:),0.2]); + plot(freqs, 180/pi*angle(squeeze(freqresp(sim_ol.G_dL(i,i), freqs, 'Hz'))), 'color', [colors(2,:),0.2]); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); +ylim([-180, 180]); +yticks([-180, -90, 0, 90, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_plates_dvf_comp_simscape.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:enc_plates_dvf_comp_simscape +#+caption: Diagonal elements of the DVF Plant +#+RESULTS: +[[file:figs/enc_plates_dvf_comp_simscape.png]] + +#+begin_src matlab :exports none +%% Off-diagonal elements of the DVF plant +freqs = 2*logspace(1, 3, 1000); + +figure; +hold on; +% Off diagonal terms +plot(frf_ol.f, abs(frf_ol.G_dL(:, 1, 2)), 'color', [colors(1,:),0.2], ... + 'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - FRF') +for i = 1:5 + for j = i+1:6 + plot(frf_ol.f, abs(frf_ol.G_dL(:, i, j)), 'color', [colors(1,:),0.2], ... + 'HandleVisibility', 'off'); + end +end +plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(1, 2), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ... + 'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - Model') +for i = 1:5 + for j = i+1:6 + plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(i, j), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ... + 'HandleVisibility', 'off'); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]'); +xlim([freqs(1), freqs(end)]); ylim([1e-8, 1e-3]); +legend('location', 'northeast'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_plates_dvf_comp_offdiag_simscape.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:enc_plates_dvf_comp_offdiag_simscape +#+caption: Off diagonal elements of the DVF Plant +#+RESULTS: +[[file:figs/enc_plates_dvf_comp_offdiag_simscape.png]] + +*** Conclusion +#+begin_important +The Simscape model is quite accurate for the transfer function matrices from $\bm{u}$ to $\bm{\tau}_m$ and from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ except at frequencies of the flexible modes of the top-plate. +The Simscape model can therefore be used to develop the control strategies. +#+end_important + +** Integral Force Feedback +:PROPERTIES: +:header-args:matlab+: :tangle matlab/scripts/iff_enc_plates.m +:END: +<> +*** Introduction :ignore: + +In this section, the Integral Force Feedback (IFF) control strategy is applied to the nano-hexapod in order to add damping to the suspension modes. + +The control architecture is shown in Figure ref:fig:control_architecture_iff: +- $\bm{\tau}_m$ is the measured voltage of the 6 force sensors +- $\bm{K}_{\text{IFF}}$ is the $6 \times 6$ diagonal controller +- $\bm{u}$ is the plant input (voltage generated by the 6 DACs) +- $\bm{u}^\prime$ is the new plant inputs with added damping + +#+begin_src latex :file control_architecture_iff.pdf +\begin{tikzpicture} + % Blocs + \node[block={3.0cm}{2.0cm}] (P) {Plant}; + \coordinate[] (inputF) at ($(P.south west)!0.5!(P.north west)$); + \coordinate[] (outputF) at ($(P.south east)!0.7!(P.north east)$); + \coordinate[] (outputL) at ($(P.south east)!0.3!(P.north east)$); + + \node[block, above=0.4 of P] (Kiff) {$\bm{K}_\text{IFF}$}; + \node[addb, left= of inputF] (addF) {}; + + % Connections and labels + \draw[->] (outputF) -- ++(1, 0) node[below left]{$\bm{\tau}_m$}; + \draw[->] (outputL) -- ++(1, 0) node[below left]{$d\bm{\mathcal{L}}_m$}; + + \draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east); + \draw[->] (Kiff.west) -| (addF.north); + \draw[->] (addF.east) -- (inputF) node[above left]{$\bm{u}$}; + \draw[<-] (addF.west) -- ++(-1, 0) node[above right]{$\bm{u}^\prime$}; +\end{tikzpicture} +#+end_src + +#+name: fig:control_architecture_iff +#+caption: Integral Force Feedback Strategy +#+RESULTS: +[[file:figs/control_architecture_iff.png]] + +- Section ref:sec:enc_struts_effect_iff_plant + +*** Matlab Init :noexport:ignore: +#+begin_src matlab +%% iff_enc_plates.m +% Apply Integral Force Feedback for the nano-hexapod with encoders +% fixed to the plates +#+end_src + +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) +<> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes +<> +#+end_src + +#+begin_src matlab :tangle no :noweb yes +<> +#+end_src + +#+begin_src matlab :eval no :noweb yes +<> +#+end_src + +#+begin_src matlab :noweb yes +<> +#+end_src + +#+begin_src matlab :noweb yes +<> +#+end_src + +#+begin_src matlab +%% Load measured FRF and extracted Simscape model +frf_ol = load('identified_plants_enc_plates.mat', 'f', 'Ts', 'G_tau', 'G_dL'); +sim_ol = load('simscape_plants_enc_plates.mat', 'G_tau', 'G_dL'); +#+end_src + +*** Effect of IFF on the plant - Simscape Model +<> + +The nano-hexapod is initialized with 2DoF APA and the encoders fixed to the struts. +#+begin_src matlab +%% Initialize the Simscape model in closed loop +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'plates', ... + 'actuator_type', '2dof', ... + 'controller_type', 'iff'); + +support.type = 1; % On top of vibration table +payload.type = 0; % No Payload +#+end_src + +The same controller as the one developed when the encoder were fixed to the struts is used. +#+begin_src matlab +%% Optimal IFF controller +load('Kiff_struts_no_payload.mat', 'Kiff'); +#+end_src + +The transfer function from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ is identified. +#+begin_src matlab +%% Identify the (damped) transfer function from u to dLm for different values of the IFF gain +clear io; io_i = 1; +io(io_i) = linio([mdl, '/du'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Plate Displacement (encoder) + +%% Transfer function from u to dL (IFF) +G_dL = exp(-s*frf_ol.Ts)*linearize(mdl, io, 0.0, options); +#+end_src + +It is first verified that the system is stable: +#+begin_src matlab :results value replace :exports both :tangle no +isstable(G_dL) +#+end_src + +#+RESULTS: +: 1 + +The identified dynamics is saved for further use. +#+begin_src matlab :exports none :tangle no +%% Save Identified Plants +save('matlab/data_frf/simscape_plants_enc_plates_iff.mat', 'G_dL'); +#+end_src + +#+begin_src matlab :eval no +save('data_frf/simscape_plants_enc_plates_iff.mat', 'G_dL'); +#+end_src + +#+begin_src matlab :exports none +%% Load the Simscape model +sim_iff = load('simscape_plants_enc_plates_iff.mat', 'G_dL'); +#+end_src + +The diagonal and off-diagonal terms of the $6 \times 6$ transfer function matrices identified are compared in Figure ref:fig:enc_plates_iff_gains_effect_dvf_plant. +It is shown, as was the case when the encoders were fixed to the struts, that the IFF control strategy is very effective in damping the suspension modes of the nano-hexapod. +#+begin_src matlab :exports none +%% Bode plot of the transfer function from u to dLm with and without IFF +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(1,1), freqs, 'Hz'))), 'color', colors(1,:), ... + 'DisplayName', 'OL - Diag'); +plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL(1,1), freqs, 'Hz'))), 'color', colors(2,:), ... + 'DisplayName', 'IFF - Diag'); +for i = 2:6 + plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(1,1), freqs, 'Hz'))), 'color', colors(1,:), ... + 'HandleVisibility', 'off'); +end +for i = 2:6 + plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL(i,i), freqs, 'Hz'))), 'color', colors(2,:), ... + 'HandleVisibility', 'off'); +end + +plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(1,2), freqs, 'Hz'))), 'color', [colors(1,:), 0.2], ... + 'DisplayName', 'OL - Off-diag') +for i = 1:5 + for j = i+1:6 + plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(i,j), freqs, 'Hz'))), 'color', [colors(1,:), 0.2], ... + 'HandleVisibility', 'off'); + end +end +plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL(1,2), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ... + 'DisplayName', 'IFF - Off-diag') +for i = 1:5 + for j = i+1:6 + plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL(i,j), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ... + 'HandleVisibility', 'off'); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); +legend('location', 'northeast', 'FontSize', 8); + +ax2 = nexttile; +hold on; +for i = 1:6 + plot(freqs, 180/pi*angle(squeeze(freqresp(sim_ol.G_dL(1,1), freqs, 'Hz'))), 'color', colors(1,:), ... + 'HandleVisibility', 'off'); + plot(freqs, 180/pi*angle(squeeze(freqresp(sim_iff.G_dL(i,i), freqs, 'Hz'))), 'color', colors(2,:), ... + 'HandleVisibility', 'off'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); +ylim([-180, 180]); +yticks([-180, -90, 0, 90, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_plates_iff_gains_effect_dvf_plant.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:enc_plates_iff_gains_effect_dvf_plant +#+caption: Effect of the IFF control strategy on the transfer function from $\bm{\tau}$ to $d\bm{\mathcal{L}}_m$ +#+RESULTS: +[[file:figs/enc_plates_iff_gains_effect_dvf_plant.png]] + +*** Effect of IFF on the plant - FRF +The IFF control strategy is experimentally implemented. +The (damped) transfer function from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ is experimentally identified. + +The identification data are loaded: +#+begin_src matlab +%% Load Identification Data +meas_iff_plates = {}; + +for i = 1:6 + meas_iff_plates(i) = {load(sprintf('frf_exc_iff_strut_%i_enc_plates_noise.mat', i), 't', 'Va', 'Vs', 'de', 'u')}; +end +#+end_src + +And the parameters used for the transfer function estimation are defined below. +#+begin_src matlab +% Sampling Time [s] +Ts = (meas_iff_plates{1}.t(end) - (meas_iff_plates{1}.t(1)))/(length(meas_iff_plates{1}.t)-1); + +% Hannning Windows +win = hanning(ceil(1/Ts)); + +% And we get the frequency vector +[~, f] = tfestimate(meas_iff_plates{1}.Va, meas_iff_plates{1}.de, win, [], [], 1/Ts); +#+end_src + +The estimation is performed using the =tfestimate= command. +#+begin_src matlab +%% Estimation of the transfer function matrix from u to dL when IFF is applied +G_dL = zeros(length(f), 6, 6); + +for i = 1:6 + G_dL(:,:,i) = tfestimate(meas_iff_plates{i}.Va, meas_iff_plates{i}.de, win, [], [], 1/Ts); +end +#+end_src + +The experimentally identified plant is saved for further use. +#+begin_src matlab :exports none:tangle no +save('matlab/data_frf/damped_plant_enc_plates.mat', 'f', 'Ts', 'G_dL') +#+end_src + +#+begin_src matlab :eval no +save('data_frf/damped_plant_enc_plates.mat', 'f', 'Ts', 'G_dL') +#+end_src + +#+begin_src matlab :exports none +frf_iff = load('damped_plant_enc_plates.mat', 'f', 'Ts', 'G_dL'); +#+end_src + +The obtained diagonal and off-diagonal elements of the transfer function from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ are shown in Figure ref:fig:enc_plant_plates_effect_iff both without and with IFF. +#+begin_src matlab :exports none +%% Bode plot of the transfer function from u to dLm with and without IFF (experimental results) +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(frf_ol.f, abs(frf_ol.G_dL(:,1,1)), 'color', colors(1,:), ... + 'DisplayName', 'OL - Diag'); +plot(frf_iff.f, abs(frf_iff.G_dL(:,1,1)), 'color', colors(2,:), ... + 'DisplayName', 'IFF - Diag'); +for i = 2:6 + plot(frf_ol.f, abs(frf_ol.G_dL(:,1,1)), 'color', colors(1,:), ... + 'HandleVisibility', 'off'); + plot(frf_iff.f, abs(frf_iff.G_dL(:,i,i)), 'color', colors(2,:), ... + 'HandleVisibility', 'off'); +end + +plot(frf_ol.f, abs(frf_ol.G_dL(:,1,2)), 'color', [colors(1,:), 0.2], ... + 'DisplayName', 'OL - Off-diag') +for i = 1:5 + for j = i+1:6 + plot(frf_ol.f, abs(frf_ol.G_dL(:,i,j)), 'color', [colors(1,:), 0.2], ... + 'HandleVisibility', 'off'); + end +end +plot(frf_iff.f, abs(frf_iff.G_dL(:,1,2)), 'color', [colors(2,:), 0.2], ... + 'DisplayName', 'IFF - Off-diag') +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(frf_iff.G_dL(:,i,j)), 'color', [colors(2,:), 0.2], ... + 'HandleVisibility', 'off'); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); + +ax2 = nexttile; +hold on; +for i = 1:6 + plot(frf_ol.f, 180/pi*angle(frf_ol.G_dL( :,i,i)), 'color', colors(1,:)) + plot(frf_iff.f, 180/pi*angle(frf_iff.G_dL(:,i,i)), 'color', colors(2,:)) +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); +ylim([-180, 180]); +yticks([-180, -90, 0, 90, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_plant_plates_effect_iff.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:enc_plant_plates_effect_iff +#+caption: Effect of the IFF control strategy on the transfer function from $\bm{\tau}$ to $d\bm{\mathcal{L}}_m$ +#+RESULTS: +[[file:figs/enc_plant_plates_effect_iff.png]] + +#+begin_important +As was predicted with the Simscape model, the IFF control strategy is very effective in damping the suspension modes of the nano-hexapod. +Little damping is also applied on the first flexible mode of the strut at 235Hz. +However, no damping is applied on other modes, such as the flexible modes of the top plate. +#+end_important + +*** Comparison of the measured FRF and the Simscape model +Let's now compare the obtained damped plants obtained experimentally with the one extracted from Simscape: +- Figure ref:fig:enc_plates_opt_iff_comp_simscape_all: the individual transfer function from $u_1^\prime$ to the six encoders are comapred +- Figure ref:fig:damped_iff_plates_plant_comp_diagonal: all the diagonal elements are compared +- Figure ref:fig:damped_iff_plates_plant_comp_off_diagonal: all the off-diagonal elements are compared + +#+begin_src matlab :exports none +%% Comparison of the plants (encoder output) when tuning the misalignment +i_input = 1; + +figure; +tiledlayout(2, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile(); +hold on; +plot(frf_iff.f, abs(frf_iff.G_dL(:, 1, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL(1, i_input), freqs, 'Hz')))); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/V]'); +title(sprintf('$d\\tau_{m1}/u_{%i}$', i_input)); + +ax2 = nexttile(); +hold on; +plot(frf_iff.f, abs(frf_iff.G_dL(:, 2, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL(2, i_input), freqs, 'Hz')))); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); +title(sprintf('$d\\tau_{m2}/u_{%i}$', i_input)); + +ax3 = nexttile(); +hold on; +plot(frf_iff.f, abs(frf_iff.G_dL(:, 3, i_input)), ... + 'DisplayName', 'Meas.'); +plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL(3, i_input), freqs, 'Hz'))), ... + 'DisplayName', 'Model'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); +legend('location', 'southeast', 'FontSize', 8); +title(sprintf('$d\\tau_{m3}/u_{%i}$', i_input)); + +ax4 = nexttile(); +hold on; +plot(frf_iff.f, abs(frf_iff.G_dL(:, 4, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL(4, i_input), freqs, 'Hz')))); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]'); +title(sprintf('$d\\tau_{m4}/u_{%i}$', i_input)); + +ax5 = nexttile(); +hold on; +plot(frf_iff.f, abs(frf_iff.G_dL(:, 5, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL(5, i_input), freqs, 'Hz')))); +hold off; +xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +title(sprintf('$d\\tau_{m5}/u_{%i}$', i_input)); + +ax6 = nexttile(); +hold on; +plot(frf_iff.f, abs(frf_iff.G_dL(:, 6, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL(6, i_input), freqs, 'Hz')))); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); +title(sprintf('$d\\tau_{m6}/u_{%i}$', i_input)); + +linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'xy'); +xlim([freqs(1), 1e3]); ylim([1e-8, 1e-4]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_plates_opt_iff_comp_simscape_all.pdf', 'width', 'full', 'height', 'tall'); +#+end_src + +#+name: fig:enc_plates_opt_iff_comp_simscape_all +#+caption: FRF from one actuator to all the encoders when the plant is damped using IFF +#+RESULTS: +[[file:figs/enc_plates_opt_iff_comp_simscape_all.png]] + +#+begin_src matlab :exports none +%% Bode plot for the transfer function from u to dLm +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +% Diagonal Elements FRF +plot(frf_iff.f, abs(frf_iff.G_dL(:,1,1)), 'color', [colors(1,:), 0.2], ... + 'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - FRF') +for i = 2:6 + plot(frf_iff.f, abs(frf_iff.G_dL(:,i,i)), 'color', [colors(1,:), 0.2], ... + 'HandleVisibility', 'off'); +end + +% Diagonal Elements Model +plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL(1,1), freqs, 'Hz'))), 'color', [colors(2,:),0.2], ... + 'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - Model') +for i = 2:6 + plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL(i,i), freqs, 'Hz'))), 'color', [colors(2,:),0.2], ... + 'HandleVisibility', 'off'); +end + +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d_e/V_{exc}$ [m/V]'); set(gca, 'XTickLabel',[]); +ylim([1e-7, 1e-4]); +legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 3); + +ax2 = nexttile; +hold on; +for i =1:6 + plot(frf_iff.f, 180/pi*angle(frf_iff.G_dL(:,i,i)), 'color', [colors(1,:), 0.2]); + plot(freqs, 180/pi*angle(squeeze(freqresp(sim_iff.G_dL(i,i), freqs, 'Hz'))), 'color', [colors(2,:),0.2]); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([20, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/damped_iff_plates_plant_comp_diagonal.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:damped_iff_plates_plant_comp_diagonal +#+caption: Comparison of the diagonal elements of the transfer functions from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ with active damping (IFF) applied with an optimal gain $g = 400$ +#+RESULTS: +[[file:figs/damped_iff_plates_plant_comp_diagonal.png]] + +#+begin_src matlab :exports none +%% Bode plot for the transfer function from u to dLm +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +% Off diagonal FRF +plot(frf_iff.f, abs(frf_iff.G_dL(:,1,2)), 'color', [colors(1,:), 0.2], ... + 'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - FRF') +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(frf_iff.G_dL(:,i,j)), 'color', [colors(1,:), 0.2], ... + 'HandleVisibility', 'off'); + end +end + +% Off diagonal Model +plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL(1,2), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ... + 'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - Model') +for i = 1:5 + for j = i+1:6 + plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL(i,j), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ... + 'HandleVisibility', 'off'); + end +end + +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); +ylim([1e-8, 1e-4]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); + +ax2 = nexttile; +hold on; +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, 180/pi*angle(frf_iff.G_dL(:,i,j)), 'color', [colors(1,:), 0.2]); + plot(freqs, 180/pi*angle(squeeze(freqresp(sim_iff.G_dL(i,j), freqs, 'Hz'))), 'color', [colors(2,:),0.2]); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([20, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/damped_iff_plates_plant_comp_off_diagonal.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:damped_iff_plates_plant_comp_off_diagonal +#+caption: Comparison of the off-diagonal elements of the transfer functions from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ with active damping (IFF) applied with an optimal gain $g = 400$ +#+RESULTS: +[[file:figs/damped_iff_plates_plant_comp_off_diagonal.png]] + +#+begin_important +From Figures ref:fig:damped_iff_plates_plant_comp_diagonal and ref:fig:damped_iff_plates_plant_comp_off_diagonal, it is clear that the Simscape model very well represents the dynamics of the nano-hexapod. +This is true to around 400Hz, then the dynamics depends on the flexible modes of the top plate which are not modelled. +#+end_important + +** Effect of Payload mass on the Dynamics +:PROPERTIES: +:header-args:matlab+: :tangle matlab/scripts/id_frf_enc_plates_effect_payload.m +:END: +<> +*** Introduction :ignore: +In this section, the encoders are fixed to the plates, and we identify the dynamics for several payloads. +The added payload are half cylinders, and three layers can be added for a total of around 40kg (Figure ref:fig:picture_added_3_masses). + +#+name: fig:picture_added_3_masses +#+caption: Picture of the nano-hexapod with added mass +#+attr_latex: :width \linewidth +[[file:figs/picture_added_3_masses.jpg]] + +First the dynamics from $\bm{u}$ to $d\mathcal{L}_m$ and $\bm{\tau}_m$ is identified. +Then, the Integral Force Feedback controller is developed and applied as shown in Figure ref:fig:nano_hexapod_signals_iff. +Finally, the dynamics from $\bm{u}^\prime$ to $d\mathcal{L}_m$ is identified and the added damping can be estimated. + +#+begin_src latex :file nano_hexapod_signals_iff.pdf +\definecolor{instrumentation}{rgb}{0, 0.447, 0.741} +\definecolor{mechanics}{rgb}{0.8500, 0.325, 0.098} +\definecolor{control}{rgb}{0.4660, 0.6740, 0.1880} + +\begin{tikzpicture} + % Blocs + \node[block={4.0cm}{3.0cm}, fill=mechanics!20!white] (nano_hexapod) {Mechanics}; + \coordinate[] (inputF) at (nano_hexapod.west); + \coordinate[] (outputL) at ($(nano_hexapod.south east)!0.8!(nano_hexapod.north east)$); + \coordinate[] (outputF) at ($(nano_hexapod.south east)!0.2!(nano_hexapod.north east)$); + + \node[block, left= 0.8 of inputF, fill=instrumentation!20!white, align=center] (F_stack) {\tiny Actuator \\ \tiny stacks}; + \node[block, left= 0.8 of F_stack, fill=instrumentation!20!white] (PD200) {PD200}; + \node[DAC, left= 0.8 of PD200, fill=instrumentation!20!white] (F_DAC) {DAC}; + \node[block, right=0.8 of outputF, fill=instrumentation!20!white, align=center] (Fm_stack){\tiny Sensor \\ \tiny stack}; + \node[ADC, right=0.8 of Fm_stack,fill=instrumentation!20!white] (Fm_ADC) {ADC}; + \node[block, right=0.8 of outputL, fill=instrumentation!20!white] (encoder) {\tiny Encoder}; + \node[addb, left= 0.8 of F_DAC, fill=control!20!white] (add_iff) {}; + \node[block, below=0.8 of add_iff, fill=control!20!white] (Kiff) {\tiny $K_{\text{IFF}}(s)$}; + + % Connections and labels + \draw[->] (add_iff.east) node[above right]{$\bm{u}$} node[below right]{$[V]$} -- node[sloped]{$/$} (F_DAC.west); + \draw[->] (F_DAC.east) -- node[midway, above]{$\tilde{\bm{u}}$}node[midway, below]{$[V]$} (PD200.west); + \draw[->] (PD200.east) -- node[midway, above]{$\bm{u}_a$}node[midway, below]{$[V]$} (F_stack.west); + \draw[->] (F_stack.east) -- (inputF) node[above left]{$\bm{\tau}$}node[below left]{$[N]$}; + + \draw[->] (outputF) -- (Fm_stack.west) node[above left]{$\bm{\epsilon}$} node[below left]{$[m]$}; + \draw[->] (Fm_stack.east) -- node[midway, above]{$\tilde{\bm{\tau}}_m$}node[midway, below]{$[V]$} (Fm_ADC.west); + \draw[->] (Fm_ADC.east) -- node[sloped]{$/$} ++(0.8, 0)coordinate(end) node[above left]{$\bm{\tau}_m$}node[below left]{$[V]$}; + + \draw[->] (outputL) -- (encoder.west) node[above left]{$d\bm{\mathcal{L}}$} node[below left]{$[m]$}; + \draw[->] (encoder.east) -- node[sloped]{$/$} (encoder-|end) node[above left]{$d\bm{\mathcal{L}}_m$}node[below left]{$[m]$}; + + \draw[->] ($(Fm_ADC.east)+(0.14,0)$) node[branch]{} -- node[sloped]{$/$} ++(0, -1.8) -| (Kiff.south); + \draw[->] (Kiff.north) -- node[sloped]{$/$} (add_iff.south); + \draw[->] ($(add_iff.west)+(-0.8,0)$) node[above right]{$\bm{u}^\prime$} node[below right]{$[V]$} -- node[sloped]{$/$} (add_iff.west); + + % Nano-Hexapod + \begin{scope}[on background layer] + \node[fit={(F_stack.west|-nano_hexapod.south) (Fm_stack.east|-nano_hexapod.north)}, fill=black!20!white, draw, inner sep=2pt] (system) {}; + \node[above] at (system.north) {Nano-Hexapod}; + \end{scope} +\end{tikzpicture} +#+end_src + +#+name: fig:nano_hexapod_signals_iff +#+caption: Block Diagram of the experimental setup and model +#+RESULTS: +[[file:figs/nano_hexapod_signals_iff.png]] + +*** Matlab Init :noexport:ignore: +#+begin_src matlab +%% id_frf_enc_plates_effect_payload.m +% Identification of the nano-hexapod dynamics from u to dL and to taum for several payloads +% Encoders are fixed to the plates +#+end_src + +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) +<> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes +<> +#+end_src + +#+begin_src matlab :tangle no :noweb yes +<> +#+end_src + +#+begin_src matlab :eval no :noweb yes +<> +#+end_src + +#+begin_src matlab :noweb yes +<> +#+end_src + +*** Measured Frequency Response Functions +The following data are loaded: +- =Va=: the excitation voltage (corresponding to $u_i$) +- =Vs=: the generated voltage by the 6 force sensors (corresponding to $\bm{\tau}_m$) +- =de=: the measured motion by the 6 encoders (corresponding to $d\bm{\mathcal{L}}_m$) +#+begin_src matlab +%% Load Identification Data +meas_added_mass = {}; + +for i_mass = i_masses + for i_strut = 1:6 + meas_added_mass(i_strut, i_mass+1) = {load(sprintf('frf_data_exc_strut_%i_realigned_vib_table_%im.mat', i_strut, i_mass), 't', 'Va', 'Vs', 'de')}; + end +end +#+end_src + +The window =win= and the frequency vector =f= are defined. +#+begin_src matlab +% Sampling Time [s] +Ts = (meas_added_mass{1,1}.t(end) - (meas_added_mass{1,1}.t(1)))/(length(meas_added_mass{1,1}.t)-1); + +% Hannning Windows +win = hanning(ceil(1/Ts)); + +% And we get the frequency vector +[~, f] = tfestimate(meas_added_mass{1,1}.Va, meas_added_mass{1,1}.de, win, [], [], 1/Ts); +#+end_src + +Finally the $6 \times 6$ transfer function matrices from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ and from $\bm{u}$ to $\bm{\tau}_m$ are identified: +#+begin_src matlab +%% DVF Plant (transfer function from u to dLm) +G_dL = {}; + +for i_mass = i_masses + G_dL(i_mass+1) = {zeros(length(f), 6, 6)}; + for i_strut = 1:6 + G_dL{i_mass+1}(:,:,i_strut) = tfestimate(meas_added_mass{i_strut, i_mass+1}.Va, meas_added_mass{i_strut, i_mass+1}.de, win, [], [], 1/Ts); + end +end + +%% IFF Plant (transfer function from u to taum) +G_tau = {}; + +for i_mass = i_masses + G_tau(i_mass+1) = {zeros(length(f), 6, 6)}; + for i_strut = 1:6 + G_tau{i_mass+1}(:,:,i_strut) = tfestimate(meas_added_mass{i_strut, i_mass+1}.Va, meas_added_mass{i_strut, i_mass+1}.Vs, win, [], [], 1/Ts); + end +end +#+end_src + +The identified dynamics are then saved for further use. +#+begin_src matlab :exports none :tangle no +save('matlab/data_frf/frf_vib_table_m.mat', 'f', 'Ts', 'G_tau', 'G_dL') +#+end_src + +#+begin_src matlab :eval no +save('data_frf/frf_vib_table_m.mat', 'f', 'Ts', 'G_tau', 'G_dL') +#+end_src + +#+begin_src matlab :exports none +frf_ol = load('frf_vib_table_m.mat', 'f', 'Ts', 'G_tau', 'G_dL'); +#+end_src + +*** Rigidification of the added payloads +- [ ] figure + +#+begin_src matlab +%% Load Identification Data +meas_added_mass = {}; + +for i_strut = 1:6 + meas_added_mass(i_strut) = {load(sprintf('frf_data_exc_strut_%i_spindle_1m_solid.mat', i_strut), 't', 'Va', 'Vs', 'de')}; +end +#+end_src + +The window =win= and the frequency vector =f= are defined. +#+begin_src matlab +% Sampling Time [s] +Ts = (meas_added_mass{1}.t(end) - (meas_added_mass{1}.t(1)))/(length(meas_added_mass{1}.t)-1); + +% Hannning Windows +win = hanning(ceil(1/Ts)); + +% And we get the frequency vector +[~, f] = tfestimate(meas_added_mass{1}.Va, meas_added_mass{1}.de, win, [], [], 1/Ts); +#+end_src + +Finally the $6 \times 6$ transfer function matrices from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ and from $\bm{u}$ to $\bm{\tau}_m$ are identified: +#+begin_src matlab +%% DVF Plant (transfer function from u to dLm) + +G_dL = zeros(length(f), 6, 6); +for i_strut = 1:6 + G_dL(:,:,i_strut) = tfestimate(meas_added_mass{i_strut}.Va, meas_added_mass{i_strut}.de, win, [], [], 1/Ts); +end + +%% IFF Plant (transfer function from u to taum) +G_tau = zeros(length(f), 6, 6); +for i_strut = 1:6 + G_tau(:,:,i_strut) = tfestimate(meas_added_mass{i_strut}.Va, meas_added_mass{i_strut}.Vs, win, [], [], 1/Ts); +end +#+end_src + +#+begin_src matlab :exports none +%% Bode plot for the transfer function from u to dLm - Several payloads +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +% Diagonal terms +for i = 1:6 + plot(frf_ol.f, abs(frf_ol.G_dL{2}(:,i, i)), 'color', colors(1,:)); + plot(f, abs(G_dL(:,i, i)), 'color', colors(2,:)); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]'); +ylim([1e-8, 1e-3]); +xlim([20, 2e3]); +#+end_src + +#+begin_src matlab :exports none +%% Bode plot for the transfer function from u to dLm +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:6 + plot(frf_ol.f, abs(frf_ol.G_dL(:,i, i)), 'color', colors(1,:)); + plot(f, abs(G_dL(:,i, i)), 'color', colors(2,:)); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]'); +ylim([1e-8, 1e-3]); +xlim([10, 1e3]); +#+end_src + + +*** Transfer function from Actuators to Encoders +The transfer functions from $\bm{u}$ to $d\bm{\mathcal{L}}_{m}$ are shown in Figure ref:fig:comp_plant_payloads_dvf. + +#+begin_src matlab :exports none +%% Bode plot for the transfer function from u to dLm - Several payloads +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i_mass = i_masses + % Diagonal terms + plot(frf_ol.f, abs(frf_ol.G_dL{i_mass+1}(:,1, 1)), 'color', colors(i_mass+1,:), ... + 'DisplayName', sprintf('$d\\mathcal{L}_{m,i}/u_i$ - %i', i_mass)); + for i = 2:6 + plot(frf_ol.f, abs(frf_ol.G_dL{i_mass+1}(:,i, i)), 'color', colors(i_mass+1,:), ... + 'HandleVisibility', 'off'); + end + % Off-Diagonal terms + for i = 1:5 + for j = i+1:6 + plot(frf_ol.f, abs(frf_ol.G_dL{i_mass+1}(:,i,j)), 'color', [colors(i_mass+1,:), 0.2], ... + 'HandleVisibility', 'off'); + end + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); +ylim([1e-8, 1e-3]); +legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); + +ax2 = nexttile; +hold on; +for i_mass = i_masses + for i =1:6 + plot(frf_ol.f, 180/pi*angle(frf_ol.G_dL{i_mass+1}(:,i, i)), 'color', colors(i_mass+1,:)); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-90, 180]) + +linkaxes([ax1,ax2],'x'); +xlim([20, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/comp_plant_payloads_dvf.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:comp_plant_payloads_dvf +#+caption: Measured Frequency Response Functions from $u_i$ to $d\mathcal{L}_{m,i}$ for all 4 payload conditions. Diagonal terms are solid lines, and shaded lines are off-diagonal terms. +#+RESULTS: +[[file:figs/comp_plant_payloads_dvf.png]] + + +#+begin_important +From Figure ref:fig:comp_plant_payloads_dvf, we can observe few things: +- The obtained dynamics is changing a lot between the case without mass and when there is at least one added mass. +- Between 1, 2 and 3 added masses, the dynamics is not much different, and it would be easier to design a controller only for these cases. +- The flexible modes of the top plate is first decreased a lot when the first mass is added (from 700Hz to 400Hz). + This is due to the fact that the added mass is composed of two half cylinders which are not fixed together. + Therefore is adds a lot of mass to the top plate without adding a lot of rigidity in one direction. + When more than 1 mass layer is added, the half cylinders are added with some angles such that rigidity are added in all directions (see Figure ref:fig:picture_added_3_masses). + In that case, the frequency of these flexible modes are increased. + In practice, the payload should be one solid body, and we should not see a massive decrease of the frequency of this flexible mode. +- Flexible modes of the top plate are becoming less problematic as masses are added. +- First flexible mode of the strut at 230Hz is not much decreased when mass is added. + However, its apparent amplitude is much decreased. +#+end_important + +*** Transfer function from Actuators to Force Sensors +The transfer functions from $\bm{u}$ to $\bm{\tau}_{m}$ are shown in Figure ref:fig:comp_plant_payloads_iff. + +#+begin_src matlab :exports none +%% Bode plot for the transfer function from u to dLm +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i_mass = i_masses + % Diagonal terms + plot(frf_ol.f, abs(frf_ol.G_tau{i_mass+1}(:,1, 1)), 'color', colors(i_mass+1,:), ... + 'DisplayName', sprintf('$\\tau_{m,i}/u_i$ - %i', i_mass)); + for i = 2:6 + plot(frf_ol.f, abs(frf_ol.G_tau{i_mass+1}(:,i, i)), 'color', colors(i_mass+1,:), ... + 'HandleVisibility', 'off'); + end + % Off-Diagonal terms + for i = 1:5 + for j = i+1:6 + plot(frf_ol.f, abs(frf_ol.G_tau{i_mass+1}(:,i,j)), 'color', [colors(i_mass+1,:), 0.2], ... + 'HandleVisibility', 'off'); + end + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]); +ylim([1e-2, 1e2]); +legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 3); + +ax2 = nexttile; +hold on; +for i_mass = i_masses + for i =1:6 + plot(frf_ol.f, 180/pi*angle(frf_ol.G_tau{i_mass+1}(:,i, i)), 'color', colors(i_mass+1,:)); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); + +linkaxes([ax1,ax2],'x'); +xlim([20, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/comp_plant_payloads_iff.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:comp_plant_payloads_iff +#+caption: Measured Frequency Response Functions from $u_i$ to $\tau_{m,i}$ for all 4 payload conditions. Diagonal terms are solid lines, and shaded lines are off-diagonal terms. +#+RESULTS: +[[file:figs/comp_plant_payloads_iff.png]] + +#+begin_important +From Figure ref:fig:comp_plant_payloads_iff, we can see that for all added payloads, the transfer function from $\bm{u}$ to $\bm{\tau}_{m}$ always has alternating poles and zeros. +#+end_important + +*** Coupling of the transfer function from Actuator to Encoders +The RGA-number, which is a measure of the interaction in the system, is computed for the transfer function matrix from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ for all the payloads. +The obtained numbers are compared in Figure ref:fig:rga_num_ol_masses. + +#+begin_src matlab :exports none +%% Decentralized RGA - Undamped Plant +RGA_num = zeros(length(frf_ol.f), length(i_masses)); +for i_mass = i_masses + for i = 1:length(frf_ol.f) + RGA_num(i, i_mass+1) = sum(sum(abs(eye(6) - squeeze(frf_ol.G_dL{i_mass+1}(i,:,:)).*inv(squeeze(frf_ol.G_dL{i_mass+1}(i,:,:))).'))); + end +end +#+end_src + +#+begin_src matlab :exports none +%% RGA for Decentralized plant +figure; +hold on; +for i_mass = i_masses + plot(frf_ol.f, RGA_num(:,i_mass+1), '-', 'color', colors(i_mass+1,:), ... + 'DisplayName', sprintf('RGA-num - %i mass', i_mass)); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('RGA Number'); +xlim([10, 1e3]); ylim([1e-2, 1e2]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/rga_num_ol_masses.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:rga_num_ol_masses +#+caption: RGA-number for the open-loop transfer function from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ +#+RESULTS: +[[file:figs/rga_num_ol_masses.png]] + +#+begin_important +From Figure ref:fig:rga_num_ol_masses, it is clear that the coupling is quite large starting from the first suspension mode of the nano-hexapod. +Therefore, is the payload's mass is increase, the coupling in the system start to become unacceptably large at lower frequencies. +#+end_important + +** Comparison with the Simscape model +:PROPERTIES: +:header-args:matlab+: :tangle matlab/scripts/id_frf_enc_plates_effect_payload_comp_simscape.m +:END: +<> +*** Introduction :ignore: +Let's now compare the identified dynamics with the Simscape model. +We wish to verify if the Simscape model is still accurate for all the tested payloads. + +*** Matlab Init :noexport:ignore: +#+begin_src matlab +%% id_frf_enc_plates_effect_payload_comp_simscape.m +% Comparison of the nano-hexapod dynamics from u to dL and to taum for several payloads - +% Measured FRF and extracted dynamics from the Simscape model +% Encoders are fixed to the plates +#+end_src + +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) +<> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes +<> +#+end_src + +#+begin_src matlab :tangle no :noweb yes +<> +#+end_src + +#+begin_src matlab :eval no :noweb yes +<> +#+end_src + +#+begin_src matlab :noweb yes +<> +<> +#+end_src + +#+begin_src matlab +%% Load the identified FRF +frf_ol_m = load('frf_vib_table_m.mat', 'f', 'Ts', 'G_tau', 'G_dL'); +#+end_src + +*** System Identification +Let's initialize the simscape model with the nano-hexapod fixed on top of the vibration table. +#+begin_src matlab +%% Initialize Nano-Hexapod +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'plates', ... + 'actuator_type', '2dof'); + +support.type = 1; % On top of vibration table +#+end_src + +First perform the identification for the transfer functions from $\bm{u}$ to $d\bm{\mathcal{L}}_m$: +#+begin_src matlab +%% Identify the DVF Plant (transfer function from u to dLm) +clear io; io_i = 1; +io(io_i) = linio([mdl, '/du'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Encoders + +%% Identification for all the added payloads +G_dL = {}; + +for i = i_masses + fprintf('i = %i\n', i) + payload.type = i; % Change the payload on the nano-hexapod + G_dL(i+1) = {exp(-s*frf_ol.Ts)*linearize(mdl, io, 0.0, options)}; +end +#+end_src + +#+begin_src matlab +%% Identify the IFF Plant (transfer function from u to taum) +clear io; io_i = 1; +io(io_i) = linio([mdl, '/du'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensors + +%% Identification for all the added payloads +G_tau = {}; + +for i = 0:3 + fprintf('i = %i\n', i) + payload.type = i; % Change the payload on the nano-hexapod + G_tau(i+1) = {exp(-s*frf_ol.Ts)*linearize(mdl, io, 0.0, options)}; +end +#+end_src + +The identified dynamics are then saved for further use. +#+begin_src matlab :exports none :tangle no +save('matlab/data_frf/sim_vib_table_m.mat', 'G_tau', 'G_dL') +#+end_src + +#+begin_src matlab :eval no +save('data_frf/sim_vib_table_m.mat', 'G_tau', 'G_dL') +#+end_src + +#+begin_src matlab :exports none +sim_ol_m = load('sim_vib_table_m.mat', 'G_tau', 'G_dL'); +#+end_src + +*** Transfer function from Actuators to Encoders +The measured FRF and the identified dynamics from $u_i$ to $d\mathcal{L}_{m,i}$ are compared in Figure ref:fig:comp_masses_model_exp_dvf. +A zoom near the "suspension" modes is shown in Figure ref:fig:comp_masses_model_exp_dvf_zoom. + +#+begin_src matlab :exports none +%% Bode plot for the transfer function from u to dLm +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +freqs = 2*logspace(1,3,1000); + +ax1 = nexttile([2,1]); +hold on; +for i_mass = i_masses + plot(frf_ol_m.f, abs(frf_ol_m.G_dL{i_mass+1}(:,1, 1)), 'color', [colors(i_mass+1,:), 0.2], ... + 'DisplayName', sprintf('$d\\mathcal{L}_{m,i}/u_i$ - FRF %i', i_mass)); + for i = 2:6 + plot(frf_ol_m.f, abs(frf_ol_m.G_dL{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.2], ... + 'HandleVisibility', 'off'); + end + set(gca, 'ColorOrderIndex', i_mass+1) + plot(freqs, abs(squeeze(freqresp(sim_ol_m.G_dL{i_mass+1}(1,1), freqs, 'Hz'))), '--', ... + 'DisplayName', sprintf('$d\\mathcal{L}_{m,i}/u_i$ - Sim %i', i_mass)); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); +ylim([1e-8, 1e-3]); +legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); + +ax2 = nexttile; +hold on; +for i_mass = i_masses + for i =1:6 + plot(frf_ol_m.f, 180/pi*angle(frf_ol_m.G_dL{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.2]); + end + set(gca, 'ColorOrderIndex', i_mass+1) + plot(freqs, 180/pi*angle(squeeze(freqresp(sim_ol_m.G_dL{i_mass+1}(1,1), freqs, 'Hz'))), '--'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:45:360); +ylim([-45, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([20, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/comp_masses_model_exp_dvf.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:comp_masses_model_exp_dvf +#+caption: Comparison of the transfer functions from $u_i$ to $d\mathcal{L}_{m,i}$ - measured FRF and identification from the Simscape model +#+RESULTS: +[[file:figs/comp_masses_model_exp_dvf.png]] + +#+begin_src matlab :exports none :tangle no +ax1.YLim = [1e-6, 5e-4]; +xlim([40, 2e2]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/comp_masses_model_exp_dvf_zoom.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:comp_masses_model_exp_dvf_zoom +#+caption: Comparison of the transfer functions from $u_i$ to $d\mathcal{L}_{m,i}$ - measured FRF and identification from the Simscape model (Zoom) +#+RESULTS: +[[file:figs/comp_masses_model_exp_dvf_zoom.png]] + +#+begin_important +The Simscape model is very accurately representing the measured dynamics up. +Only the flexible modes of the struts and of the top plate are not represented here as these elements are modelled as rigid bodies. +#+end_important + +*** Transfer function from Actuators to Force Sensors +The measured FRF and the identified dynamics from $u_i$ to $\tau_{m,i}$ are compared in Figure ref:fig:comp_masses_model_exp_iff. +A zoom near the "suspension" modes is shown in Figure ref:fig:comp_masses_model_exp_iff_zoom. + +#+begin_src matlab :exports none +%% Bode plot for the transfer function from u to dLm +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +freqs = 2*logspace(1,3,1000); + +ax1 = nexttile([2,1]); +hold on; +for i_mass = 0:3 + plot(frf_ol_m.f, abs(frf_ol_m.G_tau{i_mass+1}(:,1, 1)), 'color', [colors(i_mass+1,:), 0.2], ... + 'DisplayName', sprintf('$d\\tau_{m,i}/u_i$ - FRF %i', i_mass)); + for i = 2:6 + plot(frf_ol_m.f, abs(frf_ol_m.G_tau{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.2], ... + 'HandleVisibility', 'off'); + end + plot(freqs, abs(squeeze(freqresp(sim_ol_m.G_tau{i_mass+1}(1,1), freqs, 'Hz'))), '--', 'color', colors(i_mass+1,:), ... + 'DisplayName', sprintf('$\\tau_{m,i}/u_i$ - Sim %i', i_mass)); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]); +ylim([1e-2, 1e2]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); + +ax2 = nexttile; +hold on; +for i_mass = 0:3 + for i =1:6 + plot(frf_ol_m.f, 180/pi*angle(frf_ol_m.G_tau{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.2]); + end + plot(freqs, 180/pi*angle(squeeze(freqresp(sim_ol_m.G_tau{i_mass+1}(i,i), freqs, 'Hz'))), '--', 'color', colors(i_mass+1,:)); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); + +linkaxes([ax1,ax2],'x'); +xlim([20, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/comp_masses_model_exp_iff.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:comp_masses_model_exp_iff +#+caption: Comparison of the transfer functions from $u_i$ to $\tau_{m,i}$ - measured FRF and identification from the Simscape model +#+RESULTS: +[[file:figs/comp_masses_model_exp_iff.png]] + +#+begin_src matlab :exports none :tangle no +xlim([40, 2e2]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/comp_masses_model_exp_iff_zoom.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:comp_masses_model_exp_iff_zoom +#+caption: Comparison of the transfer functions from $u_i$ to $\tau_{m,i}$ - measured FRF and identification from the Simscape model (Zoom) +#+RESULTS: +[[file:figs/comp_masses_model_exp_iff_zoom.png]] + +** Integral Force Feedback Controller +:PROPERTIES: +:header-args:matlab+: :tangle matlab/scripts/iff_robust_enc_plates.m +:END: +<> +*** Introduction :ignore: +In this section, we wish to develop the Integral Force Feedback controller that is robust with respect to the added payload. + +*** Matlab Init :noexport:ignore: +#+begin_src matlab +%% iff_robust_enc_plates.m +% Development of IFF controller that works for all payloads +% Estimation of performances using the Simscape model and validation with measurements +#+end_src + +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) +<> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes +<> +#+end_src + +#+begin_src matlab :tangle no :noweb yes +<> +#+end_src + +#+begin_src matlab :eval no :noweb yes +<> +#+end_src + +#+begin_src matlab :noweb yes +<> +<> +#+end_src + +#+begin_src matlab +%% Load the identified FRF and Simscape model +frf_ol = load('frf_vib_table_m.mat', 'f', 'Ts', 'G_tau', 'G_dL'); +sim_ol = load('sim_vib_table_m.mat', 'G_tau', 'G_dL'); +#+end_src + +*** Robust IFF Controller +Based on the measured FRF from $\bm{u}$ to $\bm{\tau}_{m}$, the following IFF controller is developed: +#+begin_src matlab +%% IFF Controller +Kiff_g1 = -(1/(s + 2*pi*20))*... % LPF: provides integral action above 20[Hz] + (s/(s + 2*pi*20))*... % HPF: limit low frequency gain + (1/(1 + s/2/pi/400)); % LPF: more robust to high frequency resonances +#+end_src + +Then, the Root Locus plot of Figure ref:fig:iff_root_locus_masses is used to estimate the optimal gain. +This Root Locus plot is computed from the Simscape model. +#+begin_src matlab :exports none +%% Root Locus for IFF +gains = logspace(1, 3, 100); + +figure; + +hold on; +% Pure Integrator +for i_mass = 0:3 + plot(real(pole(sim_ol.G_tau{i_mass+1})), imag(pole(sim_ol.G_tau{i_mass+1})), 'x', 'color', colors(i_mass+1, :), ... + 'DisplayName', sprintf('OL Poles - %i', i_mass)); + plot(real(tzero(sim_ol.G_tau{i_mass+1})), imag(tzero(sim_ol.G_tau{i_mass+1})), 'o', 'color', colors(i_mass+1, :), ... + 'HandleVisibility', 'off'); +end + +for i_mass = 0:3 + for g = gains + clpoles = pole(feedback(sim_ol.G_tau{i_mass+1}, g*Kiff_g1*eye(6), +1)); + plot(real(clpoles), imag(clpoles), '.', 'color', colors(i_mass+1, :), ... + 'HandleVisibility', 'off'); + end +end + +g_opt = 2e2; + +clpoles = pole(feedback(sim_ol.G_tau{1}, g_opt*Kiff_g1*eye(6), +1)); +plot(real(clpoles), imag(clpoles), 'kx', ... + 'DisplayName', sprintf('$g = %.0f$', g_opt)); +for i_mass = 1:3 + clpoles = pole(feedback(sim_ol.G_tau{i_mass+1}, g_opt*Kiff_g1*eye(6), +1)); + plot(real(clpoles), imag(clpoles), 'kx', ... + 'HandleVisibility', 'off'); +end +axis square; +xlim([-700, 0]); ylim([0, 1400]); +xlabel('Real Part'); ylabel('Imaginary Part'); +legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 3); +rootLocusOverlay('xis', [0.01, 0.1, 0.2, 0.3, 0.4, 0.5], 'R', 1e3); +hold off; +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/iff_root_locus_masses.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:iff_root_locus_masses +#+caption: Root Locus for the IFF control strategy (for all payload conditions). +#+RESULTS: +[[file:figs/iff_root_locus_masses.png]] + +#+begin_src matlab :exports none :tangle no +%% Verify close-loop stability for all payloads +for i_mass = 0:3 + clpoles = isstable(feedback(sim_ol.G_tau{i_mass+1}, g_opt*Kiff_g1*eye(6), +1)); + sum(real(clpoles)>0) +end +#+end_src + +The found optimal IFF controller is: +#+begin_src matlab +%% Optimal controller +g_opt = 2e2; +Kiff = g_opt*Kiff_g1*eye(6); +#+end_src + +It is saved for further use. +#+begin_src matlab :exports none :tangle no +save('matlab/data_sim/Kiff_robust_opt.mat', 'Kiff') +#+end_src + +#+begin_src matlab :eval no +save('data_sim/Kiff_robust_opt.mat', 'Kiff') +#+end_src + +The corresponding experimental loop gains are shown in Figure ref:fig:iff_loop_gain_masses. +#+begin_src matlab :exports none +%% Bode plot for the transfer function from u to dLm +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i_mass = 0:3 + for i = 1:6 + plot(frf_ol.f, abs(squeeze(freqresp(Kiff(i,i), frf_ol.f, 'Hz')).*frf_ol.G_tau{i_mass+1}(:,i,i)), '-', 'color', [colors(i_mass+1,:), 0.2]); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Loop Gain [-]'); set(gca, 'XTickLabel',[]); +ylim([1e-2, 1e2]); + +ax2 = nexttile; +hold on; +for i_mass = 0:3 + for i = 1:6 + plot(frf_ol.f, 180/pi*angle(squeeze(freqresp(-Kiff(i,i), frf_ol.f, 'Hz')).*frf_ol.G_tau{i_mass+1}(:,i,i)), '-', 'color', [colors(i_mass+1,:), 0.2]); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([10, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/iff_loop_gain_masses.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:iff_loop_gain_masses +#+caption: Loop gain for the Integral Force Feedback controller +#+RESULTS: +[[file:figs/iff_loop_gain_masses.png]] + +#+begin_important +Based on the above analysis: +- The same IFF controller can be used to damp the suspension modes for all payload conditions +- The IFF controller should be robust +#+end_important + +*** Estimated Damped Plant from the Simscape model +Let's initialize the simscape model with the nano-hexapod fixed on top of the vibration table. +#+begin_src matlab +%% Initialize the Simscape model in closed loop +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'plates', ... + 'actuator_type', '2dof', ... + 'controller_type', 'iff'); + +support.type = 1; % On top of vibration table +#+end_src + +And Load the IFF controller. +#+begin_src matlab :exports none +%% Make sure IFF controller is loaded +load('Kiff_robust_opt.mat', 'Kiff') +#+end_src + +Finally, let's identify the damped plant from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ for all payloads. +#+begin_src matlab +%% Identify the (damped) transfer function from u to dLm +clear io; io_i = 1; +io(io_i) = linio([mdl, '/du'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Plate Displacement (encoder) + +%% Identify for all add masses +G_dL = {}; + +for i = i_masses + payload.type = i; + G_dL(i+1) = {exp(-s*frf_ol.Ts)*linearize(mdl, io, 0.0, options)}; +end +#+end_src + +The identified dynamics are then saved for further use. +#+begin_src matlab :exports none :tangle no +%% Save the identified dynamics +save('matlab/data_frf/sim_iff_vib_table_m.mat', 'G_dL'); +#+end_src + +#+begin_src matlab :eval no +save('data_frf/sim_iff_vib_table_m.mat', 'G_dL'); +#+end_src + +#+begin_src matlab +%% Load the identified dynamics +sim_iff = load('sim_iff_vib_table_m.mat', 'G_dL'); +#+end_src + +#+begin_src matlab :exports none :tangle no +%% Verify Stability +for i = i_masses + isstable(sim_iff.G_dL{i+1}) +end +#+end_src + +#+begin_src matlab :exports none +%% Bode plot for the transfer function from u to dLm - With and without IFF +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +freqs = logspace(1,3,1000); + +ax1 = nexttile([2,1]); +hold on; +for i_mass = i_masses + for i = 1 + plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL{i_mass+1}(i,i), freqs, 'Hz'))), '-', 'color', [colors(i_mass+1, :), 0.5], ... + 'DisplayName', sprintf('$d\\mathcal{L}_i/u_i$ - %i', i_mass)); + plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL{i_mass+1}(i,i), freqs, 'Hz'))), '-', 'color', colors(i_mass+1, :), ... + 'DisplayName', sprintf('$d\\mathcal{L}_i/u^\\prime_i$ - %i', i_mass)); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); +ylim([1e-7, 1e-3]); +legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 4); + +ax2 = nexttile; +hold on; +for i_mass = i_masses + for i = 1 + plot(freqs, 180/pi*angle(squeeze(freqresp(sim_ol.G_dL{i_mass+1}(1,1), freqs, 'Hz'))), '-', 'color', [colors(i_mass+1, :), 0.5]); + plot(freqs, 180/pi*angle(squeeze(freqresp(sim_iff.G_dL{i_mass+1}(1,1), freqs, 'Hz'))), '-', 'color', colors(i_mass+1, :)); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/damped_plant_model_masses.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:damped_plant_model_masses +#+caption: Transfer function from $u_i$ to $d\mathcal{L}_{m,i}$ (without active damping) and from $u^\prime_i$ to $d\mathcal{L}_{m,i}$ (with IFF) +#+RESULTS: +[[file:figs/damped_plant_model_masses.png]] + +*** Compute the identified FRF with IFF +Several experimental identifications are done in order to identify the dynamics from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ with the robust IFF controller implemented and with various payloads. + +The following data are loaded: +- =Va=: the excitation voltage for the damped plant (corresponding to $u^\prime_i$) +- =de=: the measured motion by the 6 encoders (corresponding to $d\bm{\mathcal{L}}_m$) +#+begin_src matlab +%% Load Identification Data +meas_added_mass = {}; + +for i_mass = i_masses + for i_strut = 1:6 + meas_iff_mass(i_strut, i_mass+1) = {load(sprintf('frf_data_exc_strut_%i_iff_vib_table_%im.mat', i_strut, i_mass), 't', 'Va', 'de')}; + end +end +#+end_src + +The window =win= and the frequency vector =f= are defined. +#+begin_src matlab +% Sampling Time [s] +Ts = (meas_iff_mass{1,1}.t(end) - (meas_iff_mass{1,1}.t(1)))/(length(meas_iff_mass{1,1}.t)-1); + +% Hannning Windows +win = hanning(ceil(1/Ts)); + +% And we get the frequency vector +[~, f] = tfestimate(meas_iff_mass{1,1}.Va, meas_iff_mass{1,1}.de, win, [], [], 1/Ts); +#+end_src + +Finally the $6 \times 6$ transfer function matrix from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ is estimated: +#+begin_src matlab +%% DVF Plant (transfer function from u to dLm) +G_dL = {}; + +for i_mass = i_masses + G_dL(i_mass+1) = {zeros(length(f), 6, 6)}; + for i_strut = 1:6 + G_dL{i_mass+1}(:,:,i_strut) = tfestimate(meas_iff_mass{i_strut, i_mass+1}.Va, meas_iff_mass{i_strut, i_mass+1}.de, win, [], [], 1/Ts); + end +end +#+end_src + +The identified dynamics are then saved for further use. +#+begin_src matlab :exports none :tangle no +save('matlab/data_frf/frf_iff_vib_table_m.mat', 'f', 'Ts', 'G_dL'); +#+end_src + +#+begin_src matlab :eval no +save('mat/frf_iff_vib_table_m.mat', 'f', 'Ts', 'G_dL'); +#+end_src + +*** Comparison of the measured FRF and the Simscape model +#+begin_src matlab :exports none +%% Load the Measured FRF of the damped plant +frf_iff = load('frf_iff_vib_table_m.mat', 'f', 'Ts', 'G_dL'); +#+end_src + +The following figures are computed: +- Figure ref:fig:damped_iff_plant_meas_frf: the measured damped FRF are displayed +- Figure ref:fig:comp_undamped_damped_plant_meas_frf: the open-loop and damped FRF are compared (diagonal elements) +- Figure ref:fig:comp_iff_plant_frf_sim: the obtained damped FRF is compared with the identified damped from using the Simscape model + +#+begin_src matlab :exports none +%% Diagonal and Off Diagonal elements of the damped plants +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i_mass = i_masses + plot(frf_iff.f, abs(frf_iff.G_dL{i_mass+1}(:,1,1)), 'color', colors(i_mass+1,:), ... + 'DisplayName', sprintf('$d\\mathcal{L}_{m,i}/u^\\prime_i$ - %i', i_mass)); + for i = 2:6 + plot(frf_iff.f, abs(frf_iff.G_dL{i_mass+1}(:,i,i)), 'color', colors(i_mass+1,:), ... + 'HandleVisibility', 'off'); + end + plot(frf_iff.f, abs(frf_iff.G_dL{i_mass+1}(:,1,2)), 'color', [colors(i_mass+1,:), 0.2], ... + 'DisplayName', sprintf('$d\\mathcal{L}_{m,i}/u^\\prime_j$ - %i', i_mass)); + for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(frf_iff.G_dL{i_mass+1}(:,i,j)), 'color', [colors(i_mass+1,:), 0.2], ... + 'HandleVisibility', 'off'); + end + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); +ylim([1e-9, 1e-4]); + +legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); + +ax2 = nexttile; +hold on; +for i_mass = i_masses + for i =1:6 + plot(frf_iff.f, 180/pi*angle(frf_iff.G_dL{i_mass+1}(:,i, i)), 'color', colors(i_mass+1,:)); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([10, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/damped_iff_plant_meas_frf.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:damped_iff_plant_meas_frf +#+caption: Diagonal and off-diagonal of the measured FRF matrix for the damped plant +#+RESULTS: +[[file:figs/damped_iff_plant_meas_frf.png]] + +#+begin_src matlab :exports none +%% Comparison of the OL and IFF identified FRF +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i_mass = i_masses + plot(frf_ol.f, abs(frf_ol.G_dL{i_mass+1}(:,1,1)), '-', 'color', [colors(i_mass+1, :), 0.5], ... + 'DisplayName', sprintf('$d\\mathcal{L}_i/u_i$ - %i', i_mass)); + plot(frf_iff.f, abs(frf_iff.G_dL{i_mass+1}(:,1,1)), '-', 'color', colors(i_mass+1, :), ... + 'DisplayName', sprintf('$d\\mathcal{L}_i/u^\\prime_i$ - %i', i_mass)); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +ylim([1e-7, 1e-3]); +legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); + +ax2 = nexttile; +hold on; +for i_mass = i_masses + plot(frf_ol.f, 180/pi*angle(frf_ol.G_dL{i_mass+1}(:,1,1)), '-', 'color', [colors(i_mass+1, :), 0.5]); + plot(frf_iff.f, 180/pi*angle(frf_iff.G_dL{i_mass+1}(:,1,1)), '-', 'color', colors(i_mass+1, :)); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([10, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/comp_undamped_damped_plant_meas_frf.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:comp_undamped_damped_plant_meas_frf +#+caption: Damped and Undamped measured FRF (diagonal elements) +#+RESULTS: +[[file:figs/comp_undamped_damped_plant_meas_frf.png]] + +#+begin_src matlab :exports none +%% Comparison of the measured FRF and identified TF of the damped plant +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +freqs = logspace(1,3,1000); + +ax1 = nexttile([2,1]); +hold on; +for i_mass = i_masses + plot(frf_iff.f, abs(frf_iff.G_dL{i_mass+1}(:,1, 1)), 'color', [colors(i_mass+1,:), 0.2], ... + 'DisplayName', sprintf('$d\\mathcal{L}_{m,i}/u^\\prime_i$ - FRF %i', i_mass)); + for i = 2:6 + plot(frf_iff.f, abs(frf_iff.G_dL{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.2], ... + 'HandleVisibility', 'off'); + end + set(gca, 'ColorOrderIndex', i_mass+1) + plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL{i_mass+1}(1,1), freqs, 'Hz'))), '--', ... + 'DisplayName', sprintf('$d\\mathcal{L}_{m,i}/u^\\prime_i$ - Sim %i', i_mass)); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); +ylim([1e-8, 1e-4]); +legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); + +ax2 = nexttile; +hold on; +for i_mass = i_masses + for i =1:6 + plot(frf_iff.f, 180/pi*angle(frf_iff.G_dL{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.2]); + end + set(gca, 'ColorOrderIndex', i_mass+1) + plot(freqs, 180/pi*angle(squeeze(freqresp(sim_iff.G_dL{i_mass+1}(1,1), freqs, 'Hz'))), '--'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([10, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/comp_iff_plant_frf_sim.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:comp_iff_plant_frf_sim +#+caption: Comparison of the measured FRF and the identified dynamics from the Simscape model +#+RESULTS: +[[file:figs/comp_iff_plant_frf_sim.png]] + +#+begin_important +The IFF control strategy effectively damps all the suspensions modes of the nano-hexapod whatever the payload is. +The obtained plant is easier to control (provided the flexible modes of the top platform are well damped). +#+end_important + +*** Change of coupling with IFF +Let's see how the Integral Force Feedback is changing the interaction in the system. + +To study that, the RGA-number are computed both for the undamped plant and for the damped plant using IFF. + +#+begin_src matlab :exports none +%% Decentralized RGA - Undamped Plant +RGA_num_ol = zeros(length(frf_ol.f), length(i_masses)); +for i_mass = i_masses + for i = 1:length(frf_ol.f) + RGA_num_ol(i, i_mass+1) = sum(sum(abs(eye(6) - squeeze(frf_ol.G_dL{i_mass+1}(i,:,:)).*inv(squeeze(frf_ol.G_dL{i_mass+1}(i,:,:))).'))); + end +end +#+end_src + +#+begin_src matlab :exports none +%% Decentralized RGA - Damped Plant using IFF +RGA_num_iff = zeros(length(frf_iff.f), length(i_masses)); +for i_mass = i_masses + for i = 1:length(frf_iff.f) + RGA_num_iff(i, i_mass+1) = sum(sum(abs(eye(6) - squeeze(frf_iff.G_dL{i_mass+1}(i,:,:)).*inv(squeeze(frf_iff.G_dL{i_mass+1}(i,:,:))).'))); + end +end +#+end_src + +Both are compared in Figure ref:fig:rga_num_ol_iff_masses. + +#+begin_src matlab :exports none +%% RGA for Decentralized plant - With and Without IFF +figure; +hold on; +for i_mass = i_masses + plot(frf_ol.f, RGA_num_ol(:,i_mass+1), '-', 'color', colors(i_mass+1,:), ... + 'DisplayName', sprintf('RGA-num OL - %im', i_mass)); + plot(frf_iff.f, RGA_num_iff(:,i_mass+1), '--', 'color', colors(i_mass+1,:), ... + 'DisplayName', sprintf('RGA-num IFF - %im', i_mass)); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('RGA Number'); +xlim([10, 1e3]); ylim([1e-2, 1e2]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/rga_num_ol_iff_masses.pdf', 'width', 'full', 'height', 'tall'); +#+end_src + +#+name: fig:rga_num_ol_iff_masses +#+caption: Comparison of the RGA-Number (interaction estimate) without and without IFF +#+RESULTS: +[[file:figs/rga_num_ol_iff_masses.png]] + +#+begin_important +From Figure ref:fig:picture_unbalanced_payload, it is clear that the interaction in the system is largest near the resonances. +The Integral Force Feedback controller, by reducing the amplitude at the resonances, also reduces the coupling near these resonances. +It however increases the coupling bellow the frequency of the suspension modes. +#+end_important + +** Un-Balanced mass +:PROPERTIES: +:header-args:matlab+: :tangle matlab/scripts/unbalanced_mass_enc_plates.m +:END: +*** Introduction + +In this section, we wish to see if a payload with a center of mass not aligned with the symmetry axis of the nano-hexapod could cause any issue. + +To study that, the payload shown in Figure ref:fig:picture_unbalanced_payload is used. + +#+name: fig:picture_unbalanced_payload +#+caption: Nano-Hexapod with unbalanced payload +#+attr_latex: :width \linewidth +[[file:figs/picture_unbalanced_payload.jpg]] + +*** Matlab Init :noexport:ignore: +#+begin_src matlab +%% unbalanced_mass_enc_plates.m +% Study of the effect of an un-balanced payload on the plant dynamics +#+end_src + +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) +<> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes +<> +#+end_src + +#+begin_src matlab :tangle no :noweb yes +<> +#+end_src + +#+begin_src matlab :eval no :noweb yes +<> +#+end_src + +#+begin_src matlab :noweb yes +<> +<> +#+end_src + +*** Compute the identified FRF with IFF +The following data are loaded: +- =Va=: the excitation voltage for the damped plant (corresponding to $u^\prime_i$) +- =de=: the measured motion by the 6 encoders (corresponding to $d\bm{\mathcal{L}}_m$) +#+begin_src matlab +%% Load Identification Data +meas_added_mass = {zeros(6,1)}; + +for i_strut = 1:6 + meas_iff_mass(i_strut) = {load(sprintf('frf_data_exc_strut_%i_iff_vib_table_1m_unbalanced.mat', i_strut), 't', 'Va', 'de')}; +end +#+end_src + +The window =win= and the frequency vector =f= are defined. +#+begin_src matlab +% Sampling Time [s] +Ts = (meas_iff_mass{1}.t(end) - (meas_iff_mass{1}.t(1)))/(length(meas_iff_mass{1}.t)-1); + +% Hannning Windows +win = hanning(ceil(1/Ts)); + +% And we get the frequency vector +[~, f] = tfestimate(meas_iff_mass{1}.Va, meas_iff_mass{1}.de, win, [], [], 1/Ts); +#+end_src + +Finally the $6 \times 6$ transfer function matrix from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ is estimated: +#+begin_src matlab +%% DVF Plant (transfer function from u to dLm) +G_dL = zeros(length(f), 6, 6); +for i_strut = 1:6 + G_dL(:,:,i_strut) = tfestimate(meas_iff_mass{i_strut}.Va, meas_iff_mass{i_strut}.de, win, [], [], 1/Ts); +end +#+end_src + +The identified dynamics are then saved for further use. +#+begin_src matlab :exports none :tangle no +save('matlab/data_frf/frf_iff_unbalanced_vib_table_m.mat', 'f', 'Ts', 'G_dL'); +#+end_src + +#+begin_src matlab :eval no +save('data_frf/frf_iff_unbalanced_vib_table_m.mat', 'f', 'Ts', 'G_dL'); +#+end_src + +*** Effect of an unbalanced payload +#+begin_src matlab :exports none +%% Load the Measured FRF of the damped plant +frf_unb_iff = load('frf_iff_unbalanced_vib_table_m.mat', 'f', 'Ts', 'G_dL'); +#+end_src + +The transfer functions from $u_i$ to $d\mathcal{L}_i$ are shown in Figure ref:fig:frf_damp_unbalanced_mass. +Due to the unbalanced payload, the system is not symmetrical anymore, and therefore each of the diagonal elements are not equal. +This is due to the fact that each strut is not affected by the same inertia. + +#+begin_src matlab :exports none +%% Diagonal and Off Diagonal elements of the damped plants +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:6 + plot(frf_unb_iff.f, abs(frf_unb_iff.G_dL(:,i,i)), 'color', colors(i,:), ... + 'DisplayName', sprintf('$d\\mathcal{L}_{m,%i}/u^\\prime_%i$', i, i)); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +ylim([5e-8, 3e-5]); +legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); + +ax2 = nexttile; +hold on; +for i =1:6 + plot(frf_unb_iff.f, 180/pi*angle(frf_unb_iff.G_dL(:,i, i)), 'color', colors(i,:)); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([10, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/frf_damp_unbalanced_mass.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:frf_damp_unbalanced_mass +#+caption: Transfer function from $u_i$ to $d\mathcal{L}_i$ for the nano-hexapod with an unbalanced payload +#+RESULTS: +[[file:figs/frf_damp_unbalanced_mass.png]] + + + +** Conclusion +#+begin_important +In this section, the dynamics of the nano-hexapod with the encoders fixed to the plates is studied. + +It has been found that: +- The measured dynamics is in agreement with the dynamics of the simscape model, up to the flexible modes of the top plate. + See figures ref:fig:enc_plates_iff_comp_simscape and ref:fig:enc_plates_iff_comp_offdiag_simscape for the transfer function to the force sensors and Figures ref:fig:enc_plates_dvf_comp_simscape and ref:fig:enc_plates_dvf_comp_offdiag_simscapefor the transfer functions to the encoders +- The Integral Force Feedback strategy is very effective in damping the suspension modes of the nano-hexapod (Figure ref:fig:enc_plant_plates_effect_iff). +- The transfer function from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ show nice dynamical properties and is a much better candidate for the high-authority-control than when the encoders were fixed to the struts. + At least up to the flexible modes of the top plate, the diagonal elements of the transfer function matrix have alternating poles and zeros, and the phase is moving smoothly. + Only the flexible modes of the top plates seems to be problematic for control. +#+end_important + +* Decentralized High Authority Control with Integral Force Feedback +<> + +** Introduction :ignore: + +In this section is studied the HAC-IFF architecture for the Nano-Hexapod. +More precisely: +- The LAC control is a decentralized integral force feedback as studied in Section ref:sec:enc_plates_iff +- The HAC control is a decentralized controller working in the frame of the struts + +The corresponding control architecture is shown in Figure ref:fig:control_architecture_hac_iff_struts with: +- $\bm{r}_{\mathcal{X}_n}$: the $6 \times 1$ reference signal in the cartesian frame +- $\bm{r}_{d\mathcal{L}}$: the $6 \times 1$ reference signal transformed in the frame of the struts thanks to the inverse kinematic +- $\bm{\epsilon}_{d\mathcal{L}}$: the $6 \times 1$ length error of the 6 struts +- $\bm{u}^\prime$: input of the damped plant +- $\bm{u}$: generated DAC voltages +- $\bm{\tau}_m$: measured force sensors +- $d\bm{\mathcal{L}}_m$: measured displacement of the struts by the encoders + +#+begin_src latex :file control_architecture_hac_iff_struts.pdf +\definecolor{instrumentation}{rgb}{0, 0.447, 0.741} +\definecolor{mechanics}{rgb}{0.8500, 0.325, 0.098} +\definecolor{control}{rgb}{0.4660, 0.6740, 0.1880} + +\begin{tikzpicture} + % Blocs + \node[block={3.0cm}{2.0cm}, fill=black!20!white] (P) {Plant}; + \coordinate[] (inputF) at ($(P.south west)!0.5!(P.north west)$); + \coordinate[] (outputF) at ($(P.south east)!0.2!(P.north east)$); + \coordinate[] (outputL) at ($(P.south east)!0.8!(P.north east)$); + + \node[block, below=0.4 of P, fill=control!20!white] (Kiff) {$\bm{K}_\text{IFF}$}; + \node[block, left=0.8 of inputF, fill=instrumentation!20!white] (pd200) {\tiny PD200}; + \node[addb, left=0.8 of pd200, fill=control!20!white] (addF) {}; + \node[block, left=0.8 of addF, fill=control!20!white] (K) {$\bm{K}_\mathcal{L}$}; + \node[addb={+}{}{-}{}{}, left=0.8 of K, fill=control!20!white] (subr) {}; + \node[block, align=center, left= of subr, fill=control!20!white] (J) {\tiny Inverse\\\tiny Kinematics}; + + % Connections and labels + \draw[->] (outputF) -- ++(1.0, 0) node[above left]{$\bm{\tau}_m$}; + \draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east); + \draw[->] (Kiff.west) -| (addF.south); + \draw[->] (addF.east) -- (pd200.west) node[above left]{$\bm{u}$}; + \draw[->] (pd200.east) -- (inputF) node[above left]{$\bm{u}_a$}; + + \draw[->] (outputL) -- ++(1.0, 0) node[below left]{$d\bm{\mathcal{L}_m}$}; + \draw[->] ($(outputL) + (0.6, 0)$)node[branch]{} -- ++(0, 1) -| (subr.north); + \draw[->] (subr.east) -- (K.west) node[above left]{$\bm{\epsilon}_{d\mathcal{L}}$}; + \draw[->] (K.east) -- (addF.west) node[above left]{$\bm{u}^\prime$}; + + \draw[->] (J.east) -- (subr.west) node[above left]{$\bm{r}_{d\mathcal{L}}$}; + \draw[<-] (J.west)node[above left]{$\bm{r}_{\mathcal{X}_n}$} -- ++(-1, 0); +\end{tikzpicture} +#+end_src + +#+name: fig:control_architecture_hac_iff_struts +#+caption: HAC-LAC: IFF + Control in the frame of the legs +#+RESULTS: +[[file:figs/control_architecture_hac_iff_struts.png]] + +This part is structured as follow: +- Section ref:sec:hac_iff_struts_ref_track: some reference tracking tests are performed +- Section ref:sec:hac_iff_struts_controller: the decentralized high authority controller is tuned using the Simscape model and is implemented and tested experimentally +- Section ref:sec:interaction_analysis: an interaction analysis is performed, from which the best decoupling strategy can be determined +- Section ref:sec:robust_hac_design: Robust High Authority Controller are designed + +** Reference Tracking - Trajectories +:PROPERTIES: +:header-args:matlab+: :tangle matlab/scripts/reference_tracking_paths.m +:END: +<> +*** Introduction :ignore: +In this section, several trajectories representing the wanted pose (position and orientation) of the top platform with respect to the bottom platform are defined. + +These trajectories will be used to test the HAC-LAC architecture. + +In order to transform the wanted pose to the wanted displacement of the 6 struts, the inverse kinematic is required. +As a first approximation, the Jacobian matrix $\bm{J}$ can be used instead of using the full inverse kinematic equations. + +Therefore, the control architecture with the input trajectory $\bm{r}_{\mathcal{X}_n}$ is shown in Figure ref:fig:control_architecture_hac_iff_L. + +#+begin_src latex :file control_architecture_hac_iff_struts_L.pdf +\definecolor{instrumentation}{rgb}{0, 0.447, 0.741} +\definecolor{mechanics}{rgb}{0.8500, 0.325, 0.098} +\definecolor{control}{rgb}{0.4660, 0.6740, 0.1880} + +\begin{tikzpicture} + % Blocs + \node[block={3.0cm}{2.0cm}, fill=black!20!white] (P) {Plant}; + \coordinate[] (inputF) at ($(P.south west)!0.5!(P.north west)$); + \coordinate[] (outputF) at ($(P.south east)!0.2!(P.north east)$); + \coordinate[] (outputL) at ($(P.south east)!0.8!(P.north east)$); + + \node[block, below=0.4 of P, fill=control!20!white] (Kiff) {$\bm{K}_\text{IFF}$}; + \node[block, left=0.8 of inputF, fill=instrumentation!20!white] (pd200) {\tiny PD200}; + \node[addb, left=0.8 of pd200, fill=control!20!white] (addF) {}; + \node[block, left=0.8 of addF, fill=control!20!white] (K) {$\bm{K}_\mathcal{L}$}; + \node[addb={+}{}{-}{}{}, left=0.8 of K, fill=control!20!white] (subr) {}; + \node[block, align=center, left= of subr, fill=control!20!white] (J) {$\bm{J}$}; + + % Connections and labels + \draw[->] (outputF) -- ++(1.0, 0) node[above left]{$\bm{\tau}_m$}; + \draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east); + \draw[->] (Kiff.west) -| (addF.south); + \draw[->] (addF.east) -- (pd200.west) node[above left]{$\bm{u}$}; + \draw[->] (pd200.east) -- (inputF) node[above left]{$\bm{u}_a$}; + + \draw[->] (outputL) -- ++(1.0, 0) node[below left]{$d\bm{\mathcal{L}_m}$}; + \draw[->] ($(outputL) + (0.6, 0)$)node[branch]{} -- ++(0, 1) -| (subr.north); + \draw[->] (subr.east) -- (K.west) node[above left]{$\bm{\epsilon}_{d\mathcal{L}}$}; + \draw[->] (K.east) -- (addF.west) node[above left]{$\bm{u}^\prime$}; + + \draw[->] (J.east) -- (subr.west) node[above left]{$\bm{r}_{d\mathcal{L}}$}; + \draw[<-] (J.west)node[above left]{$\bm{r}_{\mathcal{X}_n}$} -- ++(-1, 0); +\end{tikzpicture} +#+end_src + +#+name: fig:control_architecture_hac_iff_L +#+caption: HAC-LAC: IFF + Control in the frame of the legs +#+RESULTS: +[[file:figs/control_architecture_hac_iff_struts_L.png]] + +In the following sections, several reference trajectories are defined: +- Section ref:sec:yz_scans: simple scans in the Y-Z plane +- Section ref:sec:tilt_scans: scans in tilt are performed +- Section ref:sec:nass_scans: scans with X-Y-Z translations in order to draw the word "NASS" + +*** Matlab Init :noexport:ignore: +#+begin_src matlab +%% reference_tracking_paths.m +% Computation of several reference paths +#+end_src + +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) +<> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes +<> +#+end_src + +#+begin_src matlab :tangle no :noweb yes +<> +#+end_src + +#+begin_src matlab :eval no :noweb yes +<> +#+end_src + +#+begin_src matlab :noweb yes +<> +#+end_src + +*** Y-Z Scans +<> +A function =generateYZScanTrajectory= has been developed in order to easily generate scans in the Y-Z plane. + +For instance, the following generated trajectory is represented in Figure ref:fig:yz_scan_example_trajectory_yz_plane. +#+begin_src matlab +%% Generate the Y-Z trajectory scan +Rx_yz = generateYZScanTrajectory(... + 'y_tot', 4e-6, ... % Length of Y scans [m] + 'z_tot', 4e-6, ... % Total Z distance [m] + 'n', 5, ... % Number of Y scans + 'Ts', 1e-3, ... % Sampling Time [s] + 'ti', 1, ... % Time to go to initial position [s] + 'tw', 0, ... % Waiting time between each points [s] + 'ty', 0.6, ... % Time for a scan in Y [s] + 'tz', 0.2); % Time for a scan in Z [s] +#+end_src + +#+begin_src matlab :exports none +%% Plot the trajectory in the Y-Z plane +figure; +plot(Rx_yz(:,3), Rx_yz(:,4)); +xlabel('y [m]'); ylabel('z [m]'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/yz_scan_example_trajectory_yz_plane.pdf', 'width', 'normal', 'height', 'normal'); +#+end_src + +#+name: fig:yz_scan_example_trajectory_yz_plane +#+caption: Generated scan in the Y-Z plane +#+RESULTS: +[[file:figs/yz_scan_example_trajectory_yz_plane.png]] + +The Y and Z positions as a function of time are shown in Figure ref:fig:yz_scan_example_trajectory. + +#+begin_src matlab :exports none +%% Plot the Y-Z trajectory as a function of time +figure; +hold on; +plot(Rx_yz(:,1), Rx_yz(:,3), ... + 'DisplayName', 'Y motion') +plot(Rx_yz(:,1), Rx_yz(:,4), ... + 'DisplayName', 'Z motion') +hold off; +xlabel('Time [s]'); +ylabel('Displacement [m]'); +legend('location', 'northeast'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/yz_scan_example_trajectory.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:yz_scan_example_trajectory +#+caption: Y and Z trajectories as a function of time +#+RESULTS: +[[file:figs/yz_scan_example_trajectory.png]] + +Using the Jacobian matrix, it is possible to compute the wanted struts lengths as a function of time: +\begin{equation} + \bm{r}_{d\mathcal{L}} = \bm{J} \bm{r}_{\mathcal{X}_n} +\end{equation} + +#+begin_src matlab :exports none +load('jacobian.mat', 'J'); +#+end_src + +#+begin_src matlab +%% Compute the reference in the frame of the legs +dL_ref = [J*Rx_yz(:, 2:7)']'; +#+end_src + +The reference signal for the strut length is shown in Figure ref:fig:yz_scan_example_trajectory_struts. +#+begin_src matlab :exports none +%% Plot the reference in the frame of the legs +figure; +hold on; +for i=1:6 + plot(Rx_yz(:,1), dL_ref(:, i), ... + 'DisplayName', sprintf('$r_{d\\mathcal{L}_%i}$', i)) +end +xlabel('Time [s]'); ylabel('Strut Motion [m]'); +legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); +yticks(1e-6*[-5:5]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/yz_scan_example_trajectory_struts.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:yz_scan_example_trajectory_struts +#+caption: Trajectories for the 6 individual struts +#+RESULTS: +[[file:figs/yz_scan_example_trajectory_struts.png]] + +*** Tilt Scans +<> + +A function =generalSpiralAngleTrajectory= has been developed in order to easily generate $R_x,R_y$ tilt scans. + +For instance, the following generated trajectory is represented in Figure ref:fig:tilt_scan_example_trajectory. +#+begin_src matlab +%% Generate the "tilt-spiral" trajectory scan +R_tilt = generateSpiralAngleTrajectory(... + 'R_tot', 20e-6, ... % Total Tilt [ad] + 'n_turn', 5, ... % Number of scans + 'Ts', 1e-3, ... % Sampling Time [s] + 't_turn', 1, ... % Turn time [s] + 't_end', 1); % End time to go back to zero [s] +#+end_src + +#+begin_src matlab :exports none +%% Plot the trajectory +figure; +plot(1e6*R_tilt(:,5), 1e6*R_tilt(:,6)); +xlabel('$R_x$ [$\mu$rad]'); ylabel('$R_y$ [$\mu$rad]'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/tilt_scan_example_trajectory.pdf', 'width', 'normal', 'height', 'normal'); +#+end_src + +#+name: fig:tilt_scan_example_trajectory +#+caption: Generated "spiral" scan +#+RESULTS: +[[file:figs/tilt_scan_example_trajectory.png]] + +#+begin_src matlab :exports none +%% Compute the reference in the frame of the legs +load('jacobian.mat', 'J'); +dL_ref = [J*R_tilt(:, 2:7)']'; +#+end_src + +The reference signal for the strut length is shown in Figure ref:fig:tilt_scan_example_trajectory_struts. +#+begin_src matlab :exports none +%% Plot the reference in the frame of the legs +figure; +hold on; +for i=1:6 + plot(R_tilt(:,1), dL_ref(:, i), ... + 'DisplayName', sprintf('$r_{d\\mathcal{L}_%i}$', i)) +end +xlabel('Time [s]'); ylabel('Strut Motion [m]'); +legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); +yticks(1e-6*[-5:5]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/tilt_scan_example_trajectory_struts.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:tilt_scan_example_trajectory_struts +#+caption: Trajectories for the 6 individual struts - Tilt scan +#+RESULTS: +[[file:figs/tilt_scan_example_trajectory_struts.png]] + +*** "NASS" reference path +<> +In this section, a reference path that "draws" the work "NASS" is developed. + +First, a series of points representing each letter are defined. +Between each letter, a negative Z motion is performed. +#+begin_src matlab +%% List of points that draws "NASS" +ref_path = [ ... + 0, 0,0; % Initial Position + 0,0,1; 0,4,1; 3,0,1; 3,4,1; % N + 3,4,0; 4,0,0; % Transition + 4,0,1; 4,3,1; 5,4,1; 6,4,1; 7,3,1; 7,2,1; 4,2,1; 4,3,1; 5,4,1; 6,4,1; 7,3,1; 7,0,1; % A + 7,0,0; 8,0,0; % Transition + 8,0,1; 11,0,1; 11,2,1; 8,2,1; 8,4,1; 11,4,1; % S + 11,4,0; 12,0,0; % Transition + 12,0,1; 15,0,1; 15,2,1; 12,2,1; 12,4,1; 15,4,1; % S + 15,4,0; + ]; + +%% Center the trajectory arround zero +ref_path = ref_path - (max(ref_path) - min(ref_path))/2; + +%% Define the X-Y-Z cuboid dimensions containing the trajectory +X_max = 10e-6; +Y_max = 4e-6; +Z_max = 2e-6; + +ref_path = ([X_max, Y_max, Z_max]./max(ref_path)).*ref_path; % [m] +#+end_src + +Then, using the =generateXYZTrajectory= function, the $6 \times 1$ trajectory signal is computed. +#+begin_src matlab +%% Generating the trajectory +Rx_nass = generateXYZTrajectory('points', ref_path); +#+end_src + +The trajectory in the X-Y plane is shown in Figure ref:fig:ref_track_test_nass (the transitions between the letters are removed). +#+begin_src matlab :exports none +%% "NASS" trajectory in the X-Y plane +figure; +plot(1e6*Rx_nass(Rx_nass(:,4)>0, 2), 1e6*Rx_nass(Rx_nass(:,4)>0, 3), 'k.') +xlabel('X [$\mu m$]'); +ylabel('Y [$\mu m$]'); +axis equal; +xlim(1e6*[min(Rx_nass(:,2)), max(Rx_nass(:,2))]); +ylim(1e6*[min(Rx_nass(:,3)), max(Rx_nass(:,3))]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/ref_track_test_nass.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:ref_track_test_nass +#+caption: Reference path corresponding to the "NASS" acronym +#+RESULTS: +[[file:figs/ref_track_test_nass.png]] + +It can also be better viewed in a 3D representation as in Figure ref:fig:ref_track_test_nass_3d. + +#+begin_src matlab :exports none +figure; +plot3(1e6*Rx_nass(:,2), 1e6*Rx_nass(:,3), 1e6*Rx_nass(:,4), 'k-'); +xlabel('x [$\mu m$]'); ylabel('y [$\mu m$]'); zlabel('z [$\mu m$]'); +view(-13, 41) +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/ref_track_test_nass_3d.pdf', 'width', 'normal', 'height', 'normal'); +#+end_src + +#+name: fig:ref_track_test_nass_3d +#+caption: Reference path that draws "NASS" - 3D view +#+RESULTS: +[[file:figs/ref_track_test_nass_3d.png]] + +** First Basic High Authority Controller +:PROPERTIES: +:header-args:matlab+: :tangle matlab/scripts/hac_lac_first_try.m +:END: +<> +*** Introduction :ignore: +In this section, a simple decentralized high authority controller $\bm{K}_{\mathcal{L}}$ is developed to work without any payload. + +The diagonal controller is tuned using classical Loop Shaping in Section ref:sec:hac_iff_no_payload_tuning. +The stability is verified in Section ref:sec:hac_iff_no_payload_stability using the Simscape model. + +*** Matlab Init :noexport:ignore: +#+begin_src matlab +%% hac_lac_first_try.m +% Development and analysis of a first basic High Authority Controller +#+end_src + +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) +<> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes +<> +#+end_src + +#+begin_src matlab :tangle no :noweb yes +<> +#+end_src + +#+begin_src matlab :eval no :noweb yes +<> +#+end_src + +#+begin_src matlab :noweb yes +<> +<> +#+end_src + +#+begin_src matlab +%% Load the identified FRF and Simscape model +frf_iff = load('frf_iff_vib_table_m.mat', 'f', 'Ts', 'G_dL'); +sim_iff = load('sim_iff_vib_table_m.mat', 'G_dL'); +#+end_src + +*** HAC Controller +<> + +Let's first try to design a first decentralized controller with: +- a bandwidth of 100Hz +- sufficient phase margin +- simple and understandable components + +After some very basic and manual loop shaping, A diagonal controller is developed. +Each diagonal terms are identical and are composed of: +- A lead around 100Hz +- A first order low pass filter starting at 200Hz to add some robustness to high frequency modes +- A notch at 700Hz to cancel the flexible modes of the top plate +- A pure integrator + +#+begin_src matlab +%% Lead to increase phase margin +a = 2; % Amount of phase lead / width of the phase lead / high frequency gain +wc = 2*pi*100; % Frequency with the maximum phase lead [rad/s] + +H_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a))); + +%% Low Pass filter to increase robustness +H_lpf = 1/(1 + s/2/pi/200); + +%% Notch at the top-plate resonance +gm = 0.02; +xi = 0.3; +wn = 2*pi*700; + +H_notch = (s^2 + 2*gm*xi*wn*s + wn^2)/(s^2 + 2*xi*wn*s + wn^2); + +%% Decentralized HAC +Khac_iff_struts = -(1/(2.87e-5)) * ... % Gain + H_lead * ... % Lead + H_notch * ... % Notch + (2*pi*100/s) * ... % Integrator + eye(6); % 6x6 Diagonal +#+end_src + +This controller is saved for further use. +#+begin_src matlab :exports none :tangle no +save('matlab/data_sim/Khac_iff_struts.mat', 'Khac_iff_struts') +#+end_src + +#+begin_src matlab :eval no +save('data_sim/Khac_iff_struts.mat', 'Khac_iff_struts') +#+end_src + +The experimental loop gain is computed and shown in Figure ref:fig:loop_gain_hac_iff_struts. +#+begin_src matlab +L_hac_iff_struts = pagemtimes(permute(frf_iff.G_dL{1}, [2 3 1]), squeeze(freqresp(Khac_iff_struts, frf_iff.f, 'Hz'))); +#+end_src + +#+begin_src matlab :exports none +%% Bode plot of the Loop Gain +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + + +ax1 = nexttile([2,1]); +hold on; +% Diagonal Elements Model +plot(frf_iff.f, abs(squeeze(L_hac_iff_struts(1,1,:))), 'color', colors(1,:), ... + 'DisplayName', 'Diagonal'); +for i = 2:6 + plot(frf_iff.f, abs(squeeze(L_hac_iff_struts(i,i,:))), 'color', colors(1,:), ... + 'HandleVisibility', 'off'); +end +plot(frf_iff.f, abs(squeeze(L_hac_iff_struts(1,2,:))), 'color', [colors(2,:), 0.2], ... + 'DisplayName', 'Off-Diag'); +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(squeeze(L_hac_iff_struts(i,j,:))), 'color', [colors(2,:), 0.2], ... + 'HandleVisibility', 'off'); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Loop Gain [-]'); set(gca, 'XTickLabel',[]); +ylim([1e-3, 1e2]); +legend('location', 'northeast'); + +ax2 = nexttile; +hold on; +for i =1:6 + plot(frf_iff.f, 180/pi*angle(squeeze(L_hac_iff_struts(i,i,:))), 'color', colors(1,:)); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([2, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/loop_gain_hac_iff_struts.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:loop_gain_hac_iff_struts +#+caption: Diagonal and off-diagonal elements of the Loop gain for "HAC-IFF-Struts" +#+RESULTS: +[[file:figs/loop_gain_hac_iff_struts.png]] + +*** Verification of the Stability using the Simscape model +<> + +The HAC-IFF control strategy is implemented using Simscape. +#+begin_src matlab +%% Initialize the Simscape model in closed loop +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'plates', ... + 'actuator_type', 'flexible', ... + 'controller_type', 'hac-iff-struts'); +#+end_src + +#+begin_src matlab :exports none +support.type = 1; % On top of vibration table +payload.type = 3; % Payload / 1 "mass layer" + +load('Kiff_opt.mat', 'Kiff'); +#+end_src + +#+begin_src matlab +%% Identify the (damped) transfer function from u to dLm +clear io; io_i = 1; +io(io_i) = linio([mdl, '/du'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Plate Displacement (encoder) +#+end_src + +We identify the closed-loop system. +#+begin_src matlab +%% Identification +Gd_iff_hac_opt = linearize(mdl, io, 0.0, options); +#+end_src + +And verify that it is indeed stable. +#+begin_src matlab :results value replace :exports both +%% Verify the stability +isstable(Gd_iff_hac_opt) +#+end_src + +#+RESULTS: +: 1 + +*** Experimental Validation +Both the Integral Force Feedback controller (developed in Section ref:sec:enc_plates_iff) and the high authority controller working in the frame of the struts (developed in Section ref:sec:hac_iff_struts_controller) are implemented experimentally. + +Two reference tracking experiments are performed to evaluate the stability and performances of the implemented control. + +#+begin_src matlab +%% Load the experimental data +load('hac_iff_struts_yz_scans.mat', 't', 'de') +#+end_src + +#+begin_src matlab :exports none +%% Reset initial time +t = t - t(1); +#+end_src + +The position of the top-platform is estimated using the Jacobian matrix: +#+begin_src matlab +%% Pose of the top platform from the encoder values +load('jacobian.mat', 'J'); +Xe = [inv(J)*de']'; +#+end_src + +#+begin_src matlab +%% Generate the Y-Z trajectory scan +Rx_yz = generateYZScanTrajectory(... + 'y_tot', 4e-6, ... % Length of Y scans [m] + 'z_tot', 8e-6, ... % Total Z distance [m] + 'n', 5, ... % Number of Y scans + 'Ts', 1e-3, ... % Sampling Time [s] + 'ti', 1, ... % Time to go to initial position [s] + 'tw', 0, ... % Waiting time between each points [s] + 'ty', 0.6, ... % Time for a scan in Y [s] + 'tz', 0.2); % Time for a scan in Z [s] +#+end_src + +The reference path as well as the measured position are partially shown in the Y-Z plane in Figure ref:fig:yz_scans_exp_results_first_K. +#+begin_src matlab :exports none +%% Position and reference signal in the Y-Z plane +figure; +tiledlayout(1, 3, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile; +hold on; +plot(1e6*Xe(t>2,2), 1e6*Xe(t>2,3)); +plot(1e6*Rx_yz(:,3), 1e6*Rx_yz(:,4), '--'); +hold off; +xlabel('Y [$\mu m$]'); ylabel('Z [$\mu m$]'); +xlim([-2.05, 2.05]); ylim([-4.1, 4.1]); +axis equal; + +ax2 = nexttile([1,2]); +hold on; +plot(1e6*Xe(:,2), 1e6*Xe(:,3), ... + 'DisplayName', '$\mathcal{X}_n$'); +plot(1e6*Rx_yz(:,3), 1e6*Rx_yz(:,4), '--', ... + 'DisplayName', '$r_{\mathcal{X}_n}$'); +hold off; +legend('location', 'northwest'); +xlabel('Y [$\mu m$]'); ylabel('Z [$\mu m$]'); +axis equal; +xlim([1.6, 2.1]); ylim([-4.1, -3.6]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/yz_scans_exp_results_first_K.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:yz_scans_exp_results_first_K +#+caption: Measured position $\bm{\mathcal{X}}_n$ and reference signal $\bm{r}_{\mathcal{X}_n}$ in the Y-Z plane - Zoom on a change of direction +#+RESULTS: +[[file:figs/yz_scans_exp_results_first_K.png]] + +#+begin_important +It is clear from Figure ref:fig:yz_scans_exp_results_first_K that the position of the nano-hexapod effectively tracks to reference signal. +However, oscillations with amplitudes as large as 50nm can be observe. + +It turns out that the frequency of these oscillations is 100Hz which is corresponding to the crossover frequency of the High Authority Control loop. +This clearly indicates poor stability margins. +In the next section, the controller is re-designed to improve the stability margins. +#+end_important + +*** Controller with increased stability margins +The High Authority Controller is re-designed in order to improve the stability margins. +#+begin_src matlab +%% Lead +a = 5; % Amount of phase lead / width of the phase lead / high frequency gain +wc = 2*pi*110; % Frequency with the maximum phase lead [rad/s] + +H_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a))); + +%% Low Pass Filter +H_lpf = 1/(1 + s/2/pi/300); + +%% Notch +gm = 0.02; +xi = 0.5; +wn = 2*pi*700; + +H_notch = (s^2 + 2*gm*xi*wn*s + wn^2)/(s^2 + 2*xi*wn*s + wn^2); + +%% HAC Controller +Khac_iff_struts = -2.2e4 * ... % Gain + H_lead * ... % Lead + H_lpf * ... % Lead + H_notch * ... % Notch + (2*pi*100/s) * ... % Integrator + eye(6); % 6x6 Diagonal +#+end_src + +#+begin_src matlab :exports none +%% Load the FRF of the transfer function from u to dL with IFF +frf_iff = load('frf_iff_vib_table_m.mat', 'f', 'Ts', 'G_dL'); +#+end_src + +#+begin_src matlab :exports none +%% Compute the Loop Gain +L_frf = pagemtimes(permute(frf_iff.G_dL{1}, [2 3 1]), squeeze(freqresp(Khac_iff_struts, frf_iff.f, 'Hz'))); +#+end_src + +The bode plot of the new loop gain is shown in Figure ref:fig:hac_iff_plates_exp_loop_gain_redesigned_K. +#+begin_src matlab :exports none +%% Bode plot for the transfer function from u to dLm +freqs = 2*logspace(1, 3, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +% Diagonal Elements FRF +plot(frf_iff.f, abs(squeeze(L_frf(1,1,:))), 'color', colors(1,:), ... + 'DisplayName', 'Diagonal'); +for i = 2:6 + plot(frf_iff.f, abs(squeeze(L_frf(i,i,:))), 'color', colors(1,:), ... + 'HandleVisibility', 'off'); +end +plot(frf_iff.f, abs(squeeze(L_frf(1,2,:))), 'color', [colors(2,:), 0.2], ... + 'DisplayName', 'Off-Diag'); +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(squeeze(L_frf(i,j,:))), 'color', [colors(2,:), 0.2], ... + 'HandleVisibility', 'off'); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Loop Gain [-]'); set(gca, 'XTickLabel',[]); +ylim([1e-3, 1e2]); +legend('location', 'northeast'); + +ax2 = nexttile; +hold on; +for i =1:6 + plot(frf_iff.f, 180/pi*angle(squeeze(L_frf(i,i,:))), 'color', colors(1,:)); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([1, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/hac_iff_plates_exp_loop_gain_redesigned_K.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:hac_iff_plates_exp_loop_gain_redesigned_K +#+caption: Loop Gain for the updated decentralized HAC controller +#+RESULTS: +[[file:figs/hac_iff_plates_exp_loop_gain_redesigned_K.png]] + +This new controller is implemented experimentally and several tracking tests are performed. +#+begin_src matlab +%% Load Measurements +load('hac_iff_more_lead_nass_scan.mat', 't', 'de') +#+end_src + +#+begin_src matlab :exports none +%% Reset Time +t = t - t(1); +#+end_src + +The pose of the top platform is estimated from the encoder position using the Jacobian matrix. +#+begin_src matlab +%% Compute the pose of the top platform +load('jacobian.mat', 'J'); +Xe = [inv(J)*de']'; +#+end_src + +#+begin_src matlab :exports none +%% Load the reference path +load('reference_path.mat', 'Rx_nass') +#+end_src + +The measured motion as well as the trajectory are shown in Figure ref:fig:nass_scans_first_test_exp. +#+begin_src matlab :exports none +%% Plot the X-Y-Z "NASS" trajectory +figure; +hold on; +plot3(Xe(1:100:end,1), Xe(1:100:end,2), Xe(1:100:end,3)) +plot3(Rx_nass(1:100:end,2), Rx_nass(1:100:end,3), Rx_nass(1:100:end,4)) +hold off; +xlabel('x [$\mu m$]'); ylabel('y [$\mu m$]'); zlabel('z [$\mu m$]'); +view(-13, 41) +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/nass_scans_first_test_exp.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:nass_scans_first_test_exp +#+caption: Measured position $\bm{\mathcal{X}}_n$ and reference signal $\bm{r}_{\mathcal{X}_n}$ for the "NASS" trajectory +#+RESULTS: +[[file:figs/nass_scans_first_test_exp.png]] + +The trajectory and measured motion are also shown in the X-Y plane in Figure ref:fig:ref_track_nass_exp_hac_iff_struts. +#+begin_src matlab :exports none +%% Estimate when the hexpod is on top position and drawing the letters +i_top = Xe(:,3) > 1.9e-6; +i_rx = Rx_nass(:,4) > 0; +#+end_src + +#+begin_src matlab :exports none +%% Plot the reference as well as the measurement in the X-Y plane +figure; +tiledlayout(1, 3, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([1,2]); +hold on; +scatter(1e6*Xe(i_top,1), 1e6*Xe(i_top,2),'.'); +plot(1e6*Rx_nass(i_rx,2), 1e6*Rx_nass(i_rx,3), '--'); +hold off; +xlabel('X [$\mu m$]'); ylabel('Y [$\mu m$]'); +axis equal; +xlim([-10.5, 10.5]); ylim([-4.5, 4.5]); + +ax2 = nexttile; +hold on; +scatter(1e6*Xe(i_top,1), 1e6*Xe(i_top,2),'.'); +plot(1e6*Rx_nass(i_rx,2), 1e6*Rx_nass(i_rx,3), '--'); +hold off; +xlabel('X [$\mu m$]'); ylabel('Y [$\mu m$]'); +axis equal; +xlim([4.5, 4.7]); ylim([-0.15, 0.05]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/ref_track_nass_exp_hac_iff_struts.pdf', 'width', 'full', 'height', 'tall'); +#+end_src + +#+name: fig:ref_track_nass_exp_hac_iff_struts +#+caption: Reference path and measured motion in the X-Y plane +#+RESULTS: +[[file:figs/ref_track_nass_exp_hac_iff_struts.png]] + +The orientation errors during all the scans are shown in Figure ref:fig:nass_ref_rx_ry. +#+begin_src matlab :exports none +%% Orientation Errors +figure; +hold on; +plot(t(t>20&t<20.1), 1e6*Xe(t>20&t<20.1,4), '-', 'DisplayName', '$\epsilon_{\theta_x}$'); +plot(t(t>20&t<20.1), 1e6*Xe(t>20&t<20.1,5), '-', 'DisplayName', '$\epsilon_{\theta_y}$'); +plot(t(t>20&t<20.1), 1e6*Xe(t>20&t<20.1,6), '-', 'DisplayName', '$\epsilon_{\theta_z}$'); +hold off; +xlabel('Time [s]'); ylabel('Orientation Error [$\mu$ rad]'); +legend('location', 'northeast'); +#+end_src + +#+begin_src matlab :exports none +%% Orientation Errors +figure; +hold on; +plot(1e9*Xe(100000:100:end,4), 1e9*Xe(100000:100:end,5), '.'); +th = 0:pi/50:2*pi; +xunit = 90 * cos(th); +yunit = 90 * sin(th); +plot(xunit, yunit, '--'); +hold off; +xlabel('$R_x$ [nrad]'); ylabel('$R_y$ [nrad]'); +xlim([-100, 100]); +ylim([-100, 100]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/nass_ref_rx_ry.pdf', 'width', 500, 'height', 500); +#+end_src + +#+name: fig:nass_ref_rx_ry +#+caption: Orientation errors during the scan +#+RESULTS: +[[file:figs/nass_ref_rx_ry.png]] + +#+begin_important +Using the updated High Authority Controller, the nano-hexapod can follow trajectories with high accuracy (the position errors are in the order of 50nm peak to peak, and the orientation errors 300nrad peak to peak). +#+end_important + +** Interaction Analysis and Decoupling +:PROPERTIES: +:header-args:matlab+: :tangle matlab/scripts/interaction_analysis_enc_plates.m +:END: +<> +*** Introduction :ignore: + +In this section, the interaction in the identified plant is estimated using the Relative Gain Array (RGA) [[cite:skogestad07_multiv_feedb_contr][Chap. 3.4]]. + +Then, several decoupling strategies are compared for the nano-hexapod. + +The RGA Matrix is defined as follow: +\begin{equation} + \text{RGA}(G(f)) = G(f) \times (G(f)^{-1})^T +\end{equation} + +Then, the RGA number is defined: +\begin{equation} +\text{RGA-num}(f) = \| \text{I - RGA(G(f))} \|_{\text{sum}} +\end{equation} + + +In this section, the plant with 2 added mass is studied. + +*** Matlab Init :noexport:ignore: +#+begin_src matlab +%% interaction_analysis_enc_plates.m +% Interaction analysis of several decoupling strategies +#+end_src + +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) +<> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes +<> +#+end_src + +#+begin_src matlab :tangle no :noweb yes +<> +#+end_src + +#+begin_src matlab :eval no :noweb yes +<> +#+end_src + +#+begin_src matlab :noweb yes +<> +#+end_src + +#+begin_src matlab +%% Load the identified FRF and Simscape model +frf_iff = load('frf_iff_vib_table_m.mat', 'f', 'Ts', 'G_dL'); +sim_iff = load('sim_iff_vib_table_m.mat', 'G_dL'); +#+end_src + +*** Parameters +#+begin_src matlab +wc = 100; % Wanted crossover frequency [Hz] +[~, i_wc] = min(abs(frf_iff.f - wc)); % Indice corresponding to wc +#+end_src + +#+begin_src matlab +%% Plant to be decoupled +frf_coupled = frf_iff.G_dL{2}; +G_coupled = sim_iff.G_dL{2}; +#+end_src + +*** No Decoupling (Decentralized) +<> + +#+begin_src latex :file decoupling_arch_decentralized.pdf +\begin{tikzpicture} + \node[block] (G) {$\bm{G}$}; + + % Connections and labels + \draw[<-] (G.west) -- ++(-1.8, 0) node[above right]{$\bm{\tau}$}; + \draw[->] (G.east) -- ++( 1.8, 0) node[above left]{$d\bm{\mathcal{L}}$}; + + \begin{scope}[on background layer] + \node[fit={(G.south west) (G.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gdec) {}; + \node[below right] at (Gdec.north west) {$\bm{G}_{\text{dec}}$}; + \end{scope} +\end{tikzpicture} +#+end_src + +#+name: fig:decoupling_arch_decentralized +#+caption: Block diagram representing the plant. +#+RESULTS: +[[file:figs/decoupling_arch_decentralized.png]] + +#+begin_src matlab :exports none +%% Decentralized Plant +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(frf_coupled(:,i,j)), 'color', [0,0,0,0.2], ... + 'HandleVisibility', 'off'); + end +end +set(gca,'ColorOrderIndex',1) +for i = 1:6 + plot(frf_iff.f, abs(frf_coupled(:,i,i)), ... + 'DisplayName', sprintf('$y_%i/u_%i$', i, i)); +end +plot(frf_iff.f, abs(frf_coupled(:,1,2)), 'color', [0,0,0,0.2], ... + 'DisplayName', 'Coupling'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude'); set(gca, 'XTickLabel',[]); +ylim([1e-9, 1e-4]); +legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 3); + +ax2 = nexttile; +hold on; +for i =1:6 + plot(frf_iff.f, 180/pi*angle(frf_coupled(:,i,i))); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([10, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_decentralized_plant.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:interaction_decentralized_plant +#+caption: Bode Plot of the decentralized plant (diagonal and off-diagonal terms) +#+RESULTS: +[[file:figs/interaction_decentralized_plant.png]] + +#+begin_src matlab :exports none +%% Decentralized RGA +RGA_dec = zeros(size(frf_coupled)); +for i = 1:length(frf_iff.f) + RGA_dec(i,:,:) = squeeze(frf_coupled(i,:,:)).*inv(squeeze(frf_coupled(i,:,:))).'; +end + +RGA_dec_sum = zeros(length(frf_iff), 1); +for i = 1:length(frf_iff.f) + RGA_dec_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_dec(i,:,:))))); +end +#+end_src + +#+begin_src matlab :exports none +%% RGA for Decentralized plant +figure; +plot(frf_iff.f, RGA_dec_sum, 'k-'); +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('RGA Number'); +xlim([10, 1e3]); ylim([1e-2, 1e2]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_rga_decentralized.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:interaction_rga_decentralized +#+caption: RGA number for the decentralized plant +#+RESULTS: +[[file:figs/interaction_rga_decentralized.png]] + +*** Static Decoupling +<> + +#+begin_src latex :file decoupling_arch_static.pdf +\begin{tikzpicture} + \node[block] (G) {$\bm{G}$}; + \node[block, left=0.8 of G] (Ginv) {$\bm{\hat{G}}(j0)^{-1}$}; + + % Connections and labels + \draw[<-] (Ginv.west) -- ++(-1.8, 0) node[above right]{$\bm{u}$}; + \draw[->] (Ginv.east) -- (G.west) node[above left]{$\bm{\tau}$}; + \draw[->] (G.east) -- ++( 1.8, 0) node[above left]{$d\bm{\mathcal{L}}$}; + + \begin{scope}[on background layer] + \node[fit={(Ginv.south west) (G.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gx) {}; + \node[below right] at (Gx.north west) {$\bm{G}_{\text{static}}$}; + \end{scope} +\end{tikzpicture} +#+end_src + +#+name: fig:decoupling_arch_static +#+caption: Decoupling using the inverse of the DC gain of the plant +#+RESULTS: +[[file:figs/decoupling_arch_static.png]] + +The DC gain is evaluated from the model as be have bad low frequency identification. + +#+begin_src matlab :exports none +%% Compute the inverse of the DC gain +G_model = G_coupled; +G_model.outputdelay = 0; % necessary for further inversion +dc_inv = inv(dcgain(G_model)); + +%% Compute the inversed plant +G_dL_sta = zeros(size(frf_coupled)); +for i = 1:length(frf_iff.f) + G_dL_sta(i,:,:) = squeeze(frf_coupled(i,:,:))*dc_inv; +end +#+end_src + +#+begin_src matlab :exports results :results value table replace :tangle no +data2orgtable(dc_inv, {}, {}, ' %.1f '); +#+end_src + +#+RESULTS: +| -62011.5 | 3910.6 | 4299.3 | 660.7 | -4016.5 | -4373.6 | +| 3914.4 | -61991.2 | -4356.8 | -4019.2 | 640.2 | 4281.6 | +| -4020.0 | -4370.5 | -62004.5 | 3914.6 | 4295.8 | 653.8 | +| 660.9 | 4292.4 | 3903.3 | -62012.2 | -4366.5 | -4008.9 | +| 4302.8 | 655.6 | -4025.8 | -4377.8 | -62006.0 | 3919.7 | +| -4377.9 | -4013.2 | 668.6 | 4303.7 | 3906.8 | -62019.3 | + +#+begin_src matlab :exports none +%% Bode plot of the static decoupled plant +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(G_dL_sta(:,i,j)), 'color', [0,0,0,0.2], ... + 'HandleVisibility', 'off'); + end +end +set(gca,'ColorOrderIndex',1) +for i = 1:6 + plot(frf_iff.f, abs(G_dL_sta(:,i,i)), ... + 'DisplayName', sprintf('$y_%i/u_%i$', i, i)); +end +plot(frf_iff.f, abs(G_dL_sta(:,1,2)), 'color', [0,0,0,0.2], ... + 'DisplayName', 'Coupling'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude'); set(gca, 'XTickLabel',[]); +ylim([1e-3, 1e1]); +legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); + +ax2 = nexttile; +hold on; +for i =1:6 + plot(frf_iff.f, 180/pi*angle(G_dL_sta(:,i,i))); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([10, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_static_dec_plant.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:interaction_static_dec_plant +#+caption: Bode Plot of the static decoupled plant +#+RESULTS: +[[file:figs/interaction_static_dec_plant.png]] + +#+begin_src matlab :exports none +%% Compute RGA Matrix +RGA_sta = zeros(size(frf_coupled)); +for i = 1:length(frf_iff.f) + RGA_sta(i,:,:) = squeeze(G_dL_sta(i,:,:)).*inv(squeeze(G_dL_sta(i,:,:))).'; +end + +%% Compute RGA-number +RGA_sta_sum = zeros(length(frf_iff), 1); +for i = 1:size(RGA_sta, 1) + RGA_sta_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_sta(i,:,:))))); +end +#+end_src + +#+begin_src matlab :exports none +%% Plot the RGA-number for statically decoupled plant +figure; +plot(frf_iff.f, RGA_sta_sum, 'k-'); +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('RGA Number'); +xlim([10, 1e3]); ylim([1e-2, 1e2]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_rga_static_dec.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:interaction_rga_static_dec +#+caption: RGA number for the statically decoupled plant +#+RESULTS: +[[file:figs/interaction_rga_static_dec.png]] + +*** Decoupling at the Crossover +<> + +#+begin_src latex :file decoupling_arch_crossover.pdf +\begin{tikzpicture} + \node[block] (G) {$\bm{G}$}; + \node[block, left=0.8 of G] (Ginv) {$\bm{\hat{G}}(j\omega_c)^{-1}$}; + + % Connections and labels + \draw[<-] (Ginv.west) -- ++(-1.8, 0) node[above right]{$\bm{u}$}; + \draw[->] (Ginv.east) -- (G.west) node[above left]{$\bm{\tau}$}; + \draw[->] (G.east) -- ++( 1.8, 0) node[above left]{$d\bm{\mathcal{L}}$}; + + \begin{scope}[on background layer] + \node[fit={(Ginv.south west) (G.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gx) {}; + \node[below right] at (Gx.north west) {$\bm{G}_{\omega_c}$}; + \end{scope} +\end{tikzpicture} +#+end_src + +#+name: fig:decoupling_arch_crossover +#+caption: Decoupling using the inverse of a dynamical model $\bm{\hat{G}}$ of the plant dynamics $\bm{G}$ +#+RESULTS: +[[file:figs/decoupling_arch_crossover.png]] + +#+begin_src matlab :exports none +%% Take complex matrix corresponding to the plant at 100Hz +V = squeeze(frf_coupled(i_wc,:,:)); + +%% Real approximation of inv(G(100Hz)) +D = pinv(real(V'*V)); +H1 = D*real(V'*diag(exp(1j*angle(diag(V*D*V.'))/2))); + +%% Compute the decoupled plant +G_dL_wc = zeros(size(frf_coupled)); +for i = 1:length(frf_iff.f) + G_dL_wc(i,:,:) = squeeze(frf_coupled(i,:,:))*H1; +end +#+end_src + +#+begin_src matlab :exports results :results value table replace :tangle no +data2orgtable(H1, {}, {}, ' %.1f '); +#+end_src + +#+RESULTS: +| 67229.8 | 3769.3 | -13704.6 | -23084.8 | -6318.2 | 23378.7 | +| 3486.2 | 67708.9 | 23220.0 | -6314.5 | -22699.8 | -14060.6 | +| -5731.7 | 22471.7 | 66701.4 | 3070.2 | -13205.6 | -21944.6 | +| -23305.5 | -14542.6 | 2743.2 | 70097.6 | 24846.8 | -5295.0 | +| -14882.9 | -22957.8 | -5344.4 | 25786.2 | 70484.6 | 2979.9 | +| 24353.3 | -5195.2 | -22449.0 | -14459.2 | 2203.6 | 69484.2 | + +#+begin_src matlab :exports none +%% Bode plot of the plant decoupled at the crossover +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(G_dL_wc(:,i,j)), 'color', [0,0,0,0.2], ... + 'HandleVisibility', 'off'); + end +end +for i = 1:6 + plot(frf_iff.f, abs(G_dL_wc(:,i,i)), ... + 'DisplayName', sprintf('$y_%i/u_%i$', i, i)); +end +plot(frf_iff.f, abs(G_dL_wc(:,1,2)), 'color', [0,0,0,0.2], ... + 'DisplayName', 'Coupling'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +ylim([1e-3, 1e1]); +legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); + +ax2 = nexttile; +hold on; +for i =1:6 + plot(frf_iff.f, 180/pi*angle(G_dL_wc(:,i,i))); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([10, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_wc_plant.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:interaction_wc_plant +#+caption: Bode Plot of the plant decoupled at the crossover +#+RESULTS: +[[file:figs/interaction_wc_plant.png]] + +#+begin_src matlab +%% Compute RGA Matrix +RGA_wc = zeros(size(frf_coupled)); +for i = 1:length(frf_iff.f) + RGA_wc(i,:,:) = squeeze(G_dL_wc(i,:,:)).*inv(squeeze(G_dL_wc(i,:,:))).'; +end + +%% Compute RGA-number +RGA_wc_sum = zeros(size(RGA_wc, 1), 1); +for i = 1:size(RGA_wc, 1) + RGA_wc_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_wc(i,:,:))))); +end +#+end_src + +#+begin_src matlab :exports none +%% Plot the RGA-Number for the plant decoupled at crossover +figure; +plot(frf_iff.f, RGA_wc_sum, 'k-'); +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('RGA Number'); +xlim([10, 1e3]); ylim([1e-2, 1e2]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_rga_wc.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:interaction_rga_wc +#+caption: RGA number for the plant decoupled at the crossover +#+RESULTS: +[[file:figs/interaction_rga_wc.png]] + +*** SVD Decoupling +<> + +#+begin_src latex :file decoupling_arch_svd.pdf +\begin{tikzpicture} + \node[block] (G) {$\bm{G}$}; + + \node[block, left=0.8 of G.west] (V) {$V^{-T}$}; + \node[block, right=0.8 of G.east] (U) {$U^{-1}$}; + + % Connections and labels + \draw[<-] (V.west) -- ++(-1.0, 0) node[above right]{$u$}; + \draw[->] (V.east) -- (G.west) node[above left]{$\bm{\tau}$}; + \draw[->] (G.east) -- (U.west) node[above left]{$d\bm{\mathcal{L}}$}; + \draw[->] (U.east) -- ++( 1.0, 0) node[above left]{$y$}; + + \begin{scope}[on background layer] + \node[fit={(V.south west) (G.north-|U.east)}, fill=black!10!white, draw, dashed, inner sep=14pt] (Gsvd) {}; + \node[below right] at (Gsvd.north west) {$\bm{G}_{SVD}$}; + \end{scope} +\end{tikzpicture} +#+end_src + +#+name: fig:decoupling_arch_svd +#+caption: Decoupling using the Singular Value Decomposition +#+RESULTS: +[[file:figs/decoupling_arch_svd.png]] + +#+begin_src matlab :exports none +%% Take complex matrix corresponding to the plant at 100Hz +V = squeeze(frf_coupled(i_wc,:,:)); + +%% Real approximation of G(100Hz) +D = pinv(real(V'*V)); +H1 = pinv(D*real(V'*diag(exp(1j*angle(diag(V*D*V.'))/2)))); + +%% Singular Value Decomposition +[U,S,V] = svd(H1); + +%% Compute the decoupled plant using SVD +G_dL_svd = zeros(size(frf_coupled)); +for i = 1:length(frf_iff.f) + G_dL_svd(i,:,:) = inv(U)*squeeze(frf_coupled(i,:,:))*inv(V'); +end +#+end_src + +#+begin_src matlab :exports none +%% Bode Plot of the SVD decoupled plant +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(G_dL_svd(:,i,j)), 'color', [0,0,0,0.2], ... + 'HandleVisibility', 'off'); + end +end +set(gca,'ColorOrderIndex',1) +for i = 1:6 + plot(frf_iff.f, abs(G_dL_svd(:,i,i)), ... + 'DisplayName', sprintf('$y_%i/u_%i$', i, i)); +end +plot(frf_iff.f, abs(G_dL_svd(:,1,2)), 'color', [0,0,0,0.2], ... + 'DisplayName', 'Coupling'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude'); set(gca, 'XTickLabel',[]); +ylim([1e-9, 1e-4]); +legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 3); + +ax2 = nexttile; +hold on; +for i =1:6 + plot(frf_iff.f, 180/pi*angle(G_dL_svd(:,i,i))); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([10, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_svd_plant.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:interaction_svd_plant +#+caption: Bode Plot of the plant decoupled using the Singular Value Decomposition +#+RESULTS: +[[file:figs/interaction_svd_plant.png]] + +#+begin_src matlab +%% Compute the RGA matrix for the SVD decoupled plant +RGA_svd = zeros(size(frf_coupled)); +for i = 1:length(frf_iff.f) + RGA_svd(i,:,:) = squeeze(G_dL_svd(i,:,:)).*inv(squeeze(G_dL_svd(i,:,:))).'; +end + +%% Compute the RGA-number +RGA_svd_sum = zeros(size(RGA_svd, 1), 1); +for i = 1:length(frf_iff.f) + RGA_svd_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_svd(i,:,:))))); +end +#+end_src + +#+begin_src matlab +%% RGA Number for the SVD decoupled plant +figure; +plot(frf_iff.f, RGA_svd_sum, 'k-'); +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('RGA Number'); +xlim([10, 1e3]); ylim([1e-2, 1e2]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_rga_svd.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:interaction_rga_svd +#+caption: RGA number for the plant decoupled using the SVD +#+RESULTS: +[[file:figs/interaction_rga_svd.png]] + +*** Dynamic decoupling +<> + +#+begin_src latex :file decoupling_arch_dynamic.pdf +\begin{tikzpicture} + \node[block] (G) {$\bm{G}$}; + \node[block, left=0.8 of G] (Ginv) {$\bm{\hat{G}}^{-1}$}; + + % Connections and labels + \draw[<-] (Ginv.west) -- ++(-1.8, 0) node[above right]{$\bm{u}$}; + \draw[->] (Ginv.east) -- (G.west) node[above left]{$\bm{\tau}$}; + \draw[->] (G.east) -- ++( 1.8, 0) node[above left]{$d\bm{\mathcal{L}}$}; + + \begin{scope}[on background layer] + \node[fit={(Ginv.south west) (G.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gx) {}; + \node[below right] at (Gx.north west) {$\bm{G}_{\text{inv}}$}; + \end{scope} +\end{tikzpicture} +#+end_src + +#+name: fig:decoupling_arch_dynamic +#+caption: Decoupling using the inverse of a dynamical model $\bm{\hat{G}}$ of the plant dynamics $\bm{G}$ +#+RESULTS: +[[file:figs/decoupling_arch_dynamic.png]] + +#+begin_src matlab :exports none +%% Compute the plant inverse from the model +G_model = G_coupled; +G_model.outputdelay = 0; % necessary for further inversion +G_inv = inv(G_model); + +%% Compute the decoupled plant +G_dL_inv = zeros(size(frf_coupled)); +for i = 1:length(frf_iff.f) + G_dL_inv(i,:,:) = squeeze(frf_coupled(i,:,:))*squeeze(evalfr(G_inv, 1j*2*pi*frf_iff.f(i))); +end +#+end_src + +#+begin_src matlab :exports none +%% Bode plot of the decoupled plant by full inversion +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(G_dL_inv(:,i,j)), 'color', [0,0,0,0.2], ... + 'HandleVisibility', 'off'); + end +end +set(gca,'ColorOrderIndex',1) +for i = 1:6 + plot(frf_iff.f, abs(G_dL_inv(:,i,i)), ... + 'DisplayName', sprintf('$y_%i/u_%i$', i, i)); +end +plot(frf_iff.f, abs(G_dL_inv(:,1,2)), 'color', [0,0,0,0.2], ... + 'DisplayName', 'Coupling'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude'); set(gca, 'XTickLabel',[]); +ylim([1e-4, 1e1]); +legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); + +ax2 = nexttile; +hold on; +for i =1:6 + plot(frf_iff.f, 180/pi*angle(G_dL_inv(:,i,i))); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([10, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_dynamic_dec_plant.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:interaction_dynamic_dec_plant +#+caption: Bode Plot of the dynamically decoupled plant +#+RESULTS: +[[file:figs/interaction_dynamic_dec_plant.png]] + +#+begin_src matlab :exports none +%% Compute the RGA matrix for the inverse based decoupled plant +RGA_inv = zeros(size(frf_coupled)); +for i = 1:length(frf_iff.f) + RGA_inv(i,:,:) = squeeze(G_dL_inv(i,:,:)).*inv(squeeze(G_dL_inv(i,:,:))).'; +end + +%% Compute the RGA-number +RGA_inv_sum = zeros(size(RGA_inv, 1), 1); +for i = 1:size(RGA_inv, 1) + RGA_inv_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_inv(i,:,:))))); +end +#+end_src + +#+begin_src matlab :exports none +%% RGA Number for the decoupled plant using full inversion +figure; +plot(frf_iff.f, RGA_inv_sum, 'k-'); +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('RGA Number'); +xlim([10, 1e3]); ylim([1e-2, 1e2]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_rga_dynamic_dec.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:interaction_rga_dynamic_dec +#+caption: RGA number for the dynamically decoupled plant +#+RESULTS: +[[file:figs/interaction_rga_dynamic_dec.png]] + +*** Jacobian Decoupling - Center of Stiffness +<> + +#+begin_src latex :file decoupling_arch_jacobian_cok.pdf +\begin{tikzpicture} + \node[block] (G) {$\bm{G}$}; + \node[block, left=0.8 of G] (Jt) {$J_{s,\{K\}}^{-T}$}; + \node[block, right=0.8 of G] (Ja) {$J_{a,\{K\}}^{-1}$}; + + % Connections and labels + \draw[<-] (Jt.west) -- ++(-1.8, 0) node[above right]{$\bm{\mathcal{F}}_{\{K\}}$}; + \draw[->] (Jt.east) -- (G.west) node[above left]{$\bm{\tau}$}; + \draw[->] (G.east) -- (Ja.west) node[above left]{$d\bm{\mathcal{L}}$}; + \draw[->] (Ja.east) -- ++( 1.8, 0) node[above left]{$\bm{\mathcal{X}}_{\{K\}}$}; + + \begin{scope}[on background layer] + \node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gx) {}; + \node[below right] at (Gx.north west) {$\bm{G}_{\{K\}}$}; + \end{scope} +\end{tikzpicture} +#+end_src + +#+name: fig:decoupling_arch_jacobian_cok +#+caption: Decoupling using Jacobian matrices evaluated at the Center of Stiffness +#+RESULTS: +[[file:figs/decoupling_arch_jacobian_cok.png]] + +#+begin_src matlab :exports none +%% Initialize the Nano-Hexapod +n_hexapod = initializeNanoHexapodFinal('MO_B', -42e-3, ... + 'motion_sensor_type', 'plates'); + +%% Get the Jacobians +J_cok = n_hexapod.geometry.J; +Js_cok = n_hexapod.geometry.Js; + +%% Decouple plant using Jacobian (CoM) +G_dL_J_cok = zeros(size(frf_coupled)); +for i = 1:length(frf_iff.f) + G_dL_J_cok(i,:,:) = inv(Js_cok)*squeeze(frf_coupled(i,:,:))*inv(J_cok'); +end +#+end_src + +The obtained plant is shown in Figure ref:fig:interaction_J_cok_plant_not_normalized. +We can see that the stiffness in the $x$, $y$ and $z$ directions are equal, which is due to the cubic architecture of the Stewart platform. + +#+begin_src matlab :exports none +%% Bode Plot of the SVD decoupled plant +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(G_dL_J_cok(:,i,j)), 'color', [0,0,0,0.2], ... + 'HandleVisibility', 'off'); + end +end +set(gca,'ColorOrderIndex',1) +plot(frf_iff.f, abs(G_dL_J_cok(:,1,1)), ... + 'DisplayName', '$D_x/\tilde{\mathcal{F}}_x$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,2,2)), ... + 'DisplayName', '$D_y/\tilde{\mathcal{F}}_y$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,3,3)), ... + 'DisplayName', '$D_z/\tilde{\mathcal{F}}_z$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,4,4)), ... + 'DisplayName', '$R_x/\tilde{\mathcal{M}}_x$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,5,5)), ... + 'DisplayName', '$R_y/\tilde{\mathcal{M}}_y$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,6,6)), ... + 'DisplayName', '$R_z/\tilde{\mathcal{M}}_z$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,1,2)), 'color', [0,0,0,0.2], ... + 'DisplayName', 'Coupling'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude'); set(gca, 'XTickLabel',[]); +ylim([1e-8, 2e-2]); +legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 3); + +ax2 = nexttile; +hold on; +for i =1:6 + plot(frf_iff.f, 180/pi*angle(G_dL_J_cok(:,i,i))); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([10, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_J_cok_plant_not_normalized.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:interaction_J_cok_plant_not_normalized +#+caption: Bode Plot of the plant decoupled using the Jacobian evaluated at the "center of stiffness" +#+RESULTS: +[[file:figs/interaction_J_cok_plant_not_normalized.png]] + +Because the plant in translation and rotation has very different gains, we choose to normalize the plant inputs such that the gain of the diagonal term is equal to $1$ at 100Hz. + +The results is shown in Figure ref:fig:interaction_J_cok_plant. +#+begin_src matlab :exports none +%% Normalize the plant input +[~, i_100] = min(abs(frf_iff.f - 100)); +input_normalize = diag(1./diag(abs(squeeze(G_dL_J_cok(i_100,:,:))))); + +for i = 1:length(frf_iff.f) + G_dL_J_cok(i,:,:) = squeeze(G_dL_J_cok(i,:,:))*input_normalize; +end +#+end_src + +#+begin_src matlab :exports none +%% Bode Plot of the SVD decoupled plant +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(G_dL_J_cok(:,i,j)), 'color', [0,0,0,0.2], ... + 'HandleVisibility', 'off'); + end +end +set(gca,'ColorOrderIndex',1) +plot(frf_iff.f, abs(G_dL_J_cok(:,1,1)), ... + 'DisplayName', '$D_x/\tilde{\mathcal{F}}_x$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,2,2)), ... + 'DisplayName', '$D_y/\tilde{\mathcal{F}}_y$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,3,3)), ... + 'DisplayName', '$D_z/\tilde{\mathcal{F}}_z$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,4,4)), ... + 'DisplayName', '$R_x/\tilde{\mathcal{M}}_x$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,5,5)), ... + 'DisplayName', '$R_y/\tilde{\mathcal{M}}_y$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,6,6)), ... + 'DisplayName', '$R_z/\tilde{\mathcal{M}}_z$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,1,2)), 'color', [0,0,0,0.2], ... + 'DisplayName', 'Coupling'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude'); set(gca, 'XTickLabel',[]); +ylim([1e-4, 1e1]); +legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); + +ax2 = nexttile; +hold on; +for i =1:6 + plot(frf_iff.f, 180/pi*angle(G_dL_J_cok(:,i,i))); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([10, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_J_cok_plant.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:interaction_J_cok_plant +#+caption: Bode Plot of the plant decoupled using the Jacobian evaluated at the "center of stiffness" +#+RESULTS: +[[file:figs/interaction_J_cok_plant.png]] + +#+begin_src matlab :exports none +%% Compute RGA Matrix +RGA_cok = zeros(size(frf_coupled)); +for i = 1:length(frf_iff.f) + RGA_cok(i,:,:) = squeeze(G_dL_J_cok(i,:,:)).*inv(squeeze(G_dL_J_cok(i,:,:))).'; +end + +%% Compute RGA-number +RGA_cok_sum = zeros(length(frf_iff.f), 1); +for i = 1:length(frf_iff.f) + RGA_cok_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_cok(i,:,:))))); +end +#+end_src + +#+begin_src matlab :exports none +%% Plot the RGA-Number for the Jacobian (CoK) decoupled plant +figure; +plot(frf_iff.f, RGA_cok_sum, 'k-'); +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('RGA Number'); +xlim([10, 1e3]); ylim([1e-2, 1e2]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_rga_J_cok.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:interaction_rga_J_cok +#+caption: RGA number for the plant decoupled using the Jacobian evaluted at the Center of Stiffness +#+RESULTS: +[[file:figs/interaction_rga_J_cok.png]] + +*** Jacobian Decoupling - Center of Mass +<> + +#+begin_src latex :file decoupling_arch_jacobian_com.pdf +\begin{tikzpicture} + \node[block] (G) {$\bm{G}$}; + \node[block, left=0.8 of G] (Jt) {$J_{s,\{M\}}^{-T}$}; + \node[block, right=0.8 of G] (Ja) {$J_{a,\{M\}}^{-1}$}; + + % Connections and labels + \draw[<-] (Jt.west) -- ++(-1.8, 0) node[above right]{$\bm{\mathcal{F}}_{\{M\}}$}; + \draw[->] (Jt.east) -- (G.west) node[above left]{$\bm{\tau}$}; + \draw[->] (G.east) -- (Ja.west) node[above left]{$d\bm{\mathcal{L}}$}; + \draw[->] (Ja.east) -- ++( 1.8, 0) node[above left]{$\bm{\mathcal{X}}_{\{M\}}$}; + + \begin{scope}[on background layer] + \node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gx) {}; + \node[below right] at (Gx.north west) {$\bm{G}_{\{M\}}$}; + \end{scope} +\end{tikzpicture} +#+end_src + +#+name: fig:decoupling_arch_jacobian_com +#+caption: Decoupling using Jacobian matrices evaluated at the Center of Mass +#+RESULTS: +[[file:figs/decoupling_arch_jacobian_com.png]] + +#+begin_src matlab :exports none +%% Initialize the Nano-Hexapod +n_hexapod = initializeNanoHexapodFinal('MO_B', 25e-3, ... + 'motion_sensor_type', 'plates'); + +%% Get the Jacobians +J_com = n_hexapod.geometry.J; +Js_com = n_hexapod.geometry.Js; + +%% Decouple plant using Jacobian (CoM) +G_dL_J_com = zeros(size(frf_coupled)); +for i = 1:length(frf_iff.f) + G_dL_J_com(i,:,:) = inv(Js_com)*squeeze(frf_coupled(i,:,:))*inv(J_com'); +end + +%% Normalize the plant input +[~, i_100] = min(abs(frf_iff.f - 100)); +input_normalize = diag(1./diag(abs(squeeze(G_dL_J_com(i_100,:,:))))); + +for i = 1:length(frf_iff.f) + G_dL_J_com(i,:,:) = squeeze(G_dL_J_com(i,:,:))*input_normalize; +end +#+end_src + +#+begin_src matlab :exports none +%% Bode Plot of the SVD decoupled plant +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(G_dL_J_com(:,i,j)), 'color', [0,0,0,0.2], ... + 'HandleVisibility', 'off'); + end +end +set(gca,'ColorOrderIndex',1) +plot(frf_iff.f, abs(G_dL_J_com(:,1,1)), ... + 'DisplayName', '$D_x/\tilde{\mathcal{F}}_x$'); +plot(frf_iff.f, abs(G_dL_J_com(:,2,2)), ... + 'DisplayName', '$D_y/\tilde{\mathcal{F}}_y$'); +plot(frf_iff.f, abs(G_dL_J_com(:,3,3)), ... + 'DisplayName', '$D_z/\tilde{\mathcal{F}}_z$'); +plot(frf_iff.f, abs(G_dL_J_com(:,4,4)), ... + 'DisplayName', '$R_x/\tilde{\mathcal{M}}_x$'); +plot(frf_iff.f, abs(G_dL_J_com(:,5,5)), ... + 'DisplayName', '$R_y/\tilde{\mathcal{M}}_y$'); +plot(frf_iff.f, abs(G_dL_J_com(:,6,6)), ... + 'DisplayName', '$R_z/\tilde{\mathcal{M}}_z$'); +plot(frf_iff.f, abs(G_dL_J_com(:,1,2)), 'color', [0,0,0,0.2], ... + 'DisplayName', 'Coupling'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude'); set(gca, 'XTickLabel',[]); +ylim([1e-3, 1e1]); +legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); + +ax2 = nexttile; +hold on; +for i =1:6 + plot(frf_iff.f, 180/pi*angle(G_dL_J_com(:,i,i))); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([10, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_J_com_plant.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:interaction_J_com_plant +#+caption: Bode Plot of the plant decoupled using the Jacobian evaluated at the Center of Mass +#+RESULTS: +[[file:figs/interaction_J_com_plant.png]] + +#+begin_src matlab :exports none +%% Compute RGA Matrix +RGA_com = zeros(size(frf_coupled)); +for i = 1:length(frf_iff.f) + RGA_com(i,:,:) = squeeze(G_dL_J_com(i,:,:)).*inv(squeeze(G_dL_J_com(i,:,:))).'; +end + +%% Compute RGA-number +RGA_com_sum = zeros(size(RGA_com, 1), 1); +for i = 1:size(RGA_com, 1) + RGA_com_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_com(i,:,:))))); +end +#+end_src + +#+begin_src matlab :exports none +%% Plot the RGA-Number for the Jacobian (CoM) decoupled plant +figure; +plot(frf_iff.f, RGA_com_sum, 'k-'); +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('RGA Number'); +xlim([10, 1e3]); ylim([1e-2, 1e2]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_rga_J_com.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:interaction_rga_J_com +#+caption: RGA number for the plant decoupled using the Jacobian evaluted at the Center of Mass +#+RESULTS: +[[file:figs/interaction_rga_J_com.png]] + +*** Decoupling Comparison +<> + +Let's now compare all of the decoupling methods (Figure ref:fig:interaction_compare_rga_numbers). + +#+begin_important +From Figure ref:fig:interaction_compare_rga_numbers, the following remarks are made: +- *Decentralized plant*: well decoupled below suspension modes +- *Static inversion*: similar to the decentralized plant as the decentralized plant has already a good decoupling at low frequency +- *Crossover inversion*: the decoupling is improved around the crossover frequency as compared to the decentralized plant. However, the decoupling is increased at lower frequency. +- *SVD decoupling*: Very good decoupling up to 235Hz. Especially between 100Hz and 200Hz. +- *Dynamic Inversion*: the plant is very well decoupled at frequencies where the model is accurate (below 235Hz where flexible modes are not modelled). +- *Jacobian - Stiffness*: good decoupling at low frequency. The decoupling increases at the frequency of the suspension modes, but is acceptable up to the strut flexible modes (235Hz). +- *Jacobian - Mass*: bad decoupling at low frequency. Better decoupling above the frequency of the suspension modes, and acceptable decoupling up to the strut flexible modes (235Hz). +#+end_important + +#+begin_src matlab :exports none +%% Comparison of the RGA-Numbers +figure; +hold on; +plot(frf_iff.f, RGA_dec_sum, 'DisplayName', 'Decentralized'); +plot(frf_iff.f, RGA_sta_sum, 'DisplayName', 'Static inv.'); +plot(frf_iff.f, RGA_wc_sum, 'DisplayName', 'Crossover inv.'); +plot(frf_iff.f, RGA_svd_sum, 'DisplayName', 'SVD'); +plot(frf_iff.f, RGA_inv_sum, 'DisplayName', 'Dynamic inv.'); +plot(frf_iff.f, RGA_cok_sum, 'DisplayName', 'Jacobian - CoK'); +plot(frf_iff.f, RGA_com_sum, 'DisplayName', 'Jacobian - CoM'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('RGA Number'); +xlim([10, 1e3]); ylim([1e-2, 1e2]); +legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 2); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_compare_rga_numbers.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:interaction_compare_rga_numbers +#+caption: Comparison of the obtained RGA-numbers for all the decoupling methods +#+RESULTS: +[[file:figs/interaction_compare_rga_numbers.png]] + +*** Decoupling Robustness +<> + +Let's now see how the decoupling is changing when changing the payload's mass. +#+begin_src matlab +frf_new = frf_iff.G_dL{3}; +#+end_src + +#+begin_src matlab :exports none +%% Decentralized RGA +RGA_dec_b = zeros(size(frf_new)); +for i = 1:length(frf_iff.f) + RGA_dec_b(i,:,:) = squeeze(frf_new(i,:,:)).*inv(squeeze(frf_new(i,:,:))).'; +end + +RGA_dec_sum_b = zeros(length(frf_iff), 1); +for i = 1:length(frf_iff.f) + RGA_dec_sum_b(i) = sum(sum(abs(eye(6) - squeeze(RGA_dec_b(i,:,:))))); +end +#+end_src + +#+begin_src matlab :exports none +%% Static Decoupling +G_dL_sta_b = zeros(size(frf_new)); +for i = 1:length(frf_iff.f) + G_dL_sta_b(i,:,:) = squeeze(frf_new(i,:,:))*dc_inv; +end + +RGA_sta_b = zeros(size(frf_new)); +for i = 1:length(frf_iff.f) + RGA_sta_b(i,:,:) = squeeze(G_dL_sta_b(i,:,:)).*inv(squeeze(G_dL_sta_b(i,:,:))).'; +end + +RGA_sta_sum_b = zeros(size(RGA_sta_b, 1), 1); +for i = 1:size(RGA_sta_b, 1) + RGA_sta_sum_b(i) = sum(sum(abs(eye(6) - squeeze(RGA_sta_b(i,:,:))))); +end +#+end_src + +#+begin_src matlab :exports none +%% Crossover Decoupling +V = squeeze(frf_coupled(i_wc,:,:)); +D = pinv(real(V'*V)); +H1 = D*real(V'*diag(exp(1j*angle(diag(V*D*V.'))/2))); + +G_dL_wc_b = zeros(size(frf_new)); +for i = 1:length(frf_iff.f) + G_dL_wc_b(i,:,:) = squeeze(frf_new(i,:,:))*H1; +end + +RGA_wc_b = zeros(size(frf_new)); +for i = 1:length(frf_iff.f) + RGA_wc_b(i,:,:) = squeeze(G_dL_wc_b(i,:,:)).*inv(squeeze(G_dL_wc_b(i,:,:))).'; +end + +RGA_wc_sum_b = zeros(size(RGA_wc_b, 1), 1); +for i = 1:size(RGA_wc_b, 1) + RGA_wc_sum_b(i) = sum(sum(abs(eye(6) - squeeze(RGA_wc_b(i,:,:))))); +end +#+end_src + +#+begin_src matlab :exports none +%% SVD +V = squeeze(frf_coupled(i_wc,:,:)); +D = pinv(real(V'*V)); +H1 = pinv(D*real(V'*diag(exp(1j*angle(diag(V*D*V.'))/2)))); +[U,S,V] = svd(H1); + +G_dL_svd_b = zeros(size(frf_new)); +for i = 1:length(frf_iff.f) + G_dL_svd_b(i,:,:) = inv(U)*squeeze(frf_new(i,:,:))*inv(V'); +end + +RGA_svd_b = zeros(size(frf_new)); +for i = 1:length(frf_iff.f) + RGA_svd_b(i,:,:) = squeeze(G_dL_svd_b(i,:,:)).*inv(squeeze(G_dL_svd_b(i,:,:))).'; +end + +RGA_svd_sum_b = zeros(size(RGA_svd_b, 1), 1); +for i = 1:size(RGA_svd, 1) + RGA_svd_sum_b(i) = sum(sum(abs(eye(6) - squeeze(RGA_svd_b(i,:,:))))); +end +#+end_src + +#+begin_src matlab :exports none +%% Dynamic Decoupling +G_model = G_coupled; +G_model.outputdelay = 0; % necessary for further inversion +G_inv = inv(G_model); + +G_dL_inv_b = zeros(size(frf_new)); +for i = 1:length(frf_iff.f) + G_dL_inv_b(i,:,:) = squeeze(frf_new(i,:,:))*squeeze(evalfr(G_inv, 1j*2*pi*frf_iff.f(i))); +end + +RGA_inv_b = zeros(size(frf_new)); +for i = 1:length(frf_iff.f) + RGA_inv_b(i,:,:) = squeeze(G_dL_inv_b(i,:,:)).*inv(squeeze(G_dL_inv_b(i,:,:))).'; +end + +RGA_inv_sum_b = zeros(size(RGA_inv_b, 1), 1); +for i = 1:size(RGA_inv_b, 1) + RGA_inv_sum_b(i) = sum(sum(abs(eye(6) - squeeze(RGA_inv_b(i,:,:))))); +end +#+end_src + +#+begin_src matlab :exports none +%% Jacobian (CoK) +G_dL_J_cok_b = zeros(size(frf_new)); +for i = 1:length(frf_iff.f) + G_dL_J_cok_b(i,:,:) = inv(Js_cok)*squeeze(frf_new(i,:,:))*inv(J_cok'); +end + +RGA_cok_b = zeros(size(frf_new)); +for i = 1:length(frf_iff.f) + RGA_cok_b(i,:,:) = squeeze(G_dL_J_cok_b(i,:,:)).*inv(squeeze(G_dL_J_cok_b(i,:,:))).'; +end + +RGA_cok_sum_b = zeros(size(RGA_cok_b, 1), 1); +for i = 1:size(RGA_cok_b, 1) + RGA_cok_sum_b(i) = sum(sum(abs(eye(6) - squeeze(RGA_cok_b(i,:,:))))); +end +#+end_src + +#+begin_src matlab :exports none +%% Jacobian (CoM) +G_dL_J_com_b = zeros(size(frf_new)); +for i = 1:length(frf_iff.f) + G_dL_J_com_b(i,:,:) = inv(Js_com)*squeeze(frf_new(i,:,:))*inv(J_com'); +end + +RGA_com_b = zeros(size(frf_new)); +for i = 1:length(frf_iff.f) + RGA_com_b(i,:,:) = squeeze(G_dL_J_com_b(i,:,:)).*inv(squeeze(G_dL_J_com_b(i,:,:))).'; +end + +RGA_com_sum_b = zeros(size(RGA_com_b, 1), 1); +for i = 1:size(RGA_com_b, 1) + RGA_com_sum_b(i) = sum(sum(abs(eye(6) - squeeze(RGA_com_b(i,:,:))))); +end +#+end_src + +The obtained RGA-numbers are shown in Figure ref:fig:interaction_compare_rga_numbers_rob. + +#+begin_important +From Figure ref:fig:interaction_compare_rga_numbers_rob: +- The decoupling using the Jacobian evaluated at the "center of stiffness" seems to give the most robust results. +#+end_important + +#+begin_src matlab :exports none +%% Robustness of the Decoupling method +figure; +hold on; +plot(frf_iff.f, RGA_dec_sum, '-', 'DisplayName', 'Decentralized'); +plot(frf_iff.f, RGA_sta_sum, '-', 'DisplayName', 'Static inv.'); +plot(frf_iff.f, RGA_wc_sum, '-', 'DisplayName', 'Crossover inv.'); +plot(frf_iff.f, RGA_svd_sum, '-', 'DisplayName', 'SVD'); +plot(frf_iff.f, RGA_inv_sum, '-', 'DisplayName', 'Dynamic inv.'); +plot(frf_iff.f, RGA_cok_sum, '-', 'DisplayName', 'Jacobian - CoK'); +plot(frf_iff.f, RGA_com_sum, '-', 'DisplayName', 'Jacobian - CoM'); +set(gca,'ColorOrderIndex',1) +plot(frf_iff.f, RGA_dec_sum_b, '--', 'HandleVisibility', 'off'); +plot(frf_iff.f, RGA_sta_sum_b, '--', 'HandleVisibility', 'off'); +plot(frf_iff.f, RGA_wc_sum_b, '--', 'HandleVisibility', 'off'); +plot(frf_iff.f, RGA_svd_sum_b, '--', 'HandleVisibility', 'off'); +plot(frf_iff.f, RGA_inv_sum_b, '--', 'HandleVisibility', 'off'); +plot(frf_iff.f, RGA_cok_sum_b, '--', 'HandleVisibility', 'off'); +plot(frf_iff.f, RGA_com_sum_b, '--', 'HandleVisibility', 'off'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('RGA Number'); +xlim([10, 1e3]); ylim([1e-2, 1e2]); +legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 2); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_compare_rga_numbers_rob.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:interaction_compare_rga_numbers_rob +#+caption: Change of the RGA-number with a change of the payload. Indication of the robustness of the inversion method. +#+RESULTS: +[[file:figs/interaction_compare_rga_numbers_rob.png]] + +*** Conclusion + +#+begin_important +Several decoupling methods can be used: +- SVD +- Inverse +- Jacobian (CoK) +#+end_important + +#+name: tab:interaction_analysis_conclusion +#+caption: Summary of the interaction analysis and different decoupling strategies +#+attr_latex: :environment tabularx :width \linewidth :align lccc +#+attr_latex: :center t :booktabs t +| *Method* | *RGA* | *Diag Plant* | *Robustness* | +|----------------+-------+--------------+--------------| +| Decentralized | -- | Equal | ++ | +| Static dec. | -- | Equal | ++ | +| Crossover dec. | - | Equal | 0 | +| SVD | ++ | Diff | + | +| Dynamic dec. | ++ | Unity, equal | - | +| Jacobian - CoK | + | Diff | ++ | +| Jacobian - CoM | 0 | Diff | + | + +** Robust High Authority Controller +:PROPERTIES: +:header-args:matlab+: :tangle matlab/scripts/hac_lac_enc_plates_suspended_table.m +:END: +<> +*** Introduction :ignore: +In this section we wish to develop a robust High Authority Controller (HAC) that is working for all payloads. + +cite:indri20_mechat_robot + +*** Matlab Init :noexport:ignore: +#+begin_src matlab +%% hac_lac_enc_plates_suspended_table.m +% Development and analysis of a robust High Authority Controller +#+end_src + +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) +<> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes +<> +#+end_src + +#+begin_src matlab :tangle no :noweb yes +<> +#+end_src + +#+begin_src matlab :eval no :noweb yes +<> +#+end_src + +#+begin_src matlab :noweb yes +<> +#+end_src + +#+begin_src matlab +%% Load the identified FRF and Simscape model +frf_iff = load('frf_iff_vib_table_m.mat', 'f', 'Ts', 'G_dL'); +sim_iff = load('sim_iff_vib_table_m.mat', 'G_dL'); +#+end_src + +*** Using Jacobian evaluated at the center of stiffness +**** Decoupled Plant +#+begin_src matlab +G_nom = frf_iff.G_dL{2}; % Nominal Plant +#+end_src + +#+begin_src matlab :exports none +%% Initialize the Nano-Hexapod +n_hexapod = initializeNanoHexapodFinal('MO_B', -42e-3, ... + 'motion_sensor_type', 'plates'); + +%% Get the Jacobians +J_cok = n_hexapod.geometry.J; +Js_cok = n_hexapod.geometry.Js; + +%% Decouple plant using Jacobian (CoM) +G_dL_J_cok = zeros(size(G_nom)); +for i = 1:length(frf_iff.f) + G_dL_J_cok(i,:,:) = inv(Js_cok)*squeeze(G_nom(i,:,:))*inv(J_cok'); +end + +%% Normalize the plant input +[~, i_100] = min(abs(frf_iff.f - 10)); +input_normalize = diag(1./diag(abs(squeeze(G_dL_J_cok(i_100,:,:))))); + +for i = 1:length(frf_iff.f) + G_dL_J_cok(i,:,:) = squeeze(G_dL_J_cok(i,:,:))*input_normalize; +end +#+end_src + +#+begin_src matlab :exports none +%% Bode Plot of the decoupled plant +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(G_dL_J_cok(:,i,j)), 'color', [0,0,0,0.2], ... + 'HandleVisibility', 'off'); + end +end +set(gca,'ColorOrderIndex',1) +plot(frf_iff.f, abs(G_dL_J_cok(:,1,1)), ... + 'DisplayName', '$D_x/\tilde{\mathcal{F}}_x$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,2,2)), ... + 'DisplayName', '$D_y/\tilde{\mathcal{F}}_y$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,3,3)), ... + 'DisplayName', '$D_z/\tilde{\mathcal{F}}_z$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,4,4)), ... + 'DisplayName', '$R_x/\tilde{\mathcal{M}}_x$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,5,5)), ... + 'DisplayName', '$R_y/\tilde{\mathcal{M}}_y$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,6,6)), ... + 'DisplayName', '$R_z/\tilde{\mathcal{M}}_z$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,1,2)), 'color', [0,0,0,0.2], ... + 'DisplayName', 'Coupling'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude'); set(gca, 'XTickLabel',[]); +ylim([1e-3, 1e1]); +legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); + +ax2 = nexttile; +hold on; +for i =1:6 + plot(frf_iff.f, 180/pi*angle(G_dL_J_cok(:,i,i))); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([10, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/bode_plot_hac_iff_plant_jacobian_cok.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:bode_plot_hac_iff_plant_jacobian_cok +#+caption: Bode plot of the decoupled plant using the Jacobian evaluated at the Center of Stiffness +#+RESULTS: +[[file:figs/bode_plot_hac_iff_plant_jacobian_cok.png]] + +**** SISO Controller Design +As the diagonal elements of the plant are not equal, several SISO controllers are designed and then combined to form a diagonal controller. +All the diagonal terms of the controller consists of: +- A double integrator to have high gain at low frequency +- A lead around the crossover frequency to increase stability margins +- Two second order low pass filters above the crossover frequency to increase the robustness to high frequency modes + +#+begin_src matlab :exports none +%% Controller Ry,Rz + +% Wanted crossover frequency +wc_Rxy = 2*pi*80; + +% Lead +a = 8.0; % Amount of phase lead / width of the phase lead / high frequency gain +wc = wc_Rxy; % Frequency with the maximum phase lead [rad/s] +Kd_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)))/sqrt(a); + +% Integrator +w0_int = wc_Rxy/2; % [rad/s] +xi_int = 0.3; + +Kd_int = (1 + 2*xi_int/w0_int*s + s^2/w0_int^2)/(s^2/w0_int^2); + +% Low Pass Filter (High frequency robustness) +w0_lpf = wc_Rxy*2; % Cut-off frequency [rad/s] +xi_lpf = 0.6; % Damping Ratio + +Kd_lpf = 1/(1 + 2*xi_lpf/w0_lpf*s + s^2/w0_lpf^2); + +w0_lpf_b = wc_Rxy*4; % Cut-off frequency [rad/s] +xi_lpf_b = 0.7; % Damping Ratio + +Kd_lpf_b = 1/(1 + 2*xi_lpf_b/w0_lpf_b*s + s^2/w0_lpf_b^2); + +% Unity Gain frequency +[~, i_80] = min(abs(frf_iff.f - wc_Rxy/2/pi)); + +% Combination of all the elements +Kd_Rxy = ... + -1/abs(G_dL_J_cok(i_80,4,4)) * ... + Kd_lead/abs(evalfr(Kd_lead, 1j*wc_Rxy)) * ... % Lead (gain of 1 at wc) + Kd_int /abs(evalfr(Kd_int, 1j*wc_Rxy)) * ... + Kd_lpf_b/abs(evalfr(Kd_lpf_b, 1j*wc_Rxy)) * ... + Kd_lpf /abs(evalfr(Kd_lpf, 1j*wc_Rxy)); % Low Pass Filter +#+end_src + +#+begin_src matlab :exports none +%% Controller Dx,Dy,Rz + +% Wanted crossover frequency +wc_Dxy = 2*pi*100; + +% Lead +a = 8.0; % Amount of phase lead / width of the phase lead / high frequency gain +wc = wc_Dxy; % Frequency with the maximum phase lead [rad/s] +Kd_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)))/sqrt(a); + +% Integrator +w0_int = wc_Dxy/2; % [rad/s] +xi_int = 0.3; + +Kd_int = (1 + 2*xi_int/w0_int*s + s^2/w0_int^2)/(s^2/w0_int^2); + +% Low Pass Filter (High frequency robustness) +w0_lpf = wc_Dxy*2; % Cut-off frequency [rad/s] +xi_lpf = 0.6; % Damping Ratio + +Kd_lpf = 1/(1 + 2*xi_lpf/w0_lpf*s + s^2/w0_lpf^2); + +w0_lpf_b = wc_Dxy*4; % Cut-off frequency [rad/s] +xi_lpf_b = 0.7; % Damping Ratio + +Kd_lpf_b = 1/(1 + 2*xi_lpf_b/w0_lpf_b*s + s^2/w0_lpf_b^2); + +% Unity Gain frequency +[~, i_100] = min(abs(frf_iff.f - wc_Dxy/2/pi)); + +% Combination of all the elements +Kd_Dyx_Rz = ... + -1/abs(G_dL_J_cok(i_100,1,1)) * ... + Kd_int /abs(evalfr(Kd_int, 1j*wc_Dxy)) * ... % Integrator + Kd_lead/abs(evalfr(Kd_lead, 1j*wc_Dxy)) * ... % Lead (gain of 1 at wc) + Kd_lpf_b/abs(evalfr(Kd_lpf_b, 1j*wc_Dxy)) * ... % Lead (gain of 1 at wc) + Kd_lpf /abs(evalfr(Kd_lpf, 1j*wc_Dxy)); % Low Pass Filter +#+end_src + +#+begin_src matlab :exports none +%% Controller Dz + +% Wanted crossover frequency +wc_Dz = 2*pi*100; + +% Lead +a = 8.0; % Amount of phase lead / width of the phase lead / high frequency gain +wc = wc_Dz; % Frequency with the maximum phase lead [rad/s] +Kd_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)))/sqrt(a); + +% Integrator +w0_int = wc_Dz/2; % [rad/s] +xi_int = 0.3; + +Kd_int = (1 + 2*xi_int/w0_int*s + s^2/w0_int^2)/(s^2/w0_int^2); + +% Low Pass Filter (High frequency robustness) +w0_lpf = wc_Dz*2; % Cut-off frequency [rad/s] +xi_lpf = 0.6; % Damping Ratio + +Kd_lpf = 1/(1 + 2*xi_lpf/w0_lpf*s + s^2/w0_lpf^2); + +w0_lpf_b = wc_Dz*4; % Cut-off frequency [rad/s] +xi_lpf_b = 0.7; % Damping Ratio + +Kd_lpf_b = 1/(1 + 2*xi_lpf_b/w0_lpf_b*s + s^2/w0_lpf_b^2); + +% Unity Gain frequency +[~, i_100] = min(abs(frf_iff.f - wc_Dz/2/pi)); + +% Combination of all the elements +Kd_Dz = ... + -1/abs(G_dL_J_cok(i_100,3,3)) * ... + Kd_int /abs(evalfr(Kd_int, 1j*wc_Dz)) * ... % Integrator + Kd_lead/abs(evalfr(Kd_lead, 1j*wc_Dz)) * ... % Lead (gain of 1 at wc) + Kd_lpf_b/abs(evalfr(Kd_lpf_b, 1j*wc_Dz)) * ... % Lead (gain of 1 at wc) + Kd_lpf /abs(evalfr(Kd_lpf, 1j*wc_Dz)); % Low Pass Filter +#+end_src + +#+begin_src matlab :exports none +%% Diagonal Controller +Kd_diag = blkdiag(Kd_Dyx_Rz, Kd_Dyx_Rz, Kd_Dz, Kd_Rxy, Kd_Rxy, Kd_Dyx_Rz); +#+end_src + +**** Obtained Loop Gain +#+begin_src matlab :exports none +%% Experimental Loop Gain +Lmimo = permute(pagemtimes(permute(G_dL_J_cok, [2,3,1]), squeeze(freqresp(Kd_diag, frf_iff.f, 'Hz'))), [3,1,2]); +#+end_src + +#+begin_src matlab :exports none +%% Bode plot of the experimental Loop Gain +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:6 + plot(frf_iff.f, abs(Lmimo(:,i,i)), '-'); +end +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(squeeze(Lmimo(:,i,j))), 'color', [0,0,0,0.2]); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Loop Gain'); set(gca, 'XTickLabel',[]); +ylim([1e-3, 1e+3]); + +ax2 = nexttile; +hold on; +for i = 1:6 + plot(frf_iff.f, 180/pi*angle(Lmimo(:,i,i)), '-'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:45:360); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([1, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/bode_plot_hac_iff_loop_gain_jacobian_cok.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:bode_plot_hac_iff_loop_gain_jacobian_cok +#+caption: Bode plot of the Loop Gain when using the Jacobian evaluated at the Center of Stiffness to decouple the system +#+RESULTS: +[[file:figs/bode_plot_hac_iff_loop_gain_jacobian_cok.png]] + +#+begin_src matlab +%% Controller to be implemented +Kd = inv(J_cok')*input_normalize*ss(Kd_diag)*inv(Js_cok); +#+end_src + +**** Verification of the Stability +Now the stability of the feedback loop is verified using the generalized Nyquist criteria. + +#+begin_src matlab :exports none +%% Compute the Eigenvalues of the loop gain +Ldet = zeros(3, 6, length(frf_iff.f)); + +for i_mass = 1:3 + % Loop gain + Lmimo = pagemtimes(permute(frf_iff.G_dL{i_mass}, [2,3,1]),squeeze(freqresp(Kd, frf_iff.f, 'Hz'))); + for i_f = 2:length(frf_iff.f) + Ldet(i,:, i_f) = eig(squeeze(Lmimo(:,:,i_f))); + end +end +#+end_src + +#+begin_src matlab :exports none +%% Plot of the eigenvalues of L in the complex plane +figure; +hold on; +for i_mass = 2:3 + plot(real(squeeze(Ldet(i_mass, 1,:))), imag(squeeze(Ldet(i_mass, 1,:))), ... + '.', 'color', colors(i_mass+1, :), ... + 'DisplayName', sprintf('%i masses', i_mass)); + plot(real(squeeze(Ldet(i_mass, 1,:))), -imag(squeeze(Ldet(i_mass, 1,:))), ... + '.', 'color', colors(i_mass+1, :), ... + 'HandleVisibility', 'off'); + for i = 1:6 + plot(real(squeeze(Ldet(i_mass, i,:))), imag(squeeze(Ldet(i_mass, i,:))), ... + '.', 'color', colors(i_mass+1, :), ... + 'HandleVisibility', 'off'); + plot(real(squeeze(Ldet(i_mass, i,:))), -imag(squeeze(Ldet(i_mass, i,:))), ... + '.', 'color', colors(i_mass+1, :), ... + 'HandleVisibility', 'off'); + end +end +plot(-1, 0, 'kx', 'HandleVisibility', 'off'); +hold off; +set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin'); +xlabel('Real'); ylabel('Imag'); +legend('location', 'southeast'); +xlim([-3, 1]); ylim([-2, 2]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/loci_hac_iff_loop_gain_jacobian_cok.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:loci_hac_iff_loop_gain_jacobian_cok +#+caption: Loci of $L(j\omega)$ in the complex plane. +#+RESULTS: +[[file:figs/loci_hac_iff_loop_gain_jacobian_cok.png]] + +**** Save for further analysis +#+begin_src matlab :exports none :tangle no +save('matlab/data_sim/Khac_iff_struts_jacobian_cok.mat', 'Kd') +#+end_src + +#+begin_src matlab :eval no +save('data_sim/Khac_iff_struts_jacobian_cok.mat', 'Kd') +#+end_src + +**** Sensitivity transfer function from the model +#+begin_src matlab :exports none +%% Open Simulink Model +mdl = 'nano_hexapod_simscape'; + +options = linearizeOptions; +options.SampleTime = 0; + +open(mdl) + +Rx = zeros(1, 7); + +colors = colororder; +#+end_src + +#+begin_src matlab :exports none +%% Initialize the Simscape model in closed loop +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'plates', ... + 'actuator_type', '2dof', ... + 'controller_type', 'hac-iff-struts'); + +support.type = 1; % On top of vibration table +payload.type = 2; % Payload +#+end_src + +#+begin_src matlab :exports none +%% Load controllers +load('Kiff_opt.mat', 'Kiff'); +Kiff = c2d(Kiff, Ts, 'Tustin'); +load('Khac_iff_struts_jacobian_cok.mat', 'Kd') +Khac_iff_struts = c2d(Kd, Ts, 'Tustin'); +#+end_src + +#+begin_src matlab :exports none +%% Identify the (damped) transfer function from u to dLm +clear io; io_i = 1; +io(io_i) = linio([mdl, '/Rx'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/dL'], 1, 'output'); io_i = io_i + 1; % Plate Displacement (encoder) +#+end_src + +#+begin_src matlab :exports none +%% Identification of the dynamics +Gcl = linearize(mdl, io, 0.0, options); +#+end_src + +#+begin_src matlab :exports none +%% Computation of the sensitivity transfer function +S = eye(6) - inv(n_hexapod.geometry.J)*Gcl; +#+end_src + +The results are shown in Figure ref:fig:sensitivity_hac_jacobian_cok_3m_comp_model. + +#+begin_src matlab :exports none +%% Bode plot for the transfer function from u to dLm +freqs = logspace(0, 3, 1000); + +figure; +hold on; +for i =1:6 + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(S(i,i), freqs, 'Hz'))), '--', ... + 'DisplayName', sprintf('$S_{%s}$ - Model', labels{i})); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Sensitivity [-]'); +ylim([1e-4, 1e1]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); +xlim([1, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/sensitivity_hac_jacobian_cok_3m_comp_model.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:sensitivity_hac_jacobian_cok_3m_comp_model +#+caption: Estimated sensitivity transfer functions for the HAC controller using the Jacobian estimated at the Center of Stiffness +#+RESULTS: +[[file:figs/sensitivity_hac_jacobian_cok_3m_comp_model.png]] + +*** Using Singular Value Decomposition +**** Decoupled Plant +#+begin_src matlab +G_nom = frf_iff.G_dL{2}; % Nominal Plant +#+end_src + +#+begin_src matlab :exports none +%% Take complex matrix corresponding to the plant at 100Hz +wc = 100; % Wanted crossover frequency [Hz] +[~, i_wc] = min(abs(frf_iff.f - wc)); % Indice corresponding to wc + +V = squeeze(G_nom(i_wc,:,:)); + +%% Real approximation of G(100Hz) +D = pinv(real(V'*V)); +H1 = pinv(D*real(V'*diag(exp(1j*angle(diag(V*D*V.'))/2)))); + +%% Singular Value Decomposition +[U,S,V] = svd(H1); + +%% Compute the decoupled plant using SVD +G_dL_svd = zeros(size(G_nom)); +for i = 1:length(frf_iff.f) + G_dL_svd(i,:,:) = inv(U)*squeeze(G_nom(i,:,:))*inv(V'); +end +#+end_src + +#+begin_src matlab :exports none +%% Bode plot of the decoupled plant using SVD +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(G_dL_svd(:,i,j)), 'color', [0,0,0,0.2], ... + 'HandleVisibility', 'off'); + end +end +set(gca,'ColorOrderIndex',1); +for i = 1:6 + plot(frf_iff.f, abs(G_dL_svd(:,i,i)), ... + 'DisplayName', sprintf('$y_%i/u_%i$', i, i)); +end +plot(frf_iff.f, abs(G_dL_svd(:,1,2)), 'color', [0,0,0,0.2], ... + 'DisplayName', 'Coupling'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude'); set(gca, 'XTickLabel',[]); +ylim([1e-9, 1e-4]); +legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); + +ax2 = nexttile; +hold on; +for i =1:6 + plot(frf_iff.f, 180/pi*angle(G_dL_svd(:,i,i))); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([10, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/bode_plot_hac_iff_plant_svd.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:bode_plot_hac_iff_plant_svd +#+caption: Bode plot of the decoupled plant using the SVD +#+RESULTS: +[[file:figs/bode_plot_hac_iff_plant_svd.png]] + +**** Controller Design +#+begin_src matlab :exports none +%% Lead +a = 6.0; % Amount of phase lead / width of the phase lead / high frequency gain +wc = 2*pi*100; % Frequency with the maximum phase lead [rad/s] +Kd_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)))/sqrt(a); + +%% Integrator +Kd_int = ((2*pi*50 + s)/(2*pi*0.1 + s))^2; + +%% Low Pass Filter (High frequency robustness) +w0_lpf = 2*pi*200; % Cut-off frequency [rad/s] +xi_lpf = 0.3; % Damping Ratio + +Kd_lpf = 1/(1 + 2*xi_lpf/w0_lpf*s + s^2/w0_lpf^2); + +%% Normalize Gain +Kd_norm = diag(1./abs(diag(squeeze(G_dL_svd(i_wc,:,:))))); + +%% Diagonal Control +Kd_diag = ... + Kd_norm * ... % Normalize gain at 100Hz + Kd_int /abs(evalfr(Kd_int, 1j*2*pi*100)) * ... % Integrator + Kd_lead/abs(evalfr(Kd_lead, 1j*2*pi*100)) * ... % Lead (gain of 1 at wc) + Kd_lpf /abs(evalfr(Kd_lpf, 1j*2*pi*100)); % Low Pass Filter +#+end_src + +#+begin_src matlab :exports none +%% MIMO Controller +Kd = -inv(V') * ... % Output decoupling + ss(Kd_diag) * ... + inv(U); % Input decoupling +#+end_src + +**** Loop Gain +#+begin_src matlab :exports none +%% Experimental Loop Gain +Lmimo = permute(pagemtimes(permute(G_nom, [2,3,1]),squeeze(freqresp(Kd, frf_iff.f, 'Hz'))), [3,1,2]); +#+end_src + +#+begin_src matlab :exports none +%% Loop gain when using SVD +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:6 + plot(frf_iff.f, abs(Lmimo(:,i,i)), '-'); +end +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(squeeze(Lmimo(:,i,j))), 'color', [0,0,0,0.2]); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Loop Gain'); set(gca, 'XTickLabel',[]); +ylim([1e-3, 1e+3]); + +ax2 = nexttile; +hold on; +for i = 1:6 + plot(frf_iff.f, 180/pi*angle(Lmimo(:,i,i)), '-'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:30:360); +ylim([-180, 0]); + +linkaxes([ax1,ax2],'x'); +xlim([1, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/bode_plot_hac_iff_loop_gain_svd.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:bode_plot_hac_iff_loop_gain_svd +#+caption: Bode plot of Loop Gain when using the SVD +#+RESULTS: +[[file:figs/bode_plot_hac_iff_loop_gain_svd.png]] + +**** Stability Verification +#+begin_src matlab +%% Compute the Eigenvalues of the loop gain +Ldet = zeros(3, 6, length(frf_iff.f)); + +for i = 1:3 + Lmimo = pagemtimes(permute(frf_iff.G_dL{i}, [2,3,1]),squeeze(freqresp(Kd, frf_iff.f, 'Hz'))); + for i_f = 2:length(frf_iff.f) + Ldet(i,:, i_f) = eig(squeeze(Lmimo(:,:,i_f))); + end +end +#+end_src + +#+begin_src matlab :exports none +%% Plot of the eigenvalues of L in the complex plane +figure; +hold on; +for i_mass = 2:3 + plot(real(squeeze(Ldet(i_mass, 1,:))), imag(squeeze(Ldet(i_mass, 1,:))), ... + '.', 'color', colors(i_mass+1, :), ... + 'DisplayName', sprintf('%i masses', i_mass)); + plot(real(squeeze(Ldet(i_mass, 1,:))), -imag(squeeze(Ldet(i_mass, 1,:))), ... + '.', 'color', colors(i_mass+1, :), ... + 'HandleVisibility', 'off'); + for i = 1:6 + plot(real(squeeze(Ldet(i_mass, i,:))), imag(squeeze(Ldet(i_mass, i,:))), ... + '.', 'color', colors(i_mass+1, :), ... + 'HandleVisibility', 'off'); + plot(real(squeeze(Ldet(i_mass, i,:))), -imag(squeeze(Ldet(i_mass, i,:))), ... + '.', 'color', colors(i_mass+1, :), ... + 'HandleVisibility', 'off'); + end +end +plot(-1, 0, 'kx', 'HandleVisibility', 'off'); +hold off; +set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin'); +xlabel('Real'); ylabel('Imag'); +legend('location', 'southeast'); +xlim([-3, 1]); ylim([-2, 2]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/loci_hac_iff_loop_gain_svd.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:loci_hac_iff_loop_gain_svd +#+caption: Locis of $L(j\omega)$ in the complex plane. +#+RESULTS: +[[file:figs/loci_hac_iff_loop_gain_svd.png]] + +**** Save for further analysis +#+begin_src matlab :exports none :tangle no +save('matlab/data_sim/Khac_iff_struts_svd.mat', 'Kd') +#+end_src + +#+begin_src matlab :eval no +save('data_sim/Khac_iff_struts_svd.mat', 'Kd') +#+end_src + +**** Measured Sensitivity Transfer Function +The sensitivity transfer function is estimated by adding a reference signal $R_x$ consisting of a low pass filtered white noise, and measuring the position error $E_x$ at the same time. + +The transfer function from $R_x$ to $E_x$ is the sensitivity transfer function. + +In order to identify the sensitivity transfer function for all directions, six reference signals are used, one for each direction. + +#+begin_src matlab :exports none +%% Tested directions +labels = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}; +#+end_src + +#+begin_src matlab :exports none +%% Load Identification Data +meas_hac_svd_3m = {}; + +for i = 1:6 + meas_hac_svd_3m(i) = {load(sprintf('T_S_meas_%s_3m_hac_svd_iff.mat', labels{i}), 't', 'Va', 'Vs', 'de', 'Rx')}; +end +#+end_src + +#+begin_src matlab :exports none +%% Setup useful variables +% Sampling Time [s] +Ts = (meas_hac_svd_3m{1}.t(end) - (meas_hac_svd_3m{1}.t(1)))/(length(meas_hac_svd_3m{1}.t)-1); + +% Sampling Frequency [Hz] +Fs = 1/Ts; + +% Hannning Windows +win = hanning(ceil(5*Fs)); + +% And we get the frequency vector +[~, f] = tfestimate(meas_hac_svd_3m{1}.Va, meas_hac_svd_3m{1}.de, win, [], [], 1/Ts); +#+end_src + +#+begin_src matlab :exports none +%% Load Jacobian matrix +load('jacobian.mat', 'J'); + +%% Compute position error +for i = 1:6 + meas_hac_svd_3m{i}.Xm = [inv(J)*meas_hac_svd_3m{i}.de']'; + meas_hac_svd_3m{i}.Ex = meas_hac_svd_3m{i}.Rx - meas_hac_svd_3m{i}.Xm; +end +#+end_src + +An example is shown in Figure ref:fig:ref_track_hac_svd_3m where both the reference signal and the measured position are shown for translations in the $x$ direction. + +#+begin_src matlab :exports none +figure; +hold on; +plot(meas_hac_svd_3m{1}.t, meas_hac_svd_3m{1}.Xm(:,1), 'DisplayName', 'Pos.') +plot(meas_hac_svd_3m{1}.t, meas_hac_svd_3m{1}.Rx(:,1), 'DisplayName', 'Ref.') +hold off; +xlabel('Time [s]'); ylabel('Dx motion [m]'); +xlim([20, 22]); +legend('location', 'northeast'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/ref_track_hac_svd_3m.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:ref_track_hac_svd_3m +#+caption: Reference position and measured position +#+RESULTS: +[[file:figs/ref_track_hac_svd_3m.png]] + +#+begin_src matlab :exports none +%% Transfer function estimate of S +S_hac_svd_3m = zeros(length(f), 6, 6); + +for i = 1:6 + S_hac_svd_3m(:,:,i) = tfestimate(meas_hac_svd_3m{i}.Rx, meas_hac_svd_3m{i}.Ex, win, [], [], 1/Ts); +end +#+end_src + +The sensitivity transfer functions estimated for all directions are shown in Figure ref:fig:sensitivity_hac_svd_3m. + +#+begin_src matlab :exports none +%% Bode plot for the transfer function from u to dLm +figure; +hold on; +for i =1:6 + plot(f, abs(S_hac_svd_3m(:,i,i)), ... + 'DisplayName', sprintf('$S_{%s}$', labels{i})); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Sensitivity [-]'); +ylim([1e-4, 1e1]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); +xlim([0.5, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/sensitivity_hac_svd_3m.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:sensitivity_hac_svd_3m +#+caption: Measured diagonal elements of the sensitivity transfer function matrix. +#+RESULTS: +[[file:figs/sensitivity_hac_svd_3m.png]] + +#+begin_important +From Figure ref:fig:sensitivity_hac_svd_3m: +- The sensitivity transfer functions are similar for all directions +- The disturbance attenuation at 1Hz is almost a factor 1000 as wanted +- The sensitivity transfer functions for $R_x$ and $R_y$ have high peak values which indicate poor stability margins. +#+end_important + +**** Sensitivity transfer function from the model +The sensitivity transfer function is now estimated using the model and compared with the one measured. + +#+begin_src matlab :exports none +%% Open Simulink Model +mdl = 'nano_hexapod_simscape'; + +options = linearizeOptions; +options.SampleTime = 0; + +open(mdl) + +Rx = zeros(1, 7); + +colors = colororder; +#+end_src + +#+begin_src matlab :exports none +%% Initialize the Simscape model in closed loop +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'plates', ... + 'actuator_type', '2dof', ... + 'controller_type', 'hac-iff-struts'); + +support.type = 1; % On top of vibration table +payload.type = 2; % Payload +#+end_src + +#+begin_src matlab :exports none +%% Load controllers +load('Kiff_opt.mat', 'Kiff'); +Kiff = c2d(Kiff, Ts, 'Tustin'); +load('Khac_iff_struts_svd.mat', 'Kd') +Khac_iff_struts = c2d(Kd, Ts, 'Tustin'); +#+end_src + +#+begin_src matlab :exports none +%% Identify the (damped) transfer function from u to dLm +clear io; io_i = 1; +io(io_i) = linio([mdl, '/Rx'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/dL'], 1, 'output'); io_i = io_i + 1; % Plate Displacement (encoder) +#+end_src + +#+begin_src matlab :exports none +%% Identification of the dynamics +Gcl = linearize(mdl, io, 0.0, options); +#+end_src + +#+begin_src matlab :exports none +%% Computation of the sensitivity transfer function +S = eye(6) - inv(n_hexapod.geometry.J)*Gcl; +#+end_src + +The results are shown in Figure ref:fig:sensitivity_hac_svd_3m_comp_model. +The model is quite effective in estimating the sensitivity transfer functions except around 60Hz were there is a peak for the measurement. + +#+begin_src matlab :exports none +%% Bode plot for the transfer function from u to dLm +freqs = logspace(0,3,1000); + +figure; +hold on; +for i =1:6 + set(gca,'ColorOrderIndex',i); + plot(f, abs(S_hac_svd_3m(:,i,i)), ... + 'DisplayName', sprintf('$S_{%s}$', labels{i})); + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(S(i,i), freqs, 'Hz'))), '--', ... + 'DisplayName', sprintf('$S_{%s}$ - Model', labels{i})); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Sensitivity [-]'); +ylim([1e-4, 1e1]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); +xlim([0.5, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/sensitivity_hac_svd_3m_comp_model.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:sensitivity_hac_svd_3m_comp_model +#+caption: Comparison of the measured sensitivity transfer functions with the model +#+RESULTS: +[[file:figs/sensitivity_hac_svd_3m_comp_model.png]] + +*** Using (diagonal) Dynamical Inverse :noexport: +**** Decoupled Plant +#+begin_src matlab +G_nom = frf_iff.G_dL{2}; % Nominal Plant +G_model = sim_iff.G_dL{2}; % Model of the Plant +#+end_src + +#+begin_src matlab :exports none +%% Simplified model of the diagonal term +balred_opts = balredOptions('FreqIntervals', 2*pi*[0, 1000], 'StateElimMethod', 'Truncate'); + +G_red = balred(G_model(1,1), 8, balred_opts); +G_red.outputdelay = 0; % necessary for further inversion +#+end_src + +#+begin_src matlab +%% Inverse +G_inv = inv(G_red); +[G_z, G_p, G_g] = zpkdata(G_inv); +p_uns = real(G_p{1}) > 0; +G_p{1}(p_uns) = -G_p{1}(p_uns); +G_inv_stable = zpk(G_z, G_p, G_g); +#+end_src + +#+begin_src matlab :exports none +%% "Uncertainty" of inversed plant +freqs = logspace(0,3,1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i_mass = i_masses + for i = 1 + plot(freqs, abs(squeeze(freqresp(G_inv_stable*sim_iff.G_dL{i_mass+1}(i,i), freqs, 'Hz'))), '-', 'color', colors(i_mass+1, :), ... + 'DisplayName', sprintf('$d\\mathcal{L}_i/u^\\prime_i$ - %i', i_mass)); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude'); set(gca, 'XTickLabel',[]); +ylim([1e-1, 1e1]); +legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 4); + +ax2 = nexttile; +hold on; +for i_mass = i_masses + for i = 1 + plot(freqs, 180/pi*angle(squeeze(freqresp(G_inv_stable*sim_iff.G_dL{i_mass+1}(1,1), freqs, 'Hz'))), '-', 'color', colors(i_mass+1, :)); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:15:360); +ylim([-45, 45]); + +linkaxes([ax1,ax2],'x'); +xlim([freqs(1), freqs(end)]); +#+end_src + +**** Controller Design +#+begin_src matlab :exports none +% Wanted crossover frequency +wc = 2*pi*80; +[~, i_wc] = min(abs(frf_iff.f - wc/2/pi)); + +%% Lead +a = 20.0; % Amount of phase lead / width of the phase lead / high frequency gain +Kd_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)))/sqrt(a); + +%% Integrator +Kd_int = ((wc)/(2*pi*0.2 + s))^2; + +%% Low Pass Filter (High frequency robustness) +w0_lpf = 2*wc; % Cut-off frequency [rad/s] +xi_lpf = 0.3; % Damping Ratio + +Kd_lpf = 1/(1 + 2*xi_lpf/w0_lpf*s + s^2/w0_lpf^2); + +w0_lpf_b = wc*4; % Cut-off frequency [rad/s] +xi_lpf_b = 0.7; % Damping Ratio + +Kd_lpf_b = 1/(1 + 2*xi_lpf_b/w0_lpf_b*s + s^2/w0_lpf_b^2); + +%% Normalize Gain +Kd_norm = diag(1./abs(diag(squeeze(G_dL_svd(i_wc,:,:))))); + +%% Diagonal Control +Kd_diag = ... + G_inv_stable * ... % Normalize gain at 100Hz + Kd_int /abs(evalfr(Kd_int, 1j*wc)) * ... % Integrator + Kd_lead/abs(evalfr(Kd_lead, 1j*wc)) * ... % Lead (gain of 1 at wc) + Kd_lpf /abs(evalfr(Kd_lpf, 1j*wc)); % Low Pass Filter +#+end_src + +#+begin_src matlab :exports none +Kd = ss(Kd_diag)*eye(6); +#+end_src + +**** Loop Gain +#+begin_src matlab :exports none +%% Experimental Loop Gain +Lmimo = permute(pagemtimes(permute(G_nom, [2,3,1]),squeeze(freqresp(Kd, frf_iff.f, 'Hz'))), [3,1,2]); +#+end_src + +#+begin_src matlab :exports none +%% Loop gain when using SVD +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:6 + plot(frf_iff.f, abs(Lmimo(:,i,i)), '-'); +end +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(squeeze(Lmimo(:,i,j))), 'color', [0,0,0,0.2]); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Loop Gain'); set(gca, 'XTickLabel',[]); +ylim([1e-3, 1e+3]); + +ax2 = nexttile; +hold on; +for i = 1:6 + plot(frf_iff.f, 180/pi*angle(Lmimo(:,i,i)), '-'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:30:360); +ylim([-180, 0]); + +linkaxes([ax1,ax2],'x'); +xlim([1, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/bode_plot_hac_iff_loop_gain_diag_inverse.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:bode_plot_hac_iff_loop_gain_diag_inverse +#+caption: Bode plot of Loop Gain when using the Diagonal inversion +#+RESULTS: +[[file:figs/bode_plot_hac_iff_loop_gain_diag_inverse.png]] + +**** Stability Verification +MIMO Nyquist with eigenvalues +#+begin_src matlab +%% Compute the Eigenvalues of the loop gain +Ldet = zeros(3, 6, length(frf_iff.f)); + +for i = 1:3 + Lmimo = pagemtimes(permute(frf_iff.G_dL{i}, [2,3,1]),squeeze(freqresp(Kd, frf_iff.f, 'Hz'))); + for i_f = 2:length(frf_iff.f) + Ldet(i,:, i_f) = eig(squeeze(Lmimo(:,:,i_f))); + end +end +#+end_src + +#+begin_src matlab :exports none +%% Plot of the eigenvalues of L in the complex plane +figure; +hold on; +for i_mass = 2:3 + plot(real(squeeze(Ldet(i_mass, 1,:))), imag(squeeze(Ldet(i_mass, 1,:))), ... + '.', 'color', colors(i_mass+1, :), ... + 'DisplayName', sprintf('%i masses', i_mass)); + plot(real(squeeze(Ldet(i_mass, 1,:))), -imag(squeeze(Ldet(i_mass, 1,:))), ... + '.', 'color', colors(i_mass+1, :), ... + 'HandleVisibility', 'off'); + for i = 1:6 + plot(real(squeeze(Ldet(i_mass, i,:))), imag(squeeze(Ldet(i_mass, i,:))), ... + '.', 'color', colors(i_mass+1, :), ... + 'HandleVisibility', 'off'); + plot(real(squeeze(Ldet(i_mass, i,:))), -imag(squeeze(Ldet(i_mass, i,:))), ... + '.', 'color', colors(i_mass+1, :), ... + 'HandleVisibility', 'off'); + end +end +plot(-1, 0, 'kx', 'HandleVisibility', 'off'); +hold off; +set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin'); +xlabel('Real'); ylabel('Imag'); +legend('location', 'southeast'); +xlim([-3, 1]); ylim([-2, 2]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/loci_hac_iff_loop_gain_diag_inverse.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:loci_hac_iff_loop_gain_diag_inverse +#+caption: Locis of $L(j\omega)$ in the complex plane. +#+RESULTS: +[[file:figs/loci_hac_iff_loop_gain_diag_inverse.png]] + +#+begin_important +Even though the loop gain seems to be fine, the closed-loop system is unstable. +This might be due to the fact that there is large interaction in the plant. +We could look at the RGA-number to verify that. +#+end_important + +**** Save for further use +#+begin_src matlab :exports none :tangle no +save('matlab/data_sim/Khac_iff_struts_diag_inverse.mat', 'Kd') +#+end_src + +#+begin_src matlab :eval no +save('data_sim/Khac_iff_struts_diag_inverse.mat', 'Kd') +#+end_src + +*** Closed Loop Stability (Model) :noexport: +Verify stability using Simscape model +#+begin_src matlab +%% Initialize the Simscape model in closed loop +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '2dof', ... + 'flex_top_type', '3dof', ... + 'motion_sensor_type', 'plates', ... + 'actuator_type', '2dof', ... + 'controller_type', 'hac-iff-struts'); +#+end_src + +#+begin_src matlab +%% IFF Controller +Kiff = -g_opt*Kiff_g1*eye(6); +Khac_iff_struts = Kd*eye(6); +#+end_src + +#+begin_src matlab +%% Identify the (damped) transfer function from u to dLm for different values of the IFF gain +clear io; io_i = 1; +io(io_i) = linio([mdl, '/du'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Plate Displacement (encoder) +#+end_src + +#+begin_src matlab +GG_cl = {}; + +for i = i_masses + payload.type = i; + GG_cl(i+1) = {exp(-s*Ts)*linearize(mdl, io, 0.0, options)}; +end +#+end_src + +#+begin_src matlab +for i = i_masses + isstable(GG_cl{i+1}) +end +#+end_src + +MIMO Nyquist +#+begin_src matlab +Kdm = Kd*eye(6); + +Ldet = zeros(3, length(fb(i_lim))); + +for i = 1:3 + Lmimo = pagemtimes(permute(G_damp_m{i}(i_lim,:,:), [2,3,1]),squeeze(freqresp(Kdm, fb(i_lim), 'Hz'))); + Ldet(i,:) = arrayfun(@(t) det(eye(6) + squeeze(Lmimo(:,:,t))), 1:size(Lmimo,3)); +end +#+end_src + +#+begin_src matlab :exports none +%% Bode plot for the transfer function from u to dLm +figure; +hold on; +for i_mass = 3 + for i = 1 + plot(real(Ldet(i_mass,:)), imag(Ldet(i_mass,:)), ... + '-', 'color', colors(i_mass+1, :)); + end +end +hold off; +set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin'); +xlabel('Real'); ylabel('Imag'); +xlim([-10, 1]); ylim([-4, 4]); +#+end_src + +MIMO Nyquist with eigenvalues +#+begin_src matlab +Kdm = Kd*eye(6); + +Ldet = zeros(3, 6, length(fb(i_lim))); + +for i = 1:3 + Lmimo = pagemtimes(permute(G_damp_m{i}(i_lim,:,:), [2,3,1]),squeeze(freqresp(Kdm, fb(i_lim), 'Hz'))); + for i_f = 1:length(fb(i_lim)) + Ldet(i,:, i_f) = eig(squeeze(Lmimo(:,:,i_f))); + end +end +#+end_src + +#+begin_src matlab :exports none +%% Bode plot for the transfer function from u to dLm +figure; +hold on; +for i_mass = 1 + for i = 1:6 + plot(real(squeeze(Ldet(i_mass, i,:))), imag(squeeze(Ldet(i_mass, i,:))), ... + '-', 'color', colors(i_mass+1, :)); + end +end +hold off; +set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin'); +xlabel('Real'); ylabel('Imag'); +xlim([-10, 1]); ylim([-4, 2]); +#+end_src +* Noise Budgeting :noexport: +** Introduction :ignore: + +Noise sources: +- PD200 => plant +- DAC => plant x 20 +- Encoder => direct output +- ADC (Force Sensor) => added when closing the loop (controller + plant) + +Disturbances Sources: +- Ground motion + +** Matlab Init :noexport:ignore: +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) +<> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes +<> +#+end_src + +#+begin_src matlab :tangle no :noweb yes +<> +#+end_src + +#+begin_src matlab :eval no :noweb yes +<> +#+end_src + +#+begin_src matlab :noweb yes +<> +#+end_src + +** Measurements +#+begin_src matlab +noise_enc = load('noise_meas_100s_20kHz.mat', 't', 'x'); +noise_enc.Ts = (noise_enc.t(end) - (noise_enc.t(1)))/(length(noise_enc.t)-1); +noise_enc.win = hanning(ceil(1/noise_enc.Ts)); +noise_enc.x = noise_enc.x - noise_enc.x(1); +[noise_enc.pxx, noise_enc.f] = pwelch(noise_enc.x, noise_enc.win, [], [], 1/noise_enc.Ts); +#+end_src + +#+begin_src matlab :exports none +noise_ol = load('noise_meas_2m_ol.mat', 't', 'Vs', 'de'); +noise_ol.Ts = (noise_ol.t(end) - (noise_ol.t(1)))/(length(noise_ol.t)-1); +noise_ol.win = hanning(ceil(1/noise_ol.Ts)); +[noise_ol.pxx, noise_ol.f] = pwelch(noise_ol.de(:,1), noise_ol.win, [], [], 1/noise_ol.Ts); +#+end_src + +#+begin_src matlab :exports none +noise_iff = load('noise_meas_2m_iff.mat', 't', 'de'); +noise_iff.Ts = (noise_iff.t(end) - (noise_iff.t(1)))/(length(noise_iff.t)-1); +noise_iff.win = hanning(ceil(1/noise_iff.Ts)); +[noise_iff.pxx, noise_iff.f] = pwelch(noise_iff.de(:,1), noise_iff.win, [], [], 1/noise_iff.Ts); +#+end_src + +#+begin_src matlab :exports none +figure; +hold on; +plot(noise_ol.f, sqrt(noise_ol.pxx), 'DisplayName', 'OL'); +plot(noise_iff.f, sqrt(noise_iff.pxx), 'DisplayName', 'IFF'); +plot(noise_enc.f, sqrt(noise_enc.pxx), 'DisplayName', 'Encoder'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('ASD [$m/\sqrt{Hz}$]'); +legend('location', 'northeast'); +xlim([1, Fs/2]); ylim([1e-11, 1e-7]); +#+end_src + + +* Feedforward Control :noexport: +<> + +** Introduction :ignore: + +#+begin_src latex :file control_architecture_iff_feedforward.pdf +\begin{tikzpicture} + % Blocs + \node[block={3.0cm}{3.0cm}] (P) {Plant}; + \coordinate[] (inputF) at ($(P.south west)!0.5!(P.north west)$); + \coordinate[] (outputF) at ($(P.south east)!0.8!(P.north east)$); + \coordinate[] (outputX) at ($(P.south east)!0.5!(P.north east)$); + \coordinate[] (outputL) at ($(P.south east)!0.2!(P.north east)$); + + \node[block, above=0.4 of P] (Kiff) {$\bm{K}_\text{IFF}$}; + \node[addb, left= of inputF] (addF) {}; + \node[block, left= of addF] (Kff) {$\bm{K}_{\mathcal{L},\text{ff}}$}; + \node[block, align=center, left= of Kff] (J) {Inverse\\Kinematics}; + + % Connections and labels + \draw[->] (outputF) -- ++(1, 0) node[below left]{$\bm{\tau}_m$}; + \draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east); + \draw[->] (Kiff.west) -| (addF.north); + \draw[->] (addF.east) -- (inputF) node[above left]{$\bm{u}$}; + + \draw[->] (outputL) -- ++(1, 0) node[above left]{$d\bm{\mathcal{L}}$}; + \draw[->] (outputX) -- ++(1, 0) node[above left]{$\bm{\mathcal{X}}$}; + + \draw[->] (Kff.east) -- (addF.west) node[above left]{$\bm{u}_{\text{ff}}$}; + \draw[->] (J.east) -- (Kff.west) node[above left]{$\bm{r}_{d\mathcal{L}}$}; + \draw[<-] (J.west)node[above left]{$\bm{r}_{\mathcal{X}}$} -- ++(-1, 0); +\end{tikzpicture} +#+end_src + +#+name: fig:control_architecture_iff_feedforward +#+caption: Feedforward control in the frame of the legs +#+RESULTS: +[[file:figs/control_architecture_iff_feedforward.png]] + +Main problems: +- Non-linearity: Creep, Hysteresis +- Variability of the plant + +** Matlab Init :noexport:ignore: +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) +<> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes +<> +#+end_src + +#+begin_src matlab :tangle no :noweb yes +<> +#+end_src + +#+begin_src matlab :eval no :noweb yes +<> +#+end_src + +#+begin_src matlab :noweb yes +<> +<> +#+end_src + +#+begin_src matlab +load('damped_plant_enc_plates.mat', 'f', 'Ts', 'G_enc_iff_opt') +#+end_src + +** Simple Feedforward Controller +Let's estimate the mean DC gain for the damped plant (diagonal elements:) +#+begin_src matlab :results value replace :exports results :tangle no +mean(diag(abs(squeeze(mean(G_enc_iff_opt(f>2 & f<4,:,:)))))) +#+end_src + +#+RESULTS: +: 1.773e-05 + +The feedforward controller is then taken as the inverse of this gain (the minus sign is there manually added as it is "removed" by the =abs= function): +#+begin_src matlab +Kff_iff_L = -1/mean(diag(abs(squeeze(mean(G_enc_iff_opt(f>2 & f<4,:,:)))))); +#+end_src + +The open-loop gain (feedforward controller times the damped plant) is shown in Figure ref:fig:open_loop_gain_feedforward_iff_struts. + +#+begin_src matlab :exports none +%% Bode plot of the transfer function from u to dLm for tested values of the IFF gain +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(f, abs(Kff_iff_L*G_enc_iff_opt(:,i,i)), 'k-'); +end + +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [-]'); set(gca, 'XTickLabel',[]); +ylim([1e-2, 1e1]); + +ax2 = nexttile; +hold on; +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(f, 180/pi*angle(Kff_iff_L*G_enc_iff_opt(:,i,i)), 'k-') +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); +ylim([-180, 180]); +yticks([-180, -90, 0, 90, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([1, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/open_loop_gain_feedforward_iff_struts.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:open_loop_gain_feedforward_iff_struts +#+caption: Diagonal elements of the "open loop gain" +#+RESULTS: +[[file:figs/open_loop_gain_feedforward_iff_struts.png]] + +And save the feedforward controller for further use: +#+begin_src matlab +Kff_iff_L = zpk(Kff_iff_L)*eye(6); +#+end_src + +#+begin_src matlab :tangle no +save('matlab/data_sim/feedforward_iff.mat', 'Kff_iff_L') +#+end_src + +#+begin_src matlab :exports none :eval no +save('data_sim/feedforward_iff.mat', 'Kff_iff_L') +#+end_src + +** Test with Simscape Model +#+begin_src matlab +load('reference_path.mat', 'Rx_yz'); +#+end_src + +** Feedback/Feedforward control in the frame of the struts +*** Introduction :ignore: + +#+begin_src latex :file control_architecture_hac_iff_L_feedforward.pdf +\begin{tikzpicture} + % Blocs + \node[block={3.0cm}{3.0cm}] (P) {Plant}; + \coordinate[] (inputF) at ($(P.south west)!0.5!(P.north west)$); + \coordinate[] (outputF) at ($(P.south east)!0.8!(P.north east)$); + \coordinate[] (outputX) at ($(P.south east)!0.5!(P.north east)$); + \coordinate[] (outputL) at ($(P.south east)!0.2!(P.north east)$); + + \node[block, above=0.4 of P] (Kiff) {$\bm{K}_\text{IFF}$}; + \node[addb, left= of inputF] (addF) {}; + \node[block, left= of addF] (K) {$\bm{K}_\mathcal{L}$}; + \node[block, above= of K] (Kff) {$\bm{K}_{\mathcal{L},\text{ff}}$}; + \node[addb, left= of K] (subr) {}; + \node[block, align=center, left= of subr] (J) {Inverse\\Kinematics}; + + % Connections and labels + \draw[->] (outputF) -- ++(1, 0) node[below left]{$\bm{\tau}_m$}; + \draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east); + \draw[->] (Kiff.west) -| (addF.north); + \draw[->] (addF.east) -- (inputF) node[above left]{$\bm{u}$}; + + \draw[->] (outputL) -- ++(1, 0) node[above left]{$d\bm{\mathcal{L}}$}; + \draw[->] ($(outputL) + (0.6, 0)$)node[branch]{} -- ++(0, -1) -| (subr.south); + \draw[->] (subr.east) -- (K.west) node[above left]{$\bm{\epsilon}_{d\mathcal{L}}$}; + \draw[->] (K.east) -- (addF.west) node[above left]{$\bm{u}^\prime$}; + + \draw[->] (outputX) -- ++(1, 0) node[above left]{$\bm{\mathcal{X}}_n$}; + + \draw[->] (J.east) -- (subr.west); + \draw[->] ($(J.east) + (0.4, 0)$)node[branch]{} node[below]{$\bm{r}_{d\mathcal{L}}$} |- (Kff.west); + \draw[->] (Kff.east) -- ++(0.5, 0) -- (addF.north west); + + \draw[<-] (J.west)node[above left]{$\bm{r}_{\mathcal{X}_n}$} -- ++(-1, 0); +\end{tikzpicture} +#+end_src + +#+name: fig:control_architecture_hac_iff_L_feedforward +#+caption: Feedback/Feedforward control in the frame of the legs +#+RESULTS: +[[file:figs/control_architecture_hac_iff_L_feedforward.png]] + + + +* Bibliography :ignore: +#+latex: \printbibliography[heading=bibintoc,title={Bibliography}] + +* Functions :noexport: +** =generateXYZTrajectory= +:PROPERTIES: +:header-args:matlab+: :tangle matlab/src/generateXYZTrajectory.m +:header-args:matlab+: :comments none :mkdirp yes :eval no +:END: +<> + +*** Function description :ignore: + +Function description: + +#+begin_src matlab -n +function [ref] = generateXYZTrajectory(args) +% generateXYZTrajectory - +% +% Syntax: [ref] = generateXYZTrajectory(args) +% +% Inputs: +% - args +% +% Outputs: +% - ref - Reference Signal +#+end_src + +*** Optional Parameters :ignore: + +Optional Parameters: + +#+begin_src matlab +n +arguments + args.points double {mustBeNumeric} = zeros(2, 3) % [m] + + args.ti (1,1) double {mustBeNumeric, mustBeNonnegative} = 1 % Time to go to first point and after last point [s] + args.tw (1,1) double {mustBeNumeric, mustBeNonnegative} = 0.5 % Time wait between each point [s] + args.tm (1,1) double {mustBeNumeric, mustBeNonnegative} = 1 % Motion time between points [s] + + args.Ts (1,1) double {mustBeNumeric, mustBePositive} = 1e-3 % Sampling Time [s] +end +#+end_src + +*** Initialize Time Vectors :ignore: + +Initialize Time Vectors: + +#+begin_src matlab +n +time_i = 0:args.Ts:args.ti; +time_w = 0:args.Ts:args.tw; +time_m = 0:args.Ts:args.tm; +#+end_src + +*** XYZ Trajectory :ignore: + +Generation of the XYZ Trajectory: + +#+begin_src matlab +n +% Go to initial position +xyz = (args.points(1,:))'*(time_i/args.ti); + +% Wait +xyz = [xyz, xyz(:,end).*ones(size(time_w))]; + +% Scans +for i = 2:size(args.points, 1) + % Go to next point + xyz = [xyz, xyz(:,end) + (args.points(i,:)' - xyz(:,end))*(time_m/args.tm)]; + % Wait a litle bit + xyz = [xyz, xyz(:,end).*ones(size(time_w))]; +end + +% End motion +xyz = [xyz, xyz(:,end) - xyz(:,end)*(time_i/args.ti)]; +#+end_src + +*** Reference Signal :ignore: + +Save the trajectory as a standard structure: + +#+begin_src matlab +n +t = 0:args.Ts:args.Ts*(length(xyz) - 1); + +ref = zeros(length(xyz), 7); + +ref(:, 1) = t; +ref(:, 2:4) = xyz'; +#+end_src + +** =generateYZScanTrajectory= +:PROPERTIES: +:header-args:matlab+: :tangle matlab/src/generateYZScanTrajectory.m +:header-args:matlab+: :comments none :mkdirp yes :eval no +:END: +<> + +*** Function description :ignore: + +Function description + +#+begin_src matlab +function [ref] = generateYZScanTrajectory(args) +% generateYZScanTrajectory - +% +% Syntax: [ref] = generateYZScanTrajectory(args) +% +% Inputs: +% - args +% +% Outputs: +% - ref - Reference Signal +#+end_src + +*** Optional Parameters :ignore: + +Optional Parameters + +#+begin_src matlab +arguments + args.y_tot (1,1) double {mustBeNumeric, mustBePositive} = 10e-6 % [m] + args.z_tot (1,1) double {mustBeNumeric, mustBePositive} = 10e-6 % [m] + + args.n (1,1) double {mustBeInteger, mustBePositive} = 10 % [-] + + args.Ts (1,1) double {mustBeNumeric, mustBePositive} = 1e-4 % [s] + + args.ti (1,1) double {mustBeNumeric, mustBeNonnegative} = 1 % [s] + args.tw (1,1) double {mustBeNumeric, mustBeNonnegative} = 1 % [s] + args.ty (1,1) double {mustBeNumeric, mustBeNonnegative} = 1 % [s] + args.tz (1,1) double {mustBeNumeric, mustBeNonnegative} = 1 % [s] +end +#+end_src + +*** Initialize Time Vectors :ignore: + +Initialize Time Vectors + +#+begin_src matlab +time_i = 0:args.Ts:args.ti; +time_w = 0:args.Ts:args.tw; +time_y = 0:args.Ts:args.ty; +time_z = 0:args.Ts:args.tz; +#+end_src + +*** Y and Z vectors :ignore: + +Y and Z vectors + +#+begin_src matlab +% Go to initial position +y = (time_i/args.ti)*(args.y_tot/2); + +% Wait +y = [y, y(end)*ones(size(time_w))]; + +% Scans +for i = 1:args.n + if mod(i,2) == 0 + y = [y, -(args.y_tot/2) + (time_y/args.ty)*args.y_tot]; + else + y = [y, (args.y_tot/2) - (time_y/args.ty)*args.y_tot]; + end + + if i < args.n + y = [y, y(end)*ones(size(time_z))]; + end +end + +% Wait a litle bit +y = [y, y(end)*ones(size(time_w))]; + +% End motion +y = [y, y(end) - y(end)*time_i/args.ti]; +#+end_src + +#+begin_src matlab +% Go to initial position +z = (time_i/args.ti)*(args.z_tot/2); + +% Wait +z = [z, z(end)*ones(size(time_w))]; + +% Scans +for i = 1:args.n + z = [z, z(end)*ones(size(time_y))]; + + if i < args.n + z = [z, z(end) - (time_z/args.tz)*args.z_tot/(args.n-1)]; + end +end + +% Wait a litle bit +z = [z, z(end)*ones(size(time_w))]; + +% End motion +z = [z, z(end) - z(end)*time_i/args.ti]; +#+end_src + +*** Reference Signal :ignore: + +Reference Signal + +#+begin_src matlab +t = 0:args.Ts:args.Ts*(length(y) - 1); + +ref = zeros(length(y), 7); + +ref(:, 1) = t; +ref(:, 3) = y; +ref(:, 4) = z; +#+end_src + +** =generateSpiralAngleTrajectory= +:PROPERTIES: +:header-args:matlab+: :tangle matlab/src/generateSpiralAngleTrajectory.m +:header-args:matlab+: :comments none :mkdirp yes :eval no +:END: +<> + +*** Function description :ignore: + +Function description + +#+begin_src matlab +function [ref] = generateSpiralAngleTrajectory(args) +% generateSpiralAngleTrajectory - +% +% Syntax: [ref] = generateSpiralAngleTrajectory(args) +% +% Inputs: +% - args +% +% Outputs: +% - ref - Reference Signal +#+end_src + +*** Optional Parameters :ignore: + +Optional Parameters + +#+begin_src matlab +arguments + args.R_tot (1,1) double {mustBeNumeric, mustBePositive} = 10e-6 % [rad] + args.n_turn (1,1) double {mustBeInteger, mustBePositive} = 5 % [-] + args.Ts (1,1) double {mustBeNumeric, mustBePositive} = 1e-3 % [s] + args.t_turn (1,1) double {mustBeNumeric, mustBePositive} = 1 % [s] + args.t_end (1,1) double {mustBeNumeric, mustBePositive} = 1 % [s] +end +#+end_src + +*** Initialize Time Vectors :ignore: + +Initialize Time Vectors + +#+begin_src matlab +time_s = 0:args.Ts:args.n_turn*args.t_turn; +time_e = 0:args.Ts:args.t_end; +#+end_src + +*** Rx and Ry vectors :ignore: + +Rx and Ry vectors + +#+begin_src matlab +Rx = sin(2*pi*time_s/args.t_turn).*(args.R_tot*time_s/(args.n_turn*args.t_turn)); +Ry = cos(2*pi*time_s/args.t_turn).*(args.R_tot*time_s/(args.n_turn*args.t_turn)); +#+end_src + +#+begin_src matlab +Rx = [Rx, 0*time_e]; +Ry = [Ry, Ry(end) - Ry(end)*time_e/args.t_end]; +#+end_src + +*** Reference Signal :ignore: + +Reference Signal + +#+begin_src matlab +t = 0:args.Ts:args.Ts*(length(Rx) - 1); + +ref = zeros(length(Rx), 7); + +ref(:, 1) = t; +ref(:, 5) = Rx; +ref(:, 6) = Ry; +#+end_src + +** =getTransformationMatrixAcc= +:PROPERTIES: +:header-args:matlab+: :tangle matlab/src/getTransformationMatrixAcc.m +:header-args:matlab+: :comments none :mkdirp yes :eval no +:END: +<> + +*** Function description :ignore: + +Function description: + +#+begin_src matlab +function [M] = getTransformationMatrixAcc(Opm, Osm) +% getTransformationMatrixAcc - +% +% Syntax: [M] = getTransformationMatrixAcc(Opm, Osm) +% +% Inputs: +% - Opm - Nx3 (N = number of accelerometer measurements) X,Y,Z position of accelerometers +% - Opm - Nx3 (N = number of accelerometer measurements) Unit vectors representing the accelerometer orientation +% +% Outputs: +% - M - Transformation Matrix +#+end_src + +*** Transformation matrix from motion of the solid body to accelerometer measurements :ignore: + +Let's try to estimate the x-y-z acceleration of any point of the solid body from the acceleration/angular acceleration of the solid body expressed in $\{O\}$. +For any point $p_i$ of the solid body (corresponding to an accelerometer), we can write: +\begin{equation} +\begin{bmatrix} +a_{i,x} \\ a_{i,y} \\ a_{i,z} +\end{bmatrix} = \begin{bmatrix} +\dot{v}_x \\ \dot{v}_y \\ \dot{v}_z +\end{bmatrix} + p_i \times \begin{bmatrix} +\dot{\omega}_x \\ \dot{\omega}_y \\ \dot{\omega}_z +\end{bmatrix} +\end{equation} + +We can write the cross product as a matrix product using the skew-symmetric transformation: +\begin{equation} +\begin{bmatrix} +a_{i,x} \\ a_{i,y} \\ a_{i,z} +\end{bmatrix} = \begin{bmatrix} +\dot{v}_x \\ \dot{v}_y \\ \dot{v}_z +\end{bmatrix} + \underbrace{\begin{bmatrix} + 0 & p_{i,z} & -p_{i,y} \\ + -p_{i,z} & 0 & p_{i,x} \\ + p_{i,y} & -p_{i,x} & 0 +\end{bmatrix}}_{P_{i,[\times]}} \cdot \begin{bmatrix} +\dot{\omega}_x \\ \dot{\omega}_y \\ \dot{\omega}_z +\end{bmatrix} +\end{equation} + +If we now want to know the (scalar) acceleration $a_i$ of the point $p_i$ in the direction of the accelerometer direction $\hat{s}_i$, we can just project the 3d acceleration on $\hat{s}_i$: +\begin{equation} +a_i = \hat{s}_i^T \cdot \begin{bmatrix} +a_{i,x} \\ a_{i,y} \\ a_{i,z} +\end{bmatrix} = \hat{s}_i^T \cdot \begin{bmatrix} +\dot{v}_x \\ \dot{v}_y \\ \dot{v}_z +\end{bmatrix} + \left( \hat{s}_i^T \cdot P_{i,[\times]} \right) \cdot \begin{bmatrix} +\dot{\omega}_x \\ \dot{\omega}_y \\ \dot{\omega}_z +\end{bmatrix} +\end{equation} + +Which is equivalent as a simple vector multiplication: +\begin{equation} +a_i = \begin{bmatrix} +\hat{s}_i^T & \hat{s}_i^T \cdot P_{i,[\times]} +\end{bmatrix} +\begin{bmatrix} +\dot{v}_x \\ \dot{v}_y \\ \dot{v}_z \\ \dot{\omega}_x \\ \dot{\omega}_y \\ \dot{\omega}_z +\end{bmatrix} = \begin{bmatrix} +\hat{s}_i^T & \hat{s}_i^T \cdot P_{i,[\times]} +\end{bmatrix} {}^O\vec{x} +\end{equation} + +And finally we can combine the 6 (line) vectors for the 6 accelerometers to write that in a matrix form. +We obtain Eq. eqref:eq:M_matrix. +#+begin_important +The transformation from solid body acceleration ${}^O\vec{x}$ from sensor measured acceleration $\vec{a}$ is: +\begin{equation} \label{eq:M_matrix} +\vec{a} = \underbrace{\begin{bmatrix} +\hat{s}_1^T & \hat{s}_1^T \cdot P_{1,[\times]} \\ +\vdots & \vdots \\ +\hat{s}_6^T & \hat{s}_6^T \cdot P_{6,[\times]} +\end{bmatrix}}_{M} {}^O\vec{x} +\end{equation} + +with $\hat{s}_i$ the unit vector representing the measured direction of the i'th accelerometer expressed in frame $\{O\}$ and $P_{i,[\times]}$ the skew-symmetric matrix representing the cross product of the position of the i'th accelerometer expressed in frame $\{O\}$. +#+end_important + +Let's define such matrix using matlab: +#+begin_src matlab +M = zeros(length(Opm), 6); + +for i = 1:length(Opm) + Ri = [0, Opm(3,i), -Opm(2,i); + -Opm(3,i), 0, Opm(1,i); + Opm(2,i), -Opm(1,i), 0]; + M(i, 1:3) = Osm(:,i)'; + M(i, 4:6) = Osm(:,i)'*Ri; +end +#+end_src + +#+begin_src matlab +end +#+end_src + +* Helping Functions :noexport: +** Initialize Path +#+NAME: m-init-path +#+BEGIN_SRC matlab +%% Path for functions, data and scripts +addpath('./matlab/data_frf/'); % Path for Computed FRF +addpath('./matlab/data_sim/'); % Path for Simulation +addpath('./matlab/data_meas/'); % Path for Measurements +addpath('./matlab/src/'); % Path for functions +addpath('./matlab/'); % Path for scripts + +%% Simscape Model - Nano Hexapod +addpath('./matlab/nass-simscape/matlab/nano_hexapod/') +addpath('./matlab/nass-simscape/STEPS/nano_hexapod/') +addpath('./matlab/nass-simscape/STEPS/png/') +addpath('./matlab/nass-simscape/src/') +addpath('./matlab/nass-simscape/mat/') + +%% Simscape Model - Vibration Table +addpath('./matlab/vibration-table/matlab/') +addpath('./matlab/vibration-table/STEPS/') +#+END_SRC + +#+NAME: m-init-path-tangle +#+BEGIN_SRC matlab +%% Path for functions, data and scripts +addpath('./data_frf/'); % Path for Computed FRF +addpath('./data_sim/'); % Path for Simulation +addpath('./data_meas/'); % Path for Measurements +addpath('./src/'); % Path for functions + +%% Simscape Model - Nano Hexapod +addpath('./nass-simscape/matlab/nano_hexapod/') +addpath('./nass-simscape/STEPS/nano_hexapod/') +addpath('./nass-simscape/STEPS/png/') +addpath('./nass-simscape/src/') +addpath('./nass-simscape/mat/') + +%% Simscape Model - Vibration Table +addpath('./vibration-table/matlab/') +addpath('./vibration-table/STEPS/') +#+END_SRC + +** Initialize Simscape Model +#+NAME: m-init-simscape +#+begin_src matlab +%% Initialize Parameters for Simscape model +support.type = 1; % On top of vibration table +payload.type = 0; % No Payload +Rx = zeros(1, 7); % Default reference path + +%% Open Simulink Model +mdl = 'nano_hexapod_simscape'; + +options = linearizeOptions; +options.SampleTime = 0; + +open(mdl) +#+end_src + +** Initialize other elements +#+NAME: m-init-other +#+BEGIN_SRC matlab +%% Colors for the figures +colors = colororder; + +%% Tested Masses +i_masses = 0:3; + +%% Frequency Vector +freqs = 2*logspace(1, 3, 1000); +#+END_SRC diff --git a/test-bench-nano-hexapod.pdf b/test-bench-nano-hexapod.pdf new file mode 100644 index 0000000..6328521 Binary files /dev/null and b/test-bench-nano-hexapod.pdf differ