Update Matlab code

.gitignore

@@ -1,37 +1,260 @@
-auto/
-.auctex-auto/
-*.tex
-
-nohup.out
-
-**/figs/*.svg
-**/figs/*.tex
-
-# 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
+mat/
+figures/
+ltximg/
 slprj/
-sccprj/
-
-# Simulink autosave extension
-*.autosave
-
-# Octave session info
-octave-workspace
-
-# Simulink Cache
+matlab/slprj/
 *.slxc
+
+
+# ============================================================
+# ============================================================
+# LATEX
+# ============================================================
+# ============================================================
+
+## Core latex/pdflatex auxiliary files:
+*.aux
+*.lof
+*.log
+*.lot
+*.fls
+*.out
+*.toc
+*.fmt
+*.fot
+*.cb
+*.cb2
+.*.lb
+
+## Intermediate documents:
+*.dvi
+*.xdv
+*-converted-to.*
+# these rules might exclude image files for figures etc.
+# *.ps
+# *.eps
+# *.pdf
+
+## Generated if empty string is given at "Please type another file name for output:"
+.pdf
+
+## Bibliography auxiliary files (bibtex/biblatex/biber):
+*.bbl
+*.bcf
+*.blg
+*-blx.aux
+*-blx.bib
+*.run.xml
+
+## Build tool auxiliary files:
+*.fdb_latexmk
+*.synctex
+*.synctex(busy)
+*.synctex.gz
+*.synctex.gz(busy)
+*.pdfsync
+
+## Build tool directories for auxiliary files
+# latexrun
+latex.out/
+
+## Auxiliary and intermediate files from other packages:
+# algorithms
+*.alg
+*.loa
+
+# achemso
+acs-*.bib
+
+# amsthm
+*.thm
+
+# beamer
+*.nav
+*.pre
+*.snm
+*.vrb
+
+# changes
+*.soc
+
+# cprotect
+*.cpt
+
+# elsarticle (documentclass of Elsevier journals)
+*.spl
+
+# endnotes
+*.ent
+
+# fixme
+*.lox
+
+# feynmf/feynmp
+*.mf
+*.mp
+*.t[1-9]
+*.t[1-9][0-9]
+*.tfm
+
+#(r)(e)ledmac/(r)(e)ledpar
+*.end
+*.?end
+*.[1-9]
+*.[1-9][0-9]
+*.[1-9][0-9][0-9]
+*.[1-9]R
+*.[1-9][0-9]R
+*.[1-9][0-9][0-9]R
+*.eledsec[1-9]
+*.eledsec[1-9]R
+*.eledsec[1-9][0-9]
+*.eledsec[1-9][0-9]R
+*.eledsec[1-9][0-9][0-9]
+*.eledsec[1-9][0-9][0-9]R
+
+# glossaries
+*.acn
+*.acr
+*.glg
+*.glo
+*.gls
+*.glsdefs
+
+# gnuplottex
+*-gnuplottex-*
+
+# gregoriotex
+*.gaux
+*.gtex
+
+# htlatex
+*.4ct
+*.4tc
+*.idv
+*.lg
+*.trc
+*.xref
+
+# hyperref
+*.brf
+
+# knitr
+*-concordance.tex
+# TODO Comment the next line if you want to keep your tikz graphics files
+*.tikz
+*-tikzDictionary
+
+# listings
+*.lol
+
+# makeidx
+*.idx
+*.ilg
+*.ind
+*.ist
+
+# minitoc
+*.maf
+*.mlf
+*.mlt
+*.mtc[0-9]*
+*.slf[0-9]*
+*.slt[0-9]*
+*.stc[0-9]*
+
+# minted
+_minted*
+*.pyg
+
+# morewrites
+*.mw
+
+# nomencl
+*.nlg
+*.nlo
+*.nls
+
+# pax
+*.pax
+
+# pdfpcnotes
+*.pdfpc
+
+# sagetex
+*.sagetex.sage
+*.sagetex.py
+*.sagetex.scmd
+
+# scrwfile
+*.wrt
+
+# sympy
+*.sout
+*.sympy
+sympy-plots-for-*.tex/
+
+# pdfcomment
+*.upa
+*.upb
+
+# pythontex
+*.pytxcode
+pythontex-files-*/
+
+# thmtools
+*.loe
+
+# TikZ & PGF
+*.dpth
+*.md5
+*.auxlock
+
+# todonotes
+*.tdo
+
+# easy-todo
+*.lod
+
+# xmpincl
+*.xmpi
+
+# xindy
+*.xdy
+
+# xypic precompiled matrices
+*.xyc
+
+# endfloat
+*.ttt
+*.fff
+
+# Latexian
+TSWLatexianTemp*
+
+## Editors:
+# WinEdt
+*.bak
+*.sav
+
+# Texpad
+.texpadtmp
+
+# LyX
+*.lyx~
+
+# Kile
+*.backup
+
+# KBibTeX
+*~[0-9]*
+
+# auto folder when using emacs and auctex
+./auto/*
+*.el
+
+# expex forward references with \gathertags
+*-tags.tex
+
+# standalone packages
+*.sta

nass-fem.bib

@@ -0,0 +1,26 @@
+@book{fleming14_desig_model_contr_nanop_system,
+  author          = {Andrew J. Fleming and Kam K. Leang},
+  title           = {Design, Modeling and Control of Nanopositioning Systems},
+  year            = 2014,
+  publisher       = {Springer International Publishing},
+  url             = {https://doi.org/10.1007/978-3-319-06617-2},
+  doi             = {10.1007/978-3-319-06617-2},
+  series          = {Advances in Industrial Control},
+}
+
+
+
+@article{souleille18_concep_activ_mount_space_applic,
+  author          = {Souleille, Adrien and Lampert, Thibault and 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},
+  keywords        = {parallel robot, iff},
+}
+

nass-fem.org

@@ -1,7 +1,5 @@
 #+TITLE: NASS - Finite Element Models with Simscape
 :DRAWER:
-#+STARTUP: overview
-
 #+LANGUAGE: en
 #+EMAIL: dehaeze.thomas@gmail.com
 #+AUTHOR: Dehaeze Thomas
@@ -12,28 +10,95 @@
 #+HTML_HEAD: <link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
 #+HTML_HEAD: <script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>
 
+#+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{nass-fem.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+ :exports both
 #+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:matlab+ :mkdirp yes
-
-#+PROPERTY: header-args:shell  :eval no-export
 
 #+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 raw replace :buffer no
+#+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
+  <hr>
+  <p>This report is also available as a <a href="./nass-fem.pdf">pdf</a>.</p>
+  <hr>
+#+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:
+
+- [ ] Convert this into a standard report
+- [ ] look at the [[file:~/Cloud/research/papers/published/brumund21_multib_simul_reduc_order_flexib_bodies_fea/paper/brumund21_multib_simul_reduc_order_flexib_bodies_fea.pdf][published paper]]
+
 * Introduction                                                        :ignore:
 
 In this document, Finite Element Models (FEM) of parts of the Nano-Hexapod are developed and integrated into Simscape for dynamical analysis.
@@ -64,6 +129,7 @@ The remote point used are also shown in this figure.
 
 #+name: fig:apa300ml_ansys
 #+caption: Ansys FEM of the APA300ML
+#+attr_latex: :width 0.7\linewidth
 [[file:figs/apa300ml_ansys.jpg]]
 
 ** Matlab Init                                             :noexport:ignore:
@@ -85,7 +151,7 @@ The remote point used are also shown in this figure.
 #+end_src
 
 #+begin_src matlab
-  open('APA300ML.slx');
+% open('APA300ML.slx');
 #+end_src
 
 ** Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
@@ -99,25 +165,31 @@ We first extract the stiffness and mass matrices.
 data2orgtable(K(1:10, 1:10), {}, {}, ' %.1g ');
 #+end_src
 
+#+name: tab:APA300ML_mat_K
 #+caption: First 10x10 elements of the Stiffness matrix
+#+attr_latex: :environment tabularx :width \linewidth :align XXXXXXXXXX
+#+attr_latex: :center t :booktabs t :font \tiny
 #+RESULTS:
-| 200000000.0 |    30000.0 |  -20000.0 |     -70.0 | 300000.0 |    40.0 |  10000000.0 |    10000.0 |   -6000.0 |     30.0 |
-|     30000.0 | 30000000.0 |    2000.0 | -200000.0 |     60.0 |   -10.0 |      4000.0 |  2000000.0 |    -500.0 |   9000.0 |
-|    -20000.0 |     2000.0 | 7000000.0 |     -10.0 |    -30.0 |    10.0 |      6000.0 |      900.0 | -500000.0 |        3 |
-|       -70.0 |  -200000.0 |     -10.0 |    1000.0 |     -0.1 |    0.08 |       -20.0 |    -9000.0 |         3 |    -30.0 |
-|    300000.0 |       60.0 |     -30.0 |      -0.1 |    900.0 |     0.1 |     30000.0 |       20.0 |     -10.0 |     0.06 |
-|        40.0 |      -10.0 |      10.0 |      0.08 |      0.1 | 10000.0 |        20.0 |          9 |        -5 |     0.03 |
-|  10000000.0 |     4000.0 |    6000.0 |     -20.0 |  30000.0 |    20.0 | 200000000.0 |    10000.0 |    9000.0 |     50.0 |
-|     10000.0 |  2000000.0 |     900.0 |   -9000.0 |     20.0 |       9 |     10000.0 | 30000000.0 |    -500.0 | 200000.0 |
-|     -6000.0 |     -500.0 | -500000.0 |         3 |    -10.0 |      -5 |      9000.0 |     -500.0 | 7000000.0 |       -2 |
-|        30.0 |     9000.0 |         3 |     -30.0 |     0.06 |    0.03 |        50.0 |   200000.0 |        -2 |   1000.0 |
+| 200000000 |    30000 |  -20000 |     -70 | 300000 |    40 |  10000000 |    10000 |   -6000 |     30 |
+|     30000 | 30000000 |    2000 | -200000 |     60 |   -10 |      4000 |  2000000 |    -500 |   9000 |
+|    -20000 |     2000 | 7000000 |     -10 |    -30 |    10 |      6000 |      900 | -500000 |      3 |
+|       -70 |  -200000 |     -10 |    1000 |   -0.1 |    08 |       -20 |    -9000 |       3 |    -30 |
+|    300000 |       60 |     -30 |    -0.1 |    900 |   0.1 |     30000 |       20 |     -10 |     06 |
+|        40 |      -10 |      10 |      08 |    0.1 | 10000 |        20 |        9 |      -5 |     03 |
+|  10000000 |     4000 |    6000 |     -20 |  30000 |    20 | 200000000 |    10000 |    9000 |     50 |
+|     10000 |  2000000 |     900 |   -9000 |     20 |     9 |     10000 | 30000000 |    -500 | 200000 |
+|     -6000 |     -500 | -500000 |       3 |    -10 |    -5 |      9000 |     -500 | 7000000 |     -2 |
+|        30 |     9000 |       3 |     -30 |     06 |    03 |        50 |   200000 |      -2 |   1000 |
 
 
 #+begin_src matlab :exports results :results value table replace :tangle no
 data2orgtable(M(1:10, 1:10), {}, {}, ' %.1g ');
 #+end_src
 
+#+name: tab:APA300ML_mat_M
 #+caption: First 10x10 elements of the Mass matrix
+#+attr_latex: :environment tabularx :width \linewidth :align XXXXXXXXXX
+#+attr_latex: :center t :booktabs t :font \tiny
 #+RESULTS:
 |    0.01 |  -2e-06 |  1e-06 |  6e-09 |  5e-05 | -5e-09 | -0.0005 |  -7e-07 |  6e-07 | -3e-09 |
 |  -2e-06 |    0.01 |  8e-07 | -2e-05 | -8e-09 |  2e-09 |  -9e-07 | -0.0002 |  1e-08 | -9e-07 |
@@ -140,23 +212,29 @@ Then, we extract the coordinates of the interface nodes.
   data2orgtable([[1:length(int_i)]', int_i, int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
 #+end_src
 
+#+name: tab:coordinate_nodes
 #+caption: Coordinates of the interface nodes
+#+attr_latex: :environment tabularx :width 0.6\linewidth :align ccccc
+#+attr_latex: :center t :booktabs t
 #+RESULTS:
 | Node i | Node Number |   x [m] | y [m] |  z [m] |
 |--------+-------------+---------+-------+--------|
-|    1.0 |    697783.0 |     0.0 |   0.0 | -0.015 |
-|    2.0 |    697784.0 |     0.0 |   0.0 |  0.015 |
-|    3.0 |    697785.0 | -0.0325 |   0.0 |    0.0 |
-|    4.0 |    697786.0 | -0.0125 |   0.0 |    0.0 |
-|    5.0 |    697787.0 | -0.0075 |   0.0 |    0.0 |
-|    6.0 |    697788.0 |  0.0125 |   0.0 |    0.0 |
-|    7.0 |    697789.0 |  0.0325 |   0.0 |    0.0 |
+|    1.0 |      697783 |     0.0 |   0.0 | -0.015 |
+|    2.0 |      697784 |     0.0 |   0.0 |  0.015 |
+|    3.0 |      697785 | -0.0325 |   0.0 |    0.0 |
+|    4.0 |      697786 | -0.0125 |   0.0 |    0.0 |
+|    5.0 |      697787 | -0.0075 |   0.0 |    0.0 |
+|    6.0 |      697788 |  0.0125 |   0.0 |    0.0 |
+|    7.0 |      697789 |  0.0325 |   0.0 |    0.0 |
 
 #+begin_src matlab :exports results :results value table replace :tangle no
   data2orgtable([length(n_i); length(int_i); size(M,1) - 6*length(int_i); size(M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f ');
 #+end_src
 
+#+name: tab:parameters_fem
 #+caption: Some extracted parameters of the FEM
+#+attr_latex: :environment tabularx :width 0.4\linewidth :align lc
+#+attr_latex: :center t :booktabs t
 #+RESULTS:
 | Total number of Nodes     |   7 |
 | Number of interface Nodes |   7 |
@@ -206,7 +284,7 @@ If we take the numerical values, we obtain:
 #+RESULTS:
 : 5.64
 
-From cite:fleming14_desig_model_contr_nanop_system (page 123), the relation between relative displacement of the sensor stack and generated voltage is:
+From [[cite:&fleming14_desig_model_contr_nanop_system]] (page 123), the relation between relative displacement of the sensor stack and generated voltage is:
 #+name: eq:piezo_strain_to_voltage
 \begin{equation}
   V_s = \frac{d_{33}}{\epsilon^T s^D n} \Delta h
@@ -561,9 +639,13 @@ The parameters are shown in the table below.
 
 #+name: fig:souleille18_model_piezo
 #+caption: Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator
+#+attr_latex: :width 0.7\linewidth
 [[file:./figs/souleille18_model_piezo.png]]
 
+#+name: tab:parameters_apa100m
 #+caption:Parameters used for the model of the APA 100M
+#+attr_latex: :environment tabularx :width 0.8\linewidth :align cl
+#+attr_latex: :center t :booktabs t
 |       | Meaning                                                        |
 |-------+----------------------------------------------------------------|
 | $k_e$ | Stiffness used to adjust the pole of the isolator              |
@@ -622,9 +704,13 @@ Then $k_e$ and $k_1$ are computed.
 #+end_src
 
 #+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
-  data2orgtable(1e-6*[ka; ke; k1], {'ka', 'ke', 'k1'}, {'Value [N/um]'}, ' %.1f ');
+  data2orgtable(1e-6*[ka; ke; k1], {'=ka=', '=ke=', '=k1='}, {'Value [N/um]'}, ' %.1f ');
 #+end_src
 
+#+name: tab:stiffnesses_apa100m
+#+caption:Obtained stiffnesses of the APA 100M
+#+attr_latex: :environment tabularx :width 0.3\linewidth :align cl
+#+attr_latex: :center t :booktabs t
 #+RESULTS:
 |    | Value [N/um] |
 |----+--------------|
@@ -729,6 +815,7 @@ The dynamics of the FEM model and the simpler model are compared in Figure [[fig
 
 #+name: fig:apa300ml_comp_simpler_model
 #+caption: Comparison of the Dynamics between the FEM model and the simplified one
+#+attr_latex: :width 1.0\linewidth
 #+RESULTS:
 [[file:figs/apa300ml_comp_simpler_model.png]]
 
@@ -1025,6 +1112,7 @@ It is the expected behavior as shown in the Figure [[fig:souleille18_results]] (
 
 #+name: fig:souleille18_results
 #+caption: Results obtained in cite:souleille18_concep_activ_mount_space_applic
+#+attr_latex: :width 1.0\linewidth
 [[file:figs/souleille18_results.png]]
 
 
@@ -1043,6 +1131,7 @@ A simplified model of the flexor is then developped.
 
 #+name: fig:flexor_id16_screenshot
 #+caption: Flexor studied
+#+attr_latex: :width 0.3\linewidth
 [[file:figs/flexor_id16_screenshot.png]]
 
 ** Matlab Init                                             :noexport:ignore:
@@ -1078,24 +1167,29 @@ We first extract the stiffness and mass matrices.
   data2orgtable(K(1:10, 1:10), {}, {}, ' %.2e ');
 #+end_src
 
+#+name: tab:APA300ML_mat_K_2MDoF
 #+caption: First 10x10 elements of the Stiffness matrix
+#+attr_latex: :environment tabularx :width \linewidth :align XXXXXXXXXX
+#+attr_latex: :center t :booktabs t :font \tiny
 #+RESULTS:
-|  11200000.0 |       195.0 |       2220.0 |    -0.719 |   -265.0 |      1.59 | -11200000.0 |      -213.0 |      -2220.0 |    0.147 |
-|       195.0 |  11400000.0 |       1290.0 |    -148.0 |   -0.188 |      2.41 |      -212.0 | -11400000.0 |      -1290.0 |    148.0 |
-|      2220.0 |      1290.0 |  119000000.0 |      1.31 |     1.49 |      1.79 |     -2220.0 |     -1290.0 | -119000000.0 |    -1.31 |
-|      -0.719 |      -148.0 |         1.31 |      33.0 | 0.000488 | -0.000977 |       0.141 |       148.0 |        -1.31 |    -33.0 |
-|      -265.0 |      -0.188 |         1.49 |  0.000488 |     33.0 |   0.00293 |       266.0 |       0.154 |        -1.49 |  0.00026 |
-|        1.59 |        2.41 |         1.79 | -0.000977 |  0.00293 |     236.0 |       -1.32 |       -2.55 |        -1.79 | 0.000379 |
-| -11200000.0 |      -212.0 |      -2220.0 |     0.141 |    266.0 |     -1.32 |  11400000.0 |     24600.0 |       1640.0 |    120.0 |
-|      -213.0 | -11400000.0 |      -1290.0 |     148.0 |    0.154 |     -2.55 |     24600.0 |  11400000.0 |       1290.0 |    -72.0 |
-|     -2220.0 |     -1290.0 | -119000000.0 |     -1.31 |    -1.49 |     -1.79 |      1640.0 |      1290.0 |  119000000.0 |     1.32 |
-|       0.147 |       148.0 |        -1.31 |     -33.0 |  0.00026 |  0.000379 |       120.0 |       -72.0 |         1.32 |     34.7 |
+|  11200000 |       195 |       2220 |  -0.719 |   -265 |    1.59 | -11200000 |     -213 |      -2220 |  0.147 |   | 195 | 11400000 | 1290 | -148 | -0.188 | 2.41 | -212 | -11400000 | -1290 | 148 |
+|      2220 |      1290 |  119000000 |    1.31 |   1.49 |    1.79 |     -2220 |    -1290 | -119000000 |  -1.31 |   |     |          |      |      |        |      |      |           |       |     |
+|    -0.719 |      -148 |       1.31 |      33 | 000488 | -000977 |     0.141 |      148 |      -1.31 |    -33 |   |     |          |      |      |        |      |      |           |       |     |
+|      -265 |    -0.188 |       1.49 |  000488 |     33 |   00293 |       266 |    0.154 |      -1.49 |  00026 |   |     |          |      |      |        |      |      |           |       |     |
+|      1.59 |      2.41 |       1.79 | -000977 |  00293 |     236 |     -1.32 |    -2.55 |      -1.79 | 000379 |   |     |          |      |      |        |      |      |           |       |     |
+| -11200000 |      -212 |      -2220 |   0.141 |    266 |   -1.32 |  11400000 |    24600 |       1640 |    120 |   |     |          |      |      |        |      |      |           |       |     |
+|      -213 | -11400000 |      -1290 |     148 |  0.154 |   -2.55 |     24600 | 11400000 |       1290 |    -72 |   |     |          |      |      |        |      |      |           |       |     |
+|     -2220 |     -1290 | -119000000 |   -1.31 |  -1.49 |   -1.79 |      1640 |     1290 |  119000000 |   1.32 |   |     |          |      |      |        |      |      |           |       |     |
+|     0.147 |       148 |      -1.31 |     -33 |  00026 |  000379 |       120 |      -72 |       1.32 |   34.7 |   |     |          |      |      |        |      |      |           |       |     |
 
 #+begin_src matlab :exports results :results value table replace :tangle no
   data2orgtable(M(1:10, 1:10), {}, {}, ' %.1g ');
 #+end_src
 
+#+name: tab:APA300ML_mat_M_2MDoF
 #+caption: First 10x10 elements of the Mass matrix
+#+attr_latex: :environment tabularx :width \linewidth :align XXXXXXXXXX
+#+attr_latex: :center t :booktabs t :font \tiny
 #+RESULTS:
 |   0.02 |   1e-09 | -4e-08 |  -1e-10 | 0.0002 | -3e-11 |  0.004 |  5e-08 |  7e-08 |  1e-10 |
 |  1e-09 |    0.02 | -3e-07 | -0.0002 | -1e-10 | -2e-09 |  2e-08 |  0.004 |  3e-07 |  1e-05 |
@@ -1108,6 +1202,20 @@ We first extract the stiffness and mass matrices.
 |  7e-08 |   3e-07 |  0.003 |  -8e-10 |  2e-09 | -1e-09 | -3e-07 | -4e-08 |   0.02 | -1e-09 |
 |  1e-10 |   1e-05 |  1e-09 |  -1e-09 | -3e-13 |  2e-12 | -2e-10 | 0.0002 | -1e-09 |  2e-06 |
 
+#+begin_src matlab :exports results :results value table replace :tangle no
+  data2orgtable([length(n_i); length(int_i); size(M,1) - 6*length(int_i); size(M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f ');
+#+end_src
+
+#+name: tab:parameters_fem_2MDoF
+#+caption: Some extracted parameters of the FEM
+#+attr_latex: :environment tabularx :width 0.4\linewidth :align lc
+#+attr_latex: :center t :booktabs t
+#+RESULTS:
+| Total number of Nodes     |  2 |
+| Number of interface Nodes |  2 |
+| Number of Modes           |  6 |
+| Size of M and K matrices  | 18 |
+
 Then, we extract the coordinates of the interface nodes.
 #+begin_src matlab
   [int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('out_nodes_3D.txt');
@@ -1117,23 +1225,16 @@ Then, we extract the coordinates of the interface nodes.
   data2orgtable([[1:length(int_i)]', int_i, int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
 #+end_src
 
+#+name: tab:coordinate_nodes_2MDoF
 #+caption: Coordinates of the interface nodes
+#+attr_latex: :environment tabularx :width 0.6\linewidth :align ccccc
+#+attr_latex: :center t :booktabs t
 #+RESULTS:
 | Node i | Node Number | x [m] | y [m] | z [m] |
 |--------+-------------+-------+-------+-------|
 |    1.0 |    181278.0 |   0.0 |   0.0 |   0.0 |
 |    2.0 |    181279.0 |   0.0 |   0.0 |  -0.0 |
 
-#+begin_src matlab :exports results :results value table replace :tangle no
-  data2orgtable([length(n_i); length(int_i); size(M,1) - 6*length(int_i); size(M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f ');
-#+end_src
-
-#+RESULTS:
-| Total number of Nodes     |  2 |
-| Number of interface Nodes |  2 |
-| Number of Modes           |  6 |
-| Size of M and K matrices  | 18 |
-
 Using =K=, =M= and =int_xyz=, we can use the =Reduced Order Flexible Solid= simscape block.
 
 ** Identification of the parameters using Simscape and looking at the Stiffness Matrix
@@ -1159,11 +1260,15 @@ The dynamics is identified from the applied force/torque to the measured displac
 And we find the same parameters as the one estimated from the Stiffness matrix.
 
 #+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
-  data2orgtable([1e-6*K(3,3), K(4,4), K(5,5), K(6,6) ; 1e-6./dcgain(G(3,3)), 1./dcgain(G(4,4)), 1./dcgain(G(5,5)), 1./dcgain(G(6,6))]', {'Axial Stiffness Dz [N/um]', 'Bending Stiffness Rx [Nm/rad]', 'Bending Stiffness Ry [Nm/rad]', 'Torsion Stiffness Rz [Nm/rad]'}, {'*Caracteristic*', '*Value*', '*Identification*'}, ' %0.f ');
+  data2orgtable([1e-6*K(3,3), K(4,4), K(5,5), K(6,6) ; 1e-6./dcgain(G(3,3)), 1./dcgain(G(4,4)), 1./dcgain(G(5,5)), 1./dcgain(G(6,6))]', {'Axial Stiffness Dz [N/um]', 'Bending Stiffness Rx [Nm/rad]', 'Bending Stiffness Ry [Nm/rad]', 'Torsion Stiffness Rz [Nm/rad]'}, {'*Characteristic*', '*Value*', '*Identification*'}, ' %0.f ');
 #+end_src
 
+#+name: tab:comp_identified_stiffnesses
+#+caption: Comparison of identified and FEM stiffnesses
+#+attr_latex: :environment tabularx :width 0.7\linewidth :align lcc
+#+attr_latex: :center t :booktabs t
 #+RESULTS:
-| *Caracteristic*               | *Value* | *Identification* |
+| *Characteristic*              | *Value* | *Identification* |
 |-------------------------------+---------+------------------|
 | Axial Stiffness Dz [N/um]     |     119 |              119 |
 | Bending Stiffness Rx [Nm/rad] |      33 |               33 |
@@ -1175,6 +1280,7 @@ Let's now model the flexible joint with a "perfect" Bushing joint as shown in Fi
 
 #+name: fig:flexible_joint_simscape
 #+caption: Bushing Joint used to model the flexible joint
+#+attr_latex: :width 0.3\linewidth
 [[file:figs/flexible_joint_simscape.png]]
 
 The parameters of the Bushing joint (stiffnesses) are estimated from the Stiffness matrix that was computed from the FEM.
@@ -1271,6 +1377,7 @@ The obtained geometry is shown in Figure [[fig:optimal_flexor]].
 
 #+name: fig:optimal_flexor
 #+caption: Flexor studied
+#+attr_latex: :width 1.0\linewidth
 [[file:figs/flexor_025_MDoF.jpg]]
 
 ** Matlab Init                                             :noexport:ignore:
@@ -1306,25 +1413,31 @@ We first extract the stiffness and mass matrices.
   data2orgtable(K(1:10, 1:10), {}, {}, ' %.2e ');
 #+end_src
 
+#+name: tab:flex_mat_K
 #+caption: First 10x10 elements of the Stiffness matrix
+#+attr_latex: :environment tabularx :width \linewidth :align XXXXXXXXXX
+#+attr_latex: :center t :booktabs t :font \tiny
 #+RESULTS:
-|  12700000.0 |       -18.5 |       -26.8 |  0.00162 |    -4.63 |      64.0 | -12700000.0 |        18.3 |        26.7 |   0.00234 |
-|       -18.5 |  12700000.0 |      -499.0 |   -132.0 |  0.00414 |    -0.495 |        18.4 | -12700000.0 |       499.0 |     132.0 |
-|       -26.8 |      -499.0 |  94000000.0 |   -470.0 |  0.00771 |    -0.855 |        26.8 |       498.0 | -94000000.0 |     470.0 |
-|     0.00162 |      -132.0 |      -470.0 |     4.83 | 2.61e-07 |  0.000123 |    -0.00163 |       132.0 |       470.0 |     -4.83 |
-|       -4.63 |     0.00414 |     0.00771 | 2.61e-07 |     4.83 |  4.43e-05 |        4.63 |    -0.00413 |    -0.00772 |  -4.3e-07 |
-|        64.0 |      -0.495 |      -0.855 | 0.000123 | 4.43e-05 |     260.0 |       -64.0 |       0.495 |       0.855 | -0.000124 |
-| -12700000.0 |        18.4 |        26.8 | -0.00163 |     4.63 |     -64.0 |  12700000.0 |       -18.2 |       -26.7 |  -0.00234 |
-|        18.3 | -12700000.0 |       498.0 |    132.0 | -0.00413 |     0.495 |       -18.2 |  12700000.0 |      -498.0 |    -132.0 |
-|        26.7 |       499.0 | -94000000.0 |    470.0 | -0.00772 |     0.855 |       -26.7 |      -498.0 |  94000000.0 |    -470.0 |
-|     0.00234 |       132.0 |       470.0 |    -4.83 | -4.3e-07 | -0.000124 |    -0.00234 |      -132.0 |      -470.0 |      4.83 |
+|  12700000 |     -18.5 |     -26.8 |    00162 |    -4.63 |       64 | -12700000 |      18.3 |      26.7 |    00234 |
+|     -18.5 |  12700000 |      -499 |     -132 |    00414 |   -0.495 |      18.4 | -12700000 |       499 |      132 |
+|     -26.8 |      -499 |  94000000 |     -470 |    00771 |   -0.855 |      26.8 |       498 | -94000000 |      470 |
+|     00162 |      -132 |      -470 |     4.83 | 2.61e-07 |   000123 |    -00163 |       132 |       470 |    -4.83 |
+|     -4.63 |     00414 |     00771 | 2.61e-07 |     4.83 | 4.43e-05 |      4.63 |    -00413 |    -00772 | -4.3e-07 |
+|        64 |    -0.495 |    -0.855 |   000123 | 4.43e-05 |      260 |       -64 |     0.495 |     0.855 |  -000124 |
+| -12700000 |      18.4 |      26.8 |   -00163 |     4.63 |      -64 |  12700000 |     -18.2 |     -26.7 |   -00234 |
+|      18.3 | -12700000 |       498 |      132 |   -00413 |    0.495 |     -18.2 |  12700000 |      -498 |     -132 |
+|      26.7 |       499 | -94000000 |      470 |   -00772 |    0.855 |     -26.7 |      -498 |  94000000 |     -470 |
+|     00234 |       132 |       470 |    -4.83 | -4.3e-07 |  -000124 |    -00234 |      -132 |      -470 |     4.83 |
 
 
 #+begin_src matlab :exports results :results value table replace :tangle no
   data2orgtable(M(1:10, 1:10), {}, {}, ' %.1g ');
 #+end_src
 
+#+name: tab:flex_mat_M
 #+caption: First 10x10 elements of the Mass matrix
+#+attr_latex: :environment tabularx :width \linewidth :align XXXXXXXXXX
+#+attr_latex: :center t :booktabs t :font \tiny
 #+RESULTS:
 |  0.006 |  8e-09 | -2e-08 | -1e-10 |  3e-05 |  3e-08 |  0.003 | -3e-09 |  9e-09 |  2e-12 |
 |  8e-09 |   0.02 |  1e-07 | -3e-05 |  1e-11 |  6e-10 |  1e-08 |  0.003 | -5e-08 |  3e-09 |
@@ -1347,6 +1460,10 @@ Then, we extract the coordinates of the interface nodes.
   data2orgtable([length(n_i); length(int_i); size(M,1) - 6*length(int_i); size(M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f ');
 #+end_src
 
+#+name: tab:coordinate_nodes_flex
+#+caption: Coordinates of the interface nodes for Flexible Joint
+#+attr_latex: :environment tabularx :width 0.6\linewidth :align ccccc
+#+attr_latex: :center t :booktabs t
 #+RESULTS:
 | Total number of Nodes     |  2 |
 | Number of interface Nodes |  2 |
@@ -1357,7 +1474,10 @@ Then, we extract the coordinates of the interface nodes.
   data2orgtable([[1:length(int_i)]', int_i, int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
 #+end_src
 
+#+name: tab:coordinate_interface_nodes_flex
 #+caption: Coordinates of the interface nodes
+#+attr_latex: :environment tabularx :width 0.6\linewidth :align ccccc
+#+attr_latex: :center t :booktabs t
 #+RESULTS:
 | Node i | Node Number | x [m] | y [m] | z [m] |
 |--------+-------------+-------+-------+-------|
@@ -1392,6 +1512,10 @@ And we find the same parameters as the one estimated from the Stiffness matrix.
   data2orgtable([1e-6*K(3,3), K(4,4), K(5,5), K(6,6) ; 1e-6./dcgain(G(3,3)), 1./dcgain(G(4,4)), 1./dcgain(G(5,5)), 1./dcgain(G(6,6))]', {'Axial Stiffness Dz [N/um]', 'Bending Stiffness Rx [Nm/rad]', 'Bending Stiffness Ry [Nm/rad]', 'Torsion Stiffness Rz [Nm/rad]'}, {'*Caracteristic*', '*Value*', '*Identification*'}, ' %.1f ');
 #+end_src
 
+#+name: tab:comp_identified_stiffnesses_flex
+#+caption: Comparison of identified and FEM stiffnesses
+#+attr_latex: :environment tabularx :width 0.7\linewidth :align lcc
+#+attr_latex: :center t :booktabs t
 #+RESULTS:
 | *Caracteristic*               | *Value* | *Identification* |
 |-------------------------------+---------+------------------|
@@ -1403,10 +1527,6 @@ And we find the same parameters as the one estimated from the Stiffness matrix.
 ** Simpler Model
 Let's now model the flexible joint with a "perfect" Bushing joint as shown in Figure [[fig:flexible_joint_simscape]].
 
-#+name: fig:flexible_joint_simscape
-#+caption: Bushing Joint used to model the flexible joint
-[[file:figs/flexible_joint_simscape.png]]
-
 The parameters of the Bushing joint (stiffnesses) are estimated from the Stiffness matrix that was computed from the FEM.
 #+begin_src matlab
   Kx = K(1,1); % [N/m]
@@ -1480,13 +1600,13 @@ The two obtained dynamics are compared in Figure
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file replace
-  exportFig('figs/flexor_ID16_compare_bushing_joint.pdf', 'width', 'wide', 'height', 'normal');
+  exportFig('figs/flexor_ID16_compare_bushing_joint_optimized.pdf', 'width', 'wide', 'height', 'normal');
 #+end_src
 
-#+name: fig:flexor_ID16_compare_bushing_joint
+#+name: fig:flexor_ID16_compare_bushing_joint_optimized
 #+caption: Comparison of the Joint compliance between the FEM model and the simpler model
 #+RESULTS:
-[[file:figs/flexor_ID16_compare_bushing_joint.png]]
+[[file:figs/flexor_ID16_compare_bushing_joint_optimized.png]]
 
 ** Comparison with a stiffer Flexible Joint
 The stiffness matrix with the flexible joint with a "hinge" size of 0.50mm is loaded.
@@ -1500,6 +1620,10 @@ Its parameters are compared with the Flexible Joint with a size of 0.25mm in the
   data2orgtable([1e-6*K(3,3), 1e-6*K(2,2), K(4,4), K(5,5), K(6,6) ; 1e-6*K_050(3,3), 1e-6*K_050(2,2), K_050(4,4), K_050(5,5), K_050(6,6)]', {'Axial Stiffness Dz [N/um]', 'Shear Stiffness [N/um]', 'Bending Stiffness Rx [Nm/rad]', 'Bending Stiffness Ry [Nm/rad]', 'Torsion Stiffness Rz [Nm/rad]'}, {'*Caracteristic*', '*0.25 mm*', '*0.50 mm*'}, ' %.1f ');
 #+end_src
 
+#+name: tab:comp_stiffnesses_flex_size
+#+caption: Comparison of flexible joint stiffnesses for 0.25 and 0.5mm
+#+attr_latex: :environment tabularx :width 0.7\linewidth :align lcc
+#+attr_latex: :center t :booktabs t
 #+RESULTS:
 | *Caracteristic*               | *0.25 mm* | *0.50 mm* |
 |-------------------------------+-----------+-----------|
@@ -1522,12 +1646,14 @@ The 3D as well as the interface nodes are shown in Figure [[fig:strut_encoder_po
 
 #+name: fig:strut_encoder_points3
 #+caption: Interface points
+#+attr_latex: :width 1.0\linewidth
 [[file:figs/strut_fem_nodes.jpg]]
 
 A side view is shown in Figure [[fig:strut_encoder_nodes_side]].
 
 #+name: fig:strut_encoder_nodes_side
 #+caption: Interface points - Side view
+#+attr_latex: :width 1.0\linewidth
 [[file:figs/strut_fem_nodes_side.jpg]]
 
 The flexible joints used have a 0.25mm width size.
@@ -1565,25 +1691,31 @@ We first extract the stiffness and mass matrices.
   data2orgtable(K(1:10, 1:10), {}, {}, ' %.1g ');
 #+end_src
 
+#+name: tab:strut_mat_K
 #+caption: First 10x10 elements of the Stiffness matrix
+#+attr_latex: :environment tabularx :width \linewidth :align XXXXXXXXXX
+#+attr_latex: :center t :booktabs t :font \tiny
 #+RESULTS:
-|  2000000.0 |  1000000.0 | -3000000.0 |  -400.0 |  300.0 | 200.0 |      -30.0 |     2000.0 |   -10000.0 |     0.3 |
-|  1000000.0 |  4000000.0 | -8000000.0 |  -900.0 |  400.0 | -50.0 |    -6000.0 |    10000.0 |   -20000.0 |       3 |
-| -3000000.0 | -8000000.0 | 20000000.0 |  2000.0 | -900.0 | 200.0 |   -10000.0 |    20000.0 |  -300000.0 |       7 |
-|     -400.0 |     -900.0 |     2000.0 |       5 |   -0.1 |  0.05 |          1 |         -3 |          6 | -0.0007 |
-|      300.0 |      400.0 |     -900.0 |    -0.1 |      5 |  0.04 |       -0.1 |        0.5 |         -3 |  0.0001 |
-|      200.0 |      -50.0 |      200.0 |    0.05 |   0.04 | 300.0 |          4 |      -0.01 |         -1 |   3e-05 |
-|      -30.0 |    -6000.0 |   -10000.0 |       1 |   -0.1 |     4 |  3000000.0 | -1000000.0 | -2000000.0 |  -300.0 |
-|     2000.0 |    10000.0 |    20000.0 |      -3 |    0.5 | -0.01 | -1000000.0 |  6000000.0 |  7000000.0 |  1000.0 |
-|   -10000.0 |   -20000.0 |  -300000.0 |       6 |     -3 |    -1 | -2000000.0 |  7000000.0 | 20000000.0 |  2000.0 |
-|        0.3 |          3 |          7 | -0.0007 | 0.0001 | 3e-05 |     -300.0 |     1000.0 |     2000.0 |       5 |
+|  2000000 |  1000000 | -3000000 |  -400 |  300 |   200 |      -30 |     2000 |   -10000 |   0.3 |
+|  1000000 |  4000000 | -8000000 |  -900 |  400 |   -50 |    -6000 |    10000 |   -20000 |     3 |
+| -3000000 | -8000000 | 20000000 |  2000 | -900 |   200 |   -10000 |    20000 |  -300000 |     7 |
+|     -400 |     -900 |     2000 |     5 | -0.1 |    05 |        1 |       -3 |        6 | -0007 |
+|      300 |      400 |     -900 |  -0.1 |    5 |    04 |     -0.1 |      0.5 |       -3 |  0001 |
+|      200 |      -50 |      200 |    05 |   04 |   300 |        4 |      -01 |       -1 | 3e-05 |
+|      -30 |    -6000 |   -10000 |     1 | -0.1 |     4 |  3000000 | -1000000 | -2000000 |  -300 |
+|     2000 |    10000 |    20000 |    -3 |  0.5 |   -01 | -1000000 |  6000000 |  7000000 |  1000 |
+|   -10000 |   -20000 |  -300000 |     6 |   -3 |    -1 | -2000000 |  7000000 | 20000000 |  2000 |
+|      0.3 |        3 |        7 | -0007 | 0001 | 3e-05 |     -300 |     1000 |     2000 |     5 |
 
 
 #+begin_src matlab :exports results :results value table replace :tangle no
   data2orgtable(M(1:10, 1:10), {}, {}, ' %.1g ');
 #+end_src
 
+#+name: tab:strut_mat_M
 #+caption: First 10x10 elements of the Mass matrix
+#+attr_latex: :environment tabularx :width \linewidth :align XXXXXXXXXX
+#+attr_latex: :center t :booktabs t :font \tiny
 #+RESULTS:
 |   0.04 |  -0.005 |   0.007 |   2e-06 | 0.0001 | -5e-07 | -1e-05 |  -9e-07 |  8e-05 | -5e-10 |
 | -0.005 |    0.03 |    0.02 | -0.0001 |  1e-06 | -3e-07 |  3e-05 | -0.0001 |  8e-05 | -3e-08 |
@@ -1606,6 +1738,10 @@ Then, we extract the coordinates of the interface nodes.
   data2orgtable([length(n_i); length(int_i); size(M,1) - 6*length(int_i); size(M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f ');
 #+end_src
 
+#+name: tab:parameters_fem_strut
+#+caption: Some extracted parameters of the FEM
+#+attr_latex: :environment tabularx :width 0.4\linewidth :align lc
+#+attr_latex: :center t :booktabs t
 #+RESULTS:
 | Total number of Nodes     |  8 |
 | Number of interface Nodes |  8 |
@@ -1616,7 +1752,10 @@ Then, we extract the coordinates of the interface nodes.
   data2orgtable([[1:length(int_i)]', int_i, int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
 #+end_src
 
+#+name: tab:coordinate_nodes_strut
 #+caption: Coordinates of the interface nodes
+#+attr_latex: :environment tabularx :width 0.6\linewidth :align ccccc
+#+attr_latex: :center t :booktabs t
 #+RESULTS:
 | Node i | Node Number |   x [m] |  y [m] |    z [m] |
 |--------+-------------+---------+--------+----------|
@@ -1849,5 +1988,4 @@ The same dynamics is identified for a payload mass of 10Kg.
 | Rz [Nm/rad]     |       118.0 |        114.803 |
 
 * Bibliography                                                        :ignore:
-bibliographystyle:unsrt
-bibliography:ref.bib
+#+latex: \printbibliography[heading=bibintoc,title={Bibliography}]

nass-fem.tex

@@ -0,0 +1,723 @@
+% Created 2024-03-19 Tue 11:08
+% Intended LaTeX compiler: pdflatex
+\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
+
+\input{preamble.tex}
+\bibliography{nass-fem.bib}
+\author{Dehaeze Thomas}
+\date{\today}
+\title{NASS - Finite Element Models with Simscape}
+\hypersetup{
+ pdfauthor={Dehaeze Thomas},
+ pdftitle={NASS - Finite Element Models with Simscape},
+ pdfkeywords={},
+ pdfsubject={},
+ pdfcreator={Emacs 29.2 (Org mode 9.7)}, 
+ pdflang={English}}
+\usepackage{biblatex}
+
+\begin{document}
+
+\maketitle
+\tableofcontents
+
+\clearpage
+In this document, Finite Element Models (FEM) of parts of the Nano-Hexapod are developed and integrated into Simscape for dynamical analysis.
+
+It is divided in the following sections:
+\begin{itemize}
+\item Section \ref{sec:APA300ML}:
+A super-element of the Amplified Piezoelectric Actuator APA300ML used for the NASS is exported using Ansys and imported in Simscape.
+The static and dynamical properties of the APA300ML are then estimated using the Simscape model.
+\item Section \ref{sec:flexor_ID16}:
+A first geometry of a Flexible joint is modelled and its characteristics are identified from the Stiffness matrix as well as from the Simscape model.
+\item Section \ref{sec:flexor_025}:
+An optimized flexible joint is developed for the Nano-Hexapod and is then imported in a Simscape model.
+\item Section \ref{sec:strut_encoder}:
+A super element of a complete strut is studied.
+\end{itemize}
+\chapter{APA300ML}
+\label{sec:APA300ML}
+In this section, the Amplified Piezoelectric Actuator APA300ML (\href{doc/APA300ML.pdf}{doc}) is modeled using a Finite Element Software.
+Then a \emph{super element} is exported and imported in Simscape where its dynamic is studied.
+
+A 3D view of the Amplified Piezoelectric Actuator (APA300ML) is shown in Figure \ref{fig:apa300ml_ansys}.
+The remote point used are also shown in this figure.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=0.7\linewidth]{figs/apa300ml_ansys.jpg}
+\caption{\label{fig:apa300ml_ansys}Ansys FEM of the APA300ML}
+\end{figure}
+\section{Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates}
+We first extract the stiffness and mass matrices.
+\begin{table}[htbp]
+\caption{\label{tab:APA300ML_mat_K}First 10x10 elements of the Stiffness matrix}
+\centering
+\tiny
+\begin{tabularx}{\linewidth}{XXXXXXXXXX}
+\toprule
+200000000 & 30000 & -20000 & -70 & 300000 & 40 & 10000000 & 10000 & -6000 & 30\\
+30000 & 30000000 & 2000 & -200000 & 60 & -10 & 4000 & 2000000 & -500 & 9000\\
+-20000 & 2000 & 7000000 & -10 & -30 & 10 & 6000 & 900 & -500000 & 3\\
+-70 & -200000 & -10 & 1000 & -0.1 & 08 & -20 & -9000 & 3 & -30\\
+300000 & 60 & -30 & -0.1 & 900 & 0.1 & 30000 & 20 & -10 & 06\\
+40 & -10 & 10 & 08 & 0.1 & 10000 & 20 & 9 & -5 & 03\\
+10000000 & 4000 & 6000 & -20 & 30000 & 20 & 200000000 & 10000 & 9000 & 50\\
+10000 & 2000000 & 900 & -9000 & 20 & 9 & 10000 & 30000000 & -500 & 200000\\
+-6000 & -500 & -500000 & 3 & -10 & -5 & 9000 & -500 & 7000000 & -2\\
+30 & 9000 & 3 & -30 & 06 & 03 & 50 & 200000 & -2 & 1000\\
+\bottomrule
+\end{tabularx}
+\end{table}
+
+
+\begin{table}[htbp]
+\caption{\label{tab:APA300ML_mat_M}First 10x10 elements of the Mass matrix}
+\centering
+\tiny
+\begin{tabularx}{\linewidth}{XXXXXXXXXX}
+\toprule
+0.01 & -2e-06 & 1e-06 & 6e-09 & 5e-05 & -5e-09 & -0.0005 & -7e-07 & 6e-07 & -3e-09\\
+-2e-06 & 0.01 & 8e-07 & -2e-05 & -8e-09 & 2e-09 & -9e-07 & -0.0002 & 1e-08 & -9e-07\\
+1e-06 & 8e-07 & 0.009 & 5e-10 & 1e-09 & -1e-09 & -5e-07 & 3e-08 & 6e-05 & 1e-10\\
+6e-09 & -2e-05 & 5e-10 & 3e-07 & 2e-11 & -3e-12 & 3e-09 & 9e-07 & -4e-10 & 3e-09\\
+5e-05 & -8e-09 & 1e-09 & 2e-11 & 6e-07 & -4e-11 & -1e-06 & -2e-09 & 1e-09 & -8e-12\\
+-5e-09 & 2e-09 & -1e-09 & -3e-12 & -4e-11 & 1e-07 & -2e-09 & -1e-09 & -4e-10 & -5e-12\\
+-0.0005 & -9e-07 & -5e-07 & 3e-09 & -1e-06 & -2e-09 & 0.01 & 1e-07 & -3e-07 & -2e-08\\
+-7e-07 & -0.0002 & 3e-08 & 9e-07 & -2e-09 & -1e-09 & 1e-07 & 0.01 & -4e-07 & 2e-05\\
+6e-07 & 1e-08 & 6e-05 & -4e-10 & 1e-09 & -4e-10 & -3e-07 & -4e-07 & 0.009 & -2e-10\\
+-3e-09 & -9e-07 & 1e-10 & 3e-09 & -8e-12 & -5e-12 & -2e-08 & 2e-05 & -2e-10 & 3e-07\\
+\bottomrule
+\end{tabularx}
+\end{table}
+
+
+Then, we extract the coordinates of the interface nodes.
+\begin{table}[htbp]
+\caption{\label{tab:coordinate_nodes}Coordinates of the interface nodes}
+\centering
+\begin{tabularx}{0.6\linewidth}{ccccc}
+\toprule
+Node i & Node Number & x [m] & y [m] & z [m]\\
+\midrule
+1.0 & 697783 & 0.0 & 0.0 & -0.015\\
+2.0 & 697784 & 0.0 & 0.0 & 0.015\\
+3.0 & 697785 & -0.0325 & 0.0 & 0.0\\
+4.0 & 697786 & -0.0125 & 0.0 & 0.0\\
+5.0 & 697787 & -0.0075 & 0.0 & 0.0\\
+6.0 & 697788 & 0.0125 & 0.0 & 0.0\\
+7.0 & 697789 & 0.0325 & 0.0 & 0.0\\
+\bottomrule
+\end{tabularx}
+\end{table}
+
+\begin{table}[htbp]
+\caption{\label{tab:parameters_fem}Some extracted parameters of the FEM}
+\centering
+\begin{tabularx}{0.4\linewidth}{lc}
+\toprule
+Total number of Nodes & 7\\
+Number of interface Nodes & 7\\
+Number of Modes & 120\\
+Size of M and K matrices & 162\\
+\bottomrule
+\end{tabularx}
+\end{table}
+
+Using \texttt{K}, \texttt{M} and \texttt{int\_xyz}, we can now use the \texttt{Reduced Order Flexible Solid} simscape block.
+\section{Piezoelectric parameters}
+In order to make the conversion from applied voltage to generated force or from the strain to the generated voltage, we need to defined some parameters corresponding to the piezoelectric material:
+PZT-4
+The ratio of the developed force to applied voltage is:
+\begin{equation}
+\label{eq:piezo_voltage_to_force}
+  F_a = g_a V_a, \quad g_a = d_{33} n k_a
+\end{equation}
+where:
+\begin{itemize}
+\item \(F_a\): developed force in [N]
+\item \(n\): number of layers of the actuator stack
+\item \(d_{33}\): strain constant in [m/V]
+\item \(k_a\): actuator stack stiffness in [N/m]
+\item \(V_a\): applied voltage in [V]
+\end{itemize}
+
+If we take the numerical values, we obtain:
+From \cite{fleming14_desig_model_contr_nanop_system} (page 123), the relation between relative displacement of the sensor stack and generated voltage is:
+\begin{equation}
+\label{eq:piezo_strain_to_voltage}
+  V_s = \frac{d_{33}}{\epsilon^T s^D n} \Delta h
+\end{equation}
+where:
+\begin{itemize}
+\item \(V_s\): measured voltage in [V]
+\item \(d_{33}\): strain constant in [m/V]
+\item \(\epsilon^T\): permittivity under constant stress in [F/m]
+\item \(s^D\): elastic compliance under constant electric displacement in [m\^{}2/N]
+\item \(n\): number of layers of the sensor stack
+\item \(\Delta h\): relative displacement in [m]
+\end{itemize}
+
+If we take the numerical values, we obtain:
+\section{Simscape Model}
+The flexible element is imported using the \texttt{Reduced Order Flexible Solid} simscape block.
+
+Let's say we use two stacks as a force sensor and one stack as an actuator:
+\begin{itemize}
+\item A \texttt{Relative Motion Sensor} block is added between the nodes A and C
+\item An \texttt{Internal Force} block is added between the remote points E and B
+\end{itemize}
+
+The interface nodes are shown in Figure \ref{fig:apa300ml_ansys}.
+
+One mass is fixed at one end of the piezo-electric stack actuator (remove point F), the other end is fixed to the world frame (remote point G).
+\section{Identification of the APA Characteristics}
+\subsection{Stiffness}
+The transfer function from vertical external force to the relative vertical displacement is identified.
+
+The inverse of its DC gain is the axial stiffness of the APA:
+The specified stiffness in the datasheet is \(k = 1.8\, [N/\mu m]\).
+\subsection{Resonance Frequency}
+The resonance frequency is specified to be between 650Hz and 840Hz.
+This is also the case for the FEM model (Figure \ref{fig:apa300ml_resonance}).
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/apa300ml_resonance.png}
+\caption{\label{fig:apa300ml_resonance}First resonance is around 800Hz}
+\end{figure}
+\subsection{Amplification factor}
+The amplification factor is the ratio of the vertical displacement to the stack displacement.
+
+The ratio of the two displacement is computed from the FEM model.
+This is actually correct and approximately corresponds to the ratio of the piezo height and length:
+\subsection{Stroke}
+
+Estimation of the actuator stroke:
+\[ \Delta H = A n \Delta L \]
+with:
+\begin{itemize}
+\item \(\Delta H\) Axial Stroke of the APA
+\item \(A\) Amplification factor (5 for the APA300ML)
+\item \(n\) Number of stack used
+\item \(\Delta L\) Stroke of the stack (0.1\% of its length)
+\end{itemize}
+
+This is exactly the specified stroke in the data-sheet.
+\subsection{Stroke BIS}
+\begin{itemize}
+\item[{$\square$}] Identified the stroke form the transfer function from V to z
+\end{itemize}
+\section{Identification of the Dynamics from actuator to replace displacement}
+We first set the mass to be approximately zero.
+The dynamics is identified from the applied force to the measured relative displacement.
+The same dynamics is identified for a payload mass of 10Kg.
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/apa300ml_plant_dynamics.png}
+\caption{\label{fig:apa300ml_plant_dynamics}Transfer function from forces applied by the stack to the axial displacement of the APA}
+\end{figure}
+
+The root locus corresponding to Direct Velocity Feedback with a mass of 10kg is shown in Figure \ref{fig:apa300ml_dvf_root_locus}.
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/apa300ml_dvf_root_locus.png}
+\caption{\label{fig:apa300ml_dvf_root_locus}Root Locus for Direct Velocity Feedback}
+\end{figure}
+\section{Identification of the Dynamics from actuator to force sensor}
+Let's use 2 stacks as a force sensor and 1 stack as force actuator.
+
+The transfer function from actuator voltage to sensor voltage is identified and shown in Figure \ref{fig:apa300ml_iff_plant}.
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/apa300ml_iff_plant.png}
+\caption{\label{fig:apa300ml_iff_plant}Transfer function from actuator to force sensor}
+\end{figure}
+
+For root locus corresponding to IFF is shown in Figure \ref{fig:apa300ml_iff_root_locus}.
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/apa300ml_iff_root_locus.png}
+\caption{\label{fig:apa300ml_iff_root_locus}Root Locus for IFF}
+\end{figure}
+\section{Identification for a simpler model}
+The goal in this section is to identify the parameters of a simple APA model from the FEM.
+This can be useful is a lower order model is to be used for simulations.
+
+The presented model is based on \cite{souleille18_concep_activ_mount_space_applic}.
+
+The model represents the Amplified Piezo Actuator (APA) from Cedrat-Technologies (Figure \ref{fig:souleille18_model_piezo}).
+The parameters are shown in the table below.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=0.7\linewidth]{./figs/souleille18_model_piezo.png}
+\caption{\label{fig:souleille18_model_piezo}Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator}
+\end{figure}
+
+\begin{table}[htbp]
+\caption{\label{tab:parameters_apa100m}Parameters used for the model of the APA 100M}
+\centering
+\begin{tabularx}{0.8\linewidth}{cl}
+\toprule
+ & Meaning\\
+\midrule
+\(k_e\) & Stiffness used to adjust the pole of the isolator\\
+\(k_1\) & Stiffness of the metallic suspension when the stack is removed\\
+\(k_a\) & Stiffness of the actuator\\
+\(c_1\) & Added viscous damping\\
+\bottomrule
+\end{tabularx}
+\end{table}
+
+The goal is to determine \(k_e\), \(k_a\) and \(k_1\) so that the simplified model fits the FEM model.
+
+\[ \alpha = \frac{x_1}{f}(\omega=0) = \frac{\frac{k_e}{k_e + k_a}}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]
+\[ \beta = \frac{x_1}{F}(\omega=0) = \frac{1}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]
+
+If we can fix \(k_a\), we can determine \(k_e\) and \(k_1\) with:
+\[ k_e = \frac{k_a}{\frac{\beta}{\alpha} - 1} \]
+\[ k_1 = \frac{1}{\beta} - \frac{k_e k_a}{k_e + k_a} \]
+
+From the identified dynamics, compute \(\alpha\) and \(\beta\)
+\(k_a\) is estimated using the following formula:
+The factor can be adjusted to better match the curves.
+
+Then \(k_e\) and \(k_1\) are computed.
+\begin{table}[htbp]
+\caption{\label{tab:stiffnesses_apa100m}Obtained stiffnesses of the APA 100M}
+\centering
+\begin{tabularx}{0.3\linewidth}{cl}
+\toprule
+ & Value [N/um]\\
+\midrule
+ka & 40.5\\
+ke & 1.5\\
+k1 & 0.4\\
+\bottomrule
+\end{tabularx}
+\end{table}
+
+The damping in the system is adjusted to match the FEM model if necessary.
+The analytical model of the simpler system is defined below:
+And the DC gain is adjusted for the force sensor:
+The dynamics of the FEM model and the simpler model are compared in Figure \ref{fig:apa300ml_comp_simpler_model}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=1.0\linewidth]{figs/apa300ml_comp_simpler_model.png}
+\caption{\label{fig:apa300ml_comp_simpler_model}Comparison of the Dynamics between the FEM model and the simplified one}
+\end{figure}
+
+The simplified model has also been implemented in Simscape.
+
+The dynamics of the Simscape simplified model is identified and compared with the FEM one in Figure \ref{fig:apa300ml_comp_simpler_simscape}.
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/apa300ml_comp_simpler_simscape.png}
+\caption{\label{fig:apa300ml_comp_simpler_simscape}Comparison of the Dynamics between the FEM model and the simplified simscape model}
+\end{figure}
+\section{Integral Force Feedback}
+In this section, Integral Force Feedback control architecture is applied on the APA300ML.
+
+First, the plant (dynamics from voltage actuator to voltage sensor is identified).
+The payload mass is set to 10kg.
+The obtained dynamics is shown in Figure \ref{fig:piezo_amplified_iff_plant}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/piezo_amplified_iff_plant.png}
+\caption{\label{fig:piezo_amplified_iff_plant}IFF Plant}
+\end{figure}
+
+The controller is defined below and the loop gain is shown in Figure \ref{fig:piezo_amplified_iff_loop_gain}.
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/piezo_amplified_iff_loop_gain.png}
+\caption{\label{fig:piezo_amplified_iff_loop_gain}IFF Loop Gain}
+\end{figure}
+
+Now the closed-loop system is identified again and compare with the open loop system in Figure \ref{fig:piezo_amplified_iff_comp}.
+
+It is the expected behavior as shown in the Figure \ref{fig:souleille18_results} (from \cite{souleille18_concep_activ_mount_space_applic}).
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/piezo_amplified_iff_comp.png}
+\caption{\label{fig:piezo_amplified_iff_comp}OL and CL transfer functions}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=1.0\linewidth]{figs/souleille18_results.png}
+\caption{\label{fig:souleille18_results}Results obtained in \cite{souleille18_concep_activ_mount_space_applic}}
+\end{figure}
+\chapter{First Flexible Joint Geometry}
+\label{sec:flexor_ID16}
+The studied flexor is shown in Figure \ref{fig:flexor_id16_screenshot}.
+
+The stiffness and mass matrices representing the dynamics of the flexor are exported from a FEM.
+It is then imported into Simscape.
+
+A simplified model of the flexor is then developped.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=0.3\linewidth]{figs/flexor_id16_screenshot.png}
+\caption{\label{fig:flexor_id16_screenshot}Flexor studied}
+\end{figure}
+\section{Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates}
+We first extract the stiffness and mass matrices.
+\begin{table}[htbp]
+\caption{\label{tab:APA300ML_mat_K_2MDoF}First 10x10 elements of the Stiffness matrix}
+\centering
+\tiny
+\begin{tabularx}{\linewidth}{XXXXXXXXXX}
+\toprule
+11200000 & 195 & 2220 & -0.719 & -265 & 1.59 & -11200000 & -213 & -2220 & 0.147 &  & 195 & 11400000 & 1290 & -148 & -0.188 & 2.41 & -212 & -11400000 & -1290 & 148\\
+2220 & 1290 & 119000000 & 1.31 & 1.49 & 1.79 & -2220 & -1290 & -119000000 & -1.31 &  &  &  &  &  &  &  &  &  &  & \\
+-0.719 & -148 & 1.31 & 33 & 000488 & -000977 & 0.141 & 148 & -1.31 & -33 &  &  &  &  &  &  &  &  &  &  & \\
+-265 & -0.188 & 1.49 & 000488 & 33 & 00293 & 266 & 0.154 & -1.49 & 00026 &  &  &  &  &  &  &  &  &  &  & \\
+1.59 & 2.41 & 1.79 & -000977 & 00293 & 236 & -1.32 & -2.55 & -1.79 & 000379 &  &  &  &  &  &  &  &  &  &  & \\
+-11200000 & -212 & -2220 & 0.141 & 266 & -1.32 & 11400000 & 24600 & 1640 & 120 &  &  &  &  &  &  &  &  &  &  & \\
+-213 & -11400000 & -1290 & 148 & 0.154 & -2.55 & 24600 & 11400000 & 1290 & -72 &  &  &  &  &  &  &  &  &  &  & \\
+-2220 & -1290 & -119000000 & -1.31 & -1.49 & -1.79 & 1640 & 1290 & 119000000 & 1.32 &  &  &  &  &  &  &  &  &  &  & \\
+0.147 & 148 & -1.31 & -33 & 00026 & 000379 & 120 & -72 & 1.32 & 34.7 &  &  &  &  &  &  &  &  &  &  & \\
+\bottomrule
+\end{tabularx}
+\end{table}
+
+\begin{table}[htbp]
+\caption{\label{tab:APA300ML_mat_M_2MDoF}First 10x10 elements of the Mass matrix}
+\centering
+\tiny
+\begin{tabularx}{\linewidth}{XXXXXXXXXX}
+\toprule
+0.02 & 1e-09 & -4e-08 & -1e-10 & 0.0002 & -3e-11 & 0.004 & 5e-08 & 7e-08 & 1e-10\\
+1e-09 & 0.02 & -3e-07 & -0.0002 & -1e-10 & -2e-09 & 2e-08 & 0.004 & 3e-07 & 1e-05\\
+-4e-08 & -3e-07 & 0.02 & 7e-10 & -2e-09 & 1e-09 & 3e-07 & 7e-08 & 0.003 & 1e-09\\
+-1e-10 & -0.0002 & 7e-10 & 4e-06 & -1e-12 & -6e-13 & 2e-10 & -7e-06 & -8e-10 & -1e-09\\
+0.0002 & -1e-10 & -2e-09 & -1e-12 & 3e-06 & 2e-13 & 9e-06 & 4e-11 & 2e-09 & -3e-13\\
+-3e-11 & -2e-09 & 1e-09 & -6e-13 & 2e-13 & 4e-07 & 8e-11 & 9e-10 & -1e-09 & 2e-12\\
+0.004 & 2e-08 & 3e-07 & 2e-10 & 9e-06 & 8e-11 & 0.02 & -7e-08 & -3e-07 & -2e-10\\
+5e-08 & 0.004 & 7e-08 & -7e-06 & 4e-11 & 9e-10 & -7e-08 & 0.01 & -4e-08 & 0.0002\\
+7e-08 & 3e-07 & 0.003 & -8e-10 & 2e-09 & -1e-09 & -3e-07 & -4e-08 & 0.02 & -1e-09\\
+1e-10 & 1e-05 & 1e-09 & -1e-09 & -3e-13 & 2e-12 & -2e-10 & 0.0002 & -1e-09 & 2e-06\\
+\bottomrule
+\end{tabularx}
+\end{table}
+
+\begin{table}[htbp]
+\caption{\label{tab:parameters_fem_2MDoF}Some extracted parameters of the FEM}
+\centering
+\begin{tabularx}{0.4\linewidth}{lc}
+\toprule
+Total number of Nodes & 2\\
+Number of interface Nodes & 2\\
+Number of Modes & 6\\
+Size of M and K matrices & 18\\
+\bottomrule
+\end{tabularx}
+\end{table}
+
+Then, we extract the coordinates of the interface nodes.
+\begin{table}[htbp]
+\caption{\label{tab:coordinate_nodes_2MDoF}Coordinates of the interface nodes}
+\centering
+\begin{tabularx}{0.6\linewidth}{ccccc}
+\toprule
+Node i & Node Number & x [m] & y [m] & z [m]\\
+\midrule
+1.0 & 181278.0 & 0.0 & 0.0 & 0.0\\
+2.0 & 181279.0 & 0.0 & 0.0 & -0.0\\
+\bottomrule
+\end{tabularx}
+\end{table}
+
+Using \texttt{K}, \texttt{M} and \texttt{int\_xyz}, we can use the \texttt{Reduced Order Flexible Solid} simscape block.
+\section{Identification of the parameters using Simscape and looking at the Stiffness Matrix}
+The flexor is now imported into Simscape and its parameters are estimated using an identification.
+
+The dynamics is identified from the applied force/torque to the measured displacement/rotation of the flexor.
+And we find the same parameters as the one estimated from the Stiffness matrix.
+
+\begin{table}[htbp]
+\caption{\label{tab:comp_identified_stiffnesses}Comparison of identified and FEM stiffnesses}
+\centering
+\begin{tabularx}{0.7\linewidth}{lcc}
+\toprule
+\textbf{Characteristic} & \textbf{Value} & \textbf{Identification}\\
+\midrule
+Axial Stiffness Dz [N/um] & 119 & 119\\
+Bending Stiffness Rx [Nm/rad] & 33 & 33\\
+Bending Stiffness Ry [Nm/rad] & 33 & 33\\
+Torsion Stiffness Rz [Nm/rad] & 236 & 236\\
+\bottomrule
+\end{tabularx}
+\end{table}
+\section{Simpler Model}
+Let's now model the flexible joint with a ``perfect'' Bushing joint as shown in Figure \ref{fig:flexible_joint_simscape}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=0.3\linewidth]{figs/flexible_joint_simscape.png}
+\caption{\label{fig:flexible_joint_simscape}Bushing Joint used to model the flexible joint}
+\end{figure}
+
+The parameters of the Bushing joint (stiffnesses) are estimated from the Stiffness matrix that was computed from the FEM.
+The dynamics from the applied force/torque to the measured displacement/rotation of the flexor is identified again for this simpler model.
+The two obtained dynamics are compared in Figure
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/flexor_ID16_compare_bushing_joint.png}
+\caption{\label{fig:flexor_ID16_compare_bushing_joint}Comparison of the Joint compliance between the FEM model and the simpler model}
+\end{figure}
+\chapter{Optimized Flexible Joint}
+\label{sec:flexor_025}
+The joint geometry has been optimized using Ansys to have lower bending stiffness while keeping a large axial stiffness.
+
+The obtained geometry is shown in Figure \ref{fig:optimal_flexor}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=1.0\linewidth]{figs/flexor_025_MDoF.jpg}
+\caption{\label{fig:optimal_flexor}Flexor studied}
+\end{figure}
+\section{Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates}
+We first extract the stiffness and mass matrices.
+\begin{table}[htbp]
+\caption{\label{tab:flex_mat_K}First 10x10 elements of the Stiffness matrix}
+\centering
+\tiny
+\begin{tabularx}{\linewidth}{XXXXXXXXXX}
+\toprule
+12700000 & -18.5 & -26.8 & 00162 & -4.63 & 64 & -12700000 & 18.3 & 26.7 & 00234\\
+-18.5 & 12700000 & -499 & -132 & 00414 & -0.495 & 18.4 & -12700000 & 499 & 132\\
+-26.8 & -499 & 94000000 & -470 & 00771 & -0.855 & 26.8 & 498 & -94000000 & 470\\
+00162 & -132 & -470 & 4.83 & 2.61e-07 & 000123 & -00163 & 132 & 470 & -4.83\\
+-4.63 & 00414 & 00771 & 2.61e-07 & 4.83 & 4.43e-05 & 4.63 & -00413 & -00772 & -4.3e-07\\
+64 & -0.495 & -0.855 & 000123 & 4.43e-05 & 260 & -64 & 0.495 & 0.855 & -000124\\
+-12700000 & 18.4 & 26.8 & -00163 & 4.63 & -64 & 12700000 & -18.2 & -26.7 & -00234\\
+18.3 & -12700000 & 498 & 132 & -00413 & 0.495 & -18.2 & 12700000 & -498 & -132\\
+26.7 & 499 & -94000000 & 470 & -00772 & 0.855 & -26.7 & -498 & 94000000 & -470\\
+00234 & 132 & 470 & -4.83 & -4.3e-07 & -000124 & -00234 & -132 & -470 & 4.83\\
+\bottomrule
+\end{tabularx}
+\end{table}
+
+
+\begin{table}[htbp]
+\caption{\label{tab:flex_mat_M}First 10x10 elements of the Mass matrix}
+\centering
+\tiny
+\begin{tabularx}{\linewidth}{XXXXXXXXXX}
+\toprule
+0.006 & 8e-09 & -2e-08 & -1e-10 & 3e-05 & 3e-08 & 0.003 & -3e-09 & 9e-09 & 2e-12\\
+8e-09 & 0.02 & 1e-07 & -3e-05 & 1e-11 & 6e-10 & 1e-08 & 0.003 & -5e-08 & 3e-09\\
+-2e-08 & 1e-07 & 0.01 & -6e-08 & -6e-11 & -8e-12 & -1e-07 & 1e-08 & 0.003 & -1e-08\\
+-1e-10 & -3e-05 & -6e-08 & 1e-06 & 7e-14 & 6e-13 & 1e-10 & 1e-06 & -1e-08 & 3e-10\\
+3e-05 & 1e-11 & -6e-11 & 7e-14 & 2e-07 & 1e-10 & 3e-08 & -7e-12 & 6e-11 & -6e-16\\
+3e-08 & 6e-10 & -8e-12 & 6e-13 & 1e-10 & 5e-07 & 1e-08 & -5e-10 & -1e-11 & 1e-13\\
+0.003 & 1e-08 & -1e-07 & 1e-10 & 3e-08 & 1e-08 & 0.02 & -2e-08 & 1e-07 & -4e-12\\
+-3e-09 & 0.003 & 1e-08 & 1e-06 & -7e-12 & -5e-10 & -2e-08 & 0.006 & -8e-08 & 3e-05\\
+9e-09 & -5e-08 & 0.003 & -1e-08 & 6e-11 & -1e-11 & 1e-07 & -8e-08 & 0.01 & -6e-08\\
+2e-12 & 3e-09 & -1e-08 & 3e-10 & -6e-16 & 1e-13 & -4e-12 & 3e-05 & -6e-08 & 2e-07\\
+\bottomrule
+\end{tabularx}
+\end{table}
+
+
+Then, we extract the coordinates of the interface nodes.
+\begin{table}[htbp]
+\caption{\label{tab:coordinate_nodes_flex}Coordinates of the interface nodes for Flexible Joint}
+\centering
+\begin{tabularx}{0.6\linewidth}{ccccc}
+\toprule
+Total number of Nodes & 2\\
+Number of interface Nodes & 2\\
+Number of Modes & 6\\
+Size of M and K matrices & 18\\
+\bottomrule
+\end{tabularx}
+\end{table}
+
+\begin{table}[htbp]
+\caption{\label{tab:coordinate_interface_nodes_flex}Coordinates of the interface nodes}
+\centering
+\begin{tabularx}{0.6\linewidth}{ccccc}
+\toprule
+Node i & Node Number & x [m] & y [m] & z [m]\\
+\midrule
+1.0 & 528875.0 & 0.0 & 0.0 & 0.0\\
+2.0 & 528876.0 & 0.0 & 0.0 & -0.0\\
+\bottomrule
+\end{tabularx}
+\end{table}
+
+Using \texttt{K}, \texttt{M} and \texttt{int\_xyz}, we can use the \texttt{Reduced Order Flexible Solid} simscape block.
+\section{Identification of the parameters using Simscape}
+The flexor is now imported into Simscape and its parameters are estimated using an identification.
+
+The dynamics is identified from the applied force/torque to the measured displacement/rotation of the flexor.
+And we find the same parameters as the one estimated from the Stiffness matrix.
+
+\begin{table}[htbp]
+\caption{\label{tab:comp_identified_stiffnesses_flex}Comparison of identified and FEM stiffnesses}
+\centering
+\begin{tabularx}{0.7\linewidth}{lcc}
+\toprule
+\textbf{Caracteristic} & \textbf{Value} & \textbf{Identification}\\
+\midrule
+Axial Stiffness Dz [N/um] & 94.0 & 93.9\\
+Bending Stiffness Rx [Nm/rad] & 4.8 & 4.8\\
+Bending Stiffness Ry [Nm/rad] & 4.8 & 4.8\\
+Torsion Stiffness Rz [Nm/rad] & 260.2 & 260.2\\
+\bottomrule
+\end{tabularx}
+\end{table}
+\section{Simpler Model}
+Let's now model the flexible joint with a ``perfect'' Bushing joint as shown in Figure \ref{fig:flexible_joint_simscape}.
+
+The parameters of the Bushing joint (stiffnesses) are estimated from the Stiffness matrix that was computed from the FEM.
+The dynamics from the applied force/torque to the measured displacement/rotation of the flexor is identified again for this simpler model.
+The two obtained dynamics are compared in Figure
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/flexor_ID16_compare_bushing_joint_optimized.png}
+\caption{\label{fig:flexor_ID16_compare_bushing_joint_optimized}Comparison of the Joint compliance between the FEM model and the simpler model}
+\end{figure}
+\section{Comparison with a stiffer Flexible Joint}
+The stiffness matrix with the flexible joint with a ``hinge'' size of 0.50mm is loaded.
+Its parameters are compared with the Flexible Joint with a size of 0.25mm in the table below.
+
+\begin{table}[htbp]
+\caption{\label{tab:comp_stiffnesses_flex_size}Comparison of flexible joint stiffnesses for 0.25 and 0.5mm}
+\centering
+\begin{tabularx}{0.7\linewidth}{lcc}
+\toprule
+\textbf{Caracteristic} & \textbf{0.25 mm} & \textbf{0.50 mm}\\
+\midrule
+Axial Stiffness Dz [N/um] & 94.0 & 124.7\\
+Shear Stiffness [N/um] & 12.7 & 25.8\\
+Bending Stiffness Rx [Nm/rad] & 4.8 & 26.0\\
+Bending Stiffness Ry [Nm/rad] & 4.8 & 26.0\\
+Torsion Stiffness Rz [Nm/rad] & 260.2 & 538.0\\
+\bottomrule
+\end{tabularx}
+\end{table}
+\chapter{Complete Strut with Encoder}
+\label{sec:strut_encoder}
+\section{Introduction}
+
+Now, the full nano-hexapod strut is modelled using Ansys.
+
+The 3D as well as the interface nodes are shown in Figure \ref{fig:strut_encoder_points3}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=1.0\linewidth]{figs/strut_fem_nodes.jpg}
+\caption{\label{fig:strut_encoder_points3}Interface points}
+\end{figure}
+
+A side view is shown in Figure \ref{fig:strut_encoder_nodes_side}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=1.0\linewidth]{figs/strut_fem_nodes_side.jpg}
+\caption{\label{fig:strut_encoder_nodes_side}Interface points - Side view}
+\end{figure}
+
+The flexible joints used have a 0.25mm width size.
+\section{Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates}
+We first extract the stiffness and mass matrices.
+\begin{table}[htbp]
+\caption{\label{tab:strut_mat_K}First 10x10 elements of the Stiffness matrix}
+\centering
+\tiny
+\begin{tabularx}{\linewidth}{XXXXXXXXXX}
+\toprule
+2000000 & 1000000 & -3000000 & -400 & 300 & 200 & -30 & 2000 & -10000 & 0.3\\
+1000000 & 4000000 & -8000000 & -900 & 400 & -50 & -6000 & 10000 & -20000 & 3\\
+-3000000 & -8000000 & 20000000 & 2000 & -900 & 200 & -10000 & 20000 & -300000 & 7\\
+-400 & -900 & 2000 & 5 & -0.1 & 05 & 1 & -3 & 6 & -0007\\
+300 & 400 & -900 & -0.1 & 5 & 04 & -0.1 & 0.5 & -3 & 0001\\
+200 & -50 & 200 & 05 & 04 & 300 & 4 & -01 & -1 & 3e-05\\
+-30 & -6000 & -10000 & 1 & -0.1 & 4 & 3000000 & -1000000 & -2000000 & -300\\
+2000 & 10000 & 20000 & -3 & 0.5 & -01 & -1000000 & 6000000 & 7000000 & 1000\\
+-10000 & -20000 & -300000 & 6 & -3 & -1 & -2000000 & 7000000 & 20000000 & 2000\\
+0.3 & 3 & 7 & -0007 & 0001 & 3e-05 & -300 & 1000 & 2000 & 5\\
+\bottomrule
+\end{tabularx}
+\end{table}
+
+
+\begin{table}[htbp]
+\caption{\label{tab:strut_mat_M}First 10x10 elements of the Mass matrix}
+\centering
+\tiny
+\begin{tabularx}{\linewidth}{XXXXXXXXXX}
+\toprule
+0.04 & -0.005 & 0.007 & 2e-06 & 0.0001 & -5e-07 & -1e-05 & -9e-07 & 8e-05 & -5e-10\\
+-0.005 & 0.03 & 0.02 & -0.0001 & 1e-06 & -3e-07 & 3e-05 & -0.0001 & 8e-05 & -3e-08\\
+0.007 & 0.02 & 0.08 & -6e-06 & -5e-06 & -7e-07 & 4e-05 & -0.0001 & 0.0005 & -3e-08\\
+2e-06 & -0.0001 & -6e-06 & 2e-06 & -4e-10 & 2e-11 & -8e-09 & 3e-08 & -2e-08 & 6e-12\\
+0.0001 & 1e-06 & -5e-06 & -4e-10 & 3e-06 & 2e-10 & -3e-09 & 3e-09 & -7e-09 & 6e-13\\
+-5e-07 & -3e-07 & -7e-07 & 2e-11 & 2e-10 & 5e-07 & -2e-08 & 5e-09 & -5e-09 & 1e-12\\
+-1e-05 & 3e-05 & 4e-05 & -8e-09 & -3e-09 & -2e-08 & 0.04 & 0.004 & 0.003 & 1e-06\\
+-9e-07 & -0.0001 & -0.0001 & 3e-08 & 3e-09 & 5e-09 & 0.004 & 0.02 & -0.02 & 0.0001\\
+8e-05 & 8e-05 & 0.0005 & -2e-08 & -7e-09 & -5e-09 & 0.003 & -0.02 & 0.08 & -5e-06\\
+-5e-10 & -3e-08 & -3e-08 & 6e-12 & 6e-13 & 1e-12 & 1e-06 & 0.0001 & -5e-06 & 2e-06\\
+\bottomrule
+\end{tabularx}
+\end{table}
+
+
+Then, we extract the coordinates of the interface nodes.
+\begin{table}[htbp]
+\caption{\label{tab:parameters_fem_strut}Some extracted parameters of the FEM}
+\centering
+\begin{tabularx}{0.4\linewidth}{lc}
+\toprule
+Total number of Nodes & 8\\
+Number of interface Nodes & 8\\
+Number of Modes & 6\\
+Size of M and K matrices & 54\\
+\bottomrule
+\end{tabularx}
+\end{table}
+
+\begin{table}[htbp]
+\caption{\label{tab:coordinate_nodes_strut}Coordinates of the interface nodes}
+\centering
+\begin{tabularx}{0.6\linewidth}{ccccc}
+\toprule
+Node i & Node Number & x [m] & y [m] & z [m]\\
+\midrule
+1.0 & 504411.0 & 0.0 & 0.0 & 0.0405\\
+2.0 & 504412.0 & 0.0 & 0.0 & -0.0405\\
+3.0 & 504413.0 & -0.0325 & 0.0 & 0.0\\
+4.0 & 504414.0 & -0.0125 & 0.0 & 0.0\\
+5.0 & 504415.0 & -0.0075 & 0.0 & 0.0\\
+6.0 & 504416.0 & 0.0325 & 0.0 & 0.0\\
+7.0 & 504417.0 & 0.004 & 0.0145 & -0.00175\\
+8.0 & 504418.0 & 0.004 & 0.0166 & -0.00175\\
+\bottomrule
+\end{tabularx}
+\end{table}
+
+Using \texttt{K}, \texttt{M} and \texttt{int\_xyz}, we can use the \texttt{Reduced Order Flexible Solid} simscape block.
+\section{Piezoelectric parameters}
+Parameters for the APA300ML:
+\section{Identification of the Dynamics}
+The dynamics is identified from the applied force to the measured relative displacement.
+The same dynamics is identified for a payload mass of 10Kg.
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/dynamics_encoder_full_strut.png}
+\caption{\label{fig:dynamics_encoder_full_strut}Dynamics from the force actuator to the measured motion by the encoder}
+\end{figure}
+\printbibliography[heading=bibintoc,title={Bibliography}]
+\end{document}

preamble.tex

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ref.bib

@@ -1,21 +0,0 @@
-@article{souleille18_concep_activ_mount_space_applic,
-  author = {Souleille, Adrien and Lampert, Thibault and 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},
-}
-
-@book{fleming14_desig_model_contr_nanop_system,
-  author = {Andrew J. Fleming and Kam K. Leang},
-  title = {Design, Modeling and Control of Nanopositioning Systems},
-  year = {2014},
-  publisher = {Springer International Publishing},
-  url = {https://doi.org/10.1007/978-3-319-06617-2},
-  doi = {10.1007/978-3-319-06617-2},
-  pages = {nil},
-  series = {Advances in Industrial Control},
-}