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.gitignore
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259
.gitignore
vendored
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ltximg/
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*.autosave
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slprj/
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matlab/slprj/
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*.slxc
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# ============================================================
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# ============================================================
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||||
# LATEX
|
||||
# ============================================================
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||||
# ============================================================
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||||
|
||||
## Core latex/pdflatex auxiliary files:
|
||||
*.aux
|
||||
*.lof
|
||||
*.log
|
||||
*.lot
|
||||
*.fls
|
||||
*.out
|
||||
*.toc
|
||||
*.fmt
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||||
*.fot
|
||||
*.cb
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||||
*.cb2
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||||
.*.lb
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||||
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||||
## Intermediate documents:
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||||
*.dvi
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||||
*.xdv
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||||
*-converted-to.*
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||||
# these rules might exclude image files for figures etc.
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||||
# *.ps
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# *.eps
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||||
# *.pdf
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||||
|
||||
## Generated if empty string is given at "Please type another file name for output:"
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||||
.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
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||||
# latexrun
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||||
latex.out/
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||||
|
||||
## Auxiliary and intermediate files from other packages:
|
||||
# algorithms
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||||
*.alg
|
||||
*.loa
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|
||||
# achemso
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||||
acs-*.bib
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||||
|
||||
# amsthm
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||||
*.thm
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||||
|
||||
# beamer
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||||
*.nav
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||||
*.pre
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||||
*.snm
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||||
*.vrb
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||||
|
||||
# changes
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||||
*.soc
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||||
# cprotect
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*.cpt
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||||
# elsarticle (documentclass of Elsevier journals)
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||||
*.spl
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||||
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||||
# endnotes
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||||
*.ent
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|
||||
# fixme
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||||
*.lox
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|
||||
# feynmf/feynmp
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||||
*.mf
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||||
*.mp
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||||
*.t[1-9]
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||||
*.t[1-9][0-9]
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||||
*.tfm
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||||
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||||
#(r)(e)ledmac/(r)(e)ledpar
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*.end
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||||
*.?end
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||||
*.[1-9]
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||||
*.[1-9][0-9]
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||||
*.[1-9][0-9][0-9]
|
||||
*.[1-9]R
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||||
*.[1-9][0-9]R
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||||
*.[1-9][0-9][0-9]R
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||||
*.eledsec[1-9]
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||||
*.eledsec[1-9]R
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||||
*.eledsec[1-9][0-9]
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||||
*.eledsec[1-9][0-9]R
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||||
*.eledsec[1-9][0-9][0-9]
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||||
*.eledsec[1-9][0-9][0-9]R
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||||
|
||||
# glossaries
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||||
*.acn
|
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*.acr
|
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*.glg
|
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*.glo
|
||||
*.gls
|
||||
*.glsdefs
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||||
|
||||
# gnuplottex
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||||
*-gnuplottex-*
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||||
|
||||
# gregoriotex
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||||
*.gaux
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||||
*.gtex
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||||
|
||||
# htlatex
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||||
*.4ct
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||||
*.4tc
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||||
*.idv
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||||
*.lg
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||||
*.trc
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||||
*.xref
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||||
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||||
# hyperref
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||||
*.brf
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||||
|
||||
# knitr
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||||
*-concordance.tex
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||||
# TODO Comment the next line if you want to keep your tikz graphics files
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||||
*.tikz
|
||||
*-tikzDictionary
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||||
|
||||
# listings
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||||
*.lol
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|
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# makeidx
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||||
*.idx
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||||
*.ilg
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||||
*.ind
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||||
*.ist
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||||
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||||
# minitoc
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||||
*.maf
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||||
*.mlf
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||||
*.mlt
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||||
*.mtc[0-9]*
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||||
*.slf[0-9]*
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||||
*.slt[0-9]*
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||||
*.stc[0-9]*
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||||
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||||
# minted
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||||
_minted*
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||||
*.pyg
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||||
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||||
# morewrites
|
||||
*.mw
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||||
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||||
# nomencl
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||||
*.nlg
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||||
*.nlo
|
||||
*.nls
|
||||
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||||
# pax
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||||
*.pax
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||||
|
||||
# pdfpcnotes
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||||
*.pdfpc
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||||
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||||
# sagetex
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||||
*.sagetex.sage
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||||
*.sagetex.py
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||||
*.sagetex.scmd
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||||
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||||
# scrwfile
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||||
*.wrt
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||||
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||||
# sympy
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||||
*.sout
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||||
*.sympy
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||||
sympy-plots-for-*.tex/
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||||
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||||
# pdfcomment
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||||
*.upa
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||||
*.upb
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||||
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||||
# pythontex
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||||
*.pytxcode
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||||
pythontex-files-*/
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||||
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||||
# thmtools
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||||
*.loe
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||||
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||||
# TikZ & PGF
|
||||
*.dpth
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||||
*.md5
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||||
*.auxlock
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||||
|
||||
# todonotes
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||||
*.tdo
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||||
|
||||
# easy-todo
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||||
*.lod
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||||
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||||
# xmpincl
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||||
*.xmpi
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||||
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||||
# xindy
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||||
*.xdy
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||||
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||||
# xypic precompiled matrices
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||||
*.xyc
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||||
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||||
# endfloat
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||||
*.ttt
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||||
*.fff
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||||
|
||||
# Latexian
|
||||
TSWLatexianTemp*
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||||
|
||||
## Editors:
|
||||
# WinEdt
|
||||
*.bak
|
||||
*.sav
|
||||
|
||||
# Texpad
|
||||
.texpadtmp
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||||
|
||||
# LyX
|
||||
*.lyx~
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||||
|
||||
# Kile
|
||||
*.backup
|
||||
|
||||
# KBibTeX
|
||||
*~[0-9]*
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||||
|
||||
# auto folder when using emacs and auctex
|
||||
./auto/*
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||||
*.el
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||||
|
||||
# expex forward references with \gathertags
|
||||
*-tags.tex
|
||||
|
||||
# standalone packages
|
||||
*.sta
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||||
1
inkscape/figs
Symbolic link
1
inkscape/figs
Symbolic link
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||||
../paper/figs
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||||
216
inkscape/tikz.org
Normal file
216
inkscape/tikz.org
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@@ -0,0 +1,216 @@
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||||
#+TITLE: Decoupling Properties of the Cubic Architecture
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||||
:DRAWER:
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||||
#+LANGUAGE: en
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||||
#+EMAIL: dehaeze.thomas@gmail.com
|
||||
#+AUTHOR: Dehaeze Thomas
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|
||||
#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/tikz/org/}{config.tex}")
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||||
#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
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||||
#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
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||||
#+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")
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||||
:END:
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||||
|
||||
#+latex: \clearpage
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||||
|
||||
#+begin_src latex :file detail_kinematics_cubic_schematic_full.pdf :results file
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||||
\begin{tikzpicture}
|
||||
\begin{scope}[rotate={45}, shift={(0, 0, -4)}]
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||||
% We first define the coordinate of the points of the Cube
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||||
\coordinate[] (bot) at (0,0,4);
|
||||
\coordinate[] (top) at (4,4,0);
|
||||
\coordinate[] (A1) at (0,0,0);
|
||||
\coordinate[] (A2) at (4,0,4);
|
||||
\coordinate[] (A3) at (0,4,4);
|
||||
\coordinate[] (B1) at (4,0,0);
|
||||
\coordinate[] (B2) at (4,4,4);
|
||||
\coordinate[] (B3) at (0,4,0);
|
||||
|
||||
% Center of the Cube
|
||||
\coordinate[] (cubecenter) at ($0.5*(bot) + 0.5*(top)$);
|
||||
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||||
% We draw parts of the cube that corresponds to the Stewart platform
|
||||
\draw[] (A1)node[]{$\bullet$} -- (B1)node[]{$\bullet$} -- (A2)node[]{$\bullet$} -- (B2)node[]{$\bullet$} -- (A3)node[]{$\bullet$} -- (B3)node[]{$\bullet$} -- (A1);
|
||||
|
||||
% ai and bi are computed
|
||||
\def\lfrom{0.0}
|
||||
\def\lto{1.0}
|
||||
|
||||
\coordinate(a1) at ($(A1) - \lfrom*(A1) + \lfrom*(B1)$);
|
||||
\coordinate(b1) at ($(A1) - \lto*(A1) + \lto*(B1)$);
|
||||
\coordinate(a2) at ($(A2) - \lfrom*(A2) + \lfrom*(B1)$);
|
||||
\coordinate(b2) at ($(A2) - \lto*(A2) + \lto*(B1)$);
|
||||
\coordinate(a3) at ($(A2) - \lfrom*(A2) + \lfrom*(B2)$);
|
||||
\coordinate(b3) at ($(A2) - \lto*(A2) + \lto*(B2)$);
|
||||
\coordinate(a4) at ($(A3) - \lfrom*(A3) + \lfrom*(B2)$);
|
||||
\coordinate(b4) at ($(A3) - \lto*(A3) + \lto*(B2)$);
|
||||
\coordinate(a5) at ($(A3) - \lfrom*(A3) + \lfrom*(B3)$);
|
||||
\coordinate(b5) at ($(A3) - \lto*(A3) + \lto*(B3)$);
|
||||
\coordinate(a6) at ($(A1) - \lfrom*(A1) + \lfrom*(B3)$);
|
||||
\coordinate(b6) at ($(A1) - \lto*(A1) + \lto*(B3)$);
|
||||
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||||
% We draw the fixed and mobiles platforms
|
||||
\path[fill=colorblue, opacity=0.2] (a1) -- (a2) -- (a3) -- (a4) -- (a5) -- (a6) -- cycle;
|
||||
\path[fill=colorblue, opacity=0.2] (b1) -- (b2) -- (b3) -- (b4) -- (b5) -- (b6) -- cycle;
|
||||
\draw[color=colorblue, dashed] (a1) -- (a2) -- (a3) -- (a4) -- (a5) -- (a6) -- cycle;
|
||||
\draw[color=colorblue, dashed] (b1) -- (b2) -- (b3) -- (b4) -- (b5) -- (b6) -- cycle;
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||||
|
||||
% The legs of the hexapod are drawn
|
||||
\draw[color=colorblue] (a1)node{$\bullet$} -- (b1)node{$\bullet$};
|
||||
\draw[color=colorblue] (a2)node{$\bullet$} -- (b2)node{$\bullet$};
|
||||
\draw[color=colorblue] (a3)node{$\bullet$} -- (b3)node{$\bullet$};
|
||||
\draw[color=colorblue] (a4)node{$\bullet$} -- (b4)node{$\bullet$};
|
||||
\draw[color=colorblue] (a5)node{$\bullet$} -- (b5)node{$\bullet$};
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||||
\draw[color=colorblue] (a6)node{$\bullet$} -- (b6)node{$\bullet$};
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||||
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||||
% Unit vector
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||||
\draw[color=colorred, ->] ($0.9*(a1)+0.1*(b1)$)node{$\bullet$} -- ($0.65*(a1)+0.35*(b1)$)node[right]{$\hat{\bm{s}}_3$};
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||||
\draw[color=colorred, ->] ($0.9*(a2)+0.1*(b2)$)node{$\bullet$} -- ($0.65*(a2)+0.35*(b2)$)node[left]{$\hat{\bm{s}}_4$};
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||||
\draw[color=colorred, ->] ($0.9*(a3)+0.1*(b3)$)node{$\bullet$} -- ($0.65*(a3)+0.35*(b3)$)node[below]{$\hat{\bm{s}}_5$};
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||||
\draw[color=colorred, ->] ($0.9*(a4)+0.1*(b4)$)node{$\bullet$} -- ($0.65*(a4)+0.35*(b4)$)node[below]{$\hat{\bm{s}}_6$};
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||||
\draw[color=colorred, ->] ($0.9*(a5)+0.1*(b5)$)node{$\bullet$} -- ($0.65*(a5)+0.35*(b5)$)node[left]{$\hat{\bm{s}}_1$};
|
||||
\draw[color=colorred, ->] ($0.9*(a6)+0.1*(b6)$)node{$\bullet$} -- ($0.65*(a6)+0.35*(b6)$)node[right]{$\hat{\bm{s}}_2$};
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||||
|
||||
% Labels
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||||
\node[above=0.1 of B1] {$\tilde{\bm{b}}_3 = \tilde{\bm{b}}_4$};
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||||
\node[above=0.1 of B2] {$\tilde{\bm{b}}_5 = \tilde{\bm{b}}_6$};
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||||
\node[above=0.1 of B3] {$\tilde{\bm{b}}_1 = \tilde{\bm{b}}_2$};
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||||
\end{scope}
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||||
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||||
% Height of the Hexapod
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||||
\coordinate[] (sizepos) at ($(a2)+(0.2, 0)$);
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||||
\coordinate[] (origin) at (0,0,0);
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||||
|
||||
\draw[->, color=colorgreen] (cubecenter.center) node[above right]{$\{B\}$} -- ++(0,0,1);
|
||||
\draw[->, color=colorgreen] (cubecenter.center) -- ++(1,0,0);
|
||||
\draw[->, color=colorgreen] (cubecenter.center) -- ++(0,1,0);
|
||||
|
||||
\node[] at (cubecenter.center){$\bullet$};
|
||||
\node[above left] at (cubecenter.center){$\{C\}$};
|
||||
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||||
% Useful part of the cube
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||||
\draw[<->, dashed] ($(A2)+(0.5,0)$) -- node[midway, right]{$H_{C}$} ($(B1)+(0.5,0)$);
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||||
\end{tikzpicture}
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||||
#+end_src
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||||
|
||||
#+RESULTS:
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||||
[[file:figs/detail_kinematics_cubic_schematic_full.png]]
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||||
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||||
#+begin_src latex :file detail_kinematics_cubic_schematic.pdf :results file
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||||
\begin{tikzpicture}
|
||||
\begin{scope}[rotate={45}, shift={(0, 0, -4)}]
|
||||
% We first define the coordinate of the points of the Cube
|
||||
\coordinate[] (bot) at (0,0,4);
|
||||
\coordinate[] (top) at (4,4,0);
|
||||
\coordinate[] (A1) at (0,0,0);
|
||||
\coordinate[] (A2) at (4,0,4);
|
||||
\coordinate[] (A3) at (0,4,4);
|
||||
\coordinate[] (B1) at (4,0,0);
|
||||
\coordinate[] (B2) at (4,4,4);
|
||||
\coordinate[] (B3) at (0,4,0);
|
||||
|
||||
% Center of the Cube
|
||||
\coordinate[] (cubecenter) at ($0.5*(bot) + 0.5*(top)$);
|
||||
|
||||
% We draw parts of the cube that corresponds to the Stewart platform
|
||||
\draw[] (A1)node[]{$\bullet$} -- (B1)node[]{$\bullet$} -- (A2)node[]{$\bullet$} -- (B2)node[]{$\bullet$} -- (A3)node[]{$\bullet$} -- (B3)node[]{$\bullet$} -- (A1);
|
||||
|
||||
% ai and bi are computed
|
||||
\def\lfrom{0.2}
|
||||
\def\lto{0.8}
|
||||
|
||||
\coordinate(a1) at ($(A1) - \lfrom*(A1) + \lfrom*(B1)$);
|
||||
\coordinate(b1) at ($(A1) - \lto*(A1) + \lto*(B1)$);
|
||||
\coordinate(a2) at ($(A2) - \lfrom*(A2) + \lfrom*(B1)$);
|
||||
\coordinate(b2) at ($(A2) - \lto*(A2) + \lto*(B1)$);
|
||||
\coordinate(a3) at ($(A2) - \lfrom*(A2) + \lfrom*(B2)$);
|
||||
\coordinate(b3) at ($(A2) - \lto*(A2) + \lto*(B2)$);
|
||||
\coordinate(a4) at ($(A3) - \lfrom*(A3) + \lfrom*(B2)$);
|
||||
\coordinate(b4) at ($(A3) - \lto*(A3) + \lto*(B2)$);
|
||||
\coordinate(a5) at ($(A3) - \lfrom*(A3) + \lfrom*(B3)$);
|
||||
\coordinate(b5) at ($(A3) - \lto*(A3) + \lto*(B3)$);
|
||||
\coordinate(a6) at ($(A1) - \lfrom*(A1) + \lfrom*(B3)$);
|
||||
\coordinate(b6) at ($(A1) - \lto*(A1) + \lto*(B3)$);
|
||||
|
||||
% We draw the fixed and mobiles platforms
|
||||
\path[fill=colorblue, opacity=0.2] (a1) -- (a2) -- (a3) -- (a4) -- (a5) -- (a6) -- cycle;
|
||||
\path[fill=colorblue, opacity=0.2] (b1) -- (b2) -- (b3) -- (b4) -- (b5) -- (b6) -- cycle;
|
||||
\draw[color=colorblue, dashed] (a1) -- (a2) -- (a3) -- (a4) -- (a5) -- (a6) -- cycle;
|
||||
\draw[color=colorblue, dashed] (b1) -- (b2) -- (b3) -- (b4) -- (b5) -- (b6) -- cycle;
|
||||
|
||||
% The legs of the hexapod are drawn
|
||||
\draw[color=colorblue] (a1)node{$\bullet$} -- (b1)node{$\bullet$}node[below right]{$\bm{b}_3$};
|
||||
\draw[color=colorblue] (a2)node{$\bullet$} -- (b2)node{$\bullet$}node[right]{$\bm{b}_4$};
|
||||
\draw[color=colorblue] (a3)node{$\bullet$} -- (b3)node{$\bullet$}node[above right]{$\bm{b}_5$};
|
||||
\draw[color=colorblue] (a4)node{$\bullet$} -- (b4)node{$\bullet$}node[above left]{$\bm{b}_6$};
|
||||
\draw[color=colorblue] (a5)node{$\bullet$} -- (b5)node{$\bullet$}node[left]{$\bm{b}_1$};
|
||||
\draw[color=colorblue] (a6)node{$\bullet$} -- (b6)node{$\bullet$}node[below left]{$\bm{b}_2$};
|
||||
\end{scope}
|
||||
|
||||
% Height of the Hexapod
|
||||
\coordinate[] (sizepos) at ($(a2)+(0.2, 0)$);
|
||||
\coordinate[] (origin) at (0,0,0);
|
||||
|
||||
\draw[->, color=colorgreen] ($(cubecenter.center)+(0,2.0,0)$) node[above right]{$\{B\}$} -- ++(0,0,1);
|
||||
\draw[->, color=colorgreen] ($(cubecenter.center)+(0,2.0,0)$) -- ++(1,0,0);
|
||||
\draw[->, color=colorgreen] ($(cubecenter.center)+(0,2.0,0)$) -- ++(0,1,0);
|
||||
|
||||
\node[] at (cubecenter.center){$\bullet$};
|
||||
\node[right] at (cubecenter.center){$\{C\}$};
|
||||
|
||||
\draw[<->, dashed] (cubecenter.center) -- node[midway, right]{$H$} ($(cubecenter.center)+(0,2.0,0)$);
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
[[file:figs/detail_kinematics_cubic_schematic.png]]
|
||||
|
||||
#+begin_src latex :file detail_kinematics_centralized_control.pdf
|
||||
\begin{tikzpicture}
|
||||
\node[block] (Jt) at (0, 0) {$\bm{J}^{-\intercal}$};
|
||||
\node[block, right= of Jt] (G) {$\bm{G}$};
|
||||
\node[block, right= of G] (J) {$\bm{J}^{-1}$};
|
||||
\node[block, left= of Jt] (Kx) {$\bm{K}_{\mathcal{X}}$};
|
||||
|
||||
\draw[->] (Kx.east) -- node[midway, above]{$\bm{\mathcal{F}}$} (Jt.west);
|
||||
\draw[->] (Jt.east) -- (G.west) node[above left]{$\bm{\tau}$};
|
||||
\draw[->] (G.east) -- (J.west) node[above left]{$\bm{\mathcal{L}}$};
|
||||
\draw[->] (J.east) -- ++(1.0, 0);
|
||||
\draw[->] ($(J.east) + (0.5, 0)$)node[]{$\bullet$} node[above]{$\bm{\mathcal{X}}$} -- ++(0, -1) -| ($(Kx.west) + (-0.5, 0)$) -- (Kx.west);
|
||||
|
||||
\begin{scope}[on background layer]
|
||||
\node[fit={(Jt.south west) (J.north east)}, fill=black!20!white, draw, dashed, inner sep=4pt] (Px) {};
|
||||
\node[anchor={south}] at (Px.north){\small{Cartesian Plant}};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
[[file:figs/detail_kinematics_centralized_control.png]]
|
||||
|
||||
#+begin_src latex :file detail_kinematics_decentralized_control.pdf
|
||||
\begin{tikzpicture}
|
||||
\node[block] (G) at (0,0) {$\bm{G}$};
|
||||
|
||||
\node[block, left= of G] (Kl) {$\bm{K}_{\mathcal{L}}$};
|
||||
|
||||
\draw[->] (Kl.east) -- node[midway, above]{$\bm{\tau}$} (G.west);
|
||||
\draw[->] (G.east) -- ++(1.0, 0);
|
||||
\draw[->] ($(G.east) + (0.5, 0)$)node[]{$\bullet$} node[above]{$\bm{\mathcal{L}}$} -- ++(0, -1) -| ($(Kl.west) + (-0.5, 0)$) -- (Kl.west);
|
||||
|
||||
\begin{scope}[on background layer]
|
||||
\node[fit={(G.south west) (G.north east)}, fill=black!20!white, draw, dashed, inner sep=4pt] (Pl) {};
|
||||
\node[anchor={south}] at (Pl.north){\small{Strut Plant}};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
[[file:figs/detail_kinematics_decentralized_control.png]]
|
||||
749
matlab/dehaeze26_cubic_architecture.m
Normal file
749
matlab/dehaeze26_cubic_architecture.m
Normal file
@@ -0,0 +1,749 @@
|
||||
%% Clear Workspace and Close figures
|
||||
clear; close all; clc;
|
||||
|
||||
%% Intialize Laplace variable
|
||||
s = zpk('s');
|
||||
|
||||
%% Path for functions, data and scripts
|
||||
addpath('./src/'); % Path for functions
|
||||
addpath('./subsystems/'); % Path for Subsystems Simulink files
|
||||
|
||||
% Simulink Model name
|
||||
mdl = 'nano_hexapod_model';
|
||||
|
||||
%% Colors for the figures
|
||||
colors = colororder;
|
||||
|
||||
%% Example of a typical "cubic" architecture
|
||||
MO_B = -50e-3; % Position {B} with respect to {M} [m]
|
||||
|
||||
H = 100e-3; % Height of the Stewart platform [m]
|
||||
|
||||
Hc = 100e-3; % Size of the useful part of the cube [m]
|
||||
|
||||
FOc = H + MO_B; % Center of the cube with respect to {F}
|
||||
% here positionned at the frame {B}
|
||||
|
||||
stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
|
||||
stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 5e-3, 'MHb', 5e-3);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart, 'k', 1);
|
||||
stewart = initializeJointDynamics(stewart);
|
||||
stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 150e-3, 'Mpr', 150e-3);
|
||||
stewart = initializeCylindricalStruts(stewart);
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
|
||||
displayArchitecture(stewart, 'labels', false, 'frames', false);
|
||||
plotCube(stewart, 'Hc', Hc, 'FOc', FOc, 'color', [0,0,0,0.2], 'link_to_struts', false);
|
||||
view([40, 5]);
|
||||
|
||||
%% Example of a typical "cubic" architecture
|
||||
MO_B = -20e-3; % Position {B} with respect to {M} [m]
|
||||
|
||||
H = 40e-3; % Height of the Stewart platform [m]
|
||||
|
||||
Hc = 100e-3; % Size of the useful part of the cube [m]
|
||||
|
||||
FOc = H + MO_B; % Center of the cube with respect to {F}
|
||||
% here positionned at the frame {B}
|
||||
|
||||
stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
|
||||
stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 5e-3, 'MHb', 5e-3);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart, 'k', 1);
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 150e-3, 'Mpr', 150e-3);
|
||||
|
||||
displayArchitecture(stewart, 'labels', false, 'frames', false);
|
||||
plotCube(stewart, 'Hc', Hc, 'FOc', FOc, 'color', [0,0,0,0.2], 'link_to_struts', false);
|
||||
view([40, 5]);
|
||||
|
||||
%% Analytical formula for Stiffness matrix of the Cubic Stewart platform
|
||||
% Define symbolic variables
|
||||
syms k Hc alpha H
|
||||
|
||||
assume(k > 0); % k is positive real
|
||||
assume(Hc, 'real'); % Hc is real
|
||||
assume(H, 'real'); % H is real
|
||||
assume(alpha, 'real'); % alpha is real
|
||||
|
||||
% Define si matrix (edges of the cubes)
|
||||
si = 1/sqrt(3)*[
|
||||
[ sqrt(2), 0, 1]; ...
|
||||
[-sqrt(2)/2, -sqrt(3/2), 1]; ...
|
||||
[-sqrt(2)/2, sqrt(3/2), 1]; ...
|
||||
[ sqrt(2), 0, 1]; ...
|
||||
[-sqrt(2)/2, -sqrt(3/2), 1]; ...
|
||||
[-sqrt(2)/2, sqrt(3/2), 1] ...
|
||||
];
|
||||
|
||||
% Define ci matrix (vertices of the cubes)
|
||||
ci = Hc * [
|
||||
[1/sqrt(2), -sqrt(3)/sqrt(2), 0.5]; ...
|
||||
[1/sqrt(2), -sqrt(3)/sqrt(2), 0.5]; ...
|
||||
[1/sqrt(2), sqrt(3)/sqrt(2), 0.5]; ...
|
||||
[1/sqrt(2), sqrt(3)/sqrt(2), 0.5]; ...
|
||||
[-sqrt(2), 0, 0.5]; ...
|
||||
[-sqrt(2), 0, 0.5] ...
|
||||
];
|
||||
|
||||
% Apply vertical shift to ci
|
||||
ci = ci + H * [0, 0, 1];
|
||||
|
||||
% Calculate bi vectors (Stewart platform top joints)
|
||||
bi = ci + alpha * si;
|
||||
|
||||
% Initialize stiffness matrix
|
||||
K = sym(zeros(6,6));
|
||||
|
||||
% Calculate elements of the stiffness matrix
|
||||
for i = 1:6
|
||||
% Extract vectors for each leg
|
||||
s_i = si(i,:)';
|
||||
b_i = bi(i,:)';
|
||||
|
||||
% Calculate cross product vector
|
||||
cross_bs = cross(b_i, s_i);
|
||||
|
||||
% Build matrix blocks
|
||||
K(1:3, 4:6) = K(1:3, 4:6) + s_i * cross_bs';
|
||||
K(4:6, 1:3) = K(4:6, 1:3) + cross_bs * s_i';
|
||||
K(1:3, 1:3) = K(1:3, 1:3) + s_i * s_i';
|
||||
K(4:6, 4:6) = K(4:6, 4:6) + cross_bs * cross_bs';
|
||||
end
|
||||
|
||||
% Scale by stiffness coefficient
|
||||
K = k * K;
|
||||
|
||||
% Simplify the expressions
|
||||
K = simplify(K);
|
||||
|
||||
% Display the analytical stiffness matrix
|
||||
disp('Analytical Stiffness Matrix:');
|
||||
pretty(K);
|
||||
|
||||
%% Cubic configuration
|
||||
H = 100e-3; % Height of the Stewart platform [m]
|
||||
Hc = 100e-3; % Size of the useful part of the cube [m]
|
||||
FOc = 50e-3; % Center of the cube at the Stewart platform center
|
||||
MO_B = -50e-3; % Position {B} with respect to {M} [m]
|
||||
MHb = 0;
|
||||
|
||||
stewart_cubic = initializeStewartPlatform();
|
||||
stewart_cubic = initializeFramesPositions(stewart_cubic, 'H', H, 'MO_B', MO_B);
|
||||
stewart_cubic = generateCubicConfiguration(stewart_cubic, 'Hc', Hc, 'FOc', FOc, 'FHa', 5e-3, 'MHb', MHb);
|
||||
stewart_cubic = computeJointsPose(stewart_cubic);
|
||||
stewart_cubic = initializeStrutDynamics(stewart_cubic, 'k', 1);
|
||||
stewart_cubic = computeJacobian(stewart_cubic);
|
||||
stewart_cubic = initializeCylindricalPlatforms(stewart_cubic, 'Fpr', 150e-3, 'Mpr', 150e-3);
|
||||
|
||||
% Let's now define the actuator stroke.
|
||||
L_max = 50e-6; % [m]
|
||||
|
||||
%% Mobility of a Stewart platform with Cubic architecture - Translations
|
||||
thetas = linspace(0, pi, 100);
|
||||
phis = linspace(0, 2*pi, 200);
|
||||
rs = zeros(length(thetas), length(phis));
|
||||
|
||||
for i = 1:length(thetas)
|
||||
for j = 1:length(phis)
|
||||
Tx = sin(thetas(i))*cos(phis(j));
|
||||
Ty = sin(thetas(i))*sin(phis(j));
|
||||
Tz = cos(thetas(i));
|
||||
|
||||
dL = stewart_cubic.kinematics.J*[Tx; Ty; Tz; 0; 0; 0;]; % dL required for 1m displacement in theta/phi direction
|
||||
|
||||
rs(i, j) = L_max/max(abs(dL));
|
||||
% rs(i, j) = max(abs([dL(dL<0)*L_min; dL(dL>=0)*L_max]));
|
||||
end
|
||||
end
|
||||
|
||||
[phi_grid, theta_grid] = meshgrid(phis, thetas);
|
||||
X = 1e6 * rs .* sin(theta_grid) .* cos(phi_grid);
|
||||
Y = 1e6 * rs .* sin(theta_grid) .* sin(phi_grid);
|
||||
Z = 1e6 * rs .* cos(theta_grid);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
surf(X, Y, Z, 'FaceColor', 'white', 'EdgeColor', colors(1,:));
|
||||
quiver3(0, 0, 0, 150, 0, 0, 'k', 'LineWidth', 2, 'MaxHeadSize', 0.7);
|
||||
quiver3(0, 0, 0, 0, 150, 0, 'k', 'LineWidth', 2, 'MaxHeadSize', 0.7);
|
||||
quiver3(0, 0, 0, 0, 0, 150, 'k', 'LineWidth', 2, 'MaxHeadSize', 0.7);
|
||||
text(150, 0, 0, '$D_x$', 'FontSize', 10, 'HorizontalAlignment', 'center', 'VerticalAlignment', 'top' );
|
||||
text(0, 150, 0, '$D_y$', 'FontSize', 10, 'HorizontalAlignment', 'right', 'VerticalAlignment', 'bottom');
|
||||
text(0, 0, 150, '$D_z$', 'FontSize', 10, 'HorizontalAlignment', 'left', 'VerticalAlignment', 'top' );
|
||||
hold off;
|
||||
axis equal;
|
||||
grid off;
|
||||
axis off;
|
||||
view(105, 15);
|
||||
|
||||
%% Mobility of a Stewart platform with Cubic architecture - Rotations
|
||||
thetas = linspace(0, pi, 100);
|
||||
phis = linspace(0, 2*pi, 200);
|
||||
rs_cubic = zeros(length(thetas), length(phis));
|
||||
|
||||
for i = 1:length(thetas)
|
||||
for j = 1:length(phis)
|
||||
Rx = sin(thetas(i))*cos(phis(j));
|
||||
Ry = sin(thetas(i))*sin(phis(j));
|
||||
Rz = cos(thetas(i));
|
||||
|
||||
dL = stewart_cubic.kinematics.J*[0; 0; 0; Rx; Ry; Rz;];
|
||||
rs_cubic(i, j) = L_max/max(abs(dL));
|
||||
end
|
||||
end
|
||||
|
||||
[phi_grid, theta_grid] = meshgrid(phis, thetas);
|
||||
|
||||
X_cubic = 1e6 * rs_cubic .* sin(theta_grid) .* cos(phi_grid);
|
||||
Y_cubic = 1e6 * rs_cubic .* sin(theta_grid) .* sin(phi_grid);
|
||||
Z_cubic = 1e6 * rs_cubic .* cos(theta_grid);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
surf(X_cubic, Y_cubic, Z_cubic, 'FaceColor', 'white', 'LineWidth', 0.2, 'EdgeColor', colors(1,:));
|
||||
quiver3(0, 0, 0, 1500, 0, 0, 'k', 'LineWidth', 2, 'MaxHeadSize', 0.7);
|
||||
quiver3(0, 0, 0, 0, 1500, 0, 'k', 'LineWidth', 2, 'MaxHeadSize', 0.7);
|
||||
quiver3(0, 0, 0, 0, 0, 1500, 'k', 'LineWidth', 2, 'MaxHeadSize', 0.7);
|
||||
text(1500, 0, 0, '$R_x$', 'FontSize', 10, 'HorizontalAlignment', 'center', 'VerticalAlignment', 'top' );
|
||||
text(0, 1500, 0, '$R_y$', 'FontSize', 10, 'HorizontalAlignment', 'right', 'VerticalAlignment', 'bottom');
|
||||
text(0, 0, 1500, '$R_z$', 'FontSize', 10, 'HorizontalAlignment', 'left', 'VerticalAlignment', 'top' );
|
||||
hold off;
|
||||
axis equal;
|
||||
grid off;
|
||||
axis off;
|
||||
view(105, 15);
|
||||
|
||||
%% Input/Output definition of the Simscape model
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
|
||||
io(io_i) = linio([mdl, '/plant'], 1, 'openoutput'); io_i = io_i + 1; % External metrology [m,rad]
|
||||
|
||||
% Prepare simulation
|
||||
controller = initializeController('type', 'open-loop');
|
||||
sample = initializeSample('type', 'cylindrical', 'm', 10, 'H', 100e-3, 'R', 100e-3);
|
||||
|
||||
%% Cubic Stewart platform with payload above the top platform - B frame at the CoM
|
||||
H = 200e-3; % height of the Stewart platform [m]
|
||||
MO_B = 50e-3; % Position {B} with respect to {M} [m]
|
||||
|
||||
stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
|
||||
stewart = generateCubicConfiguration(stewart, 'Hc', H, 'FOc', H/2, 'FHa', 25e-3, 'MHb', 25e-3);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart, 'k', 1e6, 'c', 1e1);
|
||||
stewart = initializeJointDynamics(stewart, 'type_F', '2dof', 'type_M', '3dof');
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
stewart = initializeCylindricalPlatforms(stewart, ...
|
||||
'Mpm', 1e-6, ... % Massless platform
|
||||
'Fpm', 1e-6, ... % Massless platform
|
||||
'Mph', 20e-3, ... % Thin platform
|
||||
'Fph', 20e-3, ... % Thin platform
|
||||
'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)), ...
|
||||
'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)));
|
||||
stewart = initializeCylindricalStruts(stewart, ...
|
||||
'Fsm', 1e-6, ... % Massless strut
|
||||
'Msm', 1e-6, ... % Massless strut
|
||||
'Fsh', stewart.geometry.l(1)/2, ...
|
||||
'Msh', stewart.geometry.l(1)/2 ...
|
||||
);
|
||||
|
||||
% Run the linearization
|
||||
G_CoM = linearize(mdl, io)*inv(stewart.kinematics.J).';
|
||||
G_CoM.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
||||
G_CoM.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
|
||||
|
||||
%% Same geometry but B Frame at cube's center (CoK)
|
||||
MO_B = -100e-3; % Position {B} with respect to {M} [m]
|
||||
stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
|
||||
stewart = generateCubicConfiguration(stewart, 'Hc', H, 'FOc', H/2, 'FHa', 25e-3, 'MHb', 25e-3);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart, 'k', 1e6, 'c', 1e1);
|
||||
stewart = initializeJointDynamics(stewart, 'type_F', '2dof', 'type_M', '3dof');
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
stewart = initializeCylindricalPlatforms(stewart, ...
|
||||
'Mpm', 1e-6, ... % Massless platform
|
||||
'Fpm', 1e-6, ... % Massless platform
|
||||
'Mph', 20e-3, ... % Thin platform
|
||||
'Fph', 20e-3, ... % Thin platform
|
||||
'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)), ...
|
||||
'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)));
|
||||
stewart = initializeCylindricalStruts(stewart, ...
|
||||
'Fsm', 1e-6, ... % Massless strut
|
||||
'Msm', 1e-6, ... % Massless strut
|
||||
'Fsh', stewart.geometry.l(1)/2, ...
|
||||
'Msh', stewart.geometry.l(1)/2 ...
|
||||
);
|
||||
|
||||
% Run the linearization
|
||||
G_CoK = linearize(mdl, io)*inv(stewart.kinematics.J.');
|
||||
G_CoK.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
||||
G_CoK.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
|
||||
|
||||
%% Coupling in the cartesian frame for a Cubic Stewart platform - Frame {B} is at the center of mass of the payload
|
||||
freqs = logspace(0, 4, 1000);
|
||||
figure;
|
||||
tiledlayout(1, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||||
ax1 = nexttile();
|
||||
hold on;
|
||||
% for i = 1:5
|
||||
% for j = i+1:6
|
||||
% plot(freqs, abs(squeeze(freqresp(G_CoM(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
|
||||
% 'HandleVisibility', 'off');
|
||||
% end
|
||||
% end
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoM(1, 1), freqs, 'Hz'))), 'color', colors(1,:), ...
|
||||
'DisplayName', '$D_x/F_x$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoM(2, 2), freqs, 'Hz'))), 'color', colors(2,:), ...
|
||||
'DisplayName', '$D_y/F_y$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoM(3, 3), freqs, 'Hz'))), 'color', colors(3,:), ...
|
||||
'DisplayName', '$D_z/F_z$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoM(4, 4), freqs, 'Hz'))), 'color', colors(4,:), ...
|
||||
'DisplayName', '$R_x/M_x$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoM(5, 5), freqs, 'Hz'))), 'color', colors(5,:), ...
|
||||
'DisplayName', '$R_y/M_y$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoM(6, 6), freqs, 'Hz'))), 'color', colors(6,:), ...
|
||||
'DisplayName', '$R_z/M_z$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoM(4, 2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
|
||||
'DisplayName', '$R_x/F_y$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoM(5, 1), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
|
||||
'DisplayName', '$R_y/F_x$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoM(1, 5), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
|
||||
'DisplayName', '$D_x/M_y$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoM(2, 4), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
|
||||
'DisplayName', '$D_y/M_x$');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Amplitude');
|
||||
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2);
|
||||
leg.ItemTokenSize(1) = 15;
|
||||
xlim([1, 1e4]);
|
||||
ylim([1e-10, 2e-3])
|
||||
|
||||
%% Coupling in the cartesian frame for a Cubic Stewart platform - Frame {B} is at the center of the cube
|
||||
freqs = logspace(0, 4, 1000);
|
||||
figure;
|
||||
tiledlayout(1, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||||
ax1 = nexttile();
|
||||
hold on;
|
||||
% for i = 1:5
|
||||
% for j = i+1:6
|
||||
% plot(freqs, abs(squeeze(freqresp(G_CoK(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
|
||||
% 'HandleVisibility', 'off');
|
||||
% end
|
||||
% end
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoK(1, 1), freqs, 'Hz'))), 'color', colors(1,:), ...
|
||||
'DisplayName', '$D_x/F_x$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoK(2, 2), freqs, 'Hz'))), 'color', colors(2,:), ...
|
||||
'DisplayName', '$D_y/F_y$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoK(3, 3), freqs, 'Hz'))), 'color', colors(3,:), ...
|
||||
'DisplayName', '$D_z/F_z$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoK(4, 4), freqs, 'Hz'))), 'color', colors(4,:), ...
|
||||
'DisplayName', '$R_x/M_x$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoK(5, 5), freqs, 'Hz'))), 'color', colors(5,:), ...
|
||||
'DisplayName', '$R_y/M_y$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoK(6, 6), freqs, 'Hz'))), 'color', colors(6,:), ...
|
||||
'DisplayName', '$R_z/M_z$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoK(4, 2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
|
||||
'DisplayName', '$R_x/F_y$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoK(5, 1), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
|
||||
'DisplayName', '$R_y/F_x$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoK(1, 5), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
|
||||
'DisplayName', '$D_x/M_y$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoK(2, 4), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
|
||||
'DisplayName', '$D_y/M_x$');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Amplitude');
|
||||
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
|
||||
leg.ItemTokenSize(1) = 15;
|
||||
xlim([1, 1e4]);
|
||||
ylim([1e-10, 2e-3])
|
||||
|
||||
%% Cubic Stewart platform with payload above the top platform
|
||||
H = 200e-3;
|
||||
MO_B = -100e-3; % Position {B} with respect to {M} [m]
|
||||
|
||||
stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
|
||||
stewart = generateCubicConfiguration(stewart, 'Hc', H, 'FOc', H/2, 'FHa', 25e-3, 'MHb', 25e-3);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart, 'k', 1e6, 'c', 1e1);
|
||||
stewart = initializeJointDynamics(stewart, 'type_F', '2dof', 'type_M', '3dof');
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
stewart = initializeCylindricalPlatforms(stewart, ...
|
||||
'Mpm', 1e-6, ... % Massless platform
|
||||
'Fpm', 1e-6, ... % Massless platform
|
||||
'Mph', 20e-3, ... % Thin platform
|
||||
'Fph', 20e-3, ... % Thin platform
|
||||
'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)), ...
|
||||
'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)));
|
||||
stewart = initializeCylindricalStruts(stewart, ...
|
||||
'Fsm', 1e-6, ... % Massless strut
|
||||
'Msm', 1e-6, ... % Massless strut
|
||||
'Fsh', stewart.geometry.l(1)/2, ...
|
||||
'Msh', stewart.geometry.l(1)/2 ...
|
||||
);
|
||||
|
||||
% Sample at the Center of the cube
|
||||
sample = initializeSample('type', 'cylindrical', 'm', 10, 'H', 100e-3, 'H_offset', -H/2-50e-3);
|
||||
|
||||
% Run the linearization
|
||||
G_CoM_CoK = linearize(mdl, io)*inv(stewart.kinematics.J.');
|
||||
G_CoM_CoK.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
||||
G_CoM_CoK.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
|
||||
|
||||
freqs = logspace(0, 4, 1000);
|
||||
figure;
|
||||
tiledlayout(1, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||||
ax1 = nexttile();
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoM_CoK(1, 1), freqs, 'Hz'))), 'color', colors(1,:), ...
|
||||
'DisplayName', '$D_x/F_x$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoM_CoK(2, 2), freqs, 'Hz'))), 'color', colors(2,:), ...
|
||||
'DisplayName', '$D_y/F_y$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoM_CoK(3, 3), freqs, 'Hz'))), 'color', colors(3,:), ...
|
||||
'DisplayName', '$D_z/F_z$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoM_CoK(4, 4), freqs, 'Hz'))), 'color', colors(4,:), ...
|
||||
'DisplayName', '$R_x/M_x$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoM_CoK(5, 5), freqs, 'Hz'))), 'color', colors(5,:), ...
|
||||
'DisplayName', '$R_y/M_y$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoM_CoK(6, 6), freqs, 'Hz'))), 'color', colors(6,:), ...
|
||||
'DisplayName', '$R_z/M_z$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoM_CoK(4, 2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
|
||||
'DisplayName', '$R_x/F_y$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoM_CoK(5, 1), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
|
||||
'DisplayName', '$R_y/F_x$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoM_CoK(1, 5), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
|
||||
'DisplayName', '$D_x/M_y$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_CoM_CoK(2, 4), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
|
||||
'DisplayName', '$D_y/M_x$');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]');
|
||||
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2);
|
||||
leg.ItemTokenSize(1) = 15;
|
||||
xlim([1, 1e4]);
|
||||
ylim([1e-10, 2e-3])
|
||||
|
||||
%% Input/Output definition of the Simscape model
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
|
||||
io(io_i) = linio([mdl, '/plant'], 2, 'openoutput', [], 'dL'); io_i = io_i + 1; % Displacement sensors [m]
|
||||
io(io_i) = linio([mdl, '/plant'], 2, 'openoutput', [], 'fn'); io_i = io_i + 1; % Force Sensor [N]
|
||||
|
||||
% Prepare simulation : Payload above the top platform
|
||||
controller = initializeController('type', 'open-loop');
|
||||
sample = initializeSample('type', 'cylindrical', 'm', 10, 'H', 100e-3, 'R', 100e-3);
|
||||
|
||||
%% Cubic Stewart platform
|
||||
H = 200e-3; % height of the Stewart platform [m]
|
||||
MO_B = 50e-3; % Position {B} with respect to {M} [m]
|
||||
|
||||
stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
|
||||
stewart = generateCubicConfiguration(stewart, 'Hc', H, 'FOc', H/2, 'FHa', 25e-3, 'MHb', 25e-3);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart, 'k', 1e6, 'c', 1e1);
|
||||
stewart = initializeJointDynamics(stewart, 'type_F', '2dof', 'type_M', '3dof');
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
stewart = initializeCylindricalPlatforms(stewart, ...
|
||||
'Mpm', 1e-6, ... % Massless platform
|
||||
'Fpm', 1e-6, ... % Massless platform
|
||||
'Mph', 20e-3, ... % Thin platform
|
||||
'Fph', 20e-3, ... % Thin platform
|
||||
'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)), ...
|
||||
'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)));
|
||||
stewart = initializeCylindricalStruts(stewart, ...
|
||||
'Fsm', 1e-6, ... % Massless strut
|
||||
'Msm', 1e-6, ... % Massless strut
|
||||
'Fsh', stewart.geometry.l(1)/2, ...
|
||||
'Msh', stewart.geometry.l(1)/2 ...
|
||||
);
|
||||
|
||||
% Run the linearization
|
||||
G_cubic = linearize(mdl, io);
|
||||
G_cubic.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
|
||||
G_cubic.OutputName = {'dL1', 'dL2', 'dL3', 'dL4', 'dL5', 'dL6', ...
|
||||
'fn1', 'fn2', 'fn3', 'fn4', 'fn5', 'fn6'};
|
||||
|
||||
%% Non-Cubic Stewart platform
|
||||
stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
|
||||
stewart = generateCubicConfiguration(stewart, 'Hc', H, 'FOc', H/2, 'FHa', 25e-3, 'MHb', 25e-3);
|
||||
stewart = generateGeneralConfiguration(stewart, 'FH', 25e-3, 'FR', 250e-3, 'MH', 25e-3, 'MR', 250e-3, ...
|
||||
'FTh', [-22, 22, 120-22, 120+22, 240-22, 240+22]*(pi/180), ...
|
||||
'MTh', [-60+22, 60-22, 60+22, 180-22, 180+22, -60-22]*(pi/180));
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart, 'k', 1e6, 'c', 1e1);
|
||||
stewart = initializeJointDynamics(stewart, 'type_F', '2dof', 'type_M', '3dof');
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
stewart = initializeCylindricalPlatforms(stewart, ...
|
||||
'Mpm', 1e-6, ... % Massless platform
|
||||
'Fpm', 1e-6, ... % Massless platform
|
||||
'Mph', 20e-3, ... % Thin platform
|
||||
'Fph', 20e-3, ... % Thin platform
|
||||
'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)), ...
|
||||
'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)));
|
||||
stewart = initializeCylindricalStruts(stewart, ...
|
||||
'Fsm', 1e-6, ... % Massless strut
|
||||
'Msm', 1e-6, ... % Massless strut
|
||||
'Fsh', stewart.geometry.l(1)/2, ...
|
||||
'Msh', stewart.geometry.l(1)/2 ...
|
||||
);
|
||||
|
||||
% Run the linearization
|
||||
G_non_cubic = linearize(mdl, io);
|
||||
G_non_cubic.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
|
||||
G_non_cubic.OutputName = {'dL1', 'dL2', 'dL3', 'dL4', 'dL5', 'dL6', ...
|
||||
'fn1', 'fn2', 'fn3', 'fn4', 'fn5', 'fn6'};
|
||||
|
||||
%% Decentralized plant - Actuator force to Strut displacement - Cubic Architecture
|
||||
freqs = logspace(0, 4, 1000);
|
||||
figure;
|
||||
tiledlayout(1, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||||
ax1 = nexttile();
|
||||
hold on;
|
||||
for i = 1:5
|
||||
for j = i+1:6
|
||||
plot(freqs, abs(squeeze(freqresp(G_non_cubic(sprintf('dL%i',i), sprintf('f%i',j)), freqs, 'Hz'))), 'color', [colors(1,:), 0.1], ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
end
|
||||
plot(freqs, abs(squeeze(freqresp(G_non_cubic('dL1', 'f1'), freqs, 'Hz'))), 'color', [colors(1,:)], 'linewidth', 2.5, ...
|
||||
'DisplayName', '$l_i/f_i$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_non_cubic('dL2', 'f1'), freqs, 'Hz'))), 'color', [colors(1,:), 0.1], ...
|
||||
'DisplayName', '$l_i/f_j$');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
|
||||
leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
|
||||
leg.ItemTokenSize(1) = 15;
|
||||
xlim([1, 1e4]);
|
||||
ylim([1e-10, 1e-4])
|
||||
|
||||
%% Decentralized plant - Actuator force to Strut displacement - Cubic Architecture
|
||||
freqs = logspace(0, 4, 1000);
|
||||
figure;
|
||||
tiledlayout(1, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||||
ax1 = nexttile();
|
||||
hold on;
|
||||
for i = 1:5
|
||||
for j = i+1:6
|
||||
plot(freqs, abs(squeeze(freqresp(G_cubic(sprintf('dL%i',i), sprintf('f%i',j)), freqs, 'Hz'))), 'color', [colors(2,:), 0.1], ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
end
|
||||
plot(freqs, abs(squeeze(freqresp(G_cubic('dL1', 'f1'), freqs, 'Hz'))), 'color', [colors(2,:)], 'linewidth', 2.5, ...
|
||||
'DisplayName', '$l_i/f_i$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_cubic('dL2', 'f1'), freqs, 'Hz'))), 'color', [colors(2,:), 0.1], ...
|
||||
'DisplayName', '$l_i/f_j$');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
|
||||
leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
|
||||
leg.ItemTokenSize(1) = 15;
|
||||
xlim([1, 1e4]);
|
||||
ylim([1e-10, 1e-4])
|
||||
|
||||
%% Decentralized plant - Actuator force to strut force sensor - Cubic Architecture
|
||||
freqs = logspace(0, 4, 1000);
|
||||
figure;
|
||||
tiledlayout(1, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||||
ax1 = nexttile();
|
||||
hold on;
|
||||
for i = 1:5
|
||||
for j = i+1:6
|
||||
plot(freqs, abs(squeeze(freqresp(G_non_cubic(sprintf('fn%i',i), sprintf('f%i',j)), freqs, 'Hz'))), 'color', [colors(1,:), 0.1], ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
end
|
||||
plot(freqs, abs(squeeze(freqresp(G_non_cubic('fn1', 'f1'), freqs, 'Hz'))), 'color', [colors(1,:)], 'linewidth', 2.5, ...
|
||||
'DisplayName', '$f_{m,i}/f_i$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_non_cubic('fn2', 'f1'), freqs, 'Hz'))), 'color', [colors(1,:), 0.1], ...
|
||||
'DisplayName', '$f_{m,i}/f_j$');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Amplitude [N/N]');
|
||||
leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
|
||||
leg.ItemTokenSize(1) = 15;
|
||||
xlim([1, 1e4]); ylim([1e-4, 1e2]);
|
||||
|
||||
%% Decentralized plant - Actuator force to strut force sensor - Cubic Architecture
|
||||
freqs = logspace(0, 4, 1000);
|
||||
figure;
|
||||
tiledlayout(1, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||||
ax1 = nexttile();
|
||||
hold on;
|
||||
for i = 1:5
|
||||
for j = i+1:6
|
||||
plot(freqs, abs(squeeze(freqresp(G_cubic(sprintf('fn%i',i), sprintf('f%i',j)), freqs, 'Hz'))), 'color', [colors(2,:), 0.1], ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
end
|
||||
plot(freqs, abs(squeeze(freqresp(G_cubic('fn1', 'f1'), freqs, 'Hz'))), 'color', [colors(2,:)], 'linewidth', 2.5, ...
|
||||
'DisplayName', '$f_{m,i}/f_i$');
|
||||
plot(freqs, abs(squeeze(freqresp(G_cubic('fn2', 'f1'), freqs, 'Hz'))), 'color', [colors(2,:), 0.1], ...
|
||||
'DisplayName', '$f_{m,i}/f_j$');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Amplitude [N/N]');
|
||||
leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
|
||||
leg.ItemTokenSize(1) = 15;
|
||||
xlim([1, 1e4]); ylim([1e-4, 1e2]);
|
||||
|
||||
%% Cubic configurations with center of the cube above the top platform
|
||||
H = 100e-3; % height of the Stewart platform [m]
|
||||
MO_B = 20e-3; % Position {B} with respect to {M} [m]
|
||||
FOc = H + MO_B; % Center of the cube with respect to {F}
|
||||
|
||||
%% Small cube
|
||||
Hc = 2*MO_B; % Size of the useful part of the cube [m]
|
||||
|
||||
stewart_small = initializeStewartPlatform();
|
||||
stewart_small = initializeFramesPositions(stewart_small, 'H', H, 'MO_B', MO_B);
|
||||
stewart_small = generateCubicConfiguration(stewart_small, 'Hc', Hc, 'FOc', FOc, 'FHa', 5e-3, 'MHb', 5e-3);
|
||||
stewart_small = computeJointsPose(stewart_small);
|
||||
stewart_small = initializeStrutDynamics(stewart_small, 'k', 1);
|
||||
stewart_small = computeJacobian(stewart_small);
|
||||
stewart_small = initializeCylindricalPlatforms(stewart_small, 'Fpr', 1.1*max(vecnorm(stewart_small.platform_F.Fa)), 'Mpr', 1.1*max(vecnorm(stewart_small.platform_M.Mb)));
|
||||
|
||||
%% ISO View
|
||||
displayArchitecture(stewart_small, 'labels', false, 'frames', false);
|
||||
plotCube(stewart_small, 'Hc', Hc, 'FOc', FOc, 'color', [0,0,0,0.2], 'link_to_struts', true);
|
||||
scatter3(0, 0, FOc, 200, 'kh');
|
||||
|
||||
%% Side view
|
||||
displayArchitecture(stewart_small, 'labels', false, 'frames', false);
|
||||
plotCube(stewart_small, 'Hc', Hc, 'FOc', FOc, 'color', [0, 0, 0, 0.2], 'link_to_struts', true);
|
||||
scatter3(0, 0, FOc, 200, 'kh');
|
||||
view([90,0])
|
||||
|
||||
%% Top view
|
||||
displayArchitecture(stewart_small, 'labels', false, 'frames', false);
|
||||
plotCube(stewart_small, 'Hc', Hc, 'FOc', FOc, 'color', [0, 0, 0, 0.2], 'link_to_struts', true);
|
||||
scatter3(0, 0, FOc, 200, 'kh');
|
||||
view([0,90])
|
||||
|
||||
%% Example of a cubic architecture with cube's center above the top platform - Medium cube size
|
||||
Hc = H + 2*MO_B; % Size of the useful part of the cube [m]
|
||||
|
||||
stewart_medium = initializeStewartPlatform();
|
||||
stewart_medium = initializeFramesPositions(stewart_medium, 'H', H, 'MO_B', MO_B);
|
||||
stewart_medium = generateCubicConfiguration(stewart_medium, 'Hc', Hc, 'FOc', FOc, 'FHa', 5e-3, 'MHb', 5e-3);
|
||||
stewart_medium = computeJointsPose(stewart_medium);
|
||||
stewart_medium = initializeStrutDynamics(stewart_medium, 'k', 1);
|
||||
stewart_medium = computeJacobian(stewart_medium);
|
||||
stewart_medium = initializeCylindricalPlatforms(stewart_medium, 'Fpr', 1.1*max(vecnorm(stewart_medium.platform_F.Fa)), 'Mpr', 1.1*max(vecnorm(stewart_medium.platform_M.Mb)));
|
||||
|
||||
%% ISO View
|
||||
displayArchitecture(stewart_medium, 'labels', false, 'frames', false);
|
||||
plotCube(stewart_medium, 'Hc', Hc, 'FOc', FOc, 'color', [0,0,0,0.2], 'link_to_struts', true);
|
||||
scatter3(0, 0, FOc, 200, 'kh');
|
||||
|
||||
%% Side view
|
||||
displayArchitecture(stewart_medium, 'labels', false, 'frames', false);
|
||||
plotCube(stewart_medium, 'Hc', Hc, 'FOc', FOc, 'color', [0, 0, 0, 0.2], 'link_to_struts', true);
|
||||
scatter3(0, 0, FOc, 200, 'kh');
|
||||
view([90,0])
|
||||
|
||||
%% Top view
|
||||
displayArchitecture(stewart_medium, 'labels', false, 'frames', false);
|
||||
plotCube(stewart_medium, 'Hc', Hc, 'FOc', FOc, 'color', [0, 0, 0, 0.2], 'link_to_struts', true);
|
||||
scatter3(0, 0, FOc, 200, 'kh');
|
||||
view([0,90])
|
||||
|
||||
%% Example of a cubic architecture with cube's center above the top platform - Large cube size
|
||||
Hc = 2*(H + MO_B); % Size of the useful part of the cube [m]
|
||||
|
||||
stewart_large = initializeStewartPlatform();
|
||||
stewart_large = initializeFramesPositions(stewart_large, 'H', H, 'MO_B', MO_B);
|
||||
stewart_large = generateCubicConfiguration(stewart_large, 'Hc', Hc, 'FOc', FOc, 'FHa', 5e-3, 'MHb', 5e-3);
|
||||
stewart_large = computeJointsPose(stewart_large);
|
||||
stewart_large = initializeStrutDynamics(stewart_large, 'k', 1);
|
||||
stewart_large = computeJacobian(stewart_large);
|
||||
stewart_large = initializeCylindricalPlatforms(stewart_large, 'Fpr', 1.1*max(vecnorm(stewart_large.platform_F.Fa)), 'Mpr', 1.1*max(vecnorm(stewart_large.platform_M.Mb)));
|
||||
|
||||
%% ISO View
|
||||
displayArchitecture(stewart_large, 'labels', false, 'frames', false);
|
||||
plotCube(stewart_large, 'Hc', Hc, 'FOc', FOc, 'color', [0,0,0,0.2], 'link_to_struts', true);
|
||||
scatter3(0, 0, FOc, 200, 'kh');
|
||||
|
||||
%% Side view
|
||||
displayArchitecture(stewart_large, 'labels', false, 'frames', false);
|
||||
plotCube(stewart_large, 'Hc', Hc, 'FOc', FOc, 'color', [0, 0, 0, 0.2], 'link_to_struts', true);
|
||||
scatter3(0, 0, FOc, 200, 'kh');
|
||||
view([90,0])
|
||||
|
||||
%% Top view
|
||||
displayArchitecture(stewart_large, 'labels', false, 'frames', false);
|
||||
plotCube(stewart_large, 'Hc', Hc, 'FOc', FOc, 'color', [0, 0, 0, 0.2], 'link_to_struts', true);
|
||||
scatter3(0, 0, FOc, 200, 'kh');
|
||||
view([0,90])
|
||||
|
||||
%% Get the analytical formula for the location of the top and bottom joints
|
||||
% Define symbolic variables
|
||||
syms k Hc Hcom alpha H
|
||||
|
||||
assume(k > 0); % k is positive real
|
||||
assume(Hcom > 0); % k is positive real
|
||||
assume(Hc > 0); % Hc is real
|
||||
assume(H > 0); % H is real
|
||||
assume(alpha, 'real'); % alpha is real
|
||||
|
||||
% Define si matrix (edges of the cubes)
|
||||
si = 1/sqrt(3)*[
|
||||
[ sqrt(2), 0, 1]; ...
|
||||
[-sqrt(2)/2, -sqrt(3/2), 1]; ...
|
||||
[-sqrt(2)/2, sqrt(3/2), 1]; ...
|
||||
[ sqrt(2), 0, 1]; ...
|
||||
[-sqrt(2)/2, -sqrt(3/2), 1]; ...
|
||||
[-sqrt(2)/2, sqrt(3/2), 1] ...
|
||||
];
|
||||
|
||||
% Define ci matrix (vertices of the cubes)
|
||||
ci = Hc * [
|
||||
[1/sqrt(2), -sqrt(3)/sqrt(2), 0.5]; ...
|
||||
[1/sqrt(2), -sqrt(3)/sqrt(2), 0.5]; ...
|
||||
[1/sqrt(2), sqrt(3)/sqrt(2), 0.5]; ...
|
||||
[1/sqrt(2), sqrt(3)/sqrt(2), 0.5]; ...
|
||||
[-sqrt(2), 0, 0.5]; ...
|
||||
[-sqrt(2), 0, 0.5] ...
|
||||
];
|
||||
|
||||
% Apply vertical shift to ci
|
||||
ci = ci + (H + Hcom) * [0, 0, 1];
|
||||
|
||||
% Calculate bi vectors (Stewart platform top joints)
|
||||
bi = ci + alpha * si;
|
||||
|
||||
|
||||
% Extract the z-component value from the first row of ci
|
||||
% (all rows have the same z-component)
|
||||
ci_z = ci(1, 3);
|
||||
|
||||
% The z-component of si is 1 for all rows
|
||||
si_z = si(1, 3);
|
||||
|
||||
alpha_for_0 = solve(ci_z + alpha * si_z == 0, alpha);
|
||||
alpha_for_H = solve(ci_z + alpha * si_z == H, alpha);
|
||||
|
||||
% Verify the results
|
||||
% Substitute alpha values and check the resulting bi_z values
|
||||
bi_z_0 = ci + alpha_for_0 * si;
|
||||
disp('Radius for fixed base:');
|
||||
simplify(sqrt(bi_z_0(1,1).^2 + bi_z_0(1,2).^2))
|
||||
|
||||
bi_z_H = ci + alpha_for_H * si;
|
||||
disp('Radius for mobile platform:');
|
||||
simplify(sqrt(bi_z_H(1,1).^2 + bi_z_H(1,2).^2))
|
||||
1
matlab/figs
Symbolic link
1
matlab/figs
Symbolic link
@@ -0,0 +1 @@
|
||||
../paper/figs
|
||||
1426
matlab/index.org
Normal file
1426
matlab/index.org
Normal file
File diff suppressed because it is too large
Load Diff
BIN
matlab/nano_hexapod_model.slx
Normal file
BIN
matlab/nano_hexapod_model.slx
Normal file
Binary file not shown.
35
matlab/src/computeJacobian.m
Normal file
35
matlab/src/computeJacobian.m
Normal file
@@ -0,0 +1,35 @@
|
||||
function [stewart] = computeJacobian(stewart)
|
||||
% computeJacobian -
|
||||
%
|
||||
% Syntax: [stewart] = computeJacobian(stewart)
|
||||
%
|
||||
% Inputs:
|
||||
% - stewart - With at least the following fields:
|
||||
% - geometry.As [3x6] - The 6 unit vectors for each strut expressed in {A}
|
||||
% - geometry.Ab [3x6] - The 6 position of the joints bi expressed in {A}
|
||||
% - actuators.K [6x1] - Total stiffness of the actuators
|
||||
%
|
||||
% Outputs:
|
||||
% - stewart - With the 3 added field:
|
||||
% - kinematics.J [6x6] - The Jacobian Matrix
|
||||
% - kinematics.K [6x6] - The Stiffness Matrix
|
||||
% - kinematics.C [6x6] - The Compliance Matrix
|
||||
|
||||
assert(isfield(stewart.geometry, 'As'), 'stewart.geometry should have attribute As')
|
||||
As = stewart.geometry.As;
|
||||
|
||||
assert(isfield(stewart.geometry, 'Ab'), 'stewart.geometry should have attribute Ab')
|
||||
Ab = stewart.geometry.Ab;
|
||||
|
||||
assert(isfield(stewart.actuators, 'k'), 'stewart.actuators should have attribute k')
|
||||
Ki = stewart.actuators.k*eye(6);
|
||||
|
||||
J = [As' , cross(Ab, As)'];
|
||||
|
||||
K = J'*Ki*J;
|
||||
|
||||
C = inv(K);
|
||||
|
||||
stewart.kinematics.J = J;
|
||||
stewart.kinematics.K = K;
|
||||
stewart.kinematics.C = C;
|
||||
90
matlab/src/computeJointsPose.m
Normal file
90
matlab/src/computeJointsPose.m
Normal file
@@ -0,0 +1,90 @@
|
||||
function [stewart] = computeJointsPose(stewart, args)
|
||||
% computeJointsPose -
|
||||
%
|
||||
% Syntax: [stewart] = computeJointsPose(stewart, args)
|
||||
%
|
||||
% Inputs:
|
||||
% - stewart - A structure with the following fields
|
||||
% - platform_F.Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}
|
||||
% - platform_M.Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}
|
||||
% - platform_F.FO_A [3x1] - Position of {A} with respect to {F}
|
||||
% - platform_M.MO_B [3x1] - Position of {B} with respect to {M}
|
||||
% - geometry.FO_M [3x1] - Position of {M} with respect to {F}
|
||||
% - args - Can have the following fields:
|
||||
% - AP [3x1] - The wanted position of {B} with respect to {A}
|
||||
% - ARB [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}
|
||||
%
|
||||
% Outputs:
|
||||
% - stewart - A structure with the following added fields
|
||||
% - geometry.Aa [3x6] - The i'th column is the position of ai with respect to {A}
|
||||
% - geometry.Ab [3x6] - The i'th column is the position of bi with respect to {A}
|
||||
% - geometry.Ba [3x6] - The i'th column is the position of ai with respect to {B}
|
||||
% - geometry.Bb [3x6] - The i'th column is the position of bi with respect to {B}
|
||||
% - geometry.l [6x1] - The i'th element is the initial length of strut i
|
||||
% - geometry.As [3x6] - The i'th column is the unit vector of strut i expressed in {A}
|
||||
% - geometry.Bs [3x6] - The i'th column is the unit vector of strut i expressed in {B}
|
||||
% - struts_F.l [6x1] - Length of the Fixed part of the i'th strut
|
||||
% - struts_M.l [6x1] - Length of the Mobile part of the i'th strut
|
||||
% - platform_F.FRa [3x3x6] - The i'th 3x3 array is the rotation matrix to orientate the bottom of the i'th strut from {F}
|
||||
% - platform_M.MRb [3x3x6] - The i'th 3x3 array is the rotation matrix to orientate the top of the i'th strut from {M}
|
||||
|
||||
arguments
|
||||
stewart
|
||||
args.AP (3,1) double {mustBeNumeric} = zeros(3,1)
|
||||
args.ARB (3,3) double {mustBeNumeric} = eye(3)
|
||||
end
|
||||
|
||||
assert(isfield(stewart.platform_F, 'Fa'), 'stewart.platform_F should have attribute Fa')
|
||||
Fa = stewart.platform_F.Fa;
|
||||
|
||||
assert(isfield(stewart.platform_M, 'Mb'), 'stewart.platform_M should have attribute Mb')
|
||||
Mb = stewart.platform_M.Mb;
|
||||
|
||||
assert(isfield(stewart.platform_F, 'FO_A'), 'stewart.platform_F should have attribute FO_A')
|
||||
FO_A = stewart.platform_F.FO_A;
|
||||
|
||||
assert(isfield(stewart.platform_M, 'MO_B'), 'stewart.platform_M should have attribute MO_B')
|
||||
MO_B = stewart.platform_M.MO_B;
|
||||
|
||||
assert(isfield(stewart.geometry, 'FO_M'), 'stewart.geometry should have attribute FO_M')
|
||||
FO_M = stewart.geometry.FO_M;
|
||||
|
||||
Aa = Fa - repmat(FO_A, [1, 6]);
|
||||
Bb = Mb - repmat(MO_B, [1, 6]);
|
||||
|
||||
Ab = Bb - repmat(-MO_B-FO_M+FO_A, [1, 6]);
|
||||
Ba = Aa - repmat( MO_B+FO_M-FO_A, [1, 6]);
|
||||
|
||||
Ab = args.ARB *Bb - repmat(-args.AP, [1, 6]);
|
||||
Ba = args.ARB'*Aa - repmat( args.AP, [1, 6]);
|
||||
|
||||
As = (Ab - Aa)./vecnorm(Ab - Aa); % As_i is the i'th vector of As
|
||||
|
||||
l = vecnorm(Ab - Aa)';
|
||||
|
||||
Bs = (Bb - Ba)./vecnorm(Bb - Ba);
|
||||
|
||||
FRa = zeros(3,3,6);
|
||||
MRb = zeros(3,3,6);
|
||||
|
||||
for i = 1:6
|
||||
FRa(:,:,i) = [cross([0;1;0], As(:,i)) , cross(As(:,i), cross([0;1;0], As(:,i))) , As(:,i)];
|
||||
FRa(:,:,i) = FRa(:,:,i)./vecnorm(FRa(:,:,i));
|
||||
|
||||
MRb(:,:,i) = [cross([0;1;0], Bs(:,i)) , cross(Bs(:,i), cross([0;1;0], Bs(:,i))) , Bs(:,i)];
|
||||
MRb(:,:,i) = MRb(:,:,i)./vecnorm(MRb(:,:,i));
|
||||
end
|
||||
|
||||
stewart.geometry.Aa = Aa;
|
||||
stewart.geometry.Ab = Ab;
|
||||
stewart.geometry.Ba = Ba;
|
||||
stewart.geometry.Bb = Bb;
|
||||
stewart.geometry.As = As;
|
||||
stewart.geometry.Bs = Bs;
|
||||
stewart.geometry.l = l;
|
||||
|
||||
stewart.struts_F.l = l/2;
|
||||
stewart.struts_M.l = l/2;
|
||||
|
||||
stewart.platform_F.FRa = FRa;
|
||||
stewart.platform_M.MRb = MRb;
|
||||
79
matlab/src/describeStewartPlatform.m
Normal file
79
matlab/src/describeStewartPlatform.m
Normal file
@@ -0,0 +1,79 @@
|
||||
function [] = describeStewartPlatform(stewart)
|
||||
% describeStewartPlatform - Display some text describing the current defined Stewart Platform
|
||||
%
|
||||
% Syntax: [] = describeStewartPlatform(args)
|
||||
%
|
||||
% Inputs:
|
||||
% - stewart
|
||||
%
|
||||
% Outputs:
|
||||
|
||||
arguments
|
||||
stewart
|
||||
end
|
||||
|
||||
fprintf('GEOMETRY:\n')
|
||||
fprintf('- The height between the fixed based and the top platform is %.3g [mm].\n', 1e3*stewart.geometry.H)
|
||||
|
||||
if stewart.platform_M.MO_B(3) > 0
|
||||
fprintf('- Frame {A} is located %.3g [mm] above the top platform.\n', 1e3*stewart.platform_M.MO_B(3))
|
||||
else
|
||||
fprintf('- Frame {A} is located %.3g [mm] below the top platform.\n', - 1e3*stewart.platform_M.MO_B(3))
|
||||
end
|
||||
|
||||
fprintf('- The initial length of the struts are:\n')
|
||||
fprintf('\t %.3g, %.3g, %.3g, %.3g, %.3g, %.3g [mm]\n', 1e3*stewart.geometry.l)
|
||||
fprintf('\n')
|
||||
|
||||
fprintf('ACTUATORS:\n')
|
||||
if stewart.actuators.type == 1
|
||||
fprintf('- The actuators are classical.\n')
|
||||
fprintf('- The Stiffness and Damping of each actuators is:\n')
|
||||
fprintf('\t k = %.0e [N/m] \t c = %.0e [N/(m/s)]\n', stewart.actuators.k, stewart.actuators.c)
|
||||
elseif stewart.actuators.type == 2
|
||||
fprintf('- The actuators are mechanicaly amplified.\n')
|
||||
end
|
||||
fprintf('\n')
|
||||
|
||||
fprintf('JOINTS:\n')
|
||||
|
||||
switch stewart.joints_F.type
|
||||
case 1
|
||||
fprintf('- The joints on the fixed based are universal joints\n')
|
||||
case 2
|
||||
fprintf('- The joints on the fixed based are spherical joints\n')
|
||||
case 3
|
||||
fprintf('- The joints on the fixed based are perfect universal joints\n')
|
||||
case 4
|
||||
fprintf('- The joints on the fixed based are perfect spherical joints\n')
|
||||
end
|
||||
|
||||
switch stewart.joints_M.type
|
||||
case 1
|
||||
fprintf('- The joints on the mobile based are universal joints\n')
|
||||
case 2
|
||||
fprintf('- The joints on the mobile based are spherical joints\n')
|
||||
case 3
|
||||
fprintf('- The joints on the mobile based are perfect universal joints\n')
|
||||
case 4
|
||||
fprintf('- The joints on the mobile based are perfect spherical joints\n')
|
||||
end
|
||||
|
||||
fprintf('- The position of the joints on the fixed based with respect to {F} are (in [mm]):\n')
|
||||
fprintf('\t % .3g \t % .3g \t % .3g\n', 1e3*stewart.platform_F.Fa)
|
||||
|
||||
fprintf('- The position of the joints on the mobile based with respect to {M} are (in [mm]):\n')
|
||||
fprintf('\t % .3g \t % .3g \t % .3g\n', 1e3*stewart.platform_M.Mb)
|
||||
fprintf('\n')
|
||||
|
||||
fprintf('KINEMATICS:\n')
|
||||
|
||||
if isfield(stewart.kinematics, 'K')
|
||||
fprintf('- The Stiffness matrix K is (in [N/m]):\n')
|
||||
fprintf('\t % .0e \t % .0e \t % .0e \t % .0e \t % .0e \t % .0e\n', stewart.kinematics.K)
|
||||
end
|
||||
|
||||
if isfield(stewart.kinematics, 'C')
|
||||
fprintf('- The Damping matrix C is (in [m/N]):\n')
|
||||
fprintf('\t % .0e \t % .0e \t % .0e \t % .0e \t % .0e \t % .0e\n', stewart.kinematics.C)
|
||||
end
|
||||
252
matlab/src/displayArchitecture.m
Normal file
252
matlab/src/displayArchitecture.m
Normal file
@@ -0,0 +1,252 @@
|
||||
function [] = displayArchitecture(stewart, args)
|
||||
% displayArchitecture - 3D plot of the Stewart platform architecture
|
||||
%
|
||||
% Syntax: [] = displayArchitecture(args)
|
||||
%
|
||||
% Inputs:
|
||||
% - stewart
|
||||
% - args - Structure with the following fields:
|
||||
% - AP [3x1] - The wanted position of {B} with respect to {A}
|
||||
% - ARB [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}
|
||||
% - ARB [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}
|
||||
% - F_color [color] - Color used for the Fixed elements
|
||||
% - M_color [color] - Color used for the Mobile elements
|
||||
% - L_color [color] - Color used for the Legs elements
|
||||
% - frames [true/false] - Display the Frames
|
||||
% - legs [true/false] - Display the Legs
|
||||
% - joints [true/false] - Display the Joints
|
||||
% - labels [true/false] - Display the Labels
|
||||
% - platforms [true/false] - Display the Platforms
|
||||
% - views ['all', 'xy', 'yz', 'xz', 'default'] -
|
||||
%
|
||||
% Outputs:
|
||||
|
||||
arguments
|
||||
stewart
|
||||
args.AP (3,1) double {mustBeNumeric} = zeros(3,1)
|
||||
args.ARB (3,3) double {mustBeNumeric} = eye(3)
|
||||
args.F_color = [0 0.4470 0.7410]
|
||||
args.M_color = [0.8500 0.3250 0.0980]
|
||||
args.L_color = [0 0 0]
|
||||
args.frames logical {mustBeNumericOrLogical} = true
|
||||
args.legs logical {mustBeNumericOrLogical} = true
|
||||
args.joints logical {mustBeNumericOrLogical} = true
|
||||
args.labels logical {mustBeNumericOrLogical} = true
|
||||
args.platforms logical {mustBeNumericOrLogical} = true
|
||||
args.views char {mustBeMember(args.views,{'all', 'xy', 'xz', 'yz', 'default'})} = 'default'
|
||||
end
|
||||
|
||||
assert(isfield(stewart.platform_F, 'FO_A'), 'stewart.platform_F should have attribute FO_A')
|
||||
FO_A = stewart.platform_F.FO_A;
|
||||
|
||||
assert(isfield(stewart.platform_M, 'MO_B'), 'stewart.platform_M should have attribute MO_B')
|
||||
MO_B = stewart.platform_M.MO_B;
|
||||
|
||||
assert(isfield(stewart.geometry, 'H'), 'stewart.geometry should have attribute H')
|
||||
H = stewart.geometry.H;
|
||||
|
||||
assert(isfield(stewart.platform_F, 'Fa'), 'stewart.platform_F should have attribute Fa')
|
||||
Fa = stewart.platform_F.Fa;
|
||||
|
||||
assert(isfield(stewart.platform_M, 'Mb'), 'stewart.platform_M should have attribute Mb')
|
||||
Mb = stewart.platform_M.Mb;
|
||||
|
||||
% The reference frame of the 3d plot corresponds to the frame $\{F\}$.
|
||||
if ~strcmp(args.views, 'all')
|
||||
figure;
|
||||
else
|
||||
f = figure('visible', 'off');
|
||||
end
|
||||
|
||||
hold on;
|
||||
|
||||
% We first compute homogeneous matrices that will be useful to position elements on the figure where the reference frame is $\{F\}$.
|
||||
FTa = [eye(3), FO_A; ...
|
||||
zeros(1,3), 1];
|
||||
ATb = [args.ARB, args.AP; ...
|
||||
zeros(1,3), 1];
|
||||
BTm = [eye(3), -MO_B; ...
|
||||
zeros(1,3), 1];
|
||||
|
||||
FTm = FTa*ATb*BTm;
|
||||
|
||||
% Let's define a parameter that define the length of the unit vectors used to display the frames.
|
||||
d_unit_vector = H/4;
|
||||
|
||||
% Let's define a parameter used to position the labels with respect to the center of the element.
|
||||
d_label = H/20;
|
||||
% Let's first plot the frame $\{F\}$.
|
||||
Ff = [0, 0, 0];
|
||||
if args.frames
|
||||
quiver3(Ff(1)*ones(1,3), Ff(2)*ones(1,3), Ff(3)*ones(1,3), ...
|
||||
[d_unit_vector 0 0], [0 d_unit_vector 0], [0 0 d_unit_vector], '-', 'Color', args.F_color)
|
||||
|
||||
if args.labels
|
||||
text(Ff(1) + d_label, ...
|
||||
Ff(2) + d_label, ...
|
||||
Ff(3) + d_label, '$\{F\}$', 'Color', args.F_color);
|
||||
end
|
||||
end
|
||||
|
||||
% Now plot the frame $\{A\}$ fixed to the Base.
|
||||
if args.frames
|
||||
quiver3(FO_A(1)*ones(1,3), FO_A(2)*ones(1,3), FO_A(3)*ones(1,3), ...
|
||||
[d_unit_vector 0 0], [0 d_unit_vector 0], [0 0 d_unit_vector], '-', 'Color', args.F_color)
|
||||
|
||||
if args.labels
|
||||
text(FO_A(1) + d_label, ...
|
||||
FO_A(2) + d_label, ...
|
||||
FO_A(3) + d_label, '$\{A\}$', 'Color', args.F_color);
|
||||
end
|
||||
end
|
||||
|
||||
% Let's then plot the circle corresponding to the shape of the Fixed base.
|
||||
if args.platforms && stewart.platform_F.type == 1
|
||||
theta = [0:0.1:2*pi+0.1]; % Angles [rad]
|
||||
v = null([0; 0; 1]'); % Two vectors that are perpendicular to the circle normal
|
||||
center = [0; 0; 0]; % Center of the circle
|
||||
radius = stewart.platform_F.R; % Radius of the circle [m]
|
||||
|
||||
points = center*ones(1, length(theta)) + radius*(v(:,1)*cos(theta) + v(:,2)*sin(theta));
|
||||
|
||||
plot3(points(1,:), ...
|
||||
points(2,:), ...
|
||||
points(3,:), '-', 'Color', args.F_color);
|
||||
end
|
||||
|
||||
% Let's now plot the position and labels of the Fixed Joints
|
||||
if args.joints
|
||||
scatter3(Fa(1,:), ...
|
||||
Fa(2,:), ...
|
||||
Fa(3,:), 'MarkerEdgeColor', args.F_color);
|
||||
if args.labels
|
||||
for i = 1:size(Fa,2)
|
||||
text(Fa(1,i) + d_label, ...
|
||||
Fa(2,i), ...
|
||||
Fa(3,i), sprintf('$a_{%i}$', i), 'Color', args.F_color);
|
||||
end
|
||||
end
|
||||
end
|
||||
|
||||
% Plot the frame $\{M\}$.
|
||||
Fm = FTm*[0; 0; 0; 1]; % Get the position of frame {M} w.r.t. {F}
|
||||
|
||||
if args.frames
|
||||
FM_uv = FTm*[d_unit_vector*eye(3); zeros(1,3)]; % Rotated Unit vectors
|
||||
quiver3(Fm(1)*ones(1,3), Fm(2)*ones(1,3), Fm(3)*ones(1,3), ...
|
||||
FM_uv(1,1:3), FM_uv(2,1:3), FM_uv(3,1:3), '-', 'Color', args.M_color)
|
||||
|
||||
if args.labels
|
||||
text(Fm(1) + d_label, ...
|
||||
Fm(2) + d_label, ...
|
||||
Fm(3) + d_label, '$\{M\}$', 'Color', args.M_color);
|
||||
end
|
||||
end
|
||||
|
||||
% Plot the frame $\{B\}$.
|
||||
FB = FO_A + args.AP;
|
||||
|
||||
if args.frames
|
||||
FB_uv = FTm*[d_unit_vector*eye(3); zeros(1,3)]; % Rotated Unit vectors
|
||||
quiver3(FB(1)*ones(1,3), FB(2)*ones(1,3), FB(3)*ones(1,3), ...
|
||||
FB_uv(1,1:3), FB_uv(2,1:3), FB_uv(3,1:3), '-', 'Color', args.M_color)
|
||||
|
||||
if args.labels
|
||||
text(FB(1) - d_label, ...
|
||||
FB(2) + d_label, ...
|
||||
FB(3) + d_label, '$\{B\}$', 'Color', args.M_color);
|
||||
end
|
||||
end
|
||||
|
||||
% Let's then plot the circle corresponding to the shape of the Mobile platform.
|
||||
if args.platforms && stewart.platform_M.type == 1
|
||||
theta = [0:0.1:2*pi+0.1]; % Angles [rad]
|
||||
v = null((FTm(1:3,1:3)*[0;0;1])'); % Two vectors that are perpendicular to the circle normal
|
||||
center = Fm(1:3); % Center of the circle
|
||||
radius = stewart.platform_M.R; % Radius of the circle [m]
|
||||
|
||||
points = center*ones(1, length(theta)) + radius*(v(:,1)*cos(theta) + v(:,2)*sin(theta));
|
||||
|
||||
plot3(points(1,:), ...
|
||||
points(2,:), ...
|
||||
points(3,:), '-', 'Color', args.M_color);
|
||||
end
|
||||
|
||||
% Plot the position and labels of the rotation joints fixed to the mobile platform.
|
||||
if args.joints
|
||||
Fb = FTm*[Mb;ones(1,6)];
|
||||
|
||||
scatter3(Fb(1,:), ...
|
||||
Fb(2,:), ...
|
||||
Fb(3,:), 'MarkerEdgeColor', args.M_color);
|
||||
|
||||
if args.labels
|
||||
for i = 1:size(Fb,2)
|
||||
text(Fb(1,i) + d_label, ...
|
||||
Fb(2,i), ...
|
||||
Fb(3,i), sprintf('$b_{%i}$', i), 'Color', args.M_color);
|
||||
end
|
||||
end
|
||||
end
|
||||
|
||||
% Plot the legs connecting the joints of the fixed base to the joints of the mobile platform.
|
||||
if args.legs
|
||||
for i = 1:6
|
||||
plot3([Fa(1,i), Fb(1,i)], ...
|
||||
[Fa(2,i), Fb(2,i)], ...
|
||||
[Fa(3,i), Fb(3,i)], '-', 'Color', args.L_color);
|
||||
|
||||
if args.labels
|
||||
text((Fa(1,i)+Fb(1,i))/2 + d_label, ...
|
||||
(Fa(2,i)+Fb(2,i))/2, ...
|
||||
(Fa(3,i)+Fb(3,i))/2, sprintf('$%i$', i), 'Color', args.L_color);
|
||||
end
|
||||
end
|
||||
end
|
||||
|
||||
switch args.views
|
||||
case 'default'
|
||||
view([1 -0.6 0.4]);
|
||||
case 'xy'
|
||||
view([0 0 1]);
|
||||
case 'xz'
|
||||
view([0 -1 0]);
|
||||
case 'yz'
|
||||
view([1 0 0]);
|
||||
end
|
||||
axis equal;
|
||||
axis off;
|
||||
|
||||
if strcmp(args.views, 'all')
|
||||
hAx = findobj('type', 'axes');
|
||||
|
||||
figure;
|
||||
s1 = subplot(2,2,1);
|
||||
copyobj(get(hAx(1), 'Children'), s1);
|
||||
view([0 0 1]);
|
||||
axis equal;
|
||||
axis off;
|
||||
title('Top')
|
||||
|
||||
s2 = subplot(2,2,2);
|
||||
copyobj(get(hAx(1), 'Children'), s2);
|
||||
view([1 -0.6 0.4]);
|
||||
axis equal;
|
||||
axis off;
|
||||
|
||||
s3 = subplot(2,2,3);
|
||||
copyobj(get(hAx(1), 'Children'), s3);
|
||||
view([1 0 0]);
|
||||
axis equal;
|
||||
axis off;
|
||||
title('Front')
|
||||
|
||||
s4 = subplot(2,2,4);
|
||||
copyobj(get(hAx(1), 'Children'), s4);
|
||||
view([0 -1 0]);
|
||||
axis equal;
|
||||
axis off;
|
||||
title('Side')
|
||||
|
||||
close(f);
|
||||
end
|
||||
34
matlab/src/forwardKinematicsApprox.m
Normal file
34
matlab/src/forwardKinematicsApprox.m
Normal file
@@ -0,0 +1,34 @@
|
||||
function [P, R] = forwardKinematicsApprox(stewart, args)
|
||||
% forwardKinematicsApprox - Computed the approximate pose of {B} with respect to {A} from the length of each strut and using
|
||||
% the Jacobian Matrix
|
||||
%
|
||||
% Syntax: [P, R] = forwardKinematicsApprox(stewart, args)
|
||||
%
|
||||
% Inputs:
|
||||
% - stewart - A structure with the following fields
|
||||
% - kinematics.J [6x6] - The Jacobian Matrix
|
||||
% - args - Can have the following fields:
|
||||
% - dL [6x1] - Displacement of each strut [m]
|
||||
%
|
||||
% Outputs:
|
||||
% - P [3x1] - The estimated position of {B} with respect to {A}
|
||||
% - R [3x3] - The estimated rotation matrix that gives the orientation of {B} with respect to {A}
|
||||
|
||||
arguments
|
||||
stewart
|
||||
args.dL (6,1) double {mustBeNumeric} = zeros(6,1)
|
||||
end
|
||||
|
||||
assert(isfield(stewart.kinematics, 'J'), 'stewart.kinematics should have attribute J')
|
||||
J = stewart.kinematics.J;
|
||||
|
||||
X = J\args.dL;
|
||||
|
||||
P = X(1:3);
|
||||
|
||||
theta = norm(X(4:6));
|
||||
s = X(4:6)/theta;
|
||||
|
||||
R = [s(1)^2*(1-cos(theta)) + cos(theta) , s(1)*s(2)*(1-cos(theta)) - s(3)*sin(theta), s(1)*s(3)*(1-cos(theta)) + s(2)*sin(theta);
|
||||
s(2)*s(1)*(1-cos(theta)) + s(3)*sin(theta), s(2)^2*(1-cos(theta)) + cos(theta), s(2)*s(3)*(1-cos(theta)) - s(1)*sin(theta);
|
||||
s(3)*s(1)*(1-cos(theta)) - s(2)*sin(theta), s(3)*s(2)*(1-cos(theta)) + s(1)*sin(theta), s(3)^2*(1-cos(theta)) + cos(theta)];
|
||||
57
matlab/src/generateCubicConfiguration.m
Normal file
57
matlab/src/generateCubicConfiguration.m
Normal file
@@ -0,0 +1,57 @@
|
||||
function [stewart] = generateCubicConfiguration(stewart, args)
|
||||
% generateCubicConfiguration - Generate a Cubic Configuration
|
||||
%
|
||||
% Syntax: [stewart] = generateCubicConfiguration(stewart, args)
|
||||
%
|
||||
% Inputs:
|
||||
% - stewart - A structure with the following fields
|
||||
% - geometry.H [1x1] - Total height of the platform [m]
|
||||
% - args - Can have the following fields:
|
||||
% - Hc [1x1] - Height of the "useful" part of the cube [m]
|
||||
% - FOc [1x1] - Height of the center of the cube with respect to {F} [m]
|
||||
% - FHa [1x1] - Height of the plane joining the points ai with respect to the frame {F} [m]
|
||||
% - MHb [1x1] - Height of the plane joining the points bi with respect to the frame {M} [m]
|
||||
%
|
||||
% Outputs:
|
||||
|
||||
% - stewart - updated Stewart structure with the added fields:
|
||||
% - platform_F.Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}
|
||||
% - platform_M.Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}
|
||||
|
||||
arguments
|
||||
stewart
|
||||
args.Hc (1,1) double {mustBeNumeric, mustBePositive} = 60e-3
|
||||
args.FOc (1,1) double {mustBeNumeric} = 50e-3
|
||||
args.FHa (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3
|
||||
args.MHb (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3
|
||||
end
|
||||
|
||||
assert(isfield(stewart.geometry, 'H'), 'stewart.geometry should have attribute H')
|
||||
H = stewart.geometry.H;
|
||||
|
||||
% We define the useful points of the cube with respect to the Cube's center.
|
||||
% ${}^{C}C$ are the 6 vertices of the cubes expressed in a frame {C} which is located at the center of the cube and aligned with {F} and {M}.
|
||||
|
||||
sx = [ 2; -1; -1];
|
||||
sy = [ 0; 1; -1];
|
||||
sz = [ 1; 1; 1];
|
||||
|
||||
R = [sx, sy, sz]./vecnorm([sx, sy, sz]);
|
||||
|
||||
L = args.Hc*sqrt(3);
|
||||
|
||||
Cc = R'*[[0;0;L],[L;0;L],[L;0;0],[L;L;0],[0;L;0],[0;L;L]] - [0;0;1.5*args.Hc];
|
||||
|
||||
CCf = [Cc(:,1), Cc(:,3), Cc(:,3), Cc(:,5), Cc(:,5), Cc(:,1)]; % CCf(:,i) corresponds to the bottom cube's vertice corresponding to the i'th leg
|
||||
CCm = [Cc(:,2), Cc(:,2), Cc(:,4), Cc(:,4), Cc(:,6), Cc(:,6)]; % CCm(:,i) corresponds to the top cube's vertice corresponding to the i'th leg
|
||||
|
||||
% We can compute the vector of each leg ${}^{C}\hat{\bm{s}}_{i}$ (unit vector from ${}^{C}C_{f}$ to ${}^{C}C_{m}$).
|
||||
|
||||
CSi = (CCm - CCf)./vecnorm(CCm - CCf);
|
||||
|
||||
% We now which to compute the position of the joints $a_{i}$ and $b_{i}$.
|
||||
Fa = CCf + [0; 0; args.FOc] + ((args.FHa-(args.FOc-args.Hc/2))./CSi(3,:)).*CSi;
|
||||
Mb = CCf + [0; 0; args.FOc-H] + ((H-args.MHb-(args.FOc-args.Hc/2))./CSi(3,:)).*CSi;
|
||||
|
||||
stewart.platform_F.Fa = Fa;
|
||||
stewart.platform_M.Mb = Mb;
|
||||
39
matlab/src/generateGeneralConfiguration.m
Normal file
39
matlab/src/generateGeneralConfiguration.m
Normal file
@@ -0,0 +1,39 @@
|
||||
function [stewart] = generateGeneralConfiguration(stewart, args)
|
||||
% generateGeneralConfiguration - Generate a Very General Configuration
|
||||
%
|
||||
% Syntax: [stewart] = generateGeneralConfiguration(stewart, args)
|
||||
%
|
||||
% Inputs:
|
||||
% - args - Can have the following fields:
|
||||
% - FH [1x1] - Height of the position of the fixed joints with respect to the frame {F} [m]
|
||||
% - FR [1x1] - Radius of the position of the fixed joints in the X-Y [m]
|
||||
% - FTh [6x1] - Angles of the fixed joints in the X-Y plane with respect to the X axis [rad]
|
||||
% - MH [1x1] - Height of the position of the mobile joints with respect to the frame {M} [m]
|
||||
% - FR [1x1] - Radius of the position of the mobile joints in the X-Y [m]
|
||||
% - MTh [6x1] - Angles of the mobile joints in the X-Y plane with respect to the X axis [rad]
|
||||
%
|
||||
% Outputs:
|
||||
% - stewart - updated Stewart structure with the added fields:
|
||||
% - platform_F.Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}
|
||||
% - platform_M.Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}
|
||||
|
||||
arguments
|
||||
stewart
|
||||
args.FH (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3
|
||||
args.FR (1,1) double {mustBeNumeric, mustBePositive} = 115e-3;
|
||||
args.FTh (6,1) double {mustBeNumeric} = [-10, 10, 120-10, 120+10, 240-10, 240+10]*(pi/180);
|
||||
args.MH (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3
|
||||
args.MR (1,1) double {mustBeNumeric, mustBePositive} = 90e-3;
|
||||
args.MTh (6,1) double {mustBeNumeric} = [-60+10, 60-10, 60+10, 180-10, 180+10, -60-10]*(pi/180);
|
||||
end
|
||||
|
||||
Fa = zeros(3,6);
|
||||
Mb = zeros(3,6);
|
||||
|
||||
for i = 1:6
|
||||
Fa(:,i) = [args.FR*cos(args.FTh(i)); args.FR*sin(args.FTh(i)); args.FH];
|
||||
Mb(:,i) = [args.MR*cos(args.MTh(i)); args.MR*sin(args.MTh(i)); -args.MH];
|
||||
end
|
||||
|
||||
stewart.platform_F.Fa = Fa;
|
||||
stewart.platform_M.Mb = Mb;
|
||||
17
matlab/src/initializeController.m
Normal file
17
matlab/src/initializeController.m
Normal file
@@ -0,0 +1,17 @@
|
||||
function [controller] = initializeController(args)
|
||||
|
||||
arguments
|
||||
args.type char {mustBeMember(args.type,{'open-loop', 'iff'})} = 'open-loop'
|
||||
end
|
||||
|
||||
controller = struct();
|
||||
|
||||
switch args.type
|
||||
case 'open-loop'
|
||||
controller.type = 1;
|
||||
controller.name = 'Open-Loop';
|
||||
case 'iff'
|
||||
controller.type = 2;
|
||||
controller.name = 'Decentralized Integral Force Feedback';
|
||||
end
|
||||
end
|
||||
59
matlab/src/initializeCylindricalPlatforms.m
Normal file
59
matlab/src/initializeCylindricalPlatforms.m
Normal file
@@ -0,0 +1,59 @@
|
||||
function [stewart] = initializeCylindricalPlatforms(stewart, args)
|
||||
% initializeCylindricalPlatforms - Initialize the geometry of the Fixed and Mobile Platforms
|
||||
%
|
||||
% Syntax: [stewart] = initializeCylindricalPlatforms(args)
|
||||
%
|
||||
% Inputs:
|
||||
% - args - Structure with the following fields:
|
||||
% - Fpm [1x1] - Fixed Platform Mass [kg]
|
||||
% - Fph [1x1] - Fixed Platform Height [m]
|
||||
% - Fpr [1x1] - Fixed Platform Radius [m]
|
||||
% - Mpm [1x1] - Mobile Platform Mass [kg]
|
||||
% - Mph [1x1] - Mobile Platform Height [m]
|
||||
% - Mpr [1x1] - Mobile Platform Radius [m]
|
||||
%
|
||||
% Outputs:
|
||||
% - stewart - updated Stewart structure with the added fields:
|
||||
% - platform_F [struct] - structure with the following fields:
|
||||
% - type = 1
|
||||
% - M [1x1] - Fixed Platform Mass [kg]
|
||||
% - I [3x3] - Fixed Platform Inertia matrix [kg*m^2]
|
||||
% - H [1x1] - Fixed Platform Height [m]
|
||||
% - R [1x1] - Fixed Platform Radius [m]
|
||||
% - platform_M [struct] - structure with the following fields:
|
||||
% - M [1x1] - Mobile Platform Mass [kg]
|
||||
% - I [3x3] - Mobile Platform Inertia matrix [kg*m^2]
|
||||
% - H [1x1] - Mobile Platform Height [m]
|
||||
% - R [1x1] - Mobile Platform Radius [m]
|
||||
|
||||
arguments
|
||||
stewart
|
||||
args.Fpm (1,1) double {mustBeNumeric, mustBePositive} = 1
|
||||
args.Fph (1,1) double {mustBeNumeric, mustBePositive} = 10e-3
|
||||
args.Fpr (1,1) double {mustBeNumeric, mustBePositive} = 125e-3
|
||||
args.Mpm (1,1) double {mustBeNumeric, mustBePositive} = 1
|
||||
args.Mph (1,1) double {mustBeNumeric, mustBePositive} = 10e-3
|
||||
args.Mpr (1,1) double {mustBeNumeric, mustBePositive} = 100e-3
|
||||
end
|
||||
|
||||
I_F = diag([1/12*args.Fpm * (3*args.Fpr^2 + args.Fph^2), ...
|
||||
1/12*args.Fpm * (3*args.Fpr^2 + args.Fph^2), ...
|
||||
1/2 *args.Fpm * args.Fpr^2]);
|
||||
|
||||
I_M = diag([1/12*args.Mpm * (3*args.Mpr^2 + args.Mph^2), ...
|
||||
1/12*args.Mpm * (3*args.Mpr^2 + args.Mph^2), ...
|
||||
1/2 *args.Mpm * args.Mpr^2]);
|
||||
|
||||
stewart.platform_F.type = 1;
|
||||
|
||||
stewart.platform_F.I = I_F;
|
||||
stewart.platform_F.M = args.Fpm;
|
||||
stewart.platform_F.R = args.Fpr;
|
||||
stewart.platform_F.H = args.Fph;
|
||||
|
||||
stewart.platform_M.type = 1;
|
||||
|
||||
stewart.platform_M.I = I_M;
|
||||
stewart.platform_M.M = args.Mpm;
|
||||
stewart.platform_M.R = args.Mpr;
|
||||
stewart.platform_M.H = args.Mph;
|
||||
50
matlab/src/initializeCylindricalStruts.m
Normal file
50
matlab/src/initializeCylindricalStruts.m
Normal file
@@ -0,0 +1,50 @@
|
||||
function [stewart] = initializeCylindricalStruts(stewart, args)
|
||||
% initializeCylindricalStruts - Define the mass and moment of inertia of cylindrical struts
|
||||
%
|
||||
% Syntax: [stewart] = initializeCylindricalStruts(args)
|
||||
%
|
||||
% Inputs:
|
||||
% - args - Structure with the following fields:
|
||||
% - Fsm [1x1] - Mass of the Fixed part of the struts [kg]
|
||||
% - Fsh [1x1] - Height of cylinder for the Fixed part of the struts [m]
|
||||
% - Fsr [1x1] - Radius of cylinder for the Fixed part of the struts [m]
|
||||
% - Msm [1x1] - Mass of the Mobile part of the struts [kg]
|
||||
% - Msh [1x1] - Height of cylinder for the Mobile part of the struts [m]
|
||||
% - Msr [1x1] - Radius of cylinder for the Mobile part of the struts [m]
|
||||
%
|
||||
% Outputs:
|
||||
% - stewart - updated Stewart structure with the added fields:
|
||||
% - struts_F [struct] - structure with the following fields:
|
||||
% - M [6x1] - Mass of the Fixed part of the struts [kg]
|
||||
% - I [3x3x6] - Moment of Inertia for the Fixed part of the struts [kg*m^2]
|
||||
% - H [6x1] - Height of cylinder for the Fixed part of the struts [m]
|
||||
% - R [6x1] - Radius of cylinder for the Fixed part of the struts [m]
|
||||
% - struts_M [struct] - structure with the following fields:
|
||||
% - M [6x1] - Mass of the Mobile part of the struts [kg]
|
||||
% - I [3x3x6] - Moment of Inertia for the Mobile part of the struts [kg*m^2]
|
||||
% - H [6x1] - Height of cylinder for the Mobile part of the struts [m]
|
||||
% - R [6x1] - Radius of cylinder for the Mobile part of the struts [m]
|
||||
|
||||
arguments
|
||||
stewart
|
||||
args.Fsm (1,1) double {mustBeNumeric, mustBePositive} = 0.1
|
||||
args.Fsh (1,1) double {mustBeNumeric, mustBePositive} = 50e-3
|
||||
args.Fsr (1,1) double {mustBeNumeric, mustBePositive} = 5e-3
|
||||
args.Msm (1,1) double {mustBeNumeric, mustBePositive} = 0.1
|
||||
args.Msh (1,1) double {mustBeNumeric, mustBePositive} = 50e-3
|
||||
args.Msr (1,1) double {mustBeNumeric, mustBePositive} = 5e-3
|
||||
end
|
||||
|
||||
stewart.struts_M.type = 1;
|
||||
|
||||
stewart.struts_M.M = args.Msm;
|
||||
stewart.struts_M.R = args.Msr;
|
||||
stewart.struts_M.H = args.Msh;
|
||||
|
||||
stewart.struts_F.type = 1;
|
||||
|
||||
stewart.struts_F.M = args.Fsm;
|
||||
stewart.struts_F.R = args.Fsr;
|
||||
stewart.struts_F.H = args.Fsh;
|
||||
|
||||
end
|
||||
35
matlab/src/initializeFramesPositions.m
Normal file
35
matlab/src/initializeFramesPositions.m
Normal file
@@ -0,0 +1,35 @@
|
||||
function [stewart] = initializeFramesPositions(stewart, args)
|
||||
% initializeFramesPositions - Initialize the positions of frames {A}, {B}, {F} and {M}
|
||||
%
|
||||
% Syntax: [stewart] = initializeFramesPositions(stewart, args)
|
||||
%
|
||||
% Inputs:
|
||||
% - args - Can have the following fields:
|
||||
% - H [1x1] - Total Height of the Stewart Platform (height from {F} to {M}) [m]
|
||||
% - MO_B [1x1] - Height of the frame {B} with respect to {M} [m]
|
||||
%
|
||||
% Outputs:
|
||||
% - stewart - A structure with the following fields:
|
||||
% - geometry.H [1x1] - Total Height of the Stewart Platform [m]
|
||||
% - geometry.FO_M [3x1] - Position of {M} with respect to {F} [m]
|
||||
% - platform_M.MO_B [3x1] - Position of {B} with respect to {M} [m]
|
||||
% - platform_F.FO_A [3x1] - Position of {A} with respect to {F} [m]
|
||||
|
||||
arguments
|
||||
stewart
|
||||
args.H (1,1) double {mustBeNumeric, mustBePositive} = 90e-3
|
||||
args.MO_B (1,1) double {mustBeNumeric} = 50e-3
|
||||
end
|
||||
|
||||
H = args.H; % Total Height of the Stewart Platform [m]
|
||||
|
||||
FO_M = [0; 0; H]; % Position of {M} with respect to {F} [m]
|
||||
|
||||
MO_B = [0; 0; args.MO_B]; % Position of {B} with respect to {M} [m]
|
||||
|
||||
FO_A = MO_B + FO_M; % Position of {A} with respect to {F} [m]
|
||||
|
||||
stewart.geometry.H = H;
|
||||
stewart.geometry.FO_M = FO_M;
|
||||
stewart.platform_M.MO_B = MO_B;
|
||||
stewart.platform_F.FO_A = FO_A;
|
||||
129
matlab/src/initializeJointDynamics.m
Normal file
129
matlab/src/initializeJointDynamics.m
Normal file
@@ -0,0 +1,129 @@
|
||||
function [stewart] = initializeJointDynamics(stewart, args)
|
||||
% initializeJointDynamics - Add Stiffness and Damping properties for the spherical joints
|
||||
%
|
||||
% Syntax: [stewart] = initializeJointDynamics(args)
|
||||
%
|
||||
% Inputs:
|
||||
% - args - Structure with the following fields:
|
||||
% - type_F - 'universal', 'spherical', 'universal_p', 'spherical_p'
|
||||
% - type_M - 'universal', 'spherical', 'universal_p', 'spherical_p'
|
||||
% - Kf_M [6x1] - Bending (Rx, Ry) Stiffness for each top joints [(N.m)/rad]
|
||||
% - Kt_M [6x1] - Torsion (Rz) Stiffness for each top joints [(N.m)/rad]
|
||||
% - Cf_M [6x1] - Bending (Rx, Ry) Damping of each top joint [(N.m)/(rad/s)]
|
||||
% - Ct_M [6x1] - Torsion (Rz) Damping of each top joint [(N.m)/(rad/s)]
|
||||
% - Kf_F [6x1] - Bending (Rx, Ry) Stiffness for each bottom joints [(N.m)/rad]
|
||||
% - Kt_F [6x1] - Torsion (Rz) Stiffness for each bottom joints [(N.m)/rad]
|
||||
% - Cf_F [6x1] - Bending (Rx, Ry) Damping of each bottom joint [(N.m)/(rad/s)]
|
||||
% - Cf_F [6x1] - Torsion (Rz) Damping of each bottom joint [(N.m)/(rad/s)]
|
||||
%
|
||||
% Outputs:
|
||||
% - stewart - updated Stewart structure with the added fields:
|
||||
% - stewart.joints_F and stewart.joints_M:
|
||||
% - type - 1 (universal), 2 (spherical), 3 (universal perfect), 4 (spherical perfect)
|
||||
% - Kx, Ky, Kz [6x1] - Translation (Tx, Ty, Tz) Stiffness [N/m]
|
||||
% - Kf [6x1] - Flexion (Rx, Ry) Stiffness [(N.m)/rad]
|
||||
% - Kt [6x1] - Torsion (Rz) Stiffness [(N.m)/rad]
|
||||
% - Cx, Cy, Cz [6x1] - Translation (Rx, Ry) Damping [N/(m/s)]
|
||||
% - Cf [6x1] - Flexion (Rx, Ry) Damping [(N.m)/(rad/s)]
|
||||
% - Cb [6x1] - Torsion (Rz) Damping [(N.m)/(rad/s)]
|
||||
|
||||
arguments
|
||||
stewart
|
||||
args.type_F char {mustBeMember(args.type_F,{'2dof', '3dof', '4dof', '6dof', 'flexible'})} = '2dof'
|
||||
args.type_M char {mustBeMember(args.type_M,{'2dof', '3dof', '4dof', '6dof', 'flexible'})} = '3dof'
|
||||
args.Kf_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
|
||||
args.Cf_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
|
||||
args.Kt_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
|
||||
args.Ct_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
|
||||
args.Kf_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
|
||||
args.Cf_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
|
||||
args.Kt_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
|
||||
args.Ct_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
|
||||
args.Ka_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
|
||||
args.Ca_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
|
||||
args.Kr_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
|
||||
args.Cr_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
|
||||
args.Ka_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
|
||||
args.Ca_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
|
||||
args.Kr_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
|
||||
args.Cr_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
|
||||
args.K_M double {mustBeNumeric} = zeros(6,6)
|
||||
args.M_M double {mustBeNumeric} = zeros(6,6)
|
||||
args.n_xyz_M double {mustBeNumeric} = zeros(2,3)
|
||||
args.xi_M double {mustBeNumeric} = 0.1
|
||||
args.step_file_M char {} = ''
|
||||
args.K_F double {mustBeNumeric} = zeros(6,6)
|
||||
args.M_F double {mustBeNumeric} = zeros(6,6)
|
||||
args.n_xyz_F double {mustBeNumeric} = zeros(2,3)
|
||||
args.xi_F double {mustBeNumeric} = 0.1
|
||||
args.step_file_F char {} = ''
|
||||
end
|
||||
|
||||
switch args.type_F
|
||||
case '2dof'
|
||||
stewart.joints_F.type = 1;
|
||||
case '3dof'
|
||||
stewart.joints_F.type = 2;
|
||||
case '4dof'
|
||||
stewart.joints_F.type = 3;
|
||||
case '6dof'
|
||||
stewart.joints_F.type = 4;
|
||||
case 'flexible'
|
||||
stewart.joints_F.type = 5;
|
||||
otherwise
|
||||
error("joints_F are not correctly defined")
|
||||
end
|
||||
|
||||
switch args.type_M
|
||||
case '2dof'
|
||||
stewart.joints_M.type = 1;
|
||||
case '3dof'
|
||||
stewart.joints_M.type = 2;
|
||||
case '4dof'
|
||||
stewart.joints_M.type = 3;
|
||||
case '6dof'
|
||||
stewart.joints_M.type = 4;
|
||||
case 'flexible'
|
||||
stewart.joints_M.type = 5;
|
||||
otherwise
|
||||
error("joints_M are not correctly defined")
|
||||
end
|
||||
|
||||
stewart.joints_M.Ka = args.Ka_M;
|
||||
stewart.joints_M.Kr = args.Kr_M;
|
||||
|
||||
stewart.joints_F.Ka = args.Ka_F;
|
||||
stewart.joints_F.Kr = args.Kr_F;
|
||||
|
||||
stewart.joints_M.Ca = args.Ca_M;
|
||||
stewart.joints_M.Cr = args.Cr_M;
|
||||
|
||||
stewart.joints_F.Ca = args.Ca_F;
|
||||
stewart.joints_F.Cr = args.Cr_F;
|
||||
|
||||
stewart.joints_M.Kf = args.Kf_M;
|
||||
stewart.joints_M.Kt = args.Kt_M;
|
||||
|
||||
stewart.joints_F.Kf = args.Kf_F;
|
||||
stewart.joints_F.Kt = args.Kt_F;
|
||||
|
||||
stewart.joints_M.Cf = args.Cf_M;
|
||||
stewart.joints_M.Ct = args.Ct_M;
|
||||
|
||||
stewart.joints_F.Cf = args.Cf_F;
|
||||
stewart.joints_F.Ct = args.Ct_F;
|
||||
|
||||
stewart.joints_F.M = args.M_F;
|
||||
stewart.joints_F.K = args.K_F;
|
||||
stewart.joints_F.n_xyz = args.n_xyz_F;
|
||||
stewart.joints_F.xi = args.xi_F;
|
||||
stewart.joints_F.xi = args.xi_F;
|
||||
stewart.joints_F.step_file = args.step_file_F;
|
||||
|
||||
stewart.joints_M.M = args.M_M;
|
||||
stewart.joints_M.K = args.K_M;
|
||||
stewart.joints_M.n_xyz = args.n_xyz_M;
|
||||
stewart.joints_M.xi = args.xi_M;
|
||||
stewart.joints_M.step_file = args.step_file_M;
|
||||
|
||||
end
|
||||
26
matlab/src/initializeSample.m
Normal file
26
matlab/src/initializeSample.m
Normal file
@@ -0,0 +1,26 @@
|
||||
function [sample] = initializeSample(args)
|
||||
|
||||
arguments
|
||||
args.type char {mustBeMember(args.type,{'none', 'cylindrical'})} = 'none'
|
||||
args.H_offset (1,1) double {mustBeNumeric} = 0 % Vertical offset [m]
|
||||
args.H (1,1) double {mustBeNumeric, mustBePositive} = 200e-3 % Height [m]
|
||||
args.R (1,1) double {mustBeNumeric, mustBePositive} = 110e-3 % Radius [m]
|
||||
args.m (1,1) double {mustBeNumeric, mustBePositive} = 1 % Mass [kg]
|
||||
end
|
||||
|
||||
sample = struct();
|
||||
|
||||
switch args.type
|
||||
case 'none'
|
||||
sample.type = 0;
|
||||
sample.m = 0;
|
||||
case 'cylindrical'
|
||||
sample.type = 1;
|
||||
|
||||
sample.H_offset = args.H_offset;
|
||||
|
||||
sample.H = args.H;
|
||||
sample.R = args.R;
|
||||
sample.m = args.m;
|
||||
end
|
||||
end
|
||||
31
matlab/src/initializeStewartPlatform.m
Normal file
31
matlab/src/initializeStewartPlatform.m
Normal file
@@ -0,0 +1,31 @@
|
||||
function [stewart] = initializeStewartPlatform()
|
||||
% initializeStewartPlatform - Initialize the stewart structure
|
||||
%
|
||||
% Syntax: [stewart] = initializeStewartPlatform(args)
|
||||
%
|
||||
% Outputs:
|
||||
% - stewart - A structure with the following sub-structures:
|
||||
% - platform_F -
|
||||
% - platform_M -
|
||||
% - joints_F -
|
||||
% - joints_M -
|
||||
% - struts_F -
|
||||
% - struts_M -
|
||||
% - actuators -
|
||||
% - geometry -
|
||||
% - properties -
|
||||
|
||||
stewart = struct();
|
||||
stewart.platform_F = struct();
|
||||
stewart.platform_M = struct();
|
||||
stewart.joints_F = struct();
|
||||
stewart.joints_M = struct();
|
||||
stewart.struts_F = struct();
|
||||
stewart.struts_M = struct();
|
||||
stewart.actuators = struct();
|
||||
stewart.sensors = struct();
|
||||
stewart.sensors.inertial = struct();
|
||||
stewart.sensors.force = struct();
|
||||
stewart.sensors.relative = struct();
|
||||
stewart.geometry = struct();
|
||||
stewart.kinematics = struct();
|
||||
27
matlab/src/initializeStewartPose.m
Normal file
27
matlab/src/initializeStewartPose.m
Normal file
@@ -0,0 +1,27 @@
|
||||
function [stewart] = initializeStewartPose(stewart, args)
|
||||
% initializeStewartPose - Determine the initial stroke in each leg to have the wanted pose
|
||||
% It uses the inverse kinematic
|
||||
%
|
||||
% Syntax: [stewart] = initializeStewartPose(stewart, args)
|
||||
%
|
||||
% Inputs:
|
||||
% - stewart - A structure with the following fields
|
||||
% - Aa [3x6] - The positions ai expressed in {A}
|
||||
% - Bb [3x6] - The positions bi expressed in {B}
|
||||
% - args - Can have the following fields:
|
||||
% - AP [3x1] - The wanted position of {B} with respect to {A}
|
||||
% - ARB [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}
|
||||
%
|
||||
% Outputs:
|
||||
% - stewart - updated Stewart structure with the added fields:
|
||||
% - actuators.Leq [6x1] - The 6 needed displacement of the struts from the initial position in [m] to have the wanted pose of {B} w.r.t. {A}
|
||||
|
||||
arguments
|
||||
stewart
|
||||
args.AP (3,1) double {mustBeNumeric} = zeros(3,1)
|
||||
args.ARB (3,3) double {mustBeNumeric} = eye(3)
|
||||
end
|
||||
|
||||
[Li, dLi] = inverseKinematics(stewart, 'AP', args.AP, 'ARB', args.ARB);
|
||||
|
||||
stewart.actuators.Leq = dLi;
|
||||
60
matlab/src/initializeStrutDynamics.m
Normal file
60
matlab/src/initializeStrutDynamics.m
Normal file
@@ -0,0 +1,60 @@
|
||||
function [stewart] = initializeStrutDynamics(stewart, args)
|
||||
% initializeStrutDynamics - Add Stiffness and Damping properties of each strut
|
||||
%
|
||||
% Syntax: [stewart] = initializeStrutDynamics(args)
|
||||
%
|
||||
% Inputs:
|
||||
% - args - Structure with the following fields:
|
||||
% - K [6x1] - Stiffness of each strut [N/m]
|
||||
% - C [6x1] - Damping of each strut [N/(m/s)]
|
||||
%
|
||||
% Outputs:
|
||||
% - stewart - updated Stewart structure with the added fields:
|
||||
% - actuators.type = 1
|
||||
% - actuators.K [6x1] - Stiffness of each strut [N/m]
|
||||
% - actuators.C [6x1] - Damping of each strut [N/(m/s)]
|
||||
|
||||
arguments
|
||||
stewart
|
||||
args.type char {mustBeMember(args.type,{'1dof', '2dof', 'flexible'})} = '1dof'
|
||||
args.k (1,1) double {mustBeNumeric, mustBeNonnegative} = 20e6
|
||||
args.kp (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
|
||||
args.ke (1,1) double {mustBeNumeric, mustBeNonnegative} = 5e6
|
||||
args.ka (1,1) double {mustBeNumeric, mustBeNonnegative} = 60e6
|
||||
args.c (1,1) double {mustBeNumeric, mustBeNonnegative} = 2e1
|
||||
args.cp (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
|
||||
args.ce (1,1) double {mustBeNumeric, mustBeNonnegative} = 1e6
|
||||
args.ca (1,1) double {mustBeNumeric, mustBeNonnegative} = 10
|
||||
|
||||
args.F_gain (1,1) double {mustBeNumeric} = 1
|
||||
args.me (1,1) double {mustBeNumeric} = 0.01
|
||||
args.ma (1,1) double {mustBeNumeric} = 0.01
|
||||
end
|
||||
|
||||
if strcmp(args.type, '1dof')
|
||||
stewart.actuators.type = 1;
|
||||
elseif strcmp(args.type, '2dof')
|
||||
stewart.actuators.type = 2;
|
||||
elseif strcmp(args.type, 'flexible')
|
||||
stewart.actuators.type = 3;
|
||||
end
|
||||
|
||||
stewart.actuators.k = args.k;
|
||||
stewart.actuators.c = args.c;
|
||||
|
||||
% Parallel stiffness
|
||||
stewart.actuators.kp = args.kp;
|
||||
stewart.actuators.cp = args.cp;
|
||||
|
||||
stewart.actuators.ka = args.ka;
|
||||
stewart.actuators.ca = args.ca;
|
||||
|
||||
stewart.actuators.ke = args.ke;
|
||||
stewart.actuators.ce = args.ce;
|
||||
|
||||
stewart.actuators.F_gain = args.F_gain;
|
||||
|
||||
stewart.actuators.ma = args.ma;
|
||||
stewart.actuators.me = args.me;
|
||||
|
||||
end
|
||||
36
matlab/src/inverseKinematics.m
Normal file
36
matlab/src/inverseKinematics.m
Normal file
@@ -0,0 +1,36 @@
|
||||
function [Li, dLi] = inverseKinematics(stewart, args)
|
||||
% inverseKinematics - Compute the needed length of each strut to have the wanted position and orientation of {B} with respect to {A}
|
||||
%
|
||||
% Syntax: [stewart] = inverseKinematics(stewart)
|
||||
%
|
||||
% Inputs:
|
||||
% - stewart - A structure with the following fields
|
||||
% - geometry.Aa [3x6] - The positions ai expressed in {A}
|
||||
% - geometry.Bb [3x6] - The positions bi expressed in {B}
|
||||
% - geometry.l [6x1] - Length of each strut
|
||||
% - args - Can have the following fields:
|
||||
% - AP [3x1] - The wanted position of {B} with respect to {A}
|
||||
% - ARB [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}
|
||||
%
|
||||
% Outputs:
|
||||
% - Li [6x1] - The 6 needed length of the struts in [m] to have the wanted pose of {B} w.r.t. {A}
|
||||
% - dLi [6x1] - The 6 needed displacement of the struts from the initial position in [m] to have the wanted pose of {B} w.r.t. {A}
|
||||
|
||||
arguments
|
||||
stewart
|
||||
args.AP (3,1) double {mustBeNumeric} = zeros(3,1)
|
||||
args.ARB (3,3) double {mustBeNumeric} = eye(3)
|
||||
end
|
||||
|
||||
assert(isfield(stewart.geometry, 'Aa'), 'stewart.geometry should have attribute Aa')
|
||||
Aa = stewart.geometry.Aa;
|
||||
|
||||
assert(isfield(stewart.geometry, 'Bb'), 'stewart.geometry should have attribute Bb')
|
||||
Bb = stewart.geometry.Bb;
|
||||
|
||||
assert(isfield(stewart.geometry, 'l'), 'stewart.geometry should have attribute l')
|
||||
l = stewart.geometry.l;
|
||||
|
||||
Li = sqrt(args.AP'*args.AP + diag(Bb'*Bb) + diag(Aa'*Aa) - (2*args.AP'*Aa)' + (2*args.AP'*(args.ARB*Bb))' - diag(2*(args.ARB*Bb)'*Aa));
|
||||
|
||||
dLi = Li-l;
|
||||
58
matlab/src/plotCube.m
Normal file
58
matlab/src/plotCube.m
Normal file
@@ -0,0 +1,58 @@
|
||||
function [] = plotCube(stewart, args)
|
||||
|
||||
arguments
|
||||
stewart
|
||||
args.Hc (1,1) double {mustBeNumeric, mustBePositive} = 60e-3
|
||||
args.FOc (1,1) double {mustBeNumeric} = 50e-3
|
||||
args.color (4,1) double {mustBeNumeric} = [0,0,0,0.5]
|
||||
args.linewidth (1,1) double {mustBeNumeric, mustBePositive} = 2.5
|
||||
args.link_to_struts logical {mustBeNumericOrLogical} = false
|
||||
end
|
||||
|
||||
sx = [ 2; -1; -1];
|
||||
sy = [ 0; 1; -1];
|
||||
sz = [ 1; 1; 1];
|
||||
|
||||
R = [sx, sy, sz]./vecnorm([sx, sy, sz]);
|
||||
|
||||
L = args.Hc*sqrt(3);
|
||||
|
||||
p_xyz = R'*[[0;0;0],[L;0;0],[L;L;0],[0;L;0],[0;0;L],[L;0;L],[L;L;L],[0;L;L]] - [0;0;1.5*args.Hc];
|
||||
|
||||
% Position center of the cube
|
||||
p_xyz = p_xyz + args.FOc*[0;0;1]*ones(1,8);
|
||||
|
||||
edges_order = [1 2 3 4 1];
|
||||
plot3(p_xyz(1,edges_order), p_xyz(2,edges_order), p_xyz(3,edges_order), '-', 'color', args.color, 'linewidth', args.linewidth);
|
||||
edges_order = [5 6 7 8 5];
|
||||
plot3(p_xyz(1,edges_order), p_xyz(2,edges_order), p_xyz(3,edges_order), '-', 'color', args.color, 'linewidth', args.linewidth);
|
||||
edges_order = [1 5];
|
||||
plot3(p_xyz(1,edges_order), p_xyz(2,edges_order), p_xyz(3,edges_order), '-', 'color', args.color, 'linewidth', args.linewidth);
|
||||
edges_order = [2 6];
|
||||
plot3(p_xyz(1,edges_order), p_xyz(2,edges_order), p_xyz(3,edges_order), '-', 'color', args.color, 'linewidth', args.linewidth);
|
||||
edges_order = [3 7];
|
||||
plot3(p_xyz(1,edges_order), p_xyz(2,edges_order), p_xyz(3,edges_order), '-', 'color', args.color, 'linewidth', args.linewidth);
|
||||
edges_order = [4 8];
|
||||
plot3(p_xyz(1,edges_order), p_xyz(2,edges_order), p_xyz(3,edges_order), '-', 'color', args.color, 'linewidth', args.linewidth);
|
||||
|
||||
if args.link_to_struts
|
||||
Fb = stewart.platform_M.Mb + stewart.geometry.FO_M;
|
||||
plot3([Fb(1,1), p_xyz(1,5)],...
|
||||
[Fb(2,1), p_xyz(2,5)],...
|
||||
[Fb(3,1), p_xyz(3,5)], '--', 'color', args.color, 'linewidth', args.linewidth);
|
||||
plot3([Fb(1,2), p_xyz(1,2)],...
|
||||
[Fb(2,2), p_xyz(2,2)],...
|
||||
[Fb(3,2), p_xyz(3,2)], '--', 'color', args.color, 'linewidth', args.linewidth);
|
||||
plot3([Fb(1,3), p_xyz(1,2)],...
|
||||
[Fb(2,3), p_xyz(2,2)],...
|
||||
[Fb(3,3), p_xyz(3,2)], '--', 'color', args.color, 'linewidth', args.linewidth);
|
||||
plot3([Fb(1,4), p_xyz(1,4)],...
|
||||
[Fb(2,4), p_xyz(2,4)],...
|
||||
[Fb(3,4), p_xyz(3,4)], '--', 'color', args.color, 'linewidth', args.linewidth);
|
||||
plot3([Fb(1,5), p_xyz(1,4)],...
|
||||
[Fb(2,5), p_xyz(2,4)],...
|
||||
[Fb(3,5), p_xyz(3,4)], '--', 'color', args.color, 'linewidth', args.linewidth);
|
||||
plot3([Fb(1,6), p_xyz(1,5)],...
|
||||
[Fb(2,6), p_xyz(2,5)],...
|
||||
[Fb(3,6), p_xyz(3,5)], '--', 'color', args.color, 'linewidth', args.linewidth);
|
||||
end
|
||||
15
matlab/src/plotCylindricalPayload.m
Normal file
15
matlab/src/plotCylindricalPayload.m
Normal file
@@ -0,0 +1,15 @@
|
||||
function [] = plotCylindricalPayload(stewart, args)
|
||||
|
||||
arguments
|
||||
stewart
|
||||
args.H (1,1) double {mustBeNumeric, mustBePositive} = 100e-3
|
||||
args.R (1,1) double {mustBeNumeric, mustBePositive} = 50e-3
|
||||
args.H_offset (1,1) double {mustBeNumeric} = 0
|
||||
args.color (3,1) double {mustBeNumeric} = [0.5,0.5,0.5]
|
||||
end
|
||||
|
||||
[X,Y,Z] = cylinder(args.R);
|
||||
Z = args.H*Z + args.H_offset;
|
||||
surf(X, Y, Z, 'facecolor', args.color, 'edgecolor', 'none')
|
||||
fill3(X(1,:), Y(1,:), Z(1,:), 'k', 'facecolor', args.color)
|
||||
fill3(X(2,:), Y(2,:), Z(2,:), 'k', 'facecolor', args.color)
|
||||
BIN
matlab/subsystems/metrology_6dof.slx
Normal file
BIN
matlab/subsystems/metrology_6dof.slx
Normal file
Binary file not shown.
BIN
matlab/subsystems/nano_hexapod_fixed_base.slx
Normal file
BIN
matlab/subsystems/nano_hexapod_fixed_base.slx
Normal file
Binary file not shown.
BIN
matlab/subsystems/nano_hexapod_mobile_platform.slx
Normal file
BIN
matlab/subsystems/nano_hexapod_mobile_platform.slx
Normal file
Binary file not shown.
BIN
matlab/subsystems/nano_hexapod_simscape.slx
Normal file
BIN
matlab/subsystems/nano_hexapod_simscape.slx
Normal file
Binary file not shown.
BIN
matlab/subsystems/nano_hexapod_strut.slx
Normal file
BIN
matlab/subsystems/nano_hexapod_strut.slx
Normal file
Binary file not shown.
111
paper/.latexmkrc
Normal file
111
paper/.latexmkrc
Normal file
@@ -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=('dehaeze26_cubic_architecture.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. For that purpose, use wildcard.
|
||||
# Also delete files generated by gnuplot/pgfplots contour plots
|
||||
# (.dat, .script, .table).
|
||||
$clean_ext = "%R-*.glstex %R_contourtmp*.*";
|
||||
179
paper/dehaeze26_cubic_architecture.bib
Normal file
179
paper/dehaeze26_cubic_architecture.bib
Normal file
@@ -0,0 +1,179 @@
|
||||
@article{stewart65_platf_with_six_degrees_freed,
|
||||
author = {Stewart, D.},
|
||||
title = {A Platform With Six Degrees of Freedom},
|
||||
journal = {Proceedings of the institution of mechanical engineers},
|
||||
volume = 180,
|
||||
number = 1,
|
||||
pages = {371--386},
|
||||
year = 1965,
|
||||
publisher = {Sage Publications Sage UK: London, England},
|
||||
}
|
||||
|
||||
|
||||
|
||||
|
||||
@article{geng94_six_degree_of_freed_activ,
|
||||
author = {Geng, Z. J. and Haynes, L. S.},
|
||||
title = {Six Degree-Of-Freedom Active Vibration Control Using the
|
||||
Stewart Platforms},
|
||||
journal = {IEEE Transactions on Control Systems Technology},
|
||||
volume = 2,
|
||||
number = 1,
|
||||
pages = {45--53},
|
||||
year = 1994,
|
||||
doi = {10.1109/87.273110},
|
||||
url = {https://doi.org/10.1109/87.273110},
|
||||
keywords = {parallel robot, cubic configuration},
|
||||
}
|
||||
|
||||
|
||||
|
||||
@article{preumont07_six_axis_singl_stage_activ,
|
||||
author = {Preumont, A. and Horodinca, M. and Romanescu, I. and de
|
||||
Marneffe, B. and Avraam, M. and Deraemaeker, A. and Bossens, F. and
|
||||
Abu Hanieh, A.},
|
||||
title = {A Six-Axis Single-Stage Active Vibration Isolator Based on
|
||||
Stewart Platform},
|
||||
journal = {Journal of Sound and Vibration},
|
||||
volume = 300,
|
||||
number = {3-5},
|
||||
pages = {644--661},
|
||||
year = 2007,
|
||||
doi = {10.1016/j.jsv.2006.07.050},
|
||||
url = {https://doi.org/10.1016/j.jsv.2006.07.050},
|
||||
keywords = {parallel robot},
|
||||
}
|
||||
|
||||
|
||||
|
||||
@article{jafari03_orthog_gough_stewar_platf_microm,
|
||||
author = {Jafari, F. and McInroy, J. E.},
|
||||
title = {Orthogonal Gough-Stewart Platforms for Micromanipulation},
|
||||
journal = {IEEE Transactions on Robotics and Automation},
|
||||
volume = 19,
|
||||
number = 4,
|
||||
pages = {595--603},
|
||||
year = 2003,
|
||||
doi = {10.1109/tra.2003.814506},
|
||||
url = {https://doi.org/10.1109/tra.2003.814506},
|
||||
issn = {1042-296X},
|
||||
keywords = {parallel robot, cubic configuration},
|
||||
month = 8,
|
||||
publisher = {Institute of Electrical and Electronics Engineers (IEEE)},
|
||||
}
|
||||
|
||||
|
||||
|
||||
@phdthesis{hanieh03_activ_stewar,
|
||||
author = {Abu Hanieh, A.},
|
||||
keywords = {parallel robot},
|
||||
school = {Universit{\'e} Libre de Bruxelles, Brussels, Belgium},
|
||||
title = {Active isolation and damping of vibrations via Stewart
|
||||
platform},
|
||||
year = 2003,
|
||||
}
|
||||
|
||||
|
||||
|
||||
@book{preumont18_vibrat_contr_activ_struc_fourt_edition,
|
||||
author = {Preumont, A.},
|
||||
title = {Vibration Control of Active Structures - Fourth Edition},
|
||||
year = 2018,
|
||||
publisher = {Springer International Publishing},
|
||||
url = {https://doi.org/10.1007/978-3-319-72296-2},
|
||||
doi = {10.1007/978-3-319-72296-2},
|
||||
keywords = {favorite, parallel robot},
|
||||
series = {Solid Mechanics and Its Applications},
|
||||
}
|
||||
|
||||
|
||||
|
||||
@article{thayer02_six_axis_vibrat_isolat_system,
|
||||
author = {Thayer, D. and Campbell, M. and Vagners, J. and
|
||||
von Flotow, A.},
|
||||
title = {Six-Axis Vibration Isolation System Using Soft Actuators
|
||||
and Multiple Sensors},
|
||||
journal = {Journal of Spacecraft and Rockets},
|
||||
volume = 39,
|
||||
number = 2,
|
||||
pages = {206--212},
|
||||
year = 2002,
|
||||
doi = {10.2514/2.3821},
|
||||
url = {https://doi.org/10.2514/2.3821},
|
||||
keywords = {parallel robot},
|
||||
}
|
||||
|
||||
|
||||
|
||||
@article{mcinroy00_desig_contr_flexur_joint_hexap,
|
||||
author = {McInroy, J. E. and Hamann, J. C.},
|
||||
title = {Design and Control of Flexure Jointed Hexapods},
|
||||
journal = {IEEE Transactions on Robotics and Automation},
|
||||
volume = 16,
|
||||
number = 4,
|
||||
pages = {372--381},
|
||||
year = 2000,
|
||||
doi = {10.1109/70.864229},
|
||||
url = {https://doi.org/10.1109/70.864229},
|
||||
keywords = {parallel robot},
|
||||
}
|
||||
|
||||
|
||||
|
||||
@phdthesis{li01_simul_fault_vibrat_isolat_point,
|
||||
author = {Li, X.},
|
||||
keywords = {parallel robot},
|
||||
school = {University of Wyoming},
|
||||
title = {Simultaneous, Fault-tolerant Vibration Isolation and
|
||||
Pointing Control of Flexure Jointed Hexapods},
|
||||
year = 2001,
|
||||
}
|
||||
|
||||
|
||||
|
||||
@inproceedings{mcinroy99_dynam,
|
||||
author = {McInroy, J. E.},
|
||||
title = {Dynamic modeling of flexure jointed hexapods for control
|
||||
purposes},
|
||||
booktitle = {Proceedings of the 1999 IEEE International Conference on
|
||||
Control Applications (Cat. No.99CH36328)},
|
||||
year = 1999,
|
||||
doi = {10.1109/cca.1999.806694},
|
||||
url = {https://doi.org/10.1109/cca.1999.806694},
|
||||
keywords = {parallel robot},
|
||||
}
|
||||
|
||||
|
||||
|
||||
@article{furutani04_nanom_cuttin_machin_using_stewar,
|
||||
author = {Furutani, K. and Suzuki, M. and Kudoh, R.},
|
||||
title = {Nanometre-Cutting Machine Using a Stewart-Platform Parallel
|
||||
Mechanism},
|
||||
journal = {Measurement Science and Technology},
|
||||
volume = 15,
|
||||
number = 2,
|
||||
pages = {467--474},
|
||||
year = 2004,
|
||||
doi = {10.1088/0957-0233/15/2/022},
|
||||
url = {https://doi.org/10.1088/0957-0233/15/2/022},
|
||||
keywords = {parallel robot, cubic configuration},
|
||||
}
|
||||
|
||||
|
||||
|
||||
@article{yang19_dynam_model_decoup_contr_flexib,
|
||||
author = {Yang, X. and Wu, H. and Chen, B. and Kang, S. and Cheng, S.},
|
||||
title = {Dynamic Modeling and Decoupled Control of a Flexible
|
||||
Stewart Platform for Vibration Isolation},
|
||||
journal = {Journal of Sound and Vibration},
|
||||
volume = 439,
|
||||
pages = {398--412},
|
||||
year = 2019,
|
||||
doi = {10.1016/j.jsv.2018.10.007},
|
||||
url = {https://doi.org/10.1016/j.jsv.2018.10.007},
|
||||
issn = {0022-460X},
|
||||
keywords = {parallel robot, flexure, decoupled control},
|
||||
month = 1,
|
||||
publisher = {Elsevier BV},
|
||||
}
|
||||
|
||||
567
paper/dehaeze26_cubic_architecture.org
Normal file
567
paper/dehaeze26_cubic_architecture.org
Normal file
@@ -0,0 +1,567 @@
|
||||
#+TITLE: Decoupling Properties of the Cubic Architecture
|
||||
:DRAWER:
|
||||
#+LANGUAGE: en
|
||||
#+EMAIL: dehaeze.thomas@gmail.com
|
||||
#+AUTHOR: Dehaeze Thomas
|
||||
|
||||
#+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: \input{preamble.tex}
|
||||
#+LATEX_HEADER_EXTRA: \input{preamble_extra.tex}
|
||||
#+LATEX_HEADER_EXTRA: \bibliography{dehaeze26_cubic_architecture.bib}
|
||||
|
||||
#+BIND: org-latex-bib-compiler "biber"
|
||||
:END:
|
||||
|
||||
#+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:
|
||||
** Journal
|
||||
https://asmedigitalcollection.asme.org/mechanicaldesign
|
||||
|
||||
Guide: https://www.asme.org/publications-submissions/journals/information-for-authors/journal-guidelines/writing-a-research-paper
|
||||
|
||||
#+begin_quote
|
||||
Research papers undergo full peer review. Authors are encouraged to prepare concise manuscripts that convey clearly the significance of the work. Research Papers do not have a specified length but are usually 8,000 to 12,000 words with 5-8 figures or tables.
|
||||
#+end_quote
|
||||
|
||||
** TODO [#B] Add more content from the PhD thesis?
|
||||
|
||||
Maybe add:
|
||||
- [[file:~/Cloud/work-projects/ID31-NASS/phd-thesis-chapters/B1-nass-geometry/nass-geometry.org::*Review of Stewart platforms][Review of Stewart platforms]]
|
||||
- [[file:~/Cloud/work-projects/ID31-NASS/phd-thesis-chapters/B1-nass-geometry/nass-geometry.org::*Effect of geometry on Stewart platform properties][Effect of geometry on Stewart platform properties]]
|
||||
|
||||
* Introduction :ignore:
|
||||
|
||||
The Cubic configuration for the Stewart platform was first proposed by Dr. Gough in a comment to the original paper by Dr. Stewart [[cite:&stewart65_platf_with_six_degrees_freed]].
|
||||
This configuration is characterized by active struts arranged in a mutually orthogonal configuration connecting the corners of a cube, as shown in Figure ref:fig:detail_kinematics_cubic_architecture_example.
|
||||
|
||||
Typically, the struts have similar length to the cube's edges, as illustrated in Figure ref:fig:detail_kinematics_cubic_architecture_example.
|
||||
Practical implementations of such configurations can be observed in Figures ref:fig:detail_kinematics_jpl, ref:fig:detail_kinematics_uw_gsp and ref:fig:detail_kinematics_uqp.
|
||||
It is also possible to implement designs with strut lengths smaller than the cube's edges (Figure ref:fig:detail_kinematics_cubic_architecture_example_small), as exemplified in Figure ref:fig:detail_kinematics_ulb_pz.
|
||||
|
||||
#+name: fig:detail_kinematics_cubic_architecture_examples
|
||||
#+caption: Typical Stewart platform cubic architectures in which struts' length is similar to the cube edges's length (\subref{fig:detail_kinematics_cubic_architecture_example}) or is taking just a portion of the edge (\subref{fig:detail_kinematics_cubic_architecture_example_small}).
|
||||
#+attr_latex: :options [htbp]
|
||||
#+begin_figure
|
||||
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_architecture_example}Classical Cubic architecture}
|
||||
#+attr_latex: :options {0.49\textwidth}
|
||||
#+begin_subfigure
|
||||
#+attr_latex: :scale 1
|
||||
[[file:figs/detail_kinematics_cubic_architecture_example.png]]
|
||||
#+end_subfigure
|
||||
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_architecture_example_small}Alternative configuration}
|
||||
#+attr_latex: :options {0.49\textwidth}
|
||||
#+begin_subfigure
|
||||
#+attr_latex: :scale 1
|
||||
[[file:figs/detail_kinematics_cubic_architecture_example_small.png]]
|
||||
#+end_subfigure
|
||||
#+end_figure
|
||||
|
||||
|
||||
Several advantageous properties attributed to the cubic configuration have contributed to its widespread adoption [[cite:&geng94_six_degree_of_freed_activ;&preumont07_six_axis_singl_stage_activ;&jafari03_orthog_gough_stewar_platf_microm]]: simplified kinematics relationships and dynamical analysis [[cite:&geng94_six_degree_of_freed_activ]]; uniform stiffness in all directions [[cite:&hanieh03_activ_stewar]]; uniform mobility [[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition, chapt.8.5.2]]; and minimization of the cross coupling between actuators and sensors in different struts [[cite:&preumont07_six_axis_singl_stage_activ]].
|
||||
This minimization is attributed to the fact that the struts are orthogonal to each other, and is said to facilitate collocated sensor-actuator control system design, i.e., the implementation of decentralized control [[cite:&geng94_six_degree_of_freed_activ;&thayer02_six_axis_vibrat_isolat_system]].
|
||||
|
||||
These properties are examined in this section to assess their relevance for the nano-hexapod.
|
||||
The mobility and stiffness properties of the cubic configuration are analyzed in Section ref:ssec:detail_kinematics_cubic_static.
|
||||
Dynamical decoupling is investigated in Section ref:ssec:detail_kinematics_cubic_dynamic, while decentralized control, crucial for the NASS, is examined in Section ref:ssec:detail_kinematics_decentralized_control.
|
||||
Given that the cubic architecture imposes strict geometric constraints, alternative designs are proposed in Section ref:ssec:detail_kinematics_cubic_design.
|
||||
The ultimate objective is to determine the suitability of the cubic architecture for the nano-hexapod.
|
||||
|
||||
* Static Properties
|
||||
<<ssec:detail_kinematics_cubic_static>>
|
||||
** Stiffness matrix for the Cubic architecture
|
||||
|
||||
Consider the cubic architecture shown in Figure ref:fig:detail_kinematics_cubic_schematic_full.
|
||||
The unit vectors corresponding to the edges of the cube are described by equation eqref:eq:detail_kinematics_cubic_s.
|
||||
|
||||
\begin{equation}\label{eq:detail_kinematics_cubic_s}
|
||||
\hat{\bm{s}}_1 = \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ 0 \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
|
||||
\hat{\bm{s}}_2 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{-1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
|
||||
\hat{\bm{s}}_3 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{ 1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
|
||||
\hat{\bm{s}}_4 = \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ 0 \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
|
||||
\hat{\bm{s}}_5 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{-1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
|
||||
\hat{\bm{s}}_6 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{ 1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
#+name: fig:detail_kinematics_cubic_schematic_cases
|
||||
#+caption: Cubic architecture. Struts are represented in blue. The cube's center by a black dot. The Struts can match the cube's edges (\subref{fig:detail_kinematics_cubic_schematic_full}) or just take a portion of the edge (\subref{fig:detail_kinematics_cubic_schematic})
|
||||
#+attr_latex: :options [htbp]
|
||||
#+begin_figure
|
||||
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_schematic_full}Full cube}
|
||||
#+attr_latex: :options {0.48\textwidth}
|
||||
#+begin_subfigure
|
||||
#+attr_latex: :scale 0.9
|
||||
[[file:figs/detail_kinematics_cubic_schematic_full.png]]
|
||||
#+end_subfigure
|
||||
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_schematic}Cube's portion}
|
||||
#+attr_latex: :options {0.48\textwidth}
|
||||
#+begin_subfigure
|
||||
#+attr_latex: :scale 0.9
|
||||
[[file:figs/detail_kinematics_cubic_schematic.png]]
|
||||
#+end_subfigure
|
||||
#+end_figure
|
||||
|
||||
Coordinates of the cube's vertices relevant for the top joints, expressed with respect to the cube's center, are shown in equation eqref:eq:detail_kinematics_cubic_vertices.
|
||||
|
||||
\begin{equation}\label{eq:detail_kinematics_cubic_vertices}
|
||||
\tilde{\bm{b}}_1 = \tilde{\bm{b}}_2 = H_c \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{-\sqrt{3}}{\sqrt{2}} \\ \frac{1}{2} \end{bmatrix}, \quad
|
||||
\tilde{\bm{b}}_3 = \tilde{\bm{b}}_4 = H_c \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{ \sqrt{3}}{\sqrt{2}} \\ \frac{1}{2} \end{bmatrix}, \quad
|
||||
\tilde{\bm{b}}_5 = \tilde{\bm{b}}_6 = H_c \begin{bmatrix} \frac{-2}{\sqrt{2}} \\ 0 \\ \frac{1}{2} \end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
In the case where top joints are positioned at the cube's vertices, a diagonal stiffness matrix is obtained as shown in equation eqref:eq:detail_kinematics_cubic_stiffness.
|
||||
Translation stiffness is twice the stiffness of the struts, and rotational stiffness is proportional to the square of the cube's size $H_c$.
|
||||
|
||||
\begin{equation}\label{eq:detail_kinematics_cubic_stiffness}
|
||||
\bm{K}_{\{B\} = \{C\}} = k \begin{bmatrix}
|
||||
2 & 0 & 0 & 0 & 0 & 0 \\
|
||||
0 & 2 & 0 & 0 & 0 & 0 \\
|
||||
0 & 0 & 2 & 0 & 0 & 0 \\
|
||||
0 & 0 & 0 & \frac{3}{2} H_c^2 & 0 & 0 \\
|
||||
0 & 0 & 0 & 0 & \frac{3}{2} H_c^2 & 0 \\
|
||||
0 & 0 & 0 & 0 & 0 & 6 H_c^2 \\
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
However, typically, the top joints are not placed at the cube's vertices but at positions along the cube's edges (Figure ref:fig:detail_kinematics_cubic_schematic).
|
||||
In that case, the location of the top joints can be expressed by equation eqref:eq:detail_kinematics_cubic_edges, yet the computed stiffness matrix remains identical to Equation eqref:eq:detail_kinematics_cubic_stiffness.
|
||||
|
||||
\begin{equation}\label{eq:detail_kinematics_cubic_edges}
|
||||
\bm{b}_i = \tilde{\bm{b}}_i + \alpha \hat{\bm{s}}_i
|
||||
\end{equation}
|
||||
|
||||
|
||||
The stiffness matrix is therefore diagonal when the considered $\{B\}$ frame is located at the center of the cube (shown by frame $\{C\}$).
|
||||
This means that static forces (resp torques) applied at the cube's center will induce pure translations (resp rotations around the cube's center).
|
||||
This specific location where the stiffness matrix is diagonal is referred to as the "Center of Stiffness" (analogous to the "Center of Mass" where the mass matrix is diagonal).
|
||||
|
||||
** Effect of having frame $\{B\}$ off-centered
|
||||
|
||||
When the reference frames $\{A\}$ and $\{B\}$ are shifted from the cube's center, off-diagonal elements emerge in the stiffness matrix.
|
||||
|
||||
Considering a vertical shift as shown in Figure ref:fig:detail_kinematics_cubic_schematic, the stiffness matrix transforms into that shown in Equation eqref:eq:detail_kinematics_cubic_stiffness_off_centered.
|
||||
Off-diagonal elements increase proportionally with the height difference between the cube's center and the considered $\{B\}$ frame.
|
||||
|
||||
\begin{equation}\label{eq:detail_kinematics_cubic_stiffness_off_centered}
|
||||
\bm{K}_{\{B\} \neq \{C\}} = k \begin{bmatrix}
|
||||
2 & 0 & 0 & 0 & -2 H & 0 \\
|
||||
0 & 2 & 0 & 2 H & 0 & 0 \\
|
||||
0 & 0 & 2 & 0 & 0 & 0 \\
|
||||
0 & 2 H & 0 & \frac{3}{2} H_c^2 + 2 H^2 & 0 & 0 \\
|
||||
-2 H & 0 & 0 & 0 & \frac{3}{2} H_c^2 + 2 H^2 & 0 \\
|
||||
0 & 0 & 0 & 0 & 0 & 6 H_c^2 \\
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
This stiffness matrix structure is characteristic of Stewart platforms exhibiting symmetry, and is not an exclusive property of cubic architectures.
|
||||
Therefore, the stiffness characteristics of the cubic architecture are only distinctive when considering a reference frame located at the cube's center.
|
||||
This poses a practical limitation, as in most applications, the relevant frame (where motion is of interest and forces are applied) is located above the top platform.
|
||||
|
||||
It should be noted that for the stiffness matrix to be diagonal, the cube's center doesn't need to coincide with the geometric center of the Stewart platform.
|
||||
This observation leads to the interesting alternative architectures presented in Section ref:ssec:detail_kinematics_cubic_design.
|
||||
|
||||
** Uniform Mobility
|
||||
|
||||
The translational mobility of the Stewart platform with constant orientation was analyzed.
|
||||
Considering limited actuator stroke (elongation of each strut), the maximum achievable positions in XYZ space were estimated.
|
||||
The resulting mobility in X, Y, and Z directions for the cubic architecture is illustrated in Figure ref:fig:detail_kinematics_cubic_mobility_translations.
|
||||
|
||||
The translational workspace analysis reveals that for the cubic architecture, the achievable positions form a cube whose axes align with the struts, with the cube's edge length corresponding to the strut axial stroke.
|
||||
These findings suggest that the mobility pattern is more subtle than sometimes described in the literature [[cite:&mcinroy00_desig_contr_flexur_joint_hexap]], exhibiting uniformity primarily along directions aligned with the cube's edges rather than uniform spherical distribution in all XYZ directions.
|
||||
This configuration still offers more consistent mobility characteristics compared to alternative architectures illustrated in Figure ref:fig:detail_kinematics_mobility_trans.
|
||||
|
||||
The rotational mobility, illustrated in Figure ref:fig:detail_kinematics_cubic_mobility_rotations, exhibits greater achievable angular stroke in the $R_x$ and $R_y$ directions compared to the $R_z$ direction.
|
||||
Furthermore, an inverse relationship exists between the cube's dimension and rotational mobility, with larger cube sizes corresponding to more limited angular displacement capabilities.
|
||||
|
||||
#+name: fig:detail_kinematics_cubic_mobility
|
||||
#+caption: Mobility of a Stewart platform with Cubic architecture. Both for translations (\subref{fig:detail_kinematics_cubic_mobility_translations}) and rotations (\subref{fig:detail_kinematics_cubic_mobility_rotations})
|
||||
#+attr_latex: :options [htbp]
|
||||
#+begin_figure
|
||||
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_mobility_translations}Mobility in translation}
|
||||
#+attr_latex: :options {0.48\textwidth}
|
||||
#+begin_subfigure
|
||||
#+attr_latex: :scale 1
|
||||
[[file:figs/detail_kinematics_cubic_mobility_translations.png]]
|
||||
#+end_subfigure
|
||||
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_mobility_rotations}Mobility in rotation}
|
||||
#+attr_latex: :options {0.48\textwidth}
|
||||
#+begin_subfigure
|
||||
#+attr_latex: :scale 1
|
||||
[[file:figs/detail_kinematics_cubic_mobility_rotations.png]]
|
||||
#+end_subfigure
|
||||
#+end_figure
|
||||
|
||||
* Dynamical Decoupling
|
||||
<<ssec:detail_kinematics_cubic_dynamic>>
|
||||
** Introduction :ignore:
|
||||
|
||||
This section examines the dynamics of the cubic architecture in the Cartesian frame which corresponds to the transfer function from forces and torques $\bm{\mathcal{F}}$ to translations and rotations $\bm{\mathcal{X}}$ of the top platform.
|
||||
When relative motion sensors are integrated in each strut (measuring $\bm{\mathcal{L}}$), the pose $\bm{\mathcal{X}}$ is computed using the Jacobian matrix as shown in Figure ref:fig:detail_kinematics_centralized_control.
|
||||
|
||||
#+name: fig:detail_kinematics_centralized_control
|
||||
#+caption: Typical control architecture in the cartesian frame
|
||||
[[file:figs/detail_kinematics_centralized_control.png]]
|
||||
|
||||
** Low frequency and High frequency coupling
|
||||
|
||||
As derived during the conceptual design phase, the dynamics from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$ is described by Equation eqref:eq:detail_kinematics_transfer_function_cart.
|
||||
At low frequency, the behavior of the platform depends on the stiffness matrix eqref:eq:detail_kinematics_transfer_function_cart_low_freq.
|
||||
|
||||
\begin{equation}\label{eq:detail_kinematics_transfer_function_cart_low_freq}
|
||||
\frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(j \omega) \xrightarrow[\omega \to 0]{} \bm{K}^{-1}
|
||||
\end{equation}
|
||||
|
||||
In Section ref:ssec:detail_kinematics_cubic_static, it was demonstrated that for the cubic configuration, the stiffness matrix is diagonal if frame $\{B\}$ is positioned at the cube's center.
|
||||
In this case, the "Cartesian" plant is decoupled at low frequency.
|
||||
At high frequency, the behavior is governed by the mass matrix (evaluated at frame $\{B\}$) eqref:eq:detail_kinematics_transfer_function_high_freq.
|
||||
|
||||
\begin{equation}\label{eq:detail_kinematics_transfer_function_high_freq}
|
||||
\frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(j \omega) \xrightarrow[\omega \to \infty]{} - \omega^2 \bm{M}^{-1}
|
||||
\end{equation}
|
||||
|
||||
To achieve a diagonal mass matrix, the center of mass of the mobile components must coincide with the $\{B\}$ frame, and the principal axes of inertia must align with the axes of the $\{B\}$ frame.
|
||||
|
||||
#+name: fig:detail_kinematics_cubic_payload
|
||||
#+caption: Cubic stewart platform with top cylindrical payload
|
||||
#+attr_latex: :width 0.6\linewidth
|
||||
[[file:figs/detail_kinematics_cubic_payload.png]]
|
||||
|
||||
To verify these properties, a cubic Stewart platform with a cylindrical payload was analyzed (Figure ref:fig:detail_kinematics_cubic_payload).
|
||||
Transfer functions from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$ were computed for two specific locations of the $\{B\}$ frames.
|
||||
When the $\{B\}$ frame was positioned at the center of mass, coupling at low frequency was observed due to the non-diagonal stiffness matrix (Figure ref:fig:detail_kinematics_cubic_cart_coupling_com).
|
||||
Conversely, when positioned at the center of stiffness, coupling occurred at high frequency due to the non-diagonal mass matrix (Figure ref:fig:detail_kinematics_cubic_cart_coupling_cok).
|
||||
|
||||
#+name: fig:detail_kinematics_cubic_cart_coupling
|
||||
#+caption: Transfer functions for a Cubic Stewart platform expressed in the Cartesian frame. Two locations of the $\{B\}$ frame are considered: at the center of mass of the moving body (\subref{fig:detail_kinematics_cubic_cart_coupling_com}) and at the cube's center (\subref{fig:detail_kinematics_cubic_cart_coupling_cok}).
|
||||
#+attr_latex: :options [htbp]
|
||||
#+begin_figure
|
||||
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_cart_coupling_com}$\{B\}$ at the center of mass}
|
||||
#+attr_latex: :options {0.48\textwidth}
|
||||
#+begin_subfigure
|
||||
#+attr_latex: :width 0.95\linewidth
|
||||
[[file:figs/detail_kinematics_cubic_cart_coupling_com.png]]
|
||||
#+end_subfigure
|
||||
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_cart_coupling_cok}$\{B\}$ at the cube's center}
|
||||
#+attr_latex: :options {0.48\textwidth}
|
||||
#+begin_subfigure
|
||||
#+attr_latex: :width 0.95\linewidth
|
||||
[[file:figs/detail_kinematics_cubic_cart_coupling_cok.png]]
|
||||
#+end_subfigure
|
||||
#+end_figure
|
||||
|
||||
** Payload's CoM at the cube's center
|
||||
|
||||
An effective strategy for improving dynamical performances involves aligning the cube's center (center of stiffness) with the center of mass of the moving components [[cite:&li01_simul_fault_vibrat_isolat_point]].
|
||||
This can be achieved by positioning the payload below the top platform, such that the center of mass of the moving body coincides with the cube's center (Figure ref:fig:detail_kinematics_cubic_centered_payload).
|
||||
This approach was physically implemented in several studies [[cite:&mcinroy99_dynam;&jafari03_orthog_gough_stewar_platf_microm]], as shown in Figure ref:fig:detail_kinematics_uw_gsp.
|
||||
The resulting dynamics are indeed well-decoupled (Figure ref:fig:detail_kinematics_cubic_cart_coupling_com_cok), taking advantage from diagonal stiffness and mass matrices.
|
||||
The primary limitation of this approach is that, for many applications including the nano-hexapod, the payload must be positioned above the top platform.
|
||||
If a design similar to Figure ref:fig:detail_kinematics_cubic_centered_payload were employed for the nano-hexapod, the X-ray beam would intersect with the struts during spindle rotation.
|
||||
|
||||
#+name: fig:detail_kinematics_cubic_com_cok
|
||||
#+caption: Cubic Stewart platform with payload at the cube's center (\subref{fig:detail_kinematics_cubic_centered_payload}). Obtained cartesian plant is fully decoupled (\subref{fig:detail_kinematics_cubic_cart_coupling_com_cok})
|
||||
#+attr_latex: :options [htbp]
|
||||
#+begin_figure
|
||||
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_centered_payload}Payload at the cube's center}
|
||||
#+attr_latex: :options {0.49\textwidth}
|
||||
#+begin_subfigure
|
||||
#+attr_latex: :width 0.95\linewidth
|
||||
[[file:figs/detail_kinematics_cubic_centered_payload.png]]
|
||||
#+end_subfigure
|
||||
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_cart_coupling_com_cok}Fully decoupled cartesian plant}
|
||||
#+attr_latex: :options {0.49\textwidth}
|
||||
#+begin_subfigure
|
||||
#+attr_latex: :width 0.95\linewidth
|
||||
[[file:figs/detail_kinematics_cubic_cart_coupling_com_cok.png]]
|
||||
#+end_subfigure
|
||||
#+end_figure
|
||||
|
||||
** Conclusion
|
||||
|
||||
The analysis of dynamical properties of the cubic architecture yields several important conclusions.
|
||||
Static decoupling, characterized by a diagonal stiffness matrix, is achieved when reference frames $\{A\}$ and $\{B\}$ are positioned at the cube's center.
|
||||
Note that this property can also be obtained with non-cubic architectures that exhibit symmetrical strut arrangements.
|
||||
Dynamic decoupling requires both static decoupling and coincidence of the mobile platform's center of mass with reference frame $\{B\}$.
|
||||
While this configuration offers powerful control advantages, it requires positioning the payload at the cube's center, which is highly restrictive and often impractical.
|
||||
|
||||
* Decentralized Control
|
||||
<<ssec:detail_kinematics_decentralized_control>>
|
||||
** Introduction :ignore:
|
||||
|
||||
The orthogonal arrangement of struts in the cubic architecture suggests a potential minimization of inter-strut coupling, which could theoretically create favorable conditions for decentralized control.
|
||||
Two sensor types integrated in the struts are considered: displacement sensors and force sensors.
|
||||
The control architecture is illustrated in Figure ref:fig:detail_kinematics_decentralized_control, where $\bm{K}_{\mathcal{L}}$ represents a diagonal transfer function matrix.
|
||||
|
||||
#+name: fig:detail_kinematics_decentralized_control
|
||||
#+caption: Decentralized control in the frame of the struts.
|
||||
[[file:figs/detail_kinematics_decentralized_control.png]]
|
||||
|
||||
The obtained plant dynamics in the frame of the struts are compared for two Stewart platforms.
|
||||
The first employs a cubic architecture shown in Figure ref:fig:detail_kinematics_cubic_payload.
|
||||
The second uses a non-cubic Stewart platform shown in Figure ref:fig:detail_kinematics_non_cubic_payload, featuring identical payload and strut dynamics but with struts oriented more vertically to differentiate it from the cubic architecture.
|
||||
|
||||
#+name: fig:detail_kinematics_non_cubic_payload
|
||||
#+caption: Stewart platform with non-cubic architecture
|
||||
#+attr_latex: :width 0.6\linewidth
|
||||
[[file:figs/detail_kinematics_non_cubic_payload.png]]
|
||||
|
||||
** Relative Displacement Sensors
|
||||
|
||||
The transfer functions from actuator force in each strut to the relative motion of the struts are presented in Figure ref:fig:detail_kinematics_decentralized_dL.
|
||||
As anticipated from the equations of motion from $\bm{f}$ to $\bm{\mathcal{L}}$ eqref:eq:detail_kinematics_transfer_function_struts, the $6 \times 6$ plant is decoupled at low frequency.
|
||||
At high frequency, coupling is observed as the mass matrix projected in the strut frame is not diagonal.
|
||||
|
||||
No significant advantage is evident for the cubic architecture (Figure ref:fig:detail_kinematics_cubic_decentralized_dL) compared to the non-cubic architecture (Figure ref:fig:detail_kinematics_non_cubic_decentralized_dL).
|
||||
The resonance frequencies differ between the two cases because the more vertical strut orientation in the non-cubic architecture alters the stiffness properties of the Stewart platform, consequently shifting the frequencies of various modes.
|
||||
|
||||
#+name: fig:detail_kinematics_decentralized_dL
|
||||
#+caption: Bode plot of the transfer functions from actuator force to relative displacement sensor in each strut. Both for a non-cubic architecture (\subref{fig:detail_kinematics_non_cubic_decentralized_dL}) and for a cubic architecture (\subref{fig:detail_kinematics_cubic_decentralized_dL})
|
||||
#+attr_latex: :options [htbp]
|
||||
#+begin_figure
|
||||
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_non_cubic_decentralized_dL}Non cubic architecture}
|
||||
#+attr_latex: :options {0.48\textwidth}
|
||||
#+begin_subfigure
|
||||
#+attr_latex: :width 0.95\linewidth
|
||||
[[file:figs/detail_kinematics_non_cubic_decentralized_dL.png]]
|
||||
#+end_subfigure
|
||||
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_decentralized_dL}Cubic architecture}
|
||||
#+attr_latex: :options {0.48\textwidth}
|
||||
#+begin_subfigure
|
||||
#+attr_latex: :width 0.95\linewidth
|
||||
[[file:figs/detail_kinematics_cubic_decentralized_dL.png]]
|
||||
#+end_subfigure
|
||||
#+end_figure
|
||||
|
||||
** Force Sensors
|
||||
|
||||
Similarly, the transfer functions from actuator force to force sensors in each strut were analyzed for both cubic and non-cubic Stewart platforms.
|
||||
The results are presented in Figure ref:fig:detail_kinematics_decentralized_fn.
|
||||
The system demonstrates good decoupling at high frequency in both cases, with no clear advantage for the cubic architecture.
|
||||
|
||||
#+name: fig:detail_kinematics_decentralized_fn
|
||||
#+caption: Bode plot of the transfer functions from actuator force to force sensor in each strut. Both for a non-cubic architecture (\subref{fig:detail_kinematics_non_cubic_decentralized_fn}) and for a cubic architecture (\subref{fig:detail_kinematics_cubic_decentralized_fn})
|
||||
#+attr_latex: :options [htbp]
|
||||
#+begin_figure
|
||||
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_non_cubic_decentralized_fn}Non cubic architecture}
|
||||
#+attr_latex: :options {0.48\textwidth}
|
||||
#+begin_subfigure
|
||||
#+attr_latex: :width 0.95\linewidth
|
||||
[[file:figs/detail_kinematics_non_cubic_decentralized_fn.png]]
|
||||
#+end_subfigure
|
||||
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_decentralized_fn}Cubic architecture}
|
||||
#+attr_latex: :options {0.48\textwidth}
|
||||
#+begin_subfigure
|
||||
#+attr_latex: :width 0.95\linewidth
|
||||
[[file:figs/detail_kinematics_cubic_decentralized_fn.png]]
|
||||
#+end_subfigure
|
||||
#+end_figure
|
||||
|
||||
** Conclusion
|
||||
|
||||
The presented results do not demonstrate the pronounced decoupling advantages often associated with cubic architectures in the literature.
|
||||
Both the cubic and non-cubic configurations exhibited similar coupling characteristics, suggesting that the benefits of orthogonal strut arrangement for decentralized control is less obvious than often reported in the literature.
|
||||
|
||||
* Cubic architecture with Cube's center above the top platform
|
||||
<<ssec:detail_kinematics_cubic_design>>
|
||||
** Introduction :ignore:
|
||||
|
||||
As demonstrated in Section ref:ssec:detail_kinematics_cubic_dynamic, the cubic architecture can exhibit advantageous dynamical properties when the center of mass of the moving body coincides with the cube's center, resulting in diagonal mass and stiffness matrices.
|
||||
As shown in Section ref:ssec:detail_kinematics_cubic_static, the stiffness matrix is diagonal when the considered $\{B\}$ frame is located at the cube's center.
|
||||
However, the $\{B\}$ frame is typically positioned above the top platform where forces are applied and displacements are measured.
|
||||
|
||||
This section proposes modifications to the cubic architecture to enable positioning the payload above the top platform while still leveraging the advantageous dynamical properties of the cubic configuration.
|
||||
|
||||
Three key parameters define the geometry of the cubic Stewart platform: $H$, the height of the Stewart platform (distance from fixed base to mobile platform); $H_c$, the height of the cube, as shown in Figure ref:fig:detail_kinematics_cubic_schematic_full; and $H_{CoM}$, the height of the center of mass relative to the mobile platform (coincident with the cube's center).
|
||||
|
||||
Depending on the cube's size $H_c$ in relation to $H$ and $H_{CoM}$, different designs emerge.
|
||||
In the following examples, $H = 100\,mm$ and $H_{CoM} = 20\,mm$.
|
||||
|
||||
** Small cube
|
||||
|
||||
When the cube size $H_c$ is smaller than twice the height of the CoM $H_{CoM}$ eqref:eq:detail_kinematics_cube_small, the resulting design is shown in Figure ref:fig:detail_kinematics_cubic_above_small.
|
||||
|
||||
\begin{equation}\label{eq:detail_kinematics_cube_small}
|
||||
H_c < 2 H_{CoM}
|
||||
\end{equation}
|
||||
|
||||
# TODO - Add link to Figure ref:fig:nhexa_stewart_piezo_furutani (page pageref:fig:nhexa_stewart_piezo_furutani)
|
||||
This configuration is similar to that described in [[cite:&furutani04_nanom_cuttin_machin_using_stewar]], although they do not explicitly identify it as a cubic configuration.
|
||||
Adjacent struts are parallel to each other, differing from the typical architecture where parallel struts are positioned opposite to each other.
|
||||
|
||||
This approach yields a compact architecture, but the small cube size may result in insufficient rotational stiffness.
|
||||
|
||||
#+name: fig:detail_kinematics_cubic_above_small
|
||||
#+caption: Cubic architecture with cube's center above the top platform. A cube height of 40mm is used.
|
||||
#+attr_latex: :options [htbp]
|
||||
#+begin_figure
|
||||
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_above_small_iso}Isometric view}
|
||||
#+attr_latex: :options {0.36\textwidth}
|
||||
#+begin_subfigure
|
||||
#+attr_latex: :width 0.9\linewidth
|
||||
[[file:figs/detail_kinematics_cubic_above_small_iso.png]]
|
||||
#+end_subfigure
|
||||
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_above_small_side}Side view}
|
||||
#+attr_latex: :options {0.30\textwidth}
|
||||
#+begin_subfigure
|
||||
#+attr_latex: :width 0.9\linewidth
|
||||
[[file:figs/detail_kinematics_cubic_above_small_side.png]]
|
||||
#+end_subfigure
|
||||
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_above_small_top}Top view}
|
||||
#+attr_latex: :options {0.30\textwidth}
|
||||
#+begin_subfigure
|
||||
#+attr_latex: :width 0.9\linewidth
|
||||
[[file:figs/detail_kinematics_cubic_above_small_top.png]]
|
||||
#+end_subfigure
|
||||
#+end_figure
|
||||
|
||||
** Medium sized cube
|
||||
|
||||
Increasing the cube's size such that eqref:eq:detail_kinematics_cube_medium is verified produces an architecture with intersecting struts (Figure ref:fig:detail_kinematics_cubic_above_medium).
|
||||
|
||||
\begin{equation}\label{eq:detail_kinematics_cube_medium}
|
||||
2 H_{CoM} < H_c < 2 (H_{CoM} + H)
|
||||
\end{equation}
|
||||
|
||||
This configuration resembles the design proposed in [[cite:&yang19_dynam_model_decoup_contr_flexib]] (Figure ref:fig:detail_kinematics_yang19), although their design is not strictly cubic.
|
||||
|
||||
#+name: fig:detail_kinematics_cubic_above_medium
|
||||
#+caption: Cubic architecture with cube's center above the top platform. A cube height of 140mm is used.
|
||||
#+attr_latex: :options [htbp]
|
||||
#+begin_figure
|
||||
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_above_medium_iso}Isometric view}
|
||||
#+attr_latex: :options {0.36\textwidth}
|
||||
#+begin_subfigure
|
||||
#+attr_latex: :width 0.9\linewidth
|
||||
[[file:figs/detail_kinematics_cubic_above_medium_iso.png]]
|
||||
#+end_subfigure
|
||||
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_above_medium_side}Side view}
|
||||
#+attr_latex: :options {0.30\textwidth}
|
||||
#+begin_subfigure
|
||||
#+attr_latex: :width 0.9\linewidth
|
||||
[[file:figs/detail_kinematics_cubic_above_medium_side.png]]
|
||||
#+end_subfigure
|
||||
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_above_medium_top}Top view}
|
||||
#+attr_latex: :options {0.30\textwidth}
|
||||
#+begin_subfigure
|
||||
#+attr_latex: :width 0.9\linewidth
|
||||
[[file:figs/detail_kinematics_cubic_above_medium_top.png]]
|
||||
#+end_subfigure
|
||||
#+end_figure
|
||||
|
||||
** Large cube
|
||||
|
||||
When the cube's height exceeds twice the sum of the platform height and CoM height eqref:eq:detail_kinematics_cube_large, the architecture shown in Figure ref:fig:detail_kinematics_cubic_above_large is obtained.
|
||||
|
||||
\begin{equation}\label{eq:detail_kinematics_cube_large}
|
||||
2 (H_{CoM} + H) < H_c
|
||||
\end{equation}
|
||||
|
||||
#+name: fig:detail_kinematics_cubic_above_large
|
||||
#+caption: Cubic architecture with cube's center above the top platform. A cube height of 240mm is used.
|
||||
#+attr_latex: :options [htbp]
|
||||
#+begin_figure
|
||||
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_above_large_iso}Isometric view}
|
||||
#+attr_latex: :options {0.36\textwidth}
|
||||
#+begin_subfigure
|
||||
#+attr_latex: :width 0.9\linewidth
|
||||
[[file:figs/detail_kinematics_cubic_above_large_iso.png]]
|
||||
#+end_subfigure
|
||||
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_above_large_side}Side view}
|
||||
#+attr_latex: :options {0.30\textwidth}
|
||||
#+begin_subfigure
|
||||
#+attr_latex: :width 0.9\linewidth
|
||||
[[file:figs/detail_kinematics_cubic_above_large_side.png]]
|
||||
#+end_subfigure
|
||||
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_above_large_top}Top view}
|
||||
#+attr_latex: :options {0.30\textwidth}
|
||||
#+begin_subfigure
|
||||
#+attr_latex: :width 0.9\linewidth
|
||||
[[file:figs/detail_kinematics_cubic_above_large_top.png]]
|
||||
#+end_subfigure
|
||||
#+end_figure
|
||||
|
||||
** Platform size
|
||||
|
||||
For the proposed configuration, the top joints $\bm{b}_i$ (resp. the bottom joints $\bm{a}_i$) and are positioned on a circle with radius $R_{b_i}$ (resp. $R_{a_i}$) described by Equation eqref:eq:detail_kinematics_cube_joints.
|
||||
|
||||
\begin{subequations}\label{eq:detail_kinematics_cube_joints}
|
||||
\begin{align}
|
||||
R_{b_i} &= \sqrt{\frac{3}{2} H_c^2 + 2 H_{CoM}^2} \label{eq:detail_kinematics_cube_top_joints} \\
|
||||
R_{a_i} &= \sqrt{\frac{3}{2} H_c^2 + 2 (H_{CoM} + H)^2} \label{eq:detail_kinematics_cube_bot_joints}
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
|
||||
Since the rotational stiffness for the cubic architecture scales with the square of the cube's height eqref:eq:detail_kinematics_cubic_stiffness, the cube's size can be determined based on rotational stiffness requirements.
|
||||
Subsequently, using Equation eqref:eq:detail_kinematics_cube_joints, the dimensions of the top and bottom platforms can be calculated.
|
||||
|
||||
* Conclusion
|
||||
:PROPERTIES:
|
||||
:UNNUMBERED: t
|
||||
:END:
|
||||
|
||||
The analysis of the cubic architecture for Stewart platforms yielded several important findings.
|
||||
While the cubic configuration provides uniform stiffness in the XYZ directions, it stiffness property becomes particularly advantageous when forces and torques are applied at the cube's center.
|
||||
Under these conditions, the stiffness matrix becomes diagonal, resulting in a decoupled Cartesian plant at low frequencies.
|
||||
|
||||
Regarding mobility, the translational capabilities of the cubic configuration exhibit uniformity along the directions of the orthogonal struts, rather than complete uniformity in the Cartesian space.
|
||||
This understanding refines the characterization of cubic architecture mobility commonly presented in literature.
|
||||
|
||||
The analysis of decentralized control in the frame of the struts revealed more nuanced results than expected.
|
||||
While cubic architectures are frequently associated with reduced coupling between actuators and sensors, this study showed that these benefits may be more subtle or context-dependent than commonly described.
|
||||
Under the conditions analyzed, the coupling characteristics of cubic and non-cubic configurations, in the frame of the struts, appeared similar.
|
||||
|
||||
Fully decoupled dynamics in the Cartesian frame can be achieved when the center of mass of the moving body coincides with the cube's center.
|
||||
However, this arrangement presents practical challenges, as the cube's center is traditionally located between the top and bottom platforms, making payload placement problematic for many applications.
|
||||
|
||||
To address this limitation, modified cubic architectures have been proposed with the cube's center positioned above the top platform.
|
||||
Three distinct configurations have been identified, each with different geometric arrangements but sharing the common characteristic that the cube's center is positioned above the top platform.
|
||||
This structural modification enables the alignment of the moving body's center of mass with the center of stiffness, resulting in beneficial decoupling properties in the Cartesian frame.
|
||||
|
||||
* Bibliography :ignore:
|
||||
#+latex: \printbibliography[heading=bibintoc,title={Bibliography}]
|
||||
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paper/dehaeze26_cubic_architecture.pdf
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paper/dehaeze26_cubic_architecture.pdf
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paper/dehaeze26_cubic_architecture.tex
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paper/dehaeze26_cubic_architecture.tex
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@@ -0,0 +1,469 @@
|
||||
% Created 2025-11-26 Wed 09:39
|
||||
% Intended LaTeX compiler: pdflatex
|
||||
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
|
||||
|
||||
\input{preamble.tex}
|
||||
\input{preamble_extra.tex}
|
||||
\bibliography{dehaeze26_cubic_architecture.bib}
|
||||
\author{Dehaeze Thomas}
|
||||
\date{\today}
|
||||
\title{Decoupling Properties of the Cubic Architecture}
|
||||
\hypersetup{
|
||||
pdfauthor={Dehaeze Thomas},
|
||||
pdftitle={Decoupling Properties of the Cubic Architecture},
|
||||
pdfkeywords={},
|
||||
pdfsubject={},
|
||||
pdfcreator={Emacs 30.2 (Org mode 9.7.34)},
|
||||
pdflang={English}}
|
||||
\usepackage{biblatex}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\maketitle
|
||||
\tableofcontents
|
||||
|
||||
\clearpage
|
||||
The Cubic configuration for the Stewart platform was first proposed by Dr. Gough in a comment to the original paper by Dr. Stewart \cite{stewart65_platf_with_six_degrees_freed}.
|
||||
This configuration is characterized by active struts arranged in a mutually orthogonal configuration connecting the corners of a cube, as shown in Figure \ref{fig:detail_kinematics_cubic_architecture_example}.
|
||||
|
||||
Typically, the struts have similar length to the cube's edges, as illustrated in Figure \ref{fig:detail_kinematics_cubic_architecture_example}.
|
||||
Practical implementations of such configurations can be observed in Figures \ref{fig:detail_kinematics_jpl}, \ref{fig:detail_kinematics_uw_gsp} and \ref{fig:detail_kinematics_uqp}.
|
||||
It is also possible to implement designs with strut lengths smaller than the cube's edges (Figure \ref{fig:detail_kinematics_cubic_architecture_example_small}), as exemplified in Figure \ref{fig:detail_kinematics_ulb_pz}.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\begin{subfigure}{0.49\textwidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,scale=1]{figs/detail_kinematics_cubic_architecture_example.png}
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_kinematics_cubic_architecture_example}Classical Cubic architecture}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.49\textwidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,scale=1]{figs/detail_kinematics_cubic_architecture_example_small.png}
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_kinematics_cubic_architecture_example_small}Alternative configuration}
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:detail_kinematics_cubic_architecture_examples}Typical Stewart platform cubic architectures in which struts' length is similar to the cube edges's length (\subref{fig:detail_kinematics_cubic_architecture_example}) or is taking just a portion of the edge (\subref{fig:detail_kinematics_cubic_architecture_example_small}).}
|
||||
\end{figure}
|
||||
|
||||
|
||||
Several advantageous properties attributed to the cubic configuration have contributed to its widespread adoption \cite{geng94_six_degree_of_freed_activ,preumont07_six_axis_singl_stage_activ,jafari03_orthog_gough_stewar_platf_microm}: simplified kinematics relationships and dynamical analysis \cite{geng94_six_degree_of_freed_activ}; uniform stiffness in all directions \cite{hanieh03_activ_stewar}; uniform mobility \cite[, chapt.8.5.2]{preumont18_vibrat_contr_activ_struc_fourt_edition}; and minimization of the cross coupling between actuators and sensors in different struts \cite{preumont07_six_axis_singl_stage_activ}.
|
||||
This minimization is attributed to the fact that the struts are orthogonal to each other, and is said to facilitate collocated sensor-actuator control system design, i.e., the implementation of decentralized control \cite{geng94_six_degree_of_freed_activ,thayer02_six_axis_vibrat_isolat_system}.
|
||||
|
||||
These properties are examined in this section to assess their relevance for the nano-hexapod.
|
||||
The mobility and stiffness properties of the cubic configuration are analyzed in Section \ref{ssec:detail_kinematics_cubic_static}.
|
||||
Dynamical decoupling is investigated in Section \ref{ssec:detail_kinematics_cubic_dynamic}, while decentralized control, crucial for the NASS, is examined in Section \ref{ssec:detail_kinematics_decentralized_control}.
|
||||
Given that the cubic architecture imposes strict geometric constraints, alternative designs are proposed in Section \ref{ssec:detail_kinematics_cubic_design}.
|
||||
The ultimate objective is to determine the suitability of the cubic architecture for the nano-hexapod.
|
||||
\chapter{Static Properties}
|
||||
\label{ssec:detail_kinematics_cubic_static}
|
||||
\section{Stiffness matrix for the Cubic architecture}
|
||||
|
||||
Consider the cubic architecture shown in Figure \ref{fig:detail_kinematics_cubic_schematic_full}.
|
||||
The unit vectors corresponding to the edges of the cube are described by equation \eqref{eq:detail_kinematics_cubic_s}.
|
||||
|
||||
\begin{equation}\label{eq:detail_kinematics_cubic_s}
|
||||
\hat{\bm{s}}_1 = \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ 0 \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
|
||||
\hat{\bm{s}}_2 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{-1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
|
||||
\hat{\bm{s}}_3 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{ 1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
|
||||
\hat{\bm{s}}_4 = \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ 0 \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
|
||||
\hat{\bm{s}}_5 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{-1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
|
||||
\hat{\bm{s}}_6 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{ 1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\begin{subfigure}{0.48\textwidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,scale=0.9]{figs/detail_kinematics_cubic_schematic_full.png}
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_kinematics_cubic_schematic_full}Full cube}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.48\textwidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,scale=0.9]{figs/detail_kinematics_cubic_schematic.png}
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_kinematics_cubic_schematic}Cube's portion}
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:detail_kinematics_cubic_schematic_cases}Cubic architecture. Struts are represented in blue. The cube's center by a black dot. The Struts can match the cube's edges (\subref{fig:detail_kinematics_cubic_schematic_full}) or just take a portion of the edge (\subref{fig:detail_kinematics_cubic_schematic})}
|
||||
\end{figure}
|
||||
|
||||
Coordinates of the cube's vertices relevant for the top joints, expressed with respect to the cube's center, are shown in equation \eqref{eq:detail_kinematics_cubic_vertices}.
|
||||
|
||||
\begin{equation}\label{eq:detail_kinematics_cubic_vertices}
|
||||
\tilde{\bm{b}}_1 = \tilde{\bm{b}}_2 = H_c \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{-\sqrt{3}}{\sqrt{2}} \\ \frac{1}{2} \end{bmatrix}, \quad
|
||||
\tilde{\bm{b}}_3 = \tilde{\bm{b}}_4 = H_c \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{ \sqrt{3}}{\sqrt{2}} \\ \frac{1}{2} \end{bmatrix}, \quad
|
||||
\tilde{\bm{b}}_5 = \tilde{\bm{b}}_6 = H_c \begin{bmatrix} \frac{-2}{\sqrt{2}} \\ 0 \\ \frac{1}{2} \end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
In the case where top joints are positioned at the cube's vertices, a diagonal stiffness matrix is obtained as shown in equation \eqref{eq:detail_kinematics_cubic_stiffness}.
|
||||
Translation stiffness is twice the stiffness of the struts, and rotational stiffness is proportional to the square of the cube's size \(H_c\).
|
||||
|
||||
\begin{equation}\label{eq:detail_kinematics_cubic_stiffness}
|
||||
\bm{K}_{\{B\} = \{C\}} = k \begin{bmatrix}
|
||||
2 & 0 & 0 & 0 & 0 & 0 \\
|
||||
0 & 2 & 0 & 0 & 0 & 0 \\
|
||||
0 & 0 & 2 & 0 & 0 & 0 \\
|
||||
0 & 0 & 0 & \frac{3}{2} H_c^2 & 0 & 0 \\
|
||||
0 & 0 & 0 & 0 & \frac{3}{2} H_c^2 & 0 \\
|
||||
0 & 0 & 0 & 0 & 0 & 6 H_c^2 \\
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
However, typically, the top joints are not placed at the cube's vertices but at positions along the cube's edges (Figure \ref{fig:detail_kinematics_cubic_schematic}).
|
||||
In that case, the location of the top joints can be expressed by equation \eqref{eq:detail_kinematics_cubic_edges}, yet the computed stiffness matrix remains identical to Equation \eqref{eq:detail_kinematics_cubic_stiffness}.
|
||||
|
||||
\begin{equation}\label{eq:detail_kinematics_cubic_edges}
|
||||
\bm{b}_i = \tilde{\bm{b}}_i + \alpha \hat{\bm{s}}_i
|
||||
\end{equation}
|
||||
|
||||
|
||||
The stiffness matrix is therefore diagonal when the considered \(\{B\}\) frame is located at the center of the cube (shown by frame \(\{C\}\)).
|
||||
This means that static forces (resp torques) applied at the cube's center will induce pure translations (resp rotations around the cube's center).
|
||||
This specific location where the stiffness matrix is diagonal is referred to as the ``Center of Stiffness'' (analogous to the ``Center of Mass'' where the mass matrix is diagonal).
|
||||
\section{Effect of having frame \(\{B\}\) off-centered}
|
||||
|
||||
When the reference frames \(\{A\}\) and \(\{B\}\) are shifted from the cube's center, off-diagonal elements emerge in the stiffness matrix.
|
||||
|
||||
Considering a vertical shift as shown in Figure \ref{fig:detail_kinematics_cubic_schematic}, the stiffness matrix transforms into that shown in Equation \eqref{eq:detail_kinematics_cubic_stiffness_off_centered}.
|
||||
Off-diagonal elements increase proportionally with the height difference between the cube's center and the considered \(\{B\}\) frame.
|
||||
|
||||
\begin{equation}\label{eq:detail_kinematics_cubic_stiffness_off_centered}
|
||||
\bm{K}_{\{B\} \neq \{C\}} = k \begin{bmatrix}
|
||||
2 & 0 & 0 & 0 & -2 H & 0 \\
|
||||
0 & 2 & 0 & 2 H & 0 & 0 \\
|
||||
0 & 0 & 2 & 0 & 0 & 0 \\
|
||||
0 & 2 H & 0 & \frac{3}{2} H_c^2 + 2 H^2 & 0 & 0 \\
|
||||
-2 H & 0 & 0 & 0 & \frac{3}{2} H_c^2 + 2 H^2 & 0 \\
|
||||
0 & 0 & 0 & 0 & 0 & 6 H_c^2 \\
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
This stiffness matrix structure is characteristic of Stewart platforms exhibiting symmetry, and is not an exclusive property of cubic architectures.
|
||||
Therefore, the stiffness characteristics of the cubic architecture are only distinctive when considering a reference frame located at the cube's center.
|
||||
This poses a practical limitation, as in most applications, the relevant frame (where motion is of interest and forces are applied) is located above the top platform.
|
||||
|
||||
It should be noted that for the stiffness matrix to be diagonal, the cube's center doesn't need to coincide with the geometric center of the Stewart platform.
|
||||
This observation leads to the interesting alternative architectures presented in Section \ref{ssec:detail_kinematics_cubic_design}.
|
||||
\section{Uniform Mobility}
|
||||
|
||||
The translational mobility of the Stewart platform with constant orientation was analyzed.
|
||||
Considering limited actuator stroke (elongation of each strut), the maximum achievable positions in XYZ space were estimated.
|
||||
The resulting mobility in X, Y, and Z directions for the cubic architecture is illustrated in Figure \ref{fig:detail_kinematics_cubic_mobility_translations}.
|
||||
|
||||
The translational workspace analysis reveals that for the cubic architecture, the achievable positions form a cube whose axes align with the struts, with the cube's edge length corresponding to the strut axial stroke.
|
||||
These findings suggest that the mobility pattern is more subtle than sometimes described in the literature \cite{mcinroy00_desig_contr_flexur_joint_hexap}, exhibiting uniformity primarily along directions aligned with the cube's edges rather than uniform spherical distribution in all XYZ directions.
|
||||
This configuration still offers more consistent mobility characteristics compared to alternative architectures illustrated in Figure \ref{fig:detail_kinematics_mobility_trans}.
|
||||
|
||||
The rotational mobility, illustrated in Figure \ref{fig:detail_kinematics_cubic_mobility_rotations}, exhibits greater achievable angular stroke in the \(R_x\) and \(R_y\) directions compared to the \(R_z\) direction.
|
||||
Furthermore, an inverse relationship exists between the cube's dimension and rotational mobility, with larger cube sizes corresponding to more limited angular displacement capabilities.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\begin{subfigure}{0.48\textwidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,scale=1]{figs/detail_kinematics_cubic_mobility_translations.png}
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_kinematics_cubic_mobility_translations}Mobility in translation}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.48\textwidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,scale=1]{figs/detail_kinematics_cubic_mobility_rotations.png}
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_kinematics_cubic_mobility_rotations}Mobility in rotation}
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:detail_kinematics_cubic_mobility}Mobility of a Stewart platform with Cubic architecture. Both for translations (\subref{fig:detail_kinematics_cubic_mobility_translations}) and rotations (\subref{fig:detail_kinematics_cubic_mobility_rotations})}
|
||||
\end{figure}
|
||||
\chapter{Dynamical Decoupling}
|
||||
\label{ssec:detail_kinematics_cubic_dynamic}
|
||||
This section examines the dynamics of the cubic architecture in the Cartesian frame which corresponds to the transfer function from forces and torques \(\bm{\mathcal{F}}\) to translations and rotations \(\bm{\mathcal{X}}\) of the top platform.
|
||||
When relative motion sensors are integrated in each strut (measuring \(\bm{\mathcal{L}}\)), the pose \(\bm{\mathcal{X}}\) is computed using the Jacobian matrix as shown in Figure \ref{fig:detail_kinematics_centralized_control}.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1]{figs/detail_kinematics_centralized_control.png}
|
||||
\caption{\label{fig:detail_kinematics_centralized_control}Typical control architecture in the cartesian frame}
|
||||
\end{figure}
|
||||
\section{Low frequency and High frequency coupling}
|
||||
|
||||
As derived during the conceptual design phase, the dynamics from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\) is described by Equation \eqref{eq:detail_kinematics_transfer_function_cart}.
|
||||
At low frequency, the behavior of the platform depends on the stiffness matrix \eqref{eq:detail_kinematics_transfer_function_cart_low_freq}.
|
||||
|
||||
\begin{equation}\label{eq:detail_kinematics_transfer_function_cart_low_freq}
|
||||
\frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(j \omega) \xrightarrow[\omega \to 0]{} \bm{K}^{-1}
|
||||
\end{equation}
|
||||
|
||||
In Section \ref{ssec:detail_kinematics_cubic_static}, it was demonstrated that for the cubic configuration, the stiffness matrix is diagonal if frame \(\{B\}\) is positioned at the cube's center.
|
||||
In this case, the ``Cartesian'' plant is decoupled at low frequency.
|
||||
At high frequency, the behavior is governed by the mass matrix (evaluated at frame \(\{B\}\)) \eqref{eq:detail_kinematics_transfer_function_high_freq}.
|
||||
|
||||
\begin{equation}\label{eq:detail_kinematics_transfer_function_high_freq}
|
||||
\frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(j \omega) \xrightarrow[\omega \to \infty]{} - \omega^2 \bm{M}^{-1}
|
||||
\end{equation}
|
||||
|
||||
To achieve a diagonal mass matrix, the center of mass of the mobile components must coincide with the \(\{B\}\) frame, and the principal axes of inertia must align with the axes of the \(\{B\}\) frame.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1,width=0.6\linewidth]{figs/detail_kinematics_cubic_payload.png}
|
||||
\caption{\label{fig:detail_kinematics_cubic_payload}Cubic stewart platform with top cylindrical payload}
|
||||
\end{figure}
|
||||
|
||||
To verify these properties, a cubic Stewart platform with a cylindrical payload was analyzed (Figure \ref{fig:detail_kinematics_cubic_payload}).
|
||||
Transfer functions from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\) were computed for two specific locations of the \(\{B\}\) frames.
|
||||
When the \(\{B\}\) frame was positioned at the center of mass, coupling at low frequency was observed due to the non-diagonal stiffness matrix (Figure \ref{fig:detail_kinematics_cubic_cart_coupling_com}).
|
||||
Conversely, when positioned at the center of stiffness, coupling occurred at high frequency due to the non-diagonal mass matrix (Figure \ref{fig:detail_kinematics_cubic_cart_coupling_cok}).
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\begin{subfigure}{0.48\textwidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_cubic_cart_coupling_com.png}
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_kinematics_cubic_cart_coupling_com}$\{B\}$ at the center of mass}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.48\textwidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_cubic_cart_coupling_cok.png}
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_kinematics_cubic_cart_coupling_cok}$\{B\}$ at the cube's center}
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:detail_kinematics_cubic_cart_coupling}Transfer functions for a Cubic Stewart platform expressed in the Cartesian frame. Two locations of the \(\{B\}\) frame are considered: at the center of mass of the moving body (\subref{fig:detail_kinematics_cubic_cart_coupling_com}) and at the cube's center (\subref{fig:detail_kinematics_cubic_cart_coupling_cok}).}
|
||||
\end{figure}
|
||||
\section{Payload's CoM at the cube's center}
|
||||
|
||||
An effective strategy for improving dynamical performances involves aligning the cube's center (center of stiffness) with the center of mass of the moving components \cite{li01_simul_fault_vibrat_isolat_point}.
|
||||
This can be achieved by positioning the payload below the top platform, such that the center of mass of the moving body coincides with the cube's center (Figure \ref{fig:detail_kinematics_cubic_centered_payload}).
|
||||
This approach was physically implemented in several studies \cite{mcinroy99_dynam,jafari03_orthog_gough_stewar_platf_microm}, as shown in Figure \ref{fig:detail_kinematics_uw_gsp}.
|
||||
The resulting dynamics are indeed well-decoupled (Figure \ref{fig:detail_kinematics_cubic_cart_coupling_com_cok}), taking advantage from diagonal stiffness and mass matrices.
|
||||
The primary limitation of this approach is that, for many applications including the nano-hexapod, the payload must be positioned above the top platform.
|
||||
If a design similar to Figure \ref{fig:detail_kinematics_cubic_centered_payload} were employed for the nano-hexapod, the X-ray beam would intersect with the struts during spindle rotation.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\begin{subfigure}{0.49\textwidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_cubic_centered_payload.png}
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_kinematics_cubic_centered_payload}Payload at the cube's center}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.49\textwidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_cubic_cart_coupling_com_cok.png}
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_kinematics_cubic_cart_coupling_com_cok}Fully decoupled cartesian plant}
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:detail_kinematics_cubic_com_cok}Cubic Stewart platform with payload at the cube's center (\subref{fig:detail_kinematics_cubic_centered_payload}). Obtained cartesian plant is fully decoupled (\subref{fig:detail_kinematics_cubic_cart_coupling_com_cok})}
|
||||
\end{figure}
|
||||
\section{Conclusion}
|
||||
|
||||
The analysis of dynamical properties of the cubic architecture yields several important conclusions.
|
||||
Static decoupling, characterized by a diagonal stiffness matrix, is achieved when reference frames \(\{A\}\) and \(\{B\}\) are positioned at the cube's center.
|
||||
Note that this property can also be obtained with non-cubic architectures that exhibit symmetrical strut arrangements.
|
||||
Dynamic decoupling requires both static decoupling and coincidence of the mobile platform's center of mass with reference frame \(\{B\}\).
|
||||
While this configuration offers powerful control advantages, it requires positioning the payload at the cube's center, which is highly restrictive and often impractical.
|
||||
\chapter{Decentralized Control}
|
||||
\label{ssec:detail_kinematics_decentralized_control}
|
||||
The orthogonal arrangement of struts in the cubic architecture suggests a potential minimization of inter-strut coupling, which could theoretically create favorable conditions for decentralized control.
|
||||
Two sensor types integrated in the struts are considered: displacement sensors and force sensors.
|
||||
The control architecture is illustrated in Figure \ref{fig:detail_kinematics_decentralized_control}, where \(\bm{K}_{\mathcal{L}}\) represents a diagonal transfer function matrix.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1]{figs/detail_kinematics_decentralized_control.png}
|
||||
\caption{\label{fig:detail_kinematics_decentralized_control}Decentralized control in the frame of the struts.}
|
||||
\end{figure}
|
||||
|
||||
The obtained plant dynamics in the frame of the struts are compared for two Stewart platforms.
|
||||
The first employs a cubic architecture shown in Figure \ref{fig:detail_kinematics_cubic_payload}.
|
||||
The second uses a non-cubic Stewart platform shown in Figure \ref{fig:detail_kinematics_non_cubic_payload}, featuring identical payload and strut dynamics but with struts oriented more vertically to differentiate it from the cubic architecture.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1,width=0.6\linewidth]{figs/detail_kinematics_non_cubic_payload.png}
|
||||
\caption{\label{fig:detail_kinematics_non_cubic_payload}Stewart platform with non-cubic architecture}
|
||||
\end{figure}
|
||||
\section{Relative Displacement Sensors}
|
||||
|
||||
The transfer functions from actuator force in each strut to the relative motion of the struts are presented in Figure \ref{fig:detail_kinematics_decentralized_dL}.
|
||||
As anticipated from the equations of motion from \(\bm{f}\) to \(\bm{\mathcal{L}}\) \eqref{eq:detail_kinematics_transfer_function_struts}, the \(6 \times 6\) plant is decoupled at low frequency.
|
||||
At high frequency, coupling is observed as the mass matrix projected in the strut frame is not diagonal.
|
||||
|
||||
No significant advantage is evident for the cubic architecture (Figure \ref{fig:detail_kinematics_cubic_decentralized_dL}) compared to the non-cubic architecture (Figure \ref{fig:detail_kinematics_non_cubic_decentralized_dL}).
|
||||
The resonance frequencies differ between the two cases because the more vertical strut orientation in the non-cubic architecture alters the stiffness properties of the Stewart platform, consequently shifting the frequencies of various modes.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\begin{subfigure}{0.48\textwidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_non_cubic_decentralized_dL.png}
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_kinematics_non_cubic_decentralized_dL}Non cubic architecture}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.48\textwidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_cubic_decentralized_dL.png}
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_kinematics_cubic_decentralized_dL}Cubic architecture}
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:detail_kinematics_decentralized_dL}Bode plot of the transfer functions from actuator force to relative displacement sensor in each strut. Both for a non-cubic architecture (\subref{fig:detail_kinematics_non_cubic_decentralized_dL}) and for a cubic architecture (\subref{fig:detail_kinematics_cubic_decentralized_dL})}
|
||||
\end{figure}
|
||||
\section{Force Sensors}
|
||||
|
||||
Similarly, the transfer functions from actuator force to force sensors in each strut were analyzed for both cubic and non-cubic Stewart platforms.
|
||||
The results are presented in Figure \ref{fig:detail_kinematics_decentralized_fn}.
|
||||
The system demonstrates good decoupling at high frequency in both cases, with no clear advantage for the cubic architecture.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\begin{subfigure}{0.48\textwidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_non_cubic_decentralized_fn.png}
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_kinematics_non_cubic_decentralized_fn}Non cubic architecture}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.48\textwidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_cubic_decentralized_fn.png}
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_kinematics_cubic_decentralized_fn}Cubic architecture}
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:detail_kinematics_decentralized_fn}Bode plot of the transfer functions from actuator force to force sensor in each strut. Both for a non-cubic architecture (\subref{fig:detail_kinematics_non_cubic_decentralized_fn}) and for a cubic architecture (\subref{fig:detail_kinematics_cubic_decentralized_fn})}
|
||||
\end{figure}
|
||||
\section{Conclusion}
|
||||
|
||||
The presented results do not demonstrate the pronounced decoupling advantages often associated with cubic architectures in the literature.
|
||||
Both the cubic and non-cubic configurations exhibited similar coupling characteristics, suggesting that the benefits of orthogonal strut arrangement for decentralized control is less obvious than often reported in the literature.
|
||||
\chapter{Cubic architecture with Cube's center above the top platform}
|
||||
\label{ssec:detail_kinematics_cubic_design}
|
||||
As demonstrated in Section \ref{ssec:detail_kinematics_cubic_dynamic}, the cubic architecture can exhibit advantageous dynamical properties when the center of mass of the moving body coincides with the cube's center, resulting in diagonal mass and stiffness matrices.
|
||||
As shown in Section \ref{ssec:detail_kinematics_cubic_static}, the stiffness matrix is diagonal when the considered \(\{B\}\) frame is located at the cube's center.
|
||||
However, the \(\{B\}\) frame is typically positioned above the top platform where forces are applied and displacements are measured.
|
||||
|
||||
This section proposes modifications to the cubic architecture to enable positioning the payload above the top platform while still leveraging the advantageous dynamical properties of the cubic configuration.
|
||||
|
||||
Three key parameters define the geometry of the cubic Stewart platform: \(H\), the height of the Stewart platform (distance from fixed base to mobile platform); \(H_c\), the height of the cube, as shown in Figure \ref{fig:detail_kinematics_cubic_schematic_full}; and \(H_{CoM}\), the height of the center of mass relative to the mobile platform (coincident with the cube's center).
|
||||
|
||||
Depending on the cube's size \(H_c\) in relation to \(H\) and \(H_{CoM}\), different designs emerge.
|
||||
In the following examples, \(H = 100\,mm\) and \(H_{CoM} = 20\,mm\).
|
||||
\section{Small cube}
|
||||
|
||||
When the cube size \(H_c\) is smaller than twice the height of the CoM \(H_{CoM}\) \eqref{eq:detail_kinematics_cube_small}, the resulting design is shown in Figure \ref{fig:detail_kinematics_cubic_above_small}.
|
||||
|
||||
\begin{equation}\label{eq:detail_kinematics_cube_small}
|
||||
H_c < 2 H_{CoM}
|
||||
\end{equation}
|
||||
|
||||
This configuration is similar to that described in \cite{furutani04_nanom_cuttin_machin_using_stewar}, although they do not explicitly identify it as a cubic configuration.
|
||||
Adjacent struts are parallel to each other, differing from the typical architecture where parallel struts are positioned opposite to each other.
|
||||
|
||||
This approach yields a compact architecture, but the small cube size may result in insufficient rotational stiffness.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\begin{subfigure}{0.36\textwidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_kinematics_cubic_above_small_iso.png}
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_kinematics_cubic_above_small_iso}Isometric view}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.30\textwidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_kinematics_cubic_above_small_side.png}
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_kinematics_cubic_above_small_side}Side view}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.30\textwidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_kinematics_cubic_above_small_top.png}
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_kinematics_cubic_above_small_top}Top view}
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:detail_kinematics_cubic_above_small}Cubic architecture with cube's center above the top platform. A cube height of 40mm is used.}
|
||||
\end{figure}
|
||||
\section{Medium sized cube}
|
||||
|
||||
Increasing the cube's size such that \eqref{eq:detail_kinematics_cube_medium} is verified produces an architecture with intersecting struts (Figure \ref{fig:detail_kinematics_cubic_above_medium}).
|
||||
|
||||
\begin{equation}\label{eq:detail_kinematics_cube_medium}
|
||||
2 H_{CoM} < H_c < 2 (H_{CoM} + H)
|
||||
\end{equation}
|
||||
|
||||
This configuration resembles the design proposed in \cite{yang19_dynam_model_decoup_contr_flexib} (Figure \ref{fig:detail_kinematics_yang19}), although their design is not strictly cubic.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\begin{subfigure}{0.36\textwidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_kinematics_cubic_above_medium_iso.png}
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_kinematics_cubic_above_medium_iso}Isometric view}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.30\textwidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_kinematics_cubic_above_medium_side.png}
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_kinematics_cubic_above_medium_side}Side view}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.30\textwidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_kinematics_cubic_above_medium_top.png}
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_kinematics_cubic_above_medium_top}Top view}
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:detail_kinematics_cubic_above_medium}Cubic architecture with cube's center above the top platform. A cube height of 140mm is used.}
|
||||
\end{figure}
|
||||
\section{Large cube}
|
||||
|
||||
When the cube's height exceeds twice the sum of the platform height and CoM height \eqref{eq:detail_kinematics_cube_large}, the architecture shown in Figure \ref{fig:detail_kinematics_cubic_above_large} is obtained.
|
||||
|
||||
\begin{equation}\label{eq:detail_kinematics_cube_large}
|
||||
2 (H_{CoM} + H) < H_c
|
||||
\end{equation}
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\begin{subfigure}{0.36\textwidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_kinematics_cubic_above_large_iso.png}
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_kinematics_cubic_above_large_iso}Isometric view}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.30\textwidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_kinematics_cubic_above_large_side.png}
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_kinematics_cubic_above_large_side}Side view}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.30\textwidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_kinematics_cubic_above_large_top.png}
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_kinematics_cubic_above_large_top}Top view}
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:detail_kinematics_cubic_above_large}Cubic architecture with cube's center above the top platform. A cube height of 240mm is used.}
|
||||
\end{figure}
|
||||
\section{Platform size}
|
||||
|
||||
For the proposed configuration, the top joints \(\bm{b}_i\) (resp. the bottom joints \(\bm{a}_i\)) and are positioned on a circle with radius \(R_{b_i}\) (resp. \(R_{a_i}\)) described by Equation \eqref{eq:detail_kinematics_cube_joints}.
|
||||
|
||||
\begin{subequations}\label{eq:detail_kinematics_cube_joints}
|
||||
\begin{align}
|
||||
R_{b_i} &= \sqrt{\frac{3}{2} H_c^2 + 2 H_{CoM}^2} \label{eq:detail_kinematics_cube_top_joints} \\
|
||||
R_{a_i} &= \sqrt{\frac{3}{2} H_c^2 + 2 (H_{CoM} + H)^2} \label{eq:detail_kinematics_cube_bot_joints}
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
|
||||
Since the rotational stiffness for the cubic architecture scales with the square of the cube's height \eqref{eq:detail_kinematics_cubic_stiffness}, the cube's size can be determined based on rotational stiffness requirements.
|
||||
Subsequently, using Equation \eqref{eq:detail_kinematics_cube_joints}, the dimensions of the top and bottom platforms can be calculated.
|
||||
\chapter*{Conclusion}
|
||||
The analysis of the cubic architecture for Stewart platforms yielded several important findings.
|
||||
While the cubic configuration provides uniform stiffness in the XYZ directions, it stiffness property becomes particularly advantageous when forces and torques are applied at the cube's center.
|
||||
Under these conditions, the stiffness matrix becomes diagonal, resulting in a decoupled Cartesian plant at low frequencies.
|
||||
|
||||
Regarding mobility, the translational capabilities of the cubic configuration exhibit uniformity along the directions of the orthogonal struts, rather than complete uniformity in the Cartesian space.
|
||||
This understanding refines the characterization of cubic architecture mobility commonly presented in literature.
|
||||
|
||||
The analysis of decentralized control in the frame of the struts revealed more nuanced results than expected.
|
||||
While cubic architectures are frequently associated with reduced coupling between actuators and sensors, this study showed that these benefits may be more subtle or context-dependent than commonly described.
|
||||
Under the conditions analyzed, the coupling characteristics of cubic and non-cubic configurations, in the frame of the struts, appeared similar.
|
||||
|
||||
Fully decoupled dynamics in the Cartesian frame can be achieved when the center of mass of the moving body coincides with the cube's center.
|
||||
However, this arrangement presents practical challenges, as the cube's center is traditionally located between the top and bottom platforms, making payload placement problematic for many applications.
|
||||
|
||||
To address this limitation, modified cubic architectures have been proposed with the cube's center positioned above the top platform.
|
||||
Three distinct configurations have been identified, each with different geometric arrangements but sharing the common characteristic that the cube's center is positioned above the top platform.
|
||||
This structural modification enables the alignment of the moving body's center of mass with the center of stiffness, resulting in beneficial decoupling properties in the Cartesian frame.
|
||||
\printbibliography[heading=bibintoc,title={Bibliography}]
|
||||
\end{document}
|
||||
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paper/figs/detail_kinematics_non_cubic_decentralized_dL.pdf
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paper/figs/detail_kinematics_non_cubic_decentralized_fn.png
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paper/figs/detail_kinematics_non_cubic_payload.pdf
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paper/figs/detail_kinematics_non_cubic_payload.pdf
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35
paper/preamble.tex
Normal file
35
paper/preamble.tex
Normal file
@@ -0,0 +1,35 @@
|
||||
\usepackage[ %
|
||||
acronym, % Separate acronyms and glossary
|
||||
toc, % appear in ToC
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automake, % auto-use the makeglossaries command (requires shell-escape)
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nonumberlist, % don't back reference pages
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nogroupskip, % don't group by letter
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nopostdot % don't add a dot at the end of each element
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]{glossaries}
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\usepackage[stylemods=longextra]{glossaries-extra}
|
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\setabbreviationstyle[acronym]{long-short}
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\setglossarystyle{long-name-desc}
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\usepackage{amssymb}
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\usepackage{amsmath}
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\usepackage{cases}
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\usepackage{empheq}
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\usepackage[binary-units=true]{siunitx}
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\sisetup{%
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detect-shape = true,
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detect-weight = true,
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detect-inline-weight = math,
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}
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\DeclareSIUnit\px{px}
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\DeclareSIUnit\rms{rms}
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\makeindex
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\makeglossaries
|
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134
paper/preamble_extra.tex
Normal file
134
paper/preamble_extra.tex
Normal file
@@ -0,0 +1,134 @@
|
||||
\usepackage{float}
|
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\usepackage{enumitem}
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\usepackage{caption,tabularx,booktabs}
|
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\usepackage{bm}
|
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|
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\usepackage{xpatch} % Recommanded for biblatex
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\usepackage[ % use biblatex for bibliography
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backend=biber, % use biber backend (bibtex replacement) or bibtex
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style=ieee, % bib style
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hyperref=true, % activate hyperref support
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backref=true, % activate backrefs
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isbn=false, % don't show isbn tags
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url=false, % don't show url tags
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doi=false, % don't show doi tags
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urldate=long, % display type for dates
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||||
maxnames=3, %
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minnames=1, %
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||||
maxbibnames=5, %
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||||
minbibnames=3, %
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maxcitenames=2, %
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mincitenames=1 %
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||||
]{biblatex}
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||||
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\setlength\bibitemsep{1.1\itemsep}
|
||||
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\usepackage{caption}
|
||||
\usepackage{subcaption}
|
||||
|
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\captionsetup[figure]{labelfont=bf}
|
||||
\captionsetup[subfigure]{labelfont=bf}
|
||||
\captionsetup[listing]{labelfont=bf}
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\captionsetup[table]{labelfont=bf}
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\usepackage{xcolor}
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\definecolor{my-pale-blue}{HTML}{e6e9f9}
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\definecolor{my-pale-red}{HTML}{f9e6e6}
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\definecolor{my-green}{HTML}{6bdbb6}
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\definecolor{my-pale-green}{HTML}{e6f9f3}
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\definecolor{my-yellow}{HTML}{dbd26b}
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\definecolor{my-pale-yellow}{HTML}{f9f7e6}
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\definecolor{my-orange}{HTML}{dba76b}
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\definecolor{my-pale-orange}{HTML}{f9f0e6}
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\definecolor{my-pale-grey}{HTML}{f0f0f0}
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\definecolor{my-turq}{HTML}{6bc7db}
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\definecolor{my-pale-turq}{HTML}{e6f6f9}
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\usepackage{inconsolata}
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\usepackage[newfloat=true, chapter]{minted}
|
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\usemintedstyle{autumn}
|
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\setminted{frame=lines,breaklines=true,tabsize=4,fontsize=\scriptsize,autogobble=true,labelposition=topline,bgcolor=my-pale-grey}
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\setminted[matlab]{label=Matlab}
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\setminted[latex]{label=LaTeX}
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\setminted[python]{label=Python}
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\setminted[md]{label=Org Mode}
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\setmintedinline{fontsize=\normalsize,bgcolor=my-pale-grey}
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\usepackage[most]{tcolorbox}
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\tcbuselibrary{minted}
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\newtcolorbox{seealso}{ enhanced,breakable,colback=my-pale-grey,colframe=my-grey,fonttitle=\bfseries,title=See Also}
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\newtcolorbox{hint}{ enhanced,breakable,colback=my-pale-grey,colframe=my-grey,fonttitle=\bfseries,title=Hint}
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\newtcolorbox{definition}{enhanced,breakable,colback=my-pale-red, colframe=my-red, fonttitle=\bfseries,title=Definition}
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\newtcolorbox{exampl}[1][]{ enhanced,breakable,colback=my-pale-green,colframe=my-green,fonttitle=\bfseries,title=Example,#1}
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\newtcolorbox{my-quote}[1]{%
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grow to right by=-10mm,
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boxsep=0pt,
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breakable,
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enhanced jigsaw,
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borderline west={4pt}{0pt}{my-grey}}
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\renewenvironment{quote}{\begin{my-quote}}{\end{my-quote}}
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\newtcolorbox{my-verse}[1]{%
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colback=my-pale-grey,
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grow to right by=-10mm,
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grow to left by=-10mm,
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boxrule=0pt,
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boxsep=0pt,
|
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breakable,
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enhanced jigsaw,
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borderline west={4pt}{0pt}{my-grey}}
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|
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\renewenvironment{verse}{\begin{my-verse}}{\end{my-verse}}
|
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|
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\usepackage{environ}% http://ctan.org/pkg/environ
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\NewEnviron{aside}{%
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\marginpar{\BODY}
|
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}
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|
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\renewenvironment{verbatim}{\VerbatimEnvironment\begin{minted}[]{text}}{\end{minted}}
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|
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\usepackage{soul}
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\sethlcolor{my-pale-grey}
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|
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\let\OldTexttt\texttt
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\renewcommand{\texttt}[1]{{\ttfamily\hl{\mbox{\,#1\,}}}}
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\makeatletter
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\preto\Gin@extensions{png,}
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\DeclareGraphicsRule{.png}{pdf}{.pdf}{\noexpand\Gin@base.pdf}
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\preto\Gin@extensions{gif,}
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\DeclareGraphicsRule{.gif}{png}{.png}{\noexpand\Gin@base.png}
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\makeatother
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||||
|
||||
\usepackage{hyperref}
|
||||
\hypersetup{
|
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colorlinks = true,
|
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allcolors = my-blue
|
||||
}
|
||||
|
||||
\usepackage{hypcap}
|
||||
Reference in New Issue
Block a user