3215 lines
125 KiB
Org Mode
3215 lines
125 KiB
Org Mode
#+TITLE: Simscape Uniaxial Model
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:DRAWER:
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#+STARTUP: overview
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#+LANGUAGE: en
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#+EMAIL: dehaeze.thomas@gmail.com
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#+AUTHOR: Dehaeze Thomas
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#+HTML_LINK_HOME: ./index.html
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#+HTML_LINK_UP: ./index.html
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/zenburn.css"/>
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#+HTML_HEAD: <script type="text/javascript" src="./js/jquery.min.js"></script>
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#+HTML_HEAD: <script type="text/javascript" src="./js/bootstrap.min.js"></script>
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#+HTML_HEAD: <script type="text/javascript" src="./js/jquery.stickytableheaders.min.js"></script>
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#+HTML_HEAD: <script type="text/javascript" src="./js/readtheorg.js"></script>
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#+HTML_MATHJAX: align: center tagside: right font: TeX
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#+PROPERTY: header-args:matlab :session *MATLAB*
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#+PROPERTY: header-args:matlab+ :comments org
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#+PROPERTY: header-args:matlab+ :results none
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#+PROPERTY: header-args:matlab+ :exports both
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#+PROPERTY: header-args:matlab+ :eval no-export
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#+PROPERTY: header-args:matlab+ :output-dir figs
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#+PROPERTY: header-args:matlab+ :tangle no
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#+PROPERTY: header-args:matlab+ :mkdirp yes
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#+PROPERTY: header-args:shell :eval no-export
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#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/thesis/latex/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
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#+PROPERTY: header-args:latex+ :results raw replace :buffer no
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#+PROPERTY: header-args:latex+ :eval no-export
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#+PROPERTY: header-args:latex+ :exports both
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#+PROPERTY: header-args:latex+ :mkdirp yes
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#+PROPERTY: header-args:latex+ :output-dir figs
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:END:
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* Introduction :ignore:
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The idea is to use the same model as the full Simscape Model but to restrict the motion only in the vertical direction.
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This is done in order to more easily study the system and evaluate control techniques.
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* Simscape Model
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<<sec:simscape_model>>
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A schematic of the uniaxial model used for simulations is represented in figure [[fig:uniaxial-model-nass-flexible]].
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The perturbations $w$ are:
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- $F_s$: direct forces applied to the sample such as inertia forces and cable forces
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- $F_{rz}$: parasitic forces due to the rotation of the spindle
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- $F_{ty}$: parasitic forces due to scans with the translation stage
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- $D_w$: ground motion
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The quantity to $z$ to control is:
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- $D$: the position of the sample with respect to the granite
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The measured quantities $v$ are:
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- $D$: the position of the sample with respect to the granite
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We study the use of an additional sensor:
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- $F_n$: a force sensor located in the nano-hexapod
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- $v_n$: an absolute velocity sensor located on the top platform of the nano-hexapod
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- $d_r$: a relative motion sensor located in the nano-hexapod
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The control signal $u$ is:
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- $F$ the force applied by the nano-hexapod actuator
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#+begin_src latex :file uniaxial-model-nass-flexible.pdf :post pdf2svg(file=*this*, ext="png") :exports results
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\begin{tikzpicture}
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% ====================
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% Parameters
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% ====================
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\def\massw{2.2} % Width of the masses
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\def\massh{0.8} % Height of the masses
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\def\spaceh{1.2} % Height of the springs/dampers
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\def\dispw{0.4} % Width of the dashed line for the displacement
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\def\disph{0.3} % Height of the arrow for the displacements
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\def\bracs{0.05} % Brace spacing vertically
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\def\brach{-12pt} % Brace shift horizontaly
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\def\fsensh{0.2} % Height of the force sensor
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\def\velsize{0.2} % Size of the velocity sensor
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% ====================
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% ====================
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% Ground
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% ====================
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\draw (-0.5*\massw, 0) -- (0.5*\massw, 0);
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\draw[dashed] (0.5*\massw, 0) -- ++(1, 0);
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\draw[->, dasehd] (0.5*\massw+0.8, 0) -- ++(0, 0.5) node[right]{$D_{w}$};
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% ====================
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% ====================
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% Marble
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\begin{scope}[shift={(0, 0)}]
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% Mass
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\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{m}$};
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% Spring, Damper, and Actuator
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\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{m}$};
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\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{m}$};
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% Legend
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\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
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(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Marble};
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% Displacements
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\draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(2*\dispw, 0) coordinate(xm) -- ++(2.2*\dispw, 0) coordinate(dbot);
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\end{scope}
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% ====================
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% ====================
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% Ty
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\begin{scope}[shift={(0, \spaceh+\massh)}]
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% Mass
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\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{t}$};
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% Spring, Damper, and Actuator
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\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{t}$};
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\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{t}$};
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\draw[actuator={0.45}{0.2}] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](ft){$F_{ty}$};
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% Legend
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\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
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(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Ty};
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\end{scope}
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% ====================
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% ====================
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% Rz
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\begin{scope}[shift={(0, 2*(\spaceh+\massh))}]
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% Mass
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\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{z}$};
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% Spring, Damper, and Actuator
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\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{z}$};
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\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{z}$};
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\draw[actuator] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](F){$F_{rz}$};
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% Legend
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\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
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(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Rz};
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\end{scope}
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% ====================
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% ====================
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% Hexapod
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\begin{scope}[shift={(0, 3*(\spaceh+\massh))}]
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% Mass
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\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{h}$};
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% Spring, Damper, and Actuator
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\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{h}$};
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\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{h}$};
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% Displacements
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% \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(3*\dispw, 0) coordinate(drbot);
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% Legend
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\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
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(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Hexapod};
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\end{scope}
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% ====================
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% ====================
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% NASS
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\begin{scope}[shift={(0, 4*(\spaceh+\massh))}]
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% Mass
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\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{n}$};
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% % Force Sensor
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% \node[forcesensor={\massw}{\fsensh}] (fsensn) at (0, \spaceh-\fsensh){};
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% \node[right] at (fsensn.east) {$F_n$};
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% % Velocity Sensor
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% \node[inertialsensor={\velsize}] (veln) at (0.5*\massw, \spaceh+\massh) {};
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% \node[right] at (veln.north east) {$v_n$};
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% Spring, Damper, and Actuator
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\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{n}$};
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\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{n}$};
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\draw[actuator] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](F){$F$};
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% % Displacements
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% \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(2*\dispw, 0) coordinate(xn) -- ++(\dispw, 0) coordinate(drtop);
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% Legend
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\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
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(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{NASS};
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\end{scope}
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% ====================
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% ====================
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% sample
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\begin{scope}[shift={(0, 5*(\spaceh+\massh))}]
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% Mass
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\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{s}$};
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% External Force
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\draw[->] (0, \spaceh+\massh) node[]{$\bullet$} -- ++(0, 0.5*\massh) node[right]{$F_s$};
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% Spring, Damper, and Actuator
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\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{s}$};
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\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{s}$};
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% Displacements
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\draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(2*\dispw, 0) coordinate(xs) -- ++(2.2*\dispw, 0) coordinate(dtop);
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% Legend
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\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
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(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Sample};
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\end{scope}
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% ====================
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\draw[<->] ($(dbot)+(-0.3,0)$) --node[midway, right]{$D$} ($(dtop)+(-0.3,0)$);
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\end{tikzpicture}
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#+end_src
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#+name: fig:uniaxial-model-nass-flexible
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#+caption: Schematic of the uniaxial model used
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#+RESULTS:
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[[file:figs/uniaxial-model-nass-flexible.png]]
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Few active damping techniques will be compared in order to decide which sensor is to be included in the system.
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Schematics of the active damping techniques are displayed in figure [[fig:uniaxial-model-nass-flexible-active-damping]].
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#+begin_src latex :file uniaxial-model-nass-flexible-active-damping.pdf :post pdf2svg(file=*this*, ext="png") :exports results
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\begin{tikzpicture}
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% ====================
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% Parameters
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% ====================
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\def\massw{2.2} % Width of the masses
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\def\massh{0.8} % Height of the masses
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\def\spaceh{1.2} % Height of the springs/dampers
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\def\dispw{0.4} % Width of the dashed line for the displacement
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\def\disph{0.3} % Height of the arrow for the displacements
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\def\bracs{0.05} % Brace spacing vertically
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\def\brach{-12pt} % Brace shift horizontaly
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\def\fsensh{0.2} % Height of the force sensor
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\def\velsize{0.2} % Size of the velocity sensor
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% ====================
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\begin{scope}
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% ====================
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% Ground
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% ====================
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\draw (-0.5*\massw, 0) -- (0.5*\massw, 0);
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\draw[dashed] (0.5*\massw, 0) -- ++(1, 0);
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\draw[->, dasehd] (0.5*\massw+0.8, 0) -- ++(0, 0.5) node[right]{$D_{w}$};
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% ====================
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% ====================
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% Marble
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\begin{scope}[shift={(0, 0)}]
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% Mass
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\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{m}$};
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% Spring, Damper, and Actuator
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\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{m}$};
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\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{m}$};
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% Legend
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\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
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(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Marble};
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\end{scope}
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% ====================
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% ====================
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% Ty
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\begin{scope}[shift={(0, \spaceh+\massh)}]
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% Mass
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\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{t}$};
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% Spring, Damper, and Actuator
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\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{t}$};
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\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{t}$};
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\draw[actuator={0.45}{0.2}] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](ft){$F_{ty}$};
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% Legend
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\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
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(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Ty};
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\end{scope}
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% ====================
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% ====================
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% Rz
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\begin{scope}[shift={(0, 2*(\spaceh+\massh))}]
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% Mass
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\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{z}$};
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% Spring, Damper, and Actuator
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\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{z}$};
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\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{z}$};
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\draw[actuator] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](F){$F_{rz}$};
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% Legend
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\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
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(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Rz};
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\end{scope}
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% ====================
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% ====================
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% Hexapod
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\begin{scope}[shift={(0, 3*(\spaceh+\massh))}]
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% Mass
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\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{h}$};
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% Spring, Damper, and Actuator
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\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{h}$};
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\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{h}$};
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% Displacements
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% \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(3*\dispw, 0) coordinate(drbot);
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% Legend
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\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
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(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Hexapod};
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\end{scope}
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% ====================
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% ====================
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% NASS
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\begin{scope}[shift={(0, 4*(\spaceh+\massh))}]
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% Mass
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\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{n}$};
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% Force Sensor
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\node[forcesensor={\massw}{\fsensh}] (fsensn) at (0, \spaceh-\fsensh){};
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\node[right] (fn) at (fsensn.east) {$F_n$};
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% % Velocity Sensor
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% \node[inertialsensor={\velsize}] (veln) at (0.5*\massw, \spaceh+\massh) {};
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% \node[right] at (veln.north east) {$v_n$};
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% Spring, Damper, and Actuator
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\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh-\fsensh) node[midway, left=0.1]{$k_{n}$};
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\draw[damper] (0, 0) -- ( 0, \spaceh-\fsensh) node[midway, left=0.2]{$c_{n}$};
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\draw[actuator] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh-\fsensh) node[midway, right=0.1](F){$F$};
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% % Displacements
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% \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(2*\dispw, 0) coordinate(xn) -- ++(\dispw, 0) coordinate(drtop);
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% Legend
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\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
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(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{NASS};
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\end{scope}
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% ====================
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% ====================
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% sample
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\begin{scope}[shift={(0, 5*(\spaceh+\massh))}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{s}$};
|
|
|
|
% External Force
|
|
\draw[->] (0, \spaceh+\massh) node[]{$\bullet$} -- ++(0, 0.5*\massh) node[right]{$F_s$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{s}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{s}$};
|
|
|
|
% Legend
|
|
\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
|
|
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Sample};
|
|
\end{scope}
|
|
% ====================
|
|
|
|
% ====================
|
|
% IFF Control
|
|
\node[block={2em}{2em}, right=0.3 of fn] (iff) {$K_{\text{IFF}}$};
|
|
\draw[->] (fn.east) -- (iff.west);
|
|
\draw[->] (iff.south) |- (F.east);
|
|
% ====================
|
|
\end{scope}
|
|
|
|
|
|
|
|
\begin{scope}[shift={(5,0)}]
|
|
% ====================
|
|
% Ground
|
|
% ====================
|
|
\draw (-0.5*\massw, 0) -- (0.5*\massw, 0);
|
|
\draw[dashed] (0.5*\massw, 0) -- ++(1, 0);
|
|
\draw[->, dasehd] (0.5*\massw+0.8, 0) -- ++(0, 0.5) node[right]{$D_{w}$};
|
|
% ====================
|
|
|
|
|
|
% ====================
|
|
% Marble
|
|
\begin{scope}[shift={(0, 0)}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{m}$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{m}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{m}$};
|
|
\end{scope}
|
|
% ====================
|
|
|
|
|
|
% ====================
|
|
% Ty
|
|
\begin{scope}[shift={(0, \spaceh+\massh)}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{t}$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{t}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{t}$};
|
|
\draw[actuator={0.45}{0.2}] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](ft){$F_{ty}$};
|
|
\end{scope}
|
|
% ====================
|
|
|
|
|
|
% ====================
|
|
% Rz
|
|
\begin{scope}[shift={(0, 2*(\spaceh+\massh))}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{z}$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{z}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{z}$};
|
|
\draw[actuator] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](F){$F_{rz}$};
|
|
\end{scope}
|
|
% ====================
|
|
|
|
|
|
% ====================
|
|
% Hexapod
|
|
\begin{scope}[shift={(0, 3*(\spaceh+\massh))}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{h}$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{h}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{h}$};
|
|
|
|
% Displacements
|
|
% \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(3*\dispw, 0) coordinate(drbot);
|
|
\end{scope}
|
|
% ====================
|
|
|
|
|
|
% ====================
|
|
% NASS
|
|
\begin{scope}[shift={(0, 4*(\spaceh+\massh))}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{n}$};
|
|
|
|
% % Force Sensor
|
|
% \node[forcesensor={\massw}{\fsensh}] (fsensn) at (0, \spaceh-\fsensh){};
|
|
% \node[right] (fn) at (fsensn.east) {$F_n$};
|
|
|
|
% Velocity Sensor
|
|
\node[inertialsensor={\velsize}] (veln) at (0.5*\massw, \spaceh+\massh) {};
|
|
\node[above right] at (veln.east) {$v_n$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{n}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{n}$};
|
|
\draw[actuator] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](F){$F$};
|
|
|
|
% % Displacements
|
|
% \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(2*\dispw, 0) coordinate(xn) -- ++(\dispw, 0) coordinate(drtop);
|
|
\end{scope}
|
|
% ====================
|
|
|
|
% ====================
|
|
% sample
|
|
\begin{scope}[shift={(0, 5*(\spaceh+\massh))}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{s}$};
|
|
|
|
% External Force
|
|
\draw[->] (0, \spaceh+\massh) node[]{$\bullet$} -- ++(0, 0.5*\massh) node[right]{$F_s$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{s}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{s}$};
|
|
\end{scope}
|
|
% ====================
|
|
|
|
% ====================
|
|
% DVF Control
|
|
\node[block={2em}{2em}, below right=0.4 and 0.4 of veln] (ppf) {$K_{\text{DVF}}$};
|
|
\draw[->] (veln.east) -| (ppf.north);
|
|
\draw[->] (ppf.south) |- (F.east);
|
|
% ====================
|
|
\end{scope}
|
|
|
|
|
|
|
|
\begin{scope}[shift={(10,0)}]
|
|
% ====================
|
|
% Ground
|
|
% ====================
|
|
\draw (-0.5*\massw, 0) -- (0.5*\massw, 0);
|
|
\draw[dashed] (0.5*\massw, 0) -- ++(1, 0);
|
|
\draw[->, dasehd] (0.5*\massw+0.8, 0) -- ++(0, 0.5) node[right]{$D_{w}$};
|
|
% ====================
|
|
|
|
|
|
% ====================
|
|
% Marble
|
|
\begin{scope}[shift={(0, 0)}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{m}$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{m}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{m}$};
|
|
\end{scope}
|
|
% ====================
|
|
|
|
|
|
% ====================
|
|
% Ty
|
|
\begin{scope}[shift={(0, \spaceh+\massh)}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{t}$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{t}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{t}$};
|
|
\draw[actuator={0.45}{0.2}] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](ft){$F_{ty}$};
|
|
\end{scope}
|
|
% ====================
|
|
|
|
|
|
% ====================
|
|
% Rz
|
|
\begin{scope}[shift={(0, 2*(\spaceh+\massh))}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{z}$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{z}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{z}$};
|
|
\draw[actuator] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](F){$F_{rz}$};
|
|
\end{scope}
|
|
% ====================
|
|
|
|
|
|
% ====================
|
|
% Hexapod
|
|
\begin{scope}[shift={(0, 3*(\spaceh+\massh))}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{h}$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{h}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{h}$};
|
|
|
|
% Displacements
|
|
\draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(3*\dispw, 0) coordinate(drbot);
|
|
\end{scope}
|
|
% ====================
|
|
|
|
|
|
% ====================
|
|
% NASS
|
|
\begin{scope}[shift={(0, 4*(\spaceh+\massh))}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{n}$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{n}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{n}$};
|
|
\draw[actuator] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](F){$F$};
|
|
|
|
% Displacements
|
|
\draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(2*\dispw, 0) coordinate(xn) -- ++(\dispw, 0) coordinate(drtop);
|
|
\end{scope}
|
|
% ====================
|
|
|
|
% ====================
|
|
% sample
|
|
\begin{scope}[shift={(0, 5*(\spaceh+\massh))}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{s}$};
|
|
|
|
% External Force
|
|
\draw[->] (0, \spaceh+\massh) node[]{$\bullet$} -- ++(0, 0.5*\massh) node[right]{$F_s$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{s}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{s}$};
|
|
\end{scope}
|
|
% ====================
|
|
|
|
% ====================
|
|
% RMC Control
|
|
\draw[<->, dashed] ($(drbot)+(-0.1, 0)$) -- ($(drtop)+(-0.1, 0)$) node[right](dr){$d_r$};
|
|
\node[block={2em}{2em}, below right=0.2 and 0.2 of dr] (rmc) {$K_{\text{RMC}}$};
|
|
\draw[->] (dr.east) -| (rmc.north);
|
|
\draw[->] (rmc.south) |- (F.east);
|
|
% ====================
|
|
\end{scope}
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+name: fig:uniaxial-model-nass-flexible-active-damping
|
|
#+caption: Comparison of used active damping techniques
|
|
#+RESULTS:
|
|
[[file:figs/uniaxial-model-nass-flexible-active-damping.png]]
|
|
|
|
* Undamped System
|
|
<<sec:undamped>>
|
|
** Introduction :ignore:
|
|
Let's start by study the undamped system.
|
|
|
|
** Matlab Init :noexport:ignore:
|
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
|
<<matlab-dir>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results silent :noweb yes
|
|
<<matlab-init>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no
|
|
simulinkproject('../');
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
open('uniaxial/matlab/sim_nano_station_uniaxial.slx')
|
|
#+end_src
|
|
|
|
** Init
|
|
We initialize all the stages with the default parameters.
|
|
The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.
|
|
|
|
#+begin_src matlab :exports none
|
|
initializeGround();
|
|
initializeGranite();
|
|
initializeTy();
|
|
initializeRy();
|
|
initializeRz();
|
|
initializeMicroHexapod();
|
|
initializeAxisc();
|
|
initializeMirror();
|
|
initializeNanoHexapod('actuator', 'piezo');
|
|
initializeSample('mass', 50);
|
|
#+end_src
|
|
|
|
All the controllers are set to 0 (Open Loop).
|
|
#+begin_src matlab :exports none
|
|
K = tf(0);
|
|
save('./mat/controllers_uniaxial.mat', 'K', '-append');
|
|
K_iff = tf(0);
|
|
save('./mat/controllers_uniaxial.mat', 'K_iff', '-append');
|
|
K_rmc = tf(0);
|
|
save('./mat/controllers_uniaxial.mat', 'K_rmc', '-append');
|
|
K_dvf = tf(0);
|
|
save('./mat/controllers_uniaxial.mat', 'K_dvf', '-append');
|
|
#+end_src
|
|
|
|
** Identification
|
|
We identify the dynamics of the system.
|
|
#+begin_src matlab
|
|
%% Options for Linearized
|
|
options = linearizeOptions;
|
|
options.SampleTime = 0;
|
|
|
|
%% Name of the Simulink File
|
|
mdl = 'sim_nano_station_uniaxial';
|
|
#+end_src
|
|
|
|
The inputs and outputs are defined below and corresponds to the name of simulink blocks.
|
|
#+begin_src matlab
|
|
%% Input/Output definition
|
|
io(1) = linio([mdl, '/Dw'], 1, 'input'); % Ground Motion
|
|
io(2) = linio([mdl, '/Fs'], 1, 'input'); % Force applied on the sample
|
|
io(3) = linio([mdl, '/Fnl'], 1, 'input'); % Force applied by the NASS
|
|
io(4) = linio([mdl, '/Fdty'], 1, 'input'); % Parasitic force Ty
|
|
io(5) = linio([mdl, '/Fdrz'], 1, 'input'); % Parasitic force Rz
|
|
|
|
io(6) = linio([mdl, '/Dsm'], 1, 'output'); % Displacement of the sample
|
|
io(7) = linio([mdl, '/Fnlm'], 1, 'output'); % Force sensor in NASS's legs
|
|
io(8) = linio([mdl, '/Dnlm'], 1, 'output'); % Displacement of NASS's legs
|
|
io(9) = linio([mdl, '/Dgm'], 1, 'output'); % Absolute displacement of the granite
|
|
io(10) = linio([mdl, '/Vlm'], 1, 'output'); % Measured absolute velocity of the top NASS platform
|
|
#+end_src
|
|
|
|
Finally, we use the =linearize= Matlab function to extract a state space model from the simscape model.
|
|
#+begin_src matlab
|
|
%% Run the linearization
|
|
G = linearize(mdl, io, options);
|
|
G.InputName = {'Dw', ... % Ground Motion [m]
|
|
'Fs', ... % Force Applied on Sample [N]
|
|
'Fn', ... % Force applied by NASS [N]
|
|
'Fty', ... % Parasitic Force Ty [N]
|
|
'Frz'}; % Parasitic Force Rz [N]
|
|
G.OutputName = {'D', ... % Measured sample displacement x.r.t. granite [m]
|
|
'Fnm', ... % Force Sensor in NASS [N]
|
|
'Dnm', ... % Displacement Sensor in NASS [m]
|
|
'Dgm', ... % Asbolute displacement of Granite [m]
|
|
'Vlm'}; ... % Absolute Velocity of NASS [m/s]
|
|
#+end_src
|
|
|
|
Finally, we save the identified system dynamics for further analysis.
|
|
#+begin_src matlab
|
|
save('./mat/uniaxial_plants.mat', 'G');
|
|
#+end_src
|
|
|
|
** Sensitivity to Disturbances
|
|
We show several plots representing the sensitivity to disturbances:
|
|
- in figure [[fig:uniaxial-sensitivity-disturbances]] the transfer functions from ground motion $D_w$ to the sample position $D$ and the transfer function from direct force on the sample $F_s$ to the sample position $D$ are shown
|
|
- in figure [[fig:uniaxial-sensitivity-force-dist]], it is the effect of parasitic forces of the positioning stages ($F_{ty}$ and $F_{rz}$) on the position $D$ of the sample that are shown
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
subplot(2, 1, 1);
|
|
title('$D_w$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('D', 'Dw'), freqs, 'Hz'))), 'k-');
|
|
% plot(freqs, abs(squeeze(freqresp(G('Dgm', 'Dw'), freqs, 'Hz'))), 'k--');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
|
|
|
|
subplot(2, 1, 2);
|
|
title('$F_s$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('D', 'Fs'), freqs, 'Hz'))), 'k-');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial-sensitivity-disturbances.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial-sensitivity-disturbances
|
|
#+CAPTION: Sensitivity to disturbances ([[./figs/uniaxial-sensitivity-disturbances.png][png]], [[./figs/uniaxial-sensitivity-disturbances.pdf][pdf]])
|
|
[[file:figs/uniaxial-sensitivity-disturbances.png]]
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#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
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|
|
|
figure;
|
|
|
|
subplot(2, 1, 1);
|
|
title('$F_{ty}$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('D', 'Fty'), freqs, 'Hz'))), 'k-');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
|
|
subplot(2, 1, 2);
|
|
title('$F_{rz}$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('D', 'Frz'), freqs, 'Hz'))), 'k-');
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|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
#+end_src
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|
#+HEADER: :tangle no :exports results :results none :noweb yes
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|
#+begin_src matlab :var filepath="figs/uniaxial-sensitivity-force-dist.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
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|
<<plt-matlab>>
|
|
#+end_src
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|
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#+NAME: fig:uniaxial-sensitivity-force-dist
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|
#+CAPTION: Sensitivity to disturbances ([[./figs/uniaxial-sensitivity-force-dist.png][png]], [[./figs/uniaxial-sensitivity-force-dist.pdf][pdf]])
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|
[[file:figs/uniaxial-sensitivity-force-dist.png]]
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|
|
** Noise Budget
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|
We first load the measured PSD of the disturbance.
|
|
#+begin_src matlab
|
|
load('./mat/disturbances_dist_psd.mat', 'dist_f');
|
|
#+end_src
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The effect of these disturbances on the distance $D$ is computed below.
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|
#+begin_src matlab :exports none
|
|
% Power Spectral Density of the relative Displacement [m^2/Hz]
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psd_gm_d = dist_f.psd_gm.*abs(squeeze(freqresp(G('D', 'Dw'), dist_f.f, 'Hz'))).^2;
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|
psd_ty_d = dist_f.psd_ty.*abs(squeeze(freqresp(G('D', 'Fty'), dist_f.f, 'Hz'))).^2;
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|
psd_rz_d = dist_f.psd_rz.*abs(squeeze(freqresp(G('D', 'Frz'), dist_f.f, 'Hz'))).^2;
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|
#+end_src
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The PSD of the obtain distance $D$ due to each of the perturbation is shown in figure [[fig:uniaxial-psd-dist]] and the Cumulative Amplitude Spectrum is shown in figure [[fig:uniaxial-cas-dist]].
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The Root Mean Square value of the obtained displacement $D$ is computed below and can be determined from the figure [[fig:uniaxial-cas-dist]].
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#+begin_src matlab :results value replace :exports results
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|
cas_tot_d = sqrt(cumtrapz(dist_f.f, psd_rz_d+psd_ty_d+psd_gm_d)); cas_tot_d(end)
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#+end_src
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#+RESULTS:
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: 3.3793e-06
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#+begin_src matlab :exports none
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freqs = logspace(0, 3, 1000);
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figure;
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hold on;
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plot(dist_f.f, psd_gm_d, 'DisplayName', '$D_w$');
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plot(dist_f.f, psd_ty_d, 'DisplayName', '$T_y$');
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plot(dist_f.f, psd_rz_d, 'DisplayName', '$R_z$');
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|
hold off;
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|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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|
ylabel('PSD of the effect of disturbances on $D$ $\left[\frac{m^2}{Hz}\right]$'); xlabel('Frequency [Hz]');
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|
legend('location', 'northeast')
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|
xlim([0.5, 500]);
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|
#+end_src
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#+HEADER: :tangle no :exports results :results none :noweb yes
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|
#+begin_src matlab :var filepath="figs/uniaxial-psd-dist.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
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|
<<plt-matlab>>
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#+end_src
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#+NAME: fig:uniaxial-psd-dist
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#+CAPTION: PSD of the effect of disturbances on $D$ ([[./figs/uniaxial-psd-dist.png][png]], [[./figs/uniaxial-psd-dist.pdf][pdf]])
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|
[[file:figs/uniaxial-psd-dist.png]]
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#+begin_src matlab :exports none
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|
freqs = logspace(0, 3, 1000);
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|
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figure;
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|
hold on;
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|
plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_gm_d)))), 'DisplayName', '$D_w$');
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plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_ty_d)))), 'DisplayName', '$T_y$');
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plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_rz_d)))), 'DisplayName', '$R_z$');
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plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_gm_d+psd_ty_d+psd_rz_d)))), 'k-', 'DisplayName', 'tot');
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|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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|
ylabel('CAS of the effect of disturbances on $D$ [m]'); xlabel('Frequency [Hz]');
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|
legend('location', 'northeast')
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|
xlim([0.5, 500]); ylim([1e-12, 1e-6]);
|
|
#+end_src
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|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial-cas-dist.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial-cas-dist
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#+CAPTION: CAS of the effect of disturbances on $D$ ([[./figs/uniaxial-cas-dist.png][png]], [[./figs/uniaxial-cas-dist.pdf][pdf]])
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|
[[file:figs/uniaxial-cas-dist.png]]
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|
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** Plant
|
|
The transfer function from the force $F$ applied by the nano-hexapod to the position of the sample $D$ is shown in figure [[fig:uniaxial-plant]].
|
|
It corresponds to the plant to control.
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#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
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|
figure;
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|
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|
ax1 = subplot(2, 1, 1);
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|
plot(freqs, abs(squeeze(freqresp(G('D', 'Fn'), freqs, 'Hz'))), 'k-');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
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|
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|
ax2 = subplot(2, 1, 2);
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|
plot(freqs, 180/pi*angle(squeeze(freqresp(G('D', 'Fn'), freqs, 'Hz'))), 'k-');
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|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
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|
yticks([-180, -90, 0, 90, 180]);
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|
linkaxes([ax1,ax2],'x');
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|
#+end_src
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|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial-plant.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
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|
<<plt-matlab>>
|
|
#+end_src
|
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|
|
#+NAME: fig:uniaxial-plant
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|
#+CAPTION: Bode plot of the Plant ([[./figs/uniaxial-plant.png][png]], [[./figs/uniaxial-plant.pdf][pdf]])
|
|
[[file:figs/uniaxial-plant.png]]
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* Integral Force Feedback
|
|
<<sec:iff>>
|
|
** Introduction :ignore:
|
|
#+begin_src latex :file uniaxial-model-nass-flexible-iff.pdf :post pdf2svg(file=*this*, ext="png") :exports results
|
|
\begin{tikzpicture}
|
|
% ====================
|
|
% Parameters
|
|
% ====================
|
|
\def\massw{2.2} % Width of the masses
|
|
\def\massh{0.8} % Height of the masses
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|
\def\spaceh{1.2} % Height of the springs/dampers
|
|
\def\dispw{0.4} % Width of the dashed line for the displacement
|
|
\def\disph{0.3} % Height of the arrow for the displacements
|
|
\def\bracs{0.05} % Brace spacing vertically
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|
\def\brach{-12pt} % Brace shift horizontaly
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|
\def\fsensh{0.2} % Height of the force sensor
|
|
\def\velsize{0.2} % Size of the velocity sensor
|
|
% ====================
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|
|
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% ====================
|
|
% Ground
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|
% ====================
|
|
\draw (-0.5*\massw, 0) -- (0.5*\massw, 0);
|
|
\draw[dashed] (0.5*\massw, 0) -- ++(1, 0);
|
|
\draw[->, dasehd] (0.5*\massw+0.8, 0) -- ++(0, 0.5) node[right]{$D_{w}$};
|
|
% ====================
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|
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% ====================
|
|
% Marble
|
|
\begin{scope}[shift={(0, 0)}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{m}$};
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|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{m}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{m}$};
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|
|
|
% Legend
|
|
\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
|
|
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Marble};
|
|
\end{scope}
|
|
% ====================
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|
|
|
|
|
% ====================
|
|
% Ty
|
|
\begin{scope}[shift={(0, \spaceh+\massh)}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{t}$};
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|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{t}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{t}$};
|
|
\draw[actuator={0.45}{0.2}] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](ft){$F_{ty}$};
|
|
|
|
% Legend
|
|
\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
|
|
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Ty};
|
|
\end{scope}
|
|
% ====================
|
|
|
|
|
|
% ====================
|
|
% Rz
|
|
\begin{scope}[shift={(0, 2*(\spaceh+\massh))}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{z}$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{z}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{z}$};
|
|
\draw[actuator] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](F){$F_{rz}$};
|
|
|
|
% Legend
|
|
\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
|
|
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Rz};
|
|
\end{scope}
|
|
% ====================
|
|
|
|
|
|
% ====================
|
|
% Hexapod
|
|
\begin{scope}[shift={(0, 3*(\spaceh+\massh))}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{h}$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{h}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{h}$};
|
|
|
|
% Displacements
|
|
% \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(3*\dispw, 0) coordinate(drbot);
|
|
|
|
% Legend
|
|
\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
|
|
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Hexapod};
|
|
\end{scope}
|
|
% ====================
|
|
|
|
|
|
% ====================
|
|
% NASS
|
|
\begin{scope}[shift={(0, 4*(\spaceh+\massh))}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{n}$};
|
|
|
|
% Force Sensor
|
|
\node[forcesensor={\massw}{\fsensh}] (fsensn) at (0, \spaceh-\fsensh){};
|
|
\node[right] (fn) at (fsensn.east) {$F_n$};
|
|
|
|
% % Velocity Sensor
|
|
% \node[inertialsensor={\velsize}] (veln) at (0.5*\massw, \spaceh+\massh) {};
|
|
% \node[right] at (veln.north east) {$v_n$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh-\fsensh) node[midway, left=0.1]{$k_{n}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh-\fsensh) node[midway, left=0.2]{$c_{n}$};
|
|
\draw[actuator] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh-\fsensh) node[midway, right=0.1](F){$F$};
|
|
|
|
% % Displacements
|
|
% \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(2*\dispw, 0) coordinate(xn) -- ++(\dispw, 0) coordinate(drtop);
|
|
|
|
% Legend
|
|
\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
|
|
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{NASS};
|
|
\end{scope}
|
|
% ====================
|
|
|
|
% ====================
|
|
% sample
|
|
\begin{scope}[shift={(0, 5*(\spaceh+\massh))}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{s}$};
|
|
|
|
% External Force
|
|
\draw[->] (0, \spaceh+\massh) node[]{$\bullet$} -- ++(0, 0.5*\massh) node[right]{$F_s$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{s}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{s}$};
|
|
|
|
% Legend
|
|
\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
|
|
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Sample};
|
|
\end{scope}
|
|
% ====================
|
|
|
|
% ====================
|
|
% IFF Control
|
|
\node[block={2em}{2em}, right=0.3 of fn] (iff) {$K_{\text{IFF}}$};
|
|
\draw[->] (fn.east) -- (iff.west);
|
|
\draw[->] (iff.south) |- (F.east);
|
|
% ====================
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+name: fig:iff_schematic
|
|
#+caption: Uniaxial IFF Control Schematic
|
|
#+RESULTS:
|
|
[[file:figs/uniaxial-model-nass-flexible-iff.png]]
|
|
|
|
** Matlab Init :noexport:ignore:
|
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
|
<<matlab-dir>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results silent :noweb yes
|
|
<<matlab-init>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no
|
|
simulinkproject('../');
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
open('uniaxial/matlab/sim_nano_station_uniaxial.slx')
|
|
#+end_src
|
|
|
|
** Control Design
|
|
#+begin_src matlab
|
|
load('./mat/uniaxial_plants.mat', 'G');
|
|
#+end_src
|
|
|
|
Let's look at the transfer function from actuator forces in the nano-hexapod to the force sensor in the nano-hexapod legs for all 6 pairs of actuator/sensor.
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
plot(freqs, abs(squeeze(freqresp(G('Fnm', 'Fn'), freqs, 'Hz'))), 'k-');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G('Fnm', 'Fn'), freqs, 'Hz'))), 'k-');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial_iff_plant.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial_iff_plant
|
|
#+CAPTION: Transfer function from forces applied in the legs to force sensor ([[./figs/uniaxial_iff_plant.png][png]], [[./figs/uniaxial_iff_plant.pdf][pdf]])
|
|
[[file:figs/uniaxial_iff_plant.png]]
|
|
|
|
The controller for each pair of actuator/sensor is:
|
|
#+begin_src matlab
|
|
K_iff = -1000/s;
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
plot(freqs, abs(squeeze(freqresp(K_iff*G('Fnm', 'Fn'), freqs, 'Hz'))), 'k-');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(K_iff*G('Fnm', 'Fn'), freqs, 'Hz'))), 'k-');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial_iff_open_loop.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial_iff_open_loop
|
|
#+CAPTION: Loop Gain for the Integral Force Feedback ([[./figs/uniaxial_iff_open_loop.png][png]], [[./figs/uniaxial_iff_open_loop.pdf][pdf]])
|
|
[[file:figs/uniaxial_iff_open_loop.png]]
|
|
|
|
** Identification
|
|
Let's initialize the system prior to identification.
|
|
#+begin_src matlab
|
|
initializeGround();
|
|
initializeGranite();
|
|
initializeTy();
|
|
initializeRy();
|
|
initializeRz();
|
|
initializeMicroHexapod();
|
|
initializeAxisc();
|
|
initializeMirror();
|
|
initializeNanoHexapod('actuator', 'piezo');
|
|
initializeSample('mass', 50);
|
|
#+end_src
|
|
|
|
All the controllers are set to 0.
|
|
#+begin_src matlab
|
|
K = tf(0);
|
|
save('./mat/controllers_uniaxial.mat', 'K', '-append');
|
|
K_iff = -K_iff;
|
|
save('./mat/controllers_uniaxial.mat', 'K_iff', '-append');
|
|
K_rmc = tf(0);
|
|
save('./mat/controllers_uniaxial.mat', 'K_rmc', '-append');
|
|
K_dvf = tf(0);
|
|
save('./mat/controllers_uniaxial.mat', 'K_dvf', '-append');
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
%% Options for Linearized
|
|
options = linearizeOptions;
|
|
options.SampleTime = 0;
|
|
|
|
%% Name of the Simulink File
|
|
mdl = 'sim_nano_station_uniaxial';
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
%% Input/Output definition
|
|
io(1) = linio([mdl, '/Dw'], 1, 'input'); % Ground Motion
|
|
io(2) = linio([mdl, '/Fs'], 1, 'input'); % Force applied on the sample
|
|
io(3) = linio([mdl, '/Fnl'], 1, 'input'); % Force applied by the NASS
|
|
io(4) = linio([mdl, '/Fdty'], 1, 'input'); % Parasitic force Ty
|
|
io(5) = linio([mdl, '/Fdrz'], 1, 'input'); % Parasitic force Rz
|
|
|
|
io(6) = linio([mdl, '/Dsm'], 1, 'output'); % Displacement of the sample
|
|
io(7) = linio([mdl, '/Fnlm'], 1, 'output'); % Force sensor in NASS's legs
|
|
io(8) = linio([mdl, '/Dnlm'], 1, 'output'); % Displacement of NASS's legs
|
|
io(9) = linio([mdl, '/Dgm'], 1, 'output'); % Absolute displacement of the granite
|
|
io(10) = linio([mdl, '/Vlm'], 1, 'output'); % Measured absolute velocity of the top NASS platform
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
%% Run the linearization
|
|
G_iff = linearize(mdl, io, options);
|
|
G_iff.InputName = {'Dw', ... % Ground Motion [m]
|
|
'Fs', ... % Force Applied on Sample [N]
|
|
'Fn', ... % Force applied by NASS [N]
|
|
'Fty', ... % Parasitic Force Ty [N]
|
|
'Frz'}; % Parasitic Force Rz [N]
|
|
G_iff.OutputName = {'D', ... % Measured sample displacement x.r.t. granite [m]
|
|
'Fnm', ... % Force Sensor in NASS [N]
|
|
'Dnm', ... % Displacement Sensor in NASS [m]
|
|
'Dgm', ... % Asbolute displacement of Granite [m]
|
|
'Vlm'}; ... % Absolute Velocity of NASS [m/s]
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
save('./mat/uniaxial_plants.mat', 'G_iff', '-append');
|
|
#+end_src
|
|
|
|
** Sensitivity to Disturbance
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
subplot(2, 1, 1);
|
|
title('$D_w$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('D', 'Dw'), freqs, 'Hz'))), 'k-', 'DisplayName', 'OL');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff('D', 'Dw'), freqs, 'Hz'))), 'k--', 'DisplayName', 'IFF');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
|
|
legend('location', 'northeast');
|
|
|
|
subplot(2, 1, 2);
|
|
title('$F_s$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('D', 'Fs'), freqs, 'Hz'))), 'k-');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff('D', 'Fs'), freqs, 'Hz'))), 'k--');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial_sensitivity_dist_iff.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial_sensitivity_dist_iff
|
|
#+CAPTION: Sensitivity to disturbance once the IFF controller is applied to the system ([[./figs/uniaxial_sensitivity_dist_iff.png][png]], [[./figs/uniaxial_sensitivity_dist_iff.pdf][pdf]])
|
|
[[file:figs/uniaxial_sensitivity_dist_iff.png]]
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
subplot(2, 1, 1);
|
|
title('$F_{ty}$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('D', 'Fty'), freqs, 'Hz'))), 'k-', 'DisplayName', 'OL');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff('D', 'Fty'), freqs, 'Hz'))), 'k--', 'DisplayName', 'IFF');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
legend('location', 'northeast');
|
|
|
|
subplot(2, 1, 2);
|
|
title('$F_{rz}$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('D', 'Frz'), freqs, 'Hz'))), 'k-');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff('D', 'Frz'), freqs, 'Hz'))), 'k--');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial_sensitivity_dist_stages_iff.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial_sensitivity_dist_stages_iff
|
|
#+CAPTION: Sensitivity to force disturbances in various stages when IFF is applied ([[./figs/uniaxial_sensitivity_dist_stages_iff.png][png]], [[./figs/uniaxial_sensitivity_dist_stages_iff.pdf][pdf]])
|
|
[[file:figs/uniaxial_sensitivity_dist_stages_iff.png]]
|
|
|
|
** Damped Plant
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('D', 'Fn'), freqs, 'Hz'))), 'k-', 'DisplayName', 'OL');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff('D', 'Fn'), freqs, 'Hz'))), 'k--', 'DisplayName', 'IFF');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
legend('location', 'northeast');
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
hold on;
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G('D', 'Fn'), freqs, 'Hz'))), 'k-');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff('D', 'Fn'), freqs, 'Hz'))), 'k--');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial_plant_iff_damped.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial_plant_iff_damped
|
|
#+CAPTION: Damped Plant after IFF is applied ([[./figs/uniaxial_plant_iff_damped.png][png]], [[./figs/uniaxial_plant_iff_damped.pdf][pdf]])
|
|
[[file:figs/uniaxial_plant_iff_damped.png]]
|
|
|
|
** Conclusion
|
|
#+begin_important
|
|
Integral Force Feedback:
|
|
#+end_important
|
|
|
|
* Relative Motion Control
|
|
<<sec:rmc>>
|
|
** Introduction :ignore:
|
|
In the Relative Motion Control (RMC), a derivative feedback is applied between the measured actuator displacement to the actuator force input.
|
|
|
|
#+begin_src latex :file uniaxial-model-nass-flexible-rmc.pdf :post pdf2svg(file=*this*, ext="png") :exports results
|
|
\begin{tikzpicture}
|
|
% ====================
|
|
% Parameters
|
|
% ====================
|
|
\def\massw{2.2} % Width of the masses
|
|
\def\massh{0.8} % Height of the masses
|
|
\def\spaceh{1.2} % Height of the springs/dampers
|
|
\def\dispw{0.4} % Width of the dashed line for the displacement
|
|
\def\disph{0.3} % Height of the arrow for the displacements
|
|
\def\bracs{0.05} % Brace spacing vertically
|
|
\def\brach{-12pt} % Brace shift horizontaly
|
|
\def\fsensh{0.2} % Height of the force sensor
|
|
\def\velsize{0.2} % Size of the velocity sensor
|
|
% ====================
|
|
|
|
|
|
% ====================
|
|
% Ground
|
|
% ====================
|
|
\draw (-0.5*\massw, 0) -- (0.5*\massw, 0);
|
|
\draw[dashed] (0.5*\massw, 0) -- ++(1, 0);
|
|
\draw[->, dasehd] (0.5*\massw+0.8, 0) -- ++(0, 0.5) node[right]{$D_{w}$};
|
|
% ====================
|
|
|
|
|
|
% ====================
|
|
% Marble
|
|
\begin{scope}[shift={(0, 0)}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{m}$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{m}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{m}$};
|
|
|
|
% Legend
|
|
\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
|
|
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Marble};
|
|
\end{scope}
|
|
% ====================
|
|
|
|
|
|
% ====================
|
|
% Ty
|
|
\begin{scope}[shift={(0, \spaceh+\massh)}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{t}$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{t}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{t}$};
|
|
\draw[actuator={0.45}{0.2}] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](ft){$F_{ty}$};
|
|
|
|
% Legend
|
|
\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
|
|
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Ty};
|
|
\end{scope}
|
|
% ====================
|
|
|
|
|
|
% ====================
|
|
% Rz
|
|
\begin{scope}[shift={(0, 2*(\spaceh+\massh))}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{z}$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{z}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{z}$};
|
|
\draw[actuator] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](F){$F_{rz}$};
|
|
|
|
% Legend
|
|
\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
|
|
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Rz};
|
|
\end{scope}
|
|
% ====================
|
|
|
|
|
|
% ====================
|
|
% Hexapod
|
|
\begin{scope}[shift={(0, 3*(\spaceh+\massh))}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{h}$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{h}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{h}$};
|
|
|
|
% Displacements
|
|
\draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(3*\dispw, 0) coordinate(drbot);
|
|
|
|
% Legend
|
|
\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
|
|
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Hexapod};
|
|
\end{scope}
|
|
% ====================
|
|
|
|
|
|
% ====================
|
|
% NASS
|
|
\begin{scope}[shift={(0, 4*(\spaceh+\massh))}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{n}$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{n}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{n}$};
|
|
\draw[actuator] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](F){$F$};
|
|
|
|
% Displacements
|
|
\draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(2*\dispw, 0) coordinate(xn) -- ++(\dispw, 0) coordinate(drtop);
|
|
|
|
% Legend
|
|
\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
|
|
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{NASS};
|
|
\end{scope}
|
|
% ====================
|
|
|
|
% ====================
|
|
% sample
|
|
\begin{scope}[shift={(0, 5*(\spaceh+\massh))}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{s}$};
|
|
|
|
% External Force
|
|
\draw[->] (0, \spaceh+\massh) node[]{$\bullet$} -- ++(0, 0.5*\massh) node[right]{$F_s$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{s}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{s}$};
|
|
|
|
% Legend
|
|
\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
|
|
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Sample};
|
|
\end{scope}
|
|
% ====================
|
|
|
|
% ====================
|
|
% RMC Control
|
|
\draw[<->, dashed] ($(drbot)+(-0.1, 0)$) -- ($(drtop)+(-0.1, 0)$) node[right](dr){$d_r$};
|
|
\node[block={2em}{2em}, below right=0.2 and 0.2 of dr] (rmc) {$K_{\text{RMC}}$};
|
|
\draw[->] (dr.east) -| (rmc.north);
|
|
\draw[->] (rmc.south) |- (F.east);
|
|
% ====================
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+name: fig:rmc_schematic
|
|
#+caption: Uniaxial RMC Control Schematic
|
|
#+RESULTS:
|
|
[[file:figs/uniaxial-model-nass-flexible-rmc.png]]
|
|
|
|
** Matlab Init :noexport:ignore:
|
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
|
<<matlab-dir>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results silent :noweb yes
|
|
<<matlab-init>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no
|
|
simulinkproject('../');
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
open('uniaxial/matlab/sim_nano_station_uniaxial.slx')
|
|
#+end_src
|
|
|
|
** Control Design
|
|
#+begin_src matlab
|
|
load('./mat/uniaxial_plants.mat', 'G');
|
|
#+end_src
|
|
|
|
Let's look at the transfer function from actuator forces in the nano-hexapod to the measured displacement of the actuator for all 6 pairs of actuator/sensor.
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
plot(freqs, abs(squeeze(freqresp(G('Dnm', 'Fn'), freqs, 'Hz'))), 'k-');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G('Dnm', 'Fn'), freqs, 'Hz'))), 'k-');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial_rmc_plant.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial_rmc_plant
|
|
#+CAPTION: Transfer function from forces applied in the legs to leg displacement sensor ([[./figs/uniaxial_rmc_plant.png][png]], [[./figs/uniaxial_rmc_plant.pdf][pdf]])
|
|
[[file:figs/uniaxial_rmc_plant.png]]
|
|
|
|
The Relative Motion Controller is defined below.
|
|
A Low pass Filter is added to make the controller transfer function proper.
|
|
#+begin_src matlab
|
|
K_rmc = s*50000/(1 + s/2/pi/10000);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
plot(freqs, abs(squeeze(freqresp(K_rmc*G('Dnm', 'Fn'), freqs, 'Hz'))), 'k-');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(K_rmc*G('Dnm', 'Fn'), freqs, 'Hz'))), 'k-');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial_rmc_open_loop.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial_rmc_open_loop
|
|
#+CAPTION: Loop Gain for the Integral Force Feedback ([[./figs/uniaxial_rmc_open_loop.png][png]], [[./figs/uniaxial_rmc_open_loop.pdf][pdf]])
|
|
[[file:figs/uniaxial_rmc_open_loop.png]]
|
|
|
|
** Identification
|
|
Let's initialize the system prior to identification.
|
|
#+begin_src matlab
|
|
initializeGround();
|
|
initializeGranite();
|
|
initializeTy();
|
|
initializeRy();
|
|
initializeRz();
|
|
initializeMicroHexapod();
|
|
initializeAxisc();
|
|
initializeMirror();
|
|
initializeNanoHexapod('actuator', 'piezo');
|
|
initializeSample('mass', 50);
|
|
#+end_src
|
|
|
|
And initialize the controllers.
|
|
#+begin_src matlab
|
|
K = tf(0);
|
|
save('./mat/controllers_uniaxial.mat', 'K', '-append');
|
|
K_iff = tf(0);
|
|
save('./mat/controllers_uniaxial.mat', 'K_iff', '-append');
|
|
K_rmc = -K_rmc;
|
|
save('./mat/controllers_uniaxial.mat', 'K_rmc', '-append');
|
|
K_dvf = tf(0);
|
|
save('./mat/controllers_uniaxial.mat', 'K_dvf', '-append');
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
%% Options for Linearized
|
|
options = linearizeOptions;
|
|
options.SampleTime = 0;
|
|
|
|
%% Name of the Simulink File
|
|
mdl = 'sim_nano_station_uniaxial';
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
%% Input/Output definition
|
|
io(1) = linio([mdl, '/Dw'], 1, 'input'); % Ground Motion
|
|
io(2) = linio([mdl, '/Fs'], 1, 'input'); % Force applied on the sample
|
|
io(3) = linio([mdl, '/Fnl'], 1, 'input'); % Force applied by the NASS
|
|
io(4) = linio([mdl, '/Fdty'], 1, 'input'); % Parasitic force Ty
|
|
io(5) = linio([mdl, '/Fdrz'], 1, 'input'); % Parasitic force Rz
|
|
|
|
io(6) = linio([mdl, '/Dsm'], 1, 'output'); % Displacement of the sample
|
|
io(7) = linio([mdl, '/Fnlm'], 1, 'output'); % Force sensor in NASS's legs
|
|
io(8) = linio([mdl, '/Dnlm'], 1, 'output'); % Displacement of NASS's legs
|
|
io(9) = linio([mdl, '/Dgm'], 1, 'output'); % Absolute displacement of the granite
|
|
io(10) = linio([mdl, '/Vlm'], 1, 'output'); % Measured absolute velocity of the top NASS platform
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
%% Run the linearization
|
|
G_rmc = linearize(mdl, io, options);
|
|
G_rmc.InputName = {'Dw', ... % Ground Motion [m]
|
|
'Fs', ... % Force Applied on Sample [N]
|
|
'Fn', ... % Force applied by NASS [N]
|
|
'Fty', ... % Parasitic Force Ty [N]
|
|
'Frz'}; % Parasitic Force Rz [N]
|
|
G_rmc.OutputName = {'D', ... % Measured sample displacement x.r.t. granite [m]
|
|
'Fnm', ... % Force Sensor in NASS [N]
|
|
'Dnm', ... % Displacement Sensor in NASS [m]
|
|
'Dgm', ... % Asbolute displacement of Granite [m]
|
|
'Vlm'}; ... % Absolute Velocity of NASS [m/s]
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
save('./mat/uniaxial_plants.mat', 'G_rmc', '-append');
|
|
#+end_src
|
|
|
|
|
|
** Sensitivity to Disturbance
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
subplot(2, 1, 1);
|
|
title('$D_w$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('D', 'Dw'), freqs, 'Hz'))), 'k-', 'DisplayName', 'OL');
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc('D', 'Dw'), freqs, 'Hz'))), 'k--', 'DisplayName', 'RMC');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
|
|
legend('location', 'northeast');
|
|
|
|
subplot(2, 1, 2);
|
|
title('$F_s$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('D', 'Fs'), freqs, 'Hz'))), 'k-');
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc('D', 'Fs'), freqs, 'Hz'))), 'k--');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial_sensitivity_dist_rmc.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial_sensitivity_dist_rmc
|
|
#+CAPTION: Sensitivity to disturbance once the RMC controller is applied to the system ([[./figs/uniaxial_sensitivity_dist_rmc.png][png]], [[./figs/uniaxial_sensitivity_dist_rmc.pdf][pdf]])
|
|
[[file:figs/uniaxial_sensitivity_dist_rmc.png]]
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
subplot(2, 1, 1);
|
|
title('$F_{ty}$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('D', 'Fty'), freqs, 'Hz'))), 'k-', 'DisplayName', 'OL');
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc('D', 'Fty'), freqs, 'Hz'))), 'k--', 'DisplayName', 'RMC');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
legend('location', 'northeast');
|
|
|
|
subplot(2, 1, 2);
|
|
title('$F_{rz}$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('D', 'Frz'), freqs, 'Hz'))), 'k-');
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc('D', 'Frz'), freqs, 'Hz'))), 'k--');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial_sensitivity_dist_stages_rmc.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial_sensitivity_dist_stages_rmc
|
|
#+CAPTION: Sensitivity to force disturbances in various stages when RMC is applied ([[./figs/uniaxial_sensitivity_dist_stages_rmc.png][png]], [[./figs/uniaxial_sensitivity_dist_stages_rmc.pdf][pdf]])
|
|
[[file:figs/uniaxial_sensitivity_dist_stages_rmc.png]]
|
|
|
|
** Damped Plant
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('D', 'Fn'), freqs, 'Hz'))), 'k-', 'DisplayName', 'OL');
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc('D', 'Fn'), freqs, 'Hz'))), 'k--', 'DisplayName', 'RMC');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
legend('location', 'northeast');
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
hold on;
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G('D', 'Fn'), freqs, 'Hz'))), 'k-');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_rmc('D', 'Fn'), freqs, 'Hz'))), 'k--');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial_plant_rmc_damped.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial_plant_rmc_damped
|
|
#+CAPTION: Damped Plant after RMC is applied ([[./figs/uniaxial_plant_rmc_damped.png][png]], [[./figs/uniaxial_plant_rmc_damped.pdf][pdf]])
|
|
[[file:figs/uniaxial_plant_rmc_damped.png]]
|
|
|
|
** Conclusion
|
|
#+begin_important
|
|
Relative Motion Control:
|
|
#+end_important
|
|
|
|
* Direct Velocity Feedback
|
|
<<sec:dvf>>
|
|
** Introduction :ignore:
|
|
In the Relative Motion Control (RMC), a feedback is applied between the measured velocity of the platform to the actuator force input.
|
|
|
|
#+begin_src latex :file uniaxial-model-nass-flexible-dvf.pdf :post pdf2svg(file=*this*, ext="png") :exports results
|
|
\begin{tikzpicture}
|
|
% ====================
|
|
% Parameters
|
|
% ====================
|
|
\def\massw{2.2} % Width of the masses
|
|
\def\massh{0.8} % Height of the masses
|
|
\def\spaceh{1.2} % Height of the springs/dampers
|
|
\def\dispw{0.4} % Width of the dashed line for the displacement
|
|
\def\disph{0.3} % Height of the arrow for the displacements
|
|
\def\bracs{0.05} % Brace spacing vertically
|
|
\def\brach{-12pt} % Brace shift horizontaly
|
|
\def\fsensh{0.2} % Height of the force sensor
|
|
\def\velsize{0.2} % Size of the velocity sensor
|
|
% ====================
|
|
|
|
|
|
% ====================
|
|
% Ground
|
|
% ====================
|
|
\draw (-0.5*\massw, 0) -- (0.5*\massw, 0);
|
|
\draw[dashed] (0.5*\massw, 0) -- ++(1, 0);
|
|
\draw[->, dasehd] (0.5*\massw+0.8, 0) -- ++(0, 0.5) node[right]{$D_{w}$};
|
|
% ====================
|
|
|
|
|
|
% ====================
|
|
% Marble
|
|
\begin{scope}[shift={(0, 0)}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{m}$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{m}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{m}$};
|
|
|
|
% Legend
|
|
\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
|
|
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Marble};
|
|
\end{scope}
|
|
% ====================
|
|
|
|
|
|
% ====================
|
|
% Ty
|
|
\begin{scope}[shift={(0, \spaceh+\massh)}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{t}$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{t}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{t}$};
|
|
\draw[actuator={0.45}{0.2}] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](ft){$F_{ty}$};
|
|
|
|
% Legend
|
|
\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
|
|
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Ty};
|
|
\end{scope}
|
|
% ====================
|
|
|
|
|
|
% ====================
|
|
% Rz
|
|
\begin{scope}[shift={(0, 2*(\spaceh+\massh))}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{z}$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{z}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{z}$};
|
|
\draw[actuator] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](F){$F_{rz}$};
|
|
|
|
% Legend
|
|
\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
|
|
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Rz};
|
|
\end{scope}
|
|
% ====================
|
|
|
|
|
|
% ====================
|
|
% Hexapod
|
|
\begin{scope}[shift={(0, 3*(\spaceh+\massh))}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{h}$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{h}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{h}$};
|
|
|
|
% Displacements
|
|
% \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(3*\dispw, 0) coordinate(drbot);
|
|
|
|
% Legend
|
|
\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
|
|
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Hexapod};
|
|
\end{scope}
|
|
% ====================
|
|
|
|
|
|
% ====================
|
|
% NASS
|
|
\begin{scope}[shift={(0, 4*(\spaceh+\massh))}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{n}$};
|
|
|
|
% % Force Sensor
|
|
% \node[forcesensor={\massw}{\fsensh}] (fsensn) at (0, \spaceh-\fsensh){};
|
|
% \node[right] (fn) at (fsensn.east) {$F_n$};
|
|
|
|
% Velocity Sensor
|
|
\node[inertialsensor={\velsize}] (veln) at (0.5*\massw, \spaceh+\massh) {};
|
|
\node[above right] at (veln.east) {$v_n$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{n}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{n}$};
|
|
\draw[actuator] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](F){$F$};
|
|
|
|
% % Displacements
|
|
% \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(2*\dispw, 0) coordinate(xn) -- ++(\dispw, 0) coordinate(drtop);
|
|
|
|
% Legend
|
|
\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
|
|
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{NASS};
|
|
\end{scope}
|
|
% ====================
|
|
|
|
% ====================
|
|
% sample
|
|
\begin{scope}[shift={(0, 5*(\spaceh+\massh))}]
|
|
% Mass
|
|
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{s}$};
|
|
|
|
% External Force
|
|
\draw[->] (0, \spaceh+\massh) node[]{$\bullet$} -- ++(0, 0.5*\massh) node[right]{$F_s$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{s}$};
|
|
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{s}$};
|
|
|
|
% Legend
|
|
\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
|
|
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Sample};
|
|
\end{scope}
|
|
% ====================
|
|
|
|
% ====================
|
|
% DVF Control
|
|
\node[block={2em}{2em}, below right=0.4 and 0.4 of veln] (ppf) {$K_{\text{DVF}}$};
|
|
\draw[->] (veln.east) -| (ppf.north);
|
|
\draw[->] (ppf.south) |- (F.east);
|
|
% ====================
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+name: fig:dvf_schematic
|
|
#+caption: Uniaxial DVF Control Schematic
|
|
#+RESULTS:
|
|
[[file:figs/uniaxial-model-nass-flexible-dvf.png]]
|
|
|
|
** Matlab Init :noexport:ignore:
|
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
|
<<matlab-dir>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results silent :noweb yes
|
|
<<matlab-init>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no
|
|
simulinkproject('../');
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
open('uniaxial/matlab/sim_nano_station_uniaxial.slx')
|
|
#+end_src
|
|
|
|
** Control Design
|
|
#+begin_src matlab
|
|
load('./mat/uniaxial_plants.mat', 'G');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
plot(freqs, abs(squeeze(freqresp(G('Vlm', 'Fn'), freqs, 'Hz'))), 'k-');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G('Vlm', 'Fn'), freqs, 'Hz'))), 'k-');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial_dvf_plant.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial_dvf_plant
|
|
#+CAPTION: Transfer function from forces applied in the legs to leg velocity sensor ([[./figs/uniaxial_dvf_plant.png][png]], [[./figs/uniaxial_dvf_plant.pdf][pdf]])
|
|
[[file:figs/uniaxial_dvf_plant.png]]
|
|
|
|
#+begin_src matlab
|
|
K_dvf = tf(5e4);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
plot(freqs, abs(squeeze(freqresp(K_dvf*G('Vlm', 'Fn'), freqs, 'Hz'))), 'k-');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(K_dvf*G('Vlm', 'Fn'), freqs, 'Hz'))), 'k-');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial_dvf_loop_gain.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial_dvf_loop_gain
|
|
#+CAPTION: Transfer function from forces applied in the legs to leg velocity sensor ([[./figs/uniaxial_dvf_loop_gain.png][png]], [[./figs/uniaxial_dvf_loop_gain.pdf][pdf]])
|
|
[[file:figs/uniaxial_dvf_loop_gain.png]]
|
|
|
|
** Identification
|
|
Let's initialize the system prior to identification.
|
|
#+begin_src matlab
|
|
initializeGround();
|
|
initializeGranite();
|
|
initializeTy();
|
|
initializeRy();
|
|
initializeRz();
|
|
initializeMicroHexapod();
|
|
initializeAxisc();
|
|
initializeMirror();
|
|
initializeNanoHexapod('actuator', 'piezo');
|
|
initializeSample('mass', 50);
|
|
#+end_src
|
|
|
|
And initialize the controllers.
|
|
#+begin_src matlab
|
|
K = tf(0);
|
|
save('./mat/controllers_uniaxial.mat', 'K', '-append');
|
|
K_iff = tf(0);
|
|
save('./mat/controllers_uniaxial.mat', 'K_iff', '-append');
|
|
K_rmc = tf(0);
|
|
save('./mat/controllers_uniaxial.mat', 'K_rmc', '-append');
|
|
K_dvf = -K_dvf;
|
|
save('./mat/controllers_uniaxial.mat', 'K_dvf', '-append');
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
%% Options for Linearized
|
|
options = linearizeOptions;
|
|
options.SampleTime = 0;
|
|
|
|
%% Name of the Simulink File
|
|
mdl = 'sim_nano_station_uniaxial';
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
%% Input/Output definition
|
|
io(1) = linio([mdl, '/Dw'], 1, 'input'); % Ground Motion
|
|
io(2) = linio([mdl, '/Fs'], 1, 'input'); % Force applied on the sample
|
|
io(3) = linio([mdl, '/Fnl'], 1, 'input'); % Force applied by the NASS
|
|
io(4) = linio([mdl, '/Fdty'], 1, 'input'); % Parasitic force Ty
|
|
io(5) = linio([mdl, '/Fdrz'], 1, 'input'); % Parasitic force Rz
|
|
|
|
io(6) = linio([mdl, '/Dsm'], 1, 'output'); % Displacement of the sample
|
|
io(7) = linio([mdl, '/Fnlm'], 1, 'output'); % Force sensor in NASS's legs
|
|
io(8) = linio([mdl, '/Dnlm'], 1, 'output'); % Displacement of NASS's legs
|
|
io(9) = linio([mdl, '/Dgm'], 1, 'output'); % Absolute displacement of the granite
|
|
io(10) = linio([mdl, '/Vlm'], 1, 'output'); % Measured absolute velocity of the top NASS platform
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
%% Run the linearization
|
|
G_dvf = linearize(mdl, io, options);
|
|
G_dvf.InputName = {'Dw', ... % Ground Motion [m]
|
|
'Fs', ... % Force Applied on Sample [N]
|
|
'Fn', ... % Force applied by NASS [N]
|
|
'Fty', ... % Parasitic Force Ty [N]
|
|
'Frz'}; % Parasitic Force Rz [N]
|
|
G_dvf.OutputName = {'D', ... % Measured sample displacement x.r.t. granite [m]
|
|
'Fnm', ... % Force Sensor in NASS [N]
|
|
'Dnm', ... % Displacement Sensor in NASS [m]
|
|
'Dgm', ... % Asbolute displacement of Granite [m]
|
|
'Vlm'}; ... % Absolute Velocity of NASS [m/s]
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
save('./mat/uniaxial_plants.mat', 'G_dvf', '-append');
|
|
#+end_src
|
|
|
|
** Sensitivity to Disturbance
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
subplot(2, 1, 1);
|
|
title('$D_w$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('D', 'Dw'), freqs, 'Hz'))), 'k-', 'DisplayName', 'OL');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf('D', 'Dw'), freqs, 'Hz'))), 'k--', 'DisplayName', 'DVF');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
|
|
legend('location', 'northeast');
|
|
|
|
subplot(2, 1, 2);
|
|
title('$F_s$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('D', 'Fs'), freqs, 'Hz'))), 'k-');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf('D', 'Fs'), freqs, 'Hz'))), 'k--');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial_sensitivity_dist_dvf.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial_sensitivity_dist_dvf
|
|
#+CAPTION: Sensitivity to disturbance once the DVF controller is applied to the system ([[./figs/uniaxial_sensitivity_dist_dvf.png][png]], [[./figs/uniaxial_sensitivity_dist_dvf.pdf][pdf]])
|
|
[[file:figs/uniaxial_sensitivity_dist_dvf.png]]
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
subplot(2, 1, 1);
|
|
title('$F_{ty}$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('D', 'Fty'), freqs, 'Hz'))), 'k-', 'DisplayName', 'OL');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf('D', 'Fty'), freqs, 'Hz'))), 'k--', 'DisplayName', 'DVF');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
legend('location', 'northeast');
|
|
|
|
subplot(2, 1, 2);
|
|
title('$F_{rz}$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('D', 'Frz'), freqs, 'Hz'))), 'k-');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf('D', 'Frz'), freqs, 'Hz'))), 'k--');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial_sensitivity_dist_stages_dvf.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial_sensitivity_dist_stages_dvf
|
|
#+CAPTION: Sensitivity to force disturbances in various stages when DVF is applied ([[./figs/uniaxial_sensitivity_dist_stages_dvf.png][png]], [[./figs/uniaxial_sensitivity_dist_stages_dvf.pdf][pdf]])
|
|
[[file:figs/uniaxial_sensitivity_dist_stages_dvf.png]]
|
|
|
|
** Damped Plant
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('D', 'Fn'), freqs, 'Hz'))), 'k-', 'DisplayName', 'OL');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf('D', 'Fn'), freqs, 'Hz'))), 'k--', 'DisplayName', 'DVF');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
legend('location', 'northeast');
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
hold on;
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G('D', 'Fn'), freqs, 'Hz'))), 'k-');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_dvf('D', 'Fn'), freqs, 'Hz'))), 'k--');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial_plant_dvf_damped.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial_plant_dvf_damped
|
|
#+CAPTION: Damped Plant after DVF is applied ([[./figs/uniaxial_plant_dvf_damped.png][png]], [[./figs/uniaxial_plant_dvf_damped.pdf][pdf]])
|
|
[[file:figs/uniaxial_plant_dvf_damped.png]]
|
|
|
|
** Conclusion
|
|
#+begin_important
|
|
Direct Velocity Feedback:
|
|
#+end_important
|
|
* With Cedrat Piezo-electric Actuators
|
|
<<sec:cedrat_actuator>>
|
|
** Introduction :ignore:
|
|
The model used for the Cedrat actuator is shown in figure [[fig:cedrat_schematic]].
|
|
|
|
#+begin_src latex :file cedrat-uniaxial-actuator.pdf :post pdf2svg(file=*this*, ext="png") :exports results
|
|
\begin{tikzpicture}
|
|
% Spring, Damper, and Actuator
|
|
\draw[] (0, -0.2) -- (0, 0);
|
|
\draw[] (-1, 0) -- (1, 0);
|
|
|
|
\draw[spring] (-1, 0) -- (-1, 2) node[midway, left=0.1]{$k_{a}$};
|
|
\draw[damper] (0, 0) -- ( 0, 2) node[midway, left=0.2]{$c_{a}$};
|
|
\draw[actuator] (1, 0) -- ( 1, 2) node[midway, left=0.2]{$F$};
|
|
|
|
\draw[] (-1, 2) -- (1, 2);
|
|
\draw[] (0, 2) -- (0, 2.2);
|
|
\node[forcesensor={0.4}{0.4}] (fsensn) at (0, 2.2){};
|
|
\draw[] (0, 2.6) -- (0, 2.8);
|
|
|
|
|
|
\draw[spring] (2, -0.2) -- (2, 2.8) node[midway, left=0.2]{$k$};
|
|
|
|
\draw[] (0, -0.2) -- (2, -0.2);
|
|
\draw[] (0, 2.8) -- (2, 2.8);
|
|
|
|
\draw[] (1, 2.8) -- (1, 3);
|
|
\draw[] (1, -0.2) -- (1, -0.4);
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+name: fig:cedrat_schematic
|
|
#+caption: Schematic of the model used for the Cedrat Actuator
|
|
#+RESULTS:
|
|
[[file:figs/cedrat-uniaxial-actuator.png]]
|
|
|
|
** Matlab Init :noexport:ignore:
|
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
|
<<matlab-dir>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results silent :noweb yes
|
|
<<matlab-init>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no
|
|
simulinkproject('../');
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
open('uniaxial/matlab/sim_nano_station_uniaxial_cedrat.slx')
|
|
#+end_src
|
|
|
|
** Identification
|
|
Let's initialize the system prior to identification.
|
|
#+begin_src matlab
|
|
initializeGround();
|
|
initializeGranite();
|
|
initializeTy();
|
|
initializeRy();
|
|
initializeRz();
|
|
initializeMicroHexapod();
|
|
initializeAxisc();
|
|
initializeMirror();
|
|
initializeNanoHexapod('actuator', 'piezo');
|
|
initializeCedratPiezo();
|
|
initializeSample('mass', 50);
|
|
#+end_src
|
|
|
|
And initialize the controllers.
|
|
#+begin_src matlab
|
|
K = tf(0);
|
|
save('./mat/controllers_uniaxial.mat', 'K', '-append');
|
|
K_iff = tf(0);
|
|
save('./mat/controllers_uniaxial.mat', 'K_iff', '-append');
|
|
K_rmc = tf(0);
|
|
save('./mat/controllers_uniaxial.mat', 'K_rmc', '-append');
|
|
K_dvf = tf(0);
|
|
save('./mat/controllers_uniaxial.mat', 'K_dvf', '-append');
|
|
#+end_src
|
|
|
|
We identify the dynamics of the system.
|
|
#+begin_src matlab
|
|
%% Options for Linearized
|
|
options = linearizeOptions;
|
|
options.SampleTime = 0;
|
|
|
|
%% Name of the Simulink File
|
|
mdl = 'sim_nano_station_uniaxial_cedrat_bis';
|
|
#+end_src
|
|
|
|
The inputs and outputs are defined below and corresponds to the name of simulink blocks.
|
|
#+begin_src matlab
|
|
%% Input/Output definition
|
|
io(1) = linio([mdl, '/Dw'], 1, 'input'); % Ground Motion
|
|
io(2) = linio([mdl, '/Fs'], 1, 'input'); % Force applied on the sample
|
|
io(3) = linio([mdl, '/Fnl'], 1, 'input'); % Force applied by the NASS
|
|
io(4) = linio([mdl, '/Fdty'], 1, 'input'); % Parasitic force Ty
|
|
io(5) = linio([mdl, '/Fdrz'], 1, 'input'); % Parasitic force Rz
|
|
|
|
io(6) = linio([mdl, '/Dsm'], 1, 'output'); % Displacement of the sample
|
|
io(7) = linio([mdl, '/Fnlm'], 1, 'output'); % Force sensor in NASS's legs
|
|
io(8) = linio([mdl, '/Dnlm'], 1, 'output'); % Displacement of NASS's legs
|
|
io(9) = linio([mdl, '/Dgm'], 1, 'output'); % Absolute displacement of the granite
|
|
io(10) = linio([mdl, '/Vlm'], 1, 'output'); % Measured absolute velocity of the top NASS platform
|
|
#+end_src
|
|
|
|
Finally, we use the =linearize= Matlab function to extract a state space model from the simscape model.
|
|
#+begin_src matlab
|
|
%% Run the linearization
|
|
G = linearize(mdl, io, options);
|
|
G.InputName = {'Dw', ... % Ground Motion [m]
|
|
'Fs', ... % Force Applied on Sample [N]
|
|
'Fn', ... % Force applied by NASS [N]
|
|
'Fty', ... % Parasitic Force Ty [N]
|
|
'Frz'}; % Parasitic Force Rz [N]
|
|
G.OutputName = {'D', ... % Measured sample displacement x.r.t. granite [m]
|
|
'Fnm', ... % Force Sensor in NASS [N]
|
|
'Dnm', ... % Displacement Sensor in NASS [m]
|
|
'Dgm', ... % Asbolute displacement of Granite [m]
|
|
'Vlm'}; ... % Absolute Velocity of NASS [m/s]
|
|
#+end_src
|
|
|
|
** Control Design
|
|
Let's look at the transfer function from actuator forces in the nano-hexapod to the force sensor in the nano-hexapod legs for all 6 pairs of actuator/sensor.
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 4, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
plot(freqs, abs(squeeze(freqresp(G('Fnm', 'Fn'), freqs, 'Hz'))), 'k-');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G('Fnm', 'Fn'), freqs, 'Hz'))), 'k-');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial_cedrat_plant.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial_cedrat_plant
|
|
#+CAPTION: Transfer function from forces applied in the legs to force sensor ([[./figs/uniaxial_cedrat_plant.png][png]], [[./figs/uniaxial_cedrat_plant.pdf][pdf]])
|
|
[[file:figs/uniaxial_cedrat_plant.png]]
|
|
|
|
The controller for each pair of actuator/sensor is:
|
|
#+begin_src matlab
|
|
K_cedrat = -5000/s;
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 4, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
plot(freqs, abs(squeeze(freqresp(K_cedrat*G('Fnm', 'Fn'), freqs, 'Hz'))), 'k-');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(K_cedrat*G('Fnm', 'Fn'), freqs, 'Hz'))), 'k-');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial_cedrat_open_loop.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial_cedrat_open_loop
|
|
#+CAPTION: Loop Gain for the Integral Force Feedback ([[./figs/uniaxial_cedrat_open_loop.png][png]], [[./figs/uniaxial_cedrat_open_loop.pdf][pdf]])
|
|
[[file:figs/uniaxial_cedrat_open_loop.png]]
|
|
|
|
** Identification
|
|
Let's initialize the system prior to identification.
|
|
#+begin_src matlab
|
|
initializeGround();
|
|
initializeGranite();
|
|
initializeTy();
|
|
initializeRy();
|
|
initializeRz();
|
|
initializeMicroHexapod();
|
|
initializeAxisc();
|
|
initializeMirror();
|
|
initializeNanoHexapod('actuator', 'piezo');
|
|
initializeCedratPiezo();
|
|
initializeSample('mass', 50);
|
|
#+end_src
|
|
|
|
All the controllers are set to 0.
|
|
#+begin_src matlab
|
|
K = tf(0);
|
|
save('./mat/controllers_uniaxial.mat', 'K', '-append');
|
|
K_iff = -K_cedrat;
|
|
save('./mat/controllers_uniaxial.mat', 'K_iff', '-append');
|
|
K_rmc = tf(0);
|
|
save('./mat/controllers_uniaxial.mat', 'K_rmc', '-append');
|
|
K_dvf = tf(0);
|
|
save('./mat/controllers_uniaxial.mat', 'K_dvf', '-append');
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
%% Options for Linearized
|
|
options = linearizeOptions;
|
|
options.SampleTime = 0;
|
|
|
|
%% Name of the Simulink File
|
|
mdl = 'sim_nano_station_uniaxial_cedrat_bis';
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
%% Input/Output definition
|
|
io(1) = linio([mdl, '/Dw'], 1, 'input'); % Ground Motion
|
|
io(2) = linio([mdl, '/Fs'], 1, 'input'); % Force applied on the sample
|
|
io(3) = linio([mdl, '/Fnl'], 1, 'input'); % Force applied by the NASS
|
|
io(4) = linio([mdl, '/Fdty'], 1, 'input'); % Parasitic force Ty
|
|
io(5) = linio([mdl, '/Fdrz'], 1, 'input'); % Parasitic force Rz
|
|
|
|
io(6) = linio([mdl, '/Dsm'], 1, 'output'); % Displacement of the sample
|
|
io(7) = linio([mdl, '/Fnlm'], 1, 'output'); % Force sensor in NASS's legs
|
|
io(8) = linio([mdl, '/Dnlm'], 1, 'output'); % Displacement of NASS's legs
|
|
io(9) = linio([mdl, '/Dgm'], 1, 'output'); % Absolute displacement of the granite
|
|
io(10) = linio([mdl, '/Vlm'], 1, 'output'); % Measured absolute velocity of the top NASS platform
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
%% Run the linearization
|
|
G_cedrat = linearize(mdl, io, options);
|
|
G_cedrat.InputName = {'Dw', ... % Ground Motion [m]
|
|
'Fs', ... % Force Applied on Sample [N]
|
|
'Fn', ... % Force applied by NASS [N]
|
|
'Fty', ... % Parasitic Force Ty [N]
|
|
'Frz'}; % Parasitic Force Rz [N]
|
|
G_cedrat.OutputName = {'D', ... % Measured sample displacement x.r.t. granite [m]
|
|
'Fnm', ... % Force Sensor in NASS [N]
|
|
'Dnm', ... % Displacement Sensor in NASS [m]
|
|
'Dgm', ... % Asbolute displacement of Granite [m]
|
|
'Vlm'}; ... % Absolute Velocity of NASS [m/s]
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
% save('./mat/uniaxial_plants.mat', 'G_cedrat', '-append');
|
|
#+end_src
|
|
|
|
** Sensitivity to Disturbance
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
subplot(2, 1, 1);
|
|
title('$D_w$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('D', 'Dw'), freqs, 'Hz'))), 'k-', 'DisplayName', 'OL');
|
|
plot(freqs, abs(squeeze(freqresp(G_cedrat('D', 'Dw'), freqs, 'Hz'))), 'k--', 'DisplayName', 'CEDRAT');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
|
|
legend('location', 'northeast');
|
|
|
|
subplot(2, 1, 2);
|
|
title('$F_s$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('D', 'Fs'), freqs, 'Hz'))), 'k-');
|
|
plot(freqs, abs(squeeze(freqresp(G_cedrat('D', 'Fs'), freqs, 'Hz'))), 'k--');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial_sensitivity_dist_cedrat.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial_sensitivity_dist_cedrat
|
|
#+CAPTION: Sensitivity to disturbance once the CEDRAT controller is applied to the system ([[./figs/uniaxial_sensitivity_dist_cedrat.png][png]], [[./figs/uniaxial_sensitivity_dist_cedrat.pdf][pdf]])
|
|
[[file:figs/uniaxial_sensitivity_dist_cedrat.png]]
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
subplot(2, 1, 1);
|
|
title('$F_{ty}$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('D', 'Fty'), freqs, 'Hz'))), 'k-', 'DisplayName', 'OL');
|
|
plot(freqs, abs(squeeze(freqresp(G_cedrat('D', 'Fty'), freqs, 'Hz'))), 'k--', 'DisplayName', 'CEDRAT');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
legend('location', 'northeast');
|
|
|
|
subplot(2, 1, 2);
|
|
title('$F_{rz}$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('D', 'Frz'), freqs, 'Hz'))), 'k-');
|
|
plot(freqs, abs(squeeze(freqresp(G_cedrat('D', 'Frz'), freqs, 'Hz'))), 'k--');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial_sensitivity_dist_stages_cedrat.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial_sensitivity_dist_stages_cedrat
|
|
#+CAPTION: Sensitivity to force disturbances in various stages when CEDRAT is applied ([[./figs/uniaxial_sensitivity_dist_stages_cedrat.png][png]], [[./figs/uniaxial_sensitivity_dist_stages_cedrat.pdf][pdf]])
|
|
[[file:figs/uniaxial_sensitivity_dist_stages_cedrat.png]]
|
|
|
|
** Damped Plant
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('D', 'Fn'), freqs, 'Hz'))), 'k-', 'DisplayName', 'OL');
|
|
plot(freqs, abs(squeeze(freqresp(G_cedrat('D', 'Fn'), freqs, 'Hz'))), 'k--', 'DisplayName', 'CEDRAT');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
legend('location', 'northeast');
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
hold on;
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G('D', 'Fn'), freqs, 'Hz'))), 'k-');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_cedrat('D', 'Fn'), freqs, 'Hz'))), 'k--');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial_plant_cedrat_damped.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial_plant_cedrat_damped
|
|
#+CAPTION: Damped Plant after CEDRAT is applied ([[./figs/uniaxial_plant_cedrat_damped.png][png]], [[./figs/uniaxial_plant_cedrat_damped.pdf][pdf]])
|
|
[[file:figs/uniaxial_plant_cedrat_damped.png]]
|
|
|
|
** Conclusion
|
|
#+begin_important
|
|
This gives similar results than with a classical force sensor.
|
|
#+end_important
|
|
|
|
* Comparison of Active Damping Techniques
|
|
<<sec:comparison>>
|
|
** Matlab Init :noexport:ignore:
|
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
|
<<matlab-dir>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results silent :noweb yes
|
|
<<matlab-init>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no
|
|
simulinkproject('../');
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
open('uniaxial/matlab/sim_nano_station_uniaxial.slx')
|
|
#+end_src
|
|
|
|
** Load the plants
|
|
#+begin_src matlab
|
|
load('./mat/uniaxial_plants.mat', 'G', 'G_iff', 'G_rmc', 'G_dvf');
|
|
#+end_src
|
|
|
|
** Sensitivity to Disturbance
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
title('$D_w$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G ('D', 'Dw'), freqs, 'Hz'))), 'k-', 'DisplayName', 'OL');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff('D', 'Dw'), freqs, 'Hz'))), 'k:', 'DisplayName', 'IFF');
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc('D', 'Dw'), freqs, 'Hz'))), 'k--', 'DisplayName', 'RMC');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf('D', 'Dw'), freqs, 'Hz'))), 'k-.', 'DisplayName', 'DVF');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
|
|
legend('location', 'northeast');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial_sensitivity_ground_motion.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial_sensitivity_ground_motion
|
|
#+CAPTION: Sensitivity to Ground Motion - Comparison ([[./figs/uniaxial_sensitivity_ground_motion.png][png]], [[./figs/uniaxial_sensitivity_ground_motion.pdf][pdf]])
|
|
[[file:figs/uniaxial_sensitivity_ground_motion.png]]
|
|
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
title('$F_s$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G ('D', 'Fs'), freqs, 'Hz'))), 'k-' , 'DisplayName', 'OL');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff('D', 'Fs'), freqs, 'Hz'))), 'k:' , 'DisplayName', 'IFF');
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc('D', 'Fs'), freqs, 'Hz'))), 'k--', 'DisplayName', 'RMC');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf('D', 'Fs'), freqs, 'Hz'))), 'k-.', 'DisplayName', 'DVF');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
legend('location', 'northeast');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial_sensitivity_direct_force.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial_sensitivity_direct_force
|
|
#+CAPTION: Sensitivity to disturbance - Comparison ([[./figs/uniaxial_sensitivity_direct_force.png][png]], [[./figs/uniaxial_sensitivity_direct_force.pdf][pdf]])
|
|
[[file:figs/uniaxial_sensitivity_direct_force.png]]
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
title('$F_{ty}$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G ('D', 'Fty'), freqs, 'Hz'))), 'k-', 'DisplayName', 'OL');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff('D', 'Fty'), freqs, 'Hz'))), 'k:', 'DisplayName', 'IFF');
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc('D', 'Fty'), freqs, 'Hz'))), 'k--', 'DisplayName', 'RMC');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf('D', 'Fty'), freqs, 'Hz'))), 'k-.', 'DisplayName', 'DVF');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
legend('location', 'northeast');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial_sensitivity_fty.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial_sensitivity_fty
|
|
#+CAPTION: Sensitivity to force disturbances - Comparison ([[./figs/uniaxial_sensitivity_fty.png][png]], [[./figs/uniaxial_sensitivity_fty.pdf][pdf]])
|
|
[[file:figs/uniaxial_sensitivity_fty.png]]
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
title('$F_{rz}$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G ('D', 'Frz'), freqs, 'Hz'))), 'k-' , 'DisplayName', 'OL');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff('D', 'Frz'), freqs, 'Hz'))), 'k:' , 'DisplayName', 'IFF');
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc('D', 'Frz'), freqs, 'Hz'))), 'k--', 'DisplayName', 'RMC');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf('D', 'Frz'), freqs, 'Hz'))), 'k-.', 'DisplayName', 'DVF');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
legend('location', 'northeast');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial_sensitivity_frz.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial_sensitivity_frz
|
|
#+CAPTION: Sensitivity to force disturbances - Comparison ([[./figs/uniaxial_sensitivity_frz.png][png]], [[./figs/uniaxial_sensitivity_frz.pdf][pdf]])
|
|
[[file:figs/uniaxial_sensitivity_frz.png]]
|
|
|
|
** Noise Budget
|
|
We first load the measured PSD of the disturbance.
|
|
#+begin_src matlab
|
|
load('./mat/disturbances_dist_psd.mat', 'dist_f');
|
|
#+end_src
|
|
|
|
The effect of these disturbances on the distance $D$ is computed for all active damping techniques.
|
|
#+begin_src matlab :exports none
|
|
% Power Spectral Density of the relative Displacement [m^2/Hz]
|
|
psd_ol_gm_d = dist_f.psd_gm.*abs(squeeze(freqresp(G('D', 'Dw'), dist_f.f, 'Hz'))).^2;
|
|
psd_ol_ty_d = dist_f.psd_ty.*abs(squeeze(freqresp(G('D', 'Fty'), dist_f.f, 'Hz'))).^2;
|
|
psd_ol_rz_d = dist_f.psd_rz.*abs(squeeze(freqresp(G('D', 'Frz'), dist_f.f, 'Hz'))).^2;
|
|
|
|
psd_iff_gm_d = dist_f.psd_gm.*abs(squeeze(freqresp(G_iff('D', 'Dw'), dist_f.f, 'Hz'))).^2;
|
|
psd_iff_ty_d = dist_f.psd_ty.*abs(squeeze(freqresp(G_iff('D', 'Fty'), dist_f.f, 'Hz'))).^2;
|
|
psd_iff_rz_d = dist_f.psd_rz.*abs(squeeze(freqresp(G_iff('D', 'Frz'), dist_f.f, 'Hz'))).^2;
|
|
|
|
psd_rmc_gm_d = dist_f.psd_gm.*abs(squeeze(freqresp(G_rmc('D', 'Dw'), dist_f.f, 'Hz'))).^2;
|
|
psd_rmc_ty_d = dist_f.psd_ty.*abs(squeeze(freqresp(G_rmc('D', 'Fty'), dist_f.f, 'Hz'))).^2;
|
|
psd_rmc_rz_d = dist_f.psd_rz.*abs(squeeze(freqresp(G_rmc('D', 'Frz'), dist_f.f, 'Hz'))).^2;
|
|
|
|
psd_dvf_gm_d = dist_f.psd_gm.*abs(squeeze(freqresp(G_dvf('D', 'Dw'), dist_f.f, 'Hz'))).^2;
|
|
psd_dvf_ty_d = dist_f.psd_ty.*abs(squeeze(freqresp(G_dvf('D', 'Fty'), dist_f.f, 'Hz'))).^2;
|
|
psd_dvf_rz_d = dist_f.psd_rz.*abs(squeeze(freqresp(G_dvf('D', 'Frz'), dist_f.f, 'Hz'))).^2;
|
|
#+end_src
|
|
|
|
We then compute the Cumulative Amplitude Spectrum (figure [[fig:uniaxial-comp-cas-dist]]).
|
|
#+begin_src matlab :exports none
|
|
cas_ol_tot_d = flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_ol_rz_d+psd_ol_ty_d+psd_ol_gm_d))));
|
|
cas_iff_tot_d = flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_iff_rz_d+psd_iff_ty_d+psd_iff_gm_d))));
|
|
cas_rmc_tot_d = flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_rmc_rz_d+psd_rmc_ty_d+psd_rmc_gm_d))));
|
|
cas_dvf_tot_d = flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_dvf_rz_d+psd_dvf_ty_d+psd_dvf_gm_d))));
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
hold on;
|
|
plot(dist_f.f, cas_ol_tot_d, 'DisplayName', 'OL');
|
|
plot(dist_f.f, cas_iff_tot_d, 'DisplayName', 'IFF');
|
|
plot(dist_f.f, cas_rmc_tot_d, 'DisplayName', 'RMC');
|
|
plot(dist_f.f, cas_dvf_tot_d, 'DisplayName', 'DVF');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('CAS of the effect of disturbances on $D$ [m]'); xlabel('Frequency [Hz]');
|
|
legend('location', 'northeast')
|
|
xlim([0.5, 500]); ylim([1e-11, 1e-6]);
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial-comp-cas-dist.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial-comp-cas-dist
|
|
#+CAPTION: Comparison of the Cumulative Amplitude Spectrum of $D$ for different active damping techniques ([[./figs/uniaxial-comp-cas-dist.png][png]], [[./figs/uniaxial-comp-cas-dist.pdf][pdf]])
|
|
[[file:figs/uniaxial-comp-cas-dist.png]]
|
|
|
|
The obtained Root Mean Square Value for each active damping technique is shown below.
|
|
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
|
|
data2orgtable([cas_ol_tot_d(1); cas_iff_tot_d(1); cas_rmc_tot_d(1); cas_dvf_tot_d(1)], {'OL', 'IFF', 'RMC', 'DVF'}, {'D [m rms]'}, ' %.2e ');
|
|
#+end_src
|
|
|
|
#+name: tab:rms_d_comp
|
|
#+caption: Obtain Root Mean Square value of $D$ for each Active Damping Technique applied
|
|
#+RESULTS:
|
|
| | D [m rms] |
|
|
|-----+-----------|
|
|
| OL | 3.38e-06 |
|
|
| IFF | 3.40e-06 |
|
|
| RMC | 3.37e-06 |
|
|
| DVF | 3.38e-06 |
|
|
|
|
It is important to note that the effect of direct forces applied to the sample are not taken into account here.
|
|
|
|
** Damped Plant
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('D', 'Fn'), freqs, 'Hz'))), 'k-', 'DisplayName', 'OL');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff('D', 'Fn'), freqs, 'Hz'))), 'k:', 'DisplayName', 'IFF');
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc('D', 'Fn'), freqs, 'Hz'))), 'k--', 'DisplayName', 'RMC');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf('D', 'Fn'), freqs, 'Hz'))), 'k-.', 'DisplayName', 'DVF');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
legend('location', 'northeast');
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
hold on;
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G('D', 'Fn'), freqs, 'Hz'))), 'k-');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff('D', 'Fn'), freqs, 'Hz'))), 'k:');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_rmc('D', 'Fn'), freqs, 'Hz'))), 'k--');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_dvf('D', 'Fn'), freqs, 'Hz'))), 'k-.');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial_plant_damped_comp.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial_plant_damped_comp
|
|
#+CAPTION: Damped Plant - Comparison ([[./figs/uniaxial_plant_damped_comp.png][png]], [[./figs/uniaxial_plant_damped_comp.pdf][pdf]])
|
|
[[file:figs/uniaxial_plant_damped_comp.png]]
|
|
|
|
** Conclusion
|
|
|
|
#+name: tab:active_damping_comparison
|
|
#+caption: Comparison of proposed active damping techniques
|
|
| | IFF | RMC | DVF |
|
|
|---------------------------+-----------------+-----------------+----------|
|
|
| Sensor Type | Force sensor | Relative Motion | Inertial |
|
|
| Guaranteed Stability | + | + | - |
|
|
| Sensitivity ($D_w$) | - | + | - |
|
|
| Sensitivity ($F_s$) | - (at low freq) | + | + |
|
|
| Sensitivity ($F_{ty,rz}$) | + | - | + |
|
|
| Overall RMS of $D$ | = | = | = |
|
|
* Voice Coil
|
|
<<sec:voice_coil>>
|
|
** Matlab Init :noexport:ignore:
|
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
|
<<matlab-dir>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results silent :noweb yes
|
|
<<matlab-init>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no
|
|
simulinkproject('../');
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
open('uniaxial/matlab/sim_nano_station_uniaxial.slx')
|
|
#+end_src
|
|
|
|
** Init
|
|
We initialize all the stages with the default parameters.
|
|
The nano-hexapod is an hexapod with voice coils and the sample has a mass of 50kg.
|
|
|
|
#+begin_src matlab :exports none
|
|
initializeGround();
|
|
initializeGranite();
|
|
initializeTy();
|
|
initializeRy();
|
|
initializeRz();
|
|
initializeMicroHexapod();
|
|
initializeAxisc();
|
|
initializeMirror();
|
|
initializeNanoHexapod('actuator', 'lorentz');
|
|
initializeSample('mass', 50);
|
|
#+end_src
|
|
|
|
All the controllers are set to 0 (Open Loop).
|
|
#+begin_src matlab :exports none
|
|
K = tf(0);
|
|
save('./mat/controllers_uniaxial.mat', 'K', '-append');
|
|
K_iff = tf(0);
|
|
save('./mat/controllers_uniaxial.mat', 'K_iff', '-append');
|
|
K_rmc = tf(0);
|
|
save('./mat/controllers_uniaxial.mat', 'K_rmc', '-append');
|
|
K_dvf = tf(0);
|
|
save('./mat/controllers_uniaxial.mat', 'K_dvf', '-append');
|
|
#+end_src
|
|
|
|
** Identification
|
|
We identify the dynamics of the system.
|
|
#+begin_src matlab
|
|
%% Options for Linearized
|
|
options = linearizeOptions;
|
|
options.SampleTime = 0;
|
|
|
|
%% Name of the Simulink File
|
|
mdl = 'sim_nano_station_uniaxial';
|
|
#+end_src
|
|
|
|
The inputs and outputs are defined below and corresponds to the name of simulink blocks.
|
|
#+begin_src matlab
|
|
%% Input/Output definition
|
|
io(1) = linio([mdl, '/Dw'], 1, 'input'); % Ground Motion
|
|
io(2) = linio([mdl, '/Fs'], 1, 'input'); % Force applied on the sample
|
|
io(3) = linio([mdl, '/Fnl'], 1, 'input'); % Force applied by the NASS
|
|
io(4) = linio([mdl, '/Fdty'], 1, 'input'); % Parasitic force Ty
|
|
io(5) = linio([mdl, '/Fdrz'], 1, 'input'); % Parasitic force Rz
|
|
|
|
io(6) = linio([mdl, '/Dsm'], 1, 'output'); % Displacement of the sample
|
|
io(7) = linio([mdl, '/Fnlm'], 1, 'output'); % Force sensor in NASS's legs
|
|
io(8) = linio([mdl, '/Dnlm'], 1, 'output'); % Displacement of NASS's legs
|
|
io(9) = linio([mdl, '/Dgm'], 1, 'output'); % Absolute displacement of the granite
|
|
io(10) = linio([mdl, '/Vlm'], 1, 'output'); % Measured absolute velocity of the top NASS platform
|
|
#+end_src
|
|
|
|
Finally, we use the =linearize= Matlab function to extract a state space model from the simscape model.
|
|
#+begin_src matlab
|
|
%% Run the linearization
|
|
G_vc = linearize(mdl, io, options);
|
|
G_vc.InputName = {'Dw', ... % Ground Motion [m]
|
|
'Fs', ... % Force Applied on Sample [N]
|
|
'Fn', ... % Force applied by NASS [N]
|
|
'Fty', ... % Parasitic Force Ty [N]
|
|
'Frz'}; % Parasitic Force Rz [N]
|
|
G_vc.OutputName = {'D', ... % Measured sample displacement x.r.t. granite [m]
|
|
'Fnm', ... % Force Sensor in NASS [N]
|
|
'Dnm', ... % Displacement Sensor in NASS [m]
|
|
'Dgm', ... % Asbolute displacement of Granite [m]
|
|
'Vlm'}; ... % Absolute Velocity of NASS [m/s]
|
|
#+end_src
|
|
|
|
Finally, we save the identified system dynamics for further analysis.
|
|
#+begin_src matlab
|
|
save('./mat/uniaxial_plants.mat', 'G_vc', '-append');
|
|
#+end_src
|
|
|
|
** Sensitivity to Disturbances
|
|
We load the dynamics when using a piezo-electric nano hexapod to compare the results.
|
|
#+begin_src matlab
|
|
load('./mat/uniaxial_plants.mat', 'G');
|
|
#+end_src
|
|
|
|
We show several plots representing the sensitivity to disturbances:
|
|
- in figure [[fig:uniaxial-sensitivity-vc-disturbances]] the transfer functions from ground motion $D_w$ to the sample position $D$ and the transfer function from direct force on the sample $F_s$ to the sample position $D$ are shown
|
|
- in figure [[fig:uniaxial-sensitivity-vc-force-dist]], it is the effect of parasitic forces of the positioning stages ($F_{ty}$ and $F_{rz}$) on the position $D$ of the sample that are shown
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
subplot(2, 1, 1);
|
|
title('$D_w$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G_vc('D', 'Dw'), freqs, 'Hz'))), 'k-', 'DisplayName', 'G - VC');
|
|
plot(freqs, abs(squeeze(freqresp(G('D', 'Dw'), freqs, 'Hz'))), 'k--', 'DisplayName', 'G - PZ');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
legend('location', 'northeast');
|
|
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
|
|
|
|
subplot(2, 1, 2);
|
|
title('$F_s$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G_vc('D', 'Fs'), freqs, 'Hz'))), 'k-');
|
|
plot(freqs, abs(squeeze(freqresp(G('D', 'Fs'), freqs, 'Hz'))), 'k--');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial-sensitivity-vc-disturbances.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial-sensitivity-vc-disturbances
|
|
#+CAPTION: Sensitivity to disturbances ([[./figs/uniaxial-sensitivity-vc-disturbances.png][png]], [[./figs/uniaxial-sensitivity-vc-disturbances.pdf][pdf]])
|
|
[[file:figs/uniaxial-sensitivity-vc-disturbances.png]]
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
subplot(2, 1, 1);
|
|
title('$F_{ty}$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G_vc('D', 'Fty'), freqs, 'Hz'))), 'k-', 'DisplayName', 'G - VC');
|
|
plot(freqs, abs(squeeze(freqresp(G('D', 'Fty'), freqs, 'Hz'))), 'k--', 'DisplayName', 'G - PZ');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
legend('location', 'northeast');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
|
|
subplot(2, 1, 2);
|
|
title('$F_{rz}$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G_vc('D', 'Frz'), freqs, 'Hz'))), 'k-');
|
|
plot(freqs, abs(squeeze(freqresp(G('D', 'Frz'), freqs, 'Hz'))), 'k--');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial-sensitivity-vc-force-dist.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial-sensitivity-vc-force-dist
|
|
#+CAPTION: Sensitivity to disturbances ([[./figs/uniaxial-sensitivity-vc-force-dist.png][png]], [[./figs/uniaxial-sensitivity-vc-force-dist.pdf][pdf]])
|
|
[[file:figs/uniaxial-sensitivity-vc-force-dist.png]]
|
|
|
|
** Noise Budget
|
|
We first load the measured PSD of the disturbance.
|
|
#+begin_src matlab
|
|
load('./mat/disturbances_dist_psd.mat', 'dist_f');
|
|
#+end_src
|
|
|
|
The effect of these disturbances on the distance $D$ is computed below.
|
|
#+begin_src matlab :exports none
|
|
% Power Spectral Density of the relative Displacement [m^2/Hz]
|
|
psd_vc_gm_d = dist_f.psd_gm.*abs(squeeze(freqresp(G_vc('D', 'Dw'), dist_f.f, 'Hz'))).^2;
|
|
psd_vc_ty_d = dist_f.psd_ty.*abs(squeeze(freqresp(G_vc('D', 'Fty'), dist_f.f, 'Hz'))).^2;
|
|
psd_vc_rz_d = dist_f.psd_rz.*abs(squeeze(freqresp(G_vc('D', 'Frz'), dist_f.f, 'Hz'))).^2;
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
% Power Spectral Density of the relative Displacement [m^2/Hz]
|
|
psd_gm_d = dist_f.psd_gm.*abs(squeeze(freqresp(G('D', 'Dw'), dist_f.f, 'Hz'))).^2;
|
|
psd_ty_d = dist_f.psd_ty.*abs(squeeze(freqresp(G('D', 'Fty'), dist_f.f, 'Hz'))).^2;
|
|
psd_rz_d = dist_f.psd_rz.*abs(squeeze(freqresp(G('D', 'Frz'), dist_f.f, 'Hz'))).^2;
|
|
#+end_src
|
|
|
|
The PSD of the obtain distance $D$ due to each of the perturbation is shown in figure [[fig:uniaxial-vc-psd-dist]] and the Cumulative Amplitude Spectrum is shown in figure [[fig:uniaxial-vc-cas-dist]].
|
|
|
|
The Root Mean Square value of the obtained displacement $D$ is computed below and can be determined from the figure [[fig:uniaxial-vc-cas-dist]].
|
|
#+begin_src matlab :results value replace :exports results
|
|
cas_tot_d = sqrt(cumtrapz(dist_f.f, psd_vc_rz_d+psd_vc_ty_d+psd_vc_gm_d)); cas_tot_d(end)
|
|
#+end_src
|
|
|
|
#+RESULTS:
|
|
: 4.8793e-06
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
hold on;
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|
plot(dist_f.f, psd_vc_gm_d, 'DisplayName', '$D_w$');
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|
plot(dist_f.f, psd_vc_ty_d, 'DisplayName', '$T_y$');
|
|
plot(dist_f.f, psd_vc_rz_d, 'DisplayName', '$R_z$');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('CAS of the effect of disturbances on $D$ $\left[\frac{m^2}{Hz}\right]$'); xlabel('Frequency [Hz]');
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|
legend('location', 'northeast')
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|
xlim([0.5, 500]);
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
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|
#+begin_src matlab :var filepath="figs/uniaxial-vc-psd-dist.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
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|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial-vc-psd-dist
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|
#+CAPTION: PSD of the displacement $D$ due to disturbances ([[./figs/uniaxial-vc-psd-dist.png][png]], [[./figs/uniaxial-vc-psd-dist.pdf][pdf]])
|
|
[[file:figs/uniaxial-vc-psd-dist.png]]
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
hold on;
|
|
plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_vc_gm_d)))), 'DisplayName', '$D_w$ - VC');
|
|
plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_vc_ty_d)))), 'DisplayName', '$T_y$ - VC');
|
|
plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_vc_rz_d)))), 'DisplayName', '$R_z$ - VC');
|
|
plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_vc_gm_d+psd_vc_ty_d+psd_vc_rz_d)))), 'k-', 'DisplayName', 'tot - VC');
|
|
plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_gm_d+psd_ty_d+psd_rz_d)))), 'k--', 'DisplayName', 'tot - PZ');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('CAS of the effect of disturbances on $D$ [m]'); xlabel('Frequency [Hz]');
|
|
legend('location', 'northeast')
|
|
xlim([0.5, 500]); ylim([1e-12, 5e-6]);
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial-vc-cas-dist.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial-vc-cas-dist
|
|
#+CAPTION: CAS of the displacement $D$ due the disturbances ([[./figs/uniaxial-vc-cas-dist.png][png]], [[./figs/uniaxial-vc-cas-dist.pdf][pdf]])
|
|
[[file:figs/uniaxial-vc-cas-dist.png]]
|
|
|
|
#+begin_important
|
|
Even though the RMS value of the displacement $D$ is lower when using a piezo-electric actuator, the motion is mainly due to high frequency disturbances which are more difficult to control (an higher control bandwidth is required).
|
|
|
|
Thus, it may be desirable to use voice coil actuators.
|
|
#+end_important
|
|
** Integral Force Feedback
|
|
#+begin_src matlab
|
|
K_iff = -20/s;
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(-1, 2, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
plot(freqs, abs(squeeze(freqresp(K_iff*G_vc('Fnm', 'Fn'), freqs, 'Hz'))), 'k-');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(K_iff*G_vc('Fnm', 'Fn'), freqs, 'Hz'))), 'k-');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial_iff_vc_open_loop.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial_iff_vc_open_loop
|
|
#+CAPTION: Open Loop Transfer Function for IFF control when using a voice coil actuator ([[./figs/uniaxial_iff_vc_open_loop.png][png]], [[./figs/uniaxial_iff_vc_open_loop.pdf][pdf]])
|
|
[[file:figs/uniaxial_iff_vc_open_loop.png]]
|
|
|
|
** Identification of the Damped Plant
|
|
Let's initialize the system prior to identification.
|
|
#+begin_src matlab
|
|
initializeGround();
|
|
initializeGranite();
|
|
initializeTy();
|
|
initializeRy();
|
|
initializeRz();
|
|
initializeMicroHexapod();
|
|
initializeAxisc();
|
|
initializeMirror();
|
|
initializeNanoHexapod('actuator', 'lorentz');
|
|
initializeSample('mass', 50);
|
|
#+end_src
|
|
|
|
All the controllers are set to 0.
|
|
#+begin_src matlab
|
|
K = tf(0);
|
|
save('./mat/controllers_uniaxial.mat', 'K', '-append');
|
|
K_iff = -K_iff;
|
|
save('./mat/controllers_uniaxial.mat', 'K_iff', '-append');
|
|
K_rmc = tf(0);
|
|
save('./mat/controllers_uniaxial.mat', 'K_rmc', '-append');
|
|
K_dvf = tf(0);
|
|
save('./mat/controllers_uniaxial.mat', 'K_dvf', '-append');
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
%% Options for Linearized
|
|
options = linearizeOptions;
|
|
options.SampleTime = 0;
|
|
|
|
%% Name of the Simulink File
|
|
mdl = 'sim_nano_station_uniaxial';
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
%% Input/Output definition
|
|
io(1) = linio([mdl, '/Dw'], 1, 'input'); % Ground Motion
|
|
io(2) = linio([mdl, '/Fs'], 1, 'input'); % Force applied on the sample
|
|
io(3) = linio([mdl, '/Fnl'], 1, 'input'); % Force applied by the NASS
|
|
io(4) = linio([mdl, '/Fdty'], 1, 'input'); % Parasitic force Ty
|
|
io(5) = linio([mdl, '/Fdrz'], 1, 'input'); % Parasitic force Rz
|
|
|
|
io(6) = linio([mdl, '/Dsm'], 1, 'output'); % Displacement of the sample
|
|
io(7) = linio([mdl, '/Fnlm'], 1, 'output'); % Force sensor in NASS's legs
|
|
io(8) = linio([mdl, '/Dnlm'], 1, 'output'); % Displacement of NASS's legs
|
|
io(9) = linio([mdl, '/Dgm'], 1, 'output'); % Absolute displacement of the granite
|
|
io(10) = linio([mdl, '/Vlm'], 1, 'output'); % Measured absolute velocity of the top NASS platform
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
%% Run the linearization
|
|
G_vc_iff = linearize(mdl, io, options);
|
|
G_vc_iff.InputName = {'Dw', ... % Ground Motion [m]
|
|
'Fs', ... % Force Applied on Sample [N]
|
|
'Fn', ... % Force applied by NASS [N]
|
|
'Fty', ... % Parasitic Force Ty [N]
|
|
'Frz'}; % Parasitic Force Rz [N]
|
|
G_vc_iff.OutputName = {'D', ... % Measured sample displacement x.r.t. granite [m]
|
|
'Fnm', ... % Force Sensor in NASS [N]
|
|
'Dnm', ... % Displacement Sensor in NASS [m]
|
|
'Dgm', ... % Asbolute displacement of Granite [m]
|
|
'Vlm'}; ... % Absolute Velocity of NASS [m/s]
|
|
#+end_src
|
|
|
|
** Noise Budget
|
|
We compute the obtain PSD of the displacement $D$ when using IFF.
|
|
#+begin_src matlab :exports none
|
|
% Power Spectral Density of the relative Displacement [m^2/Hz]
|
|
psd_vc_iff_gm_d = dist_f.psd_gm.*abs(squeeze(freqresp(G_vc_iff('D', 'Dw'), dist_f.f, 'Hz'))).^2;
|
|
psd_vc_iff_ty_d = dist_f.psd_ty.*abs(squeeze(freqresp(G_vc_iff('D', 'Fty'), dist_f.f, 'Hz'))).^2;
|
|
psd_vc_iff_rz_d = dist_f.psd_rz.*abs(squeeze(freqresp(G_vc_iff('D', 'Frz'), dist_f.f, 'Hz'))).^2;
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
hold on;
|
|
plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_gm_d+psd_ty_d+psd_rz_d)))), '-', 'DisplayName', 'OL - PZ');
|
|
plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_vc_gm_d+psd_vc_ty_d+psd_vc_rz_d)))), 'k-', 'DisplayName', 'OL - VC');
|
|
plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_vc_iff_gm_d+psd_vc_iff_ty_d+psd_vc_iff_rz_d)))), 'k--', 'DisplayName', 'IFF - VC');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('CAS of the effect of disturbances on $D$ [m]'); xlabel('Frequency [Hz]');
|
|
legend('location', 'northeast')
|
|
xlim([0.5, 500]); ylim([1e-12, 5e-6]);
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/uniaxial-cas-iff-vc.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:uniaxial-cas-iff-vc
|
|
#+CAPTION: CAS of the displacement $D$ ([[./figs/uniaxial-cas-iff-vc.png][png]], [[./figs/uniaxial-cas-iff-vc.pdf][pdf]])
|
|
[[file:figs/uniaxial-cas-iff-vc.png]]
|
|
|
|
** Conclusion
|
|
#+begin_important
|
|
The use of voice coil actuators would allow a better disturbance rejection for a fixed bandwidth compared with a piezo-electric hexapod.
|
|
|
|
Similarly, it would require much lower bandwidth to attain the same level of disturbance rejection for $D$.
|
|
#+end_important
|