347 lines
11 KiB
Org Mode
347 lines
11 KiB
Org Mode
#+TITLE: Tomography Experiment
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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 matlab/modal_frf_coh.m
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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/}{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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* Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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#+begin_src matlab :tangle no
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simulinkproject('../');
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#+end_src
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#+begin_src matlab
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open 'simscape/sim_nano_station_tomo.slx'
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#+end_src
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* Initialize Experiment
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We first initialize all the stages.
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#+begin_src matlab
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initializeGround();
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initializeGranite();
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initializeTy();
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initializeRy();
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initializeRz();
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initializeMicroHexapod();
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initializeAxisc();
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initializeMirror();
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initializeNanoHexapod(struct('actuator', 'piezo'));
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initializeSample(struct('mass', 1));
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#+end_src
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We initialize the reference path for all the stages.
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#+begin_src matlab
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initializeReferences(struct('Rz_type', 'rotating', 'Rz_period', 1));
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#+end_src
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And we initialize the disturbances.
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#+begin_src matlab
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initDisturbances();
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#+end_src
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* Run the Tomography Experiment
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We first load the simulation configuration
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#+begin_src matlab
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load('simscape/conf_simscape.mat');
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#+end_src
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#+begin_src matlab
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set_param(conf_simscape, 'StopTime', '1');
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#+end_src
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#+begin_src matlab
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set_param('sim_nano_station_tomo', 'SimulationCommand', 'start');
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#+end_src
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* TODO Tests on the transformation from reference to wanted position
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- [X] Are the rotation matrix commutable? => no
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- [X] How to express the measured rotation errors? => screw axis coordinate seems nice (used in Taghirad's book)
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- [ ] Should ask Veijo how he specifies the position of the Symetrie Hexapod
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- [ ] Create functions for all distinct part and then include that in Simulink
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- [ ] How the express the orientation error?
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- [ ] If we use screw coordinate, can we add/subtract them?
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- [ ] Do some simple tests to verify that the algorithm is working fine
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** Introduction :ignore:
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#+begin_quote
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Rx = [1 0 0;
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0 cos(t) -sin(t);
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0 sin(t) cos(t)];
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Ry = [ cos(t) 0 sin(t);
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0 1 0;
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-sin(t) 0 cos(t)];
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Rz = [cos(t) -sin(t) 0;
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sin(t) cos(t) 0;
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0 0 1];
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#+end_quote
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Let's define the following frames:
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- $\{W\}$ the frame that is *fixed to the granite* and its origin at the theoretical meeting point between the X-ray and the spindle axis.
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- $\{S\}$ the frame *attached to the sample* (in reality attached to the top platform of the nano-hexapod) with its origin at 175mm above the top platform of the nano-hexapod.
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Its origin is $O_S$.
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- $\{T\}$ the theoretical wanted frame that correspond to the wanted pose of the frame $\{S\}$.
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$\{T\}$ is computed from the wanted position of each stage. It is thus theoretical and does not correspond to a real position.
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The origin of $T$ is $O_T$ and is the wanted position of the sample.
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Thus:
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- the *measurement* of the position of the sample corresponds to ${}^W O_S = \begin{bmatrix} {}^WP_{x,m} & {}^WP_{y,m} & {}^WP_{z,m} \end{bmatrix}^T$ in translation and to $\theta_m {}^W\bm{s}_m = \theta_m \cdot \begin{bmatrix} {}^Ws_{x,m} & {}^Ws_{y,m} & {}^Ws_{z,m} \end{bmatrix}^T$ in rotations
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- the *wanted position* of the sample expressed w.r.t. the granite is ${}^W O_T = \begin{bmatrix} {}^WP_{x,r} & {}^WP_{y,r} & {}^WP_{z,r} \end{bmatrix}^T$ in translation and to $\theta_r {}^W\bm{s}_r = \theta_r \cdot \begin{bmatrix} {}^Ws_{x,r} & {}^Ws_{y,r} & {}^Ws_{z,r} \end{bmatrix}^T$ in rotations
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** Wanted Position of the Sample with respect to the Granite
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Let's define the wanted position of each stage.
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#+begin_src matlab
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Ty = 0; % [m]
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Ry = 3*pi/180; % [rad]
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Rz = 180*pi/180; % [rad]
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% Hexapod (first consider only translations)
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Thx = 0; % [m]
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Thy = 0; % [m]
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Thz = 0; % [m]
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#+end_src
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Now, we compute the corresponding wanted translation and rotation of the sample with respect to the granite frame $\{W\}$.
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This corresponds to ${}^WO_T$ and $\theta_m {}^Ws_m$.
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To do so, we have to define the homogeneous transformation for each stage.
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#+begin_src matlab
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% Translation Stage
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Rty = [1 0 0 0;
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0 1 0 Ty;
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0 0 1 0;
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0 0 0 1];
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% Tilt Stage - Pure rotating aligned with Ob
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Rry = [ cos(Ry) 0 sin(Ry) 0;
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0 1 0 0;
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-sin(Ry) 0 cos(Ry) 0;
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0 0 0 1];
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% Spindle - Rotation along the Z axis
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Rrz = [cos(Rz) -sin(Rz) 0 0 ;
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sin(Rz) cos(Rz) 0 0 ;
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0 0 1 0 ;
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0 0 0 1 ];
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% Micro-Hexapod (only rotations first)
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Rh = [1 0 0 Thx ;
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0 1 0 Thy ;
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0 0 1 Thz ;
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0 0 0 1 ];
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#+end_src
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We combine the individual homogeneous transformations into one homogeneous transformation for all the station.
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#+begin_src matlab
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Ttot = Rty*Rry*Rrz*Rh;
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#+end_src
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Using this homogeneous transformation, we can compute the wanted position and orientation of the sample with respect to the granite.
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Translation.
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#+begin_src matlab
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WOr = Ttot*[0;0;0;1];
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WOr = WOr(1:3);
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#+end_src
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Rotation.
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#+begin_src matlab
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thetar = acos((trace(Ttot(1:3, 1:3))-1)/2)
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if thetar == 0
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WSr = [0; 0; 0];
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else
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[V, D] = eig(Ttot(1:3, 1:3));
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WSr = thetar*V(:, abs(diag(D) - 1) < eps(1));
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end
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#+end_src
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#+begin_src matlab
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WPr = [WOr ; WSr];
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#+end_src
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** Measured Position of the Sample with respect to the Granite
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The measurement of the position of the sample using the metrology system gives the position and orientation of the sample with respect to the granite.
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#+begin_src matlab
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% Measurements: Xm, Ym, Zm, Rx, Ry, Rz
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Dxm = 0; % [m]
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Dym = 0; % [m]
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Dzm = 0; % [m]
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Rxm = 0*pi/180; % [rad]
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Rym = 0*pi/180; % [rad]
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Rzm = 180*pi/180; % [rad]
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#+end_src
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Let's compute the corresponding orientation using screw axis.
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#+begin_src matlab
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Trxm = [1 0 0;
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0 cos(Rxm) -sin(Rxm);
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0 sin(Rxm) cos(Rxm)];
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Trym = [ cos(Rym) 0 sin(Rym);
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0 1 0;
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-sin(Rym) 0 cos(Rym)];
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Trzm = [cos(Rzm) -sin(Rzm) 0;
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sin(Rzm) cos(Rzm) 0;
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0 0 1];
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STw = [[ Trym*Trxm*Trzm , [Dxm; Dym; Dzm]]; 0 0 0 1];
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#+end_src
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We then obtain the orientation measurement in the form of screw coordinate $\theta_m ({}^Ws_{x,m},\ {}^Ws_{y,m},\ {}^Ws_{z,m})^T$ where:
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- $\theta_m = \cos^{-1} \frac{\text{Tr}(R) - 1}{2}$
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- ${}^W\bm{s}_m$ is the eigen vector of the rotation matrix $R$ corresponding to the eigen value $\lambda = 1$
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#+begin_src matlab
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thetam = acos((trace(STw(1:3, 1:3))-1)/2); % [rad]
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if thetam == 0
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WSm = [0; 0; 0];
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else
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[V, D] = eig(STw(1:3, 1:3));
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WSm = thetam*V(:, abs(diag(D) - 1) < eps(1));
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end
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#+end_src
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#+begin_src matlab
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WPm = [Dxm ; Dym ; Dzm ; WSm];
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#+end_src
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** Positioning Error with respect to the Granite
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The wanted position expressed with respect to the granite is ${}^WO_T$ and the measured position with respect to the granite is ${}^WO_S$, thus the *position error* expressed in $\{W\}$ is
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\[ {}^W E = {}^W O_T - {}^W O_S \]
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The same is true for rotations:
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\[ \theta_\epsilon {}^W\bm{s}_\epsilon = \theta_r {}^W\bm{s}_r - \theta_m {}^W\bm{s}_m \]
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#+begin_src matlab
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WPe = WPr - WPm;
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#+end_src
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#+begin_quote
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Now we want to express this error in a frame attached to the *base of the nano-hexapod* with its origin at the same point where the Jacobian of the nano-hexapod is computed (175mm above the top platform + 90mm of total height of the nano-hexapod).
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Or maybe should we want to express this error with respect to the *top platform of the nano-hexapod*?
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We are measuring the position of the top-platform, and we don't know exactly the position of the bottom platform.
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We could compute the position of the bottom platform in two ways:
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- from the encoders of each stage
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- from the measurement of the nano-hexapod top platform + the internal metrology in the nano-hexapod (capacitive sensors e.g)
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A third option is to say that the maximum stroke of the nano-hexapod is so small that the error should no change to much by the change of base.
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#+end_quote
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** Position Error Expressed in the Nano-Hexapod Frame
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We now want the position error to be expressed in $\{S\}$ (the frame attach to the sample) for control:
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\[ {}^S E = {}^S T_W \cdot {}^W E \]
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Thus we need to compute the homogeneous transformation ${}^ST_W$.
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Fortunately, this homogeneous transformation can be computed from the measurement of the sample position and orientation with respect to the granite.
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#+begin_src matlab
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Trxm = [1 0 0;
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0 cos(Rxm) -sin(Rxm);
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0 sin(Rxm) cos(Rxm)];
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Trym = [ cos(Rym) 0 sin(Rym);
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0 1 0;
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-sin(Rym) 0 cos(Rym)];
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Trzm = [cos(Rzm) -sin(Rzm) 0;
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sin(Rzm) cos(Rzm) 0;
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0 0 1];
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STw = [[ Trym*Trxm*Trzm , [Dxm; Dym; Dzm]]; 0 0 0 1];
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#+end_src
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Translation Error.
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#+begin_src matlab
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SEm = STw * [WPe(1:3); 0];
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SEm = SEm(1:3);
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#+end_src
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Rotation Error.
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#+begin_src matlab
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SEr = STw * [WPe(4:6); 0];
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SEr = SEr(1:3);
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#+end_src
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#+begin_src matlab
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Etot = [SEm ; SEr]
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#+end_src
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** Another try
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Let's denote:
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- $\{W\}$ the initial fixed frame
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- $\{R\}$ the reference frame corresponding to the wanted pose of the sample
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- $\{M\}$ the frame corresponding to the measured pose of the sample
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We have then computed:
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- ${}^WT_R$
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- ${}^WT_M$
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We have:
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\begin{align}
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{}^MT_R &= {}^MT_W {}^WT_R \\
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&= {}^WT_M^t {}^WT_R
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\end{align}
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#+begin_src matlab
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MTr = STw'*Ttot;
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#+end_src
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Position error:
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#+begin_src matlab
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MTr(1:3, 1:4)*[0; 0; 0; 1]
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#+end_src
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Orientation error:
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#+begin_src matlab
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MTr(1:3, 1:3)
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#+end_src
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** Verification
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How can we verify that the computation is correct?
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Options:
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- Test with simscape multi-body
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- Impose motion on each stage
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- Measure the position error w.r.t. the NASS
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- Compare with the computation
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