Add experimental data about stepper LUT

dcm_lookup_tables.org

@@ -50,7 +50,7 @@
 
 #+latex: \clearpage
 
-* Introduction
+* Introduction                                                        :ignore:
 
 Several Lookup Tables (LUT) are used for the DCM in order to compensate for *repeatable* errors.
 - Section [[sec:dcm_stepper_lut]]: the stepper motors are calibrated using interferometers.
@@ -62,6 +62,10 @@ Several Lookup Tables (LUT) are used for the DCM in order to compensate for *rep
 :END:
 <<sec:dcm_stepper_lut>>
 ** Introduction                                                      :ignore:
+Fast jack coarse displacement is performed with a Stepper motor and a ball screw mechanism.
+Such positioning system has some repeatable motion errors than can be calibrated using a measurement system having less errors.
+
+For the DCM, this can be done using the interferometers.
 
 ** Matlab Init                                             :noexport:ignore:
 #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@@ -86,6 +90,366 @@ Several Lookup Tables (LUT) are used for the DCM in order to compensate for *rep
 
 ** Schematic
 
+In order to measure the errors induced by the fast jacks, we have to make some scans, and measure simultaneously:
+- The wanted fast jack position: signal/step sent by the IcePAP
+- The actual (measured) position
+
+The experimental setup to perform this is shown in Figure [[fig:block_diagram_lut_stepper]].
+
+The procedure is the following:
+- A scan on the Bragg angle $\theta$ is generated from Bliss
+- Reference paths $[r_{u_r},\ r_{u_h},\ r_{d}]$ are sent to the IcePAP
+- Initially, the LUT inside the IcePAP is not changing the reference path
+- The IcePAP generates some steps $[u_{u_r},\ u_{u_h},\ u_{d}]$ that are sent to the fast jacks
+- The motion of the crystals $[d_z,\ r_y,\ r_x]$ is measured with the interferometers and computed in the Speedgoat
+- Finally, the corresponding motion $[d_{u_r},\ r_{u_h},\ r_d]$ of the fast jack is computed afterwards in BLISS
+
+The measured motion of the fast jacks $[d_{u_r},\ r_{u_h},\ r_d]$ can be compared with the IcePAP steps $[u_{u_r},\ u_{u_h},\ u_{d}]$ in order to create the LUT inside the IcePAP.
+
+#+begin_src latex :file block_diagram_lut_stepper.pdf
+\definecolor{myblue}{rgb}{0, 0.447, 0.741}
+\definecolor{myred}{rgb}{0.8500, 0.325, 0.098}
+\definecolor{mygreen}{rgb}{0.639, 0.745, 0.549}
+\definecolor{myyellow}{rgb}{0.922, 0.796, 0.545}
+
+\begin{tikzpicture}
+  % Blocks
+  \node[block={1.5cm}{2.9cm}] (traj) at (0,0){$\frac{d_{\text{off}}}{2 \cos \theta}$};
+
+  \node[block={1.5cm}{0.8cm}, right=1.1   of traj] (lut_uh) {LUT $u_h$};
+  \node[block={1.5cm}{0.8cm}, above=0.2 of lut_uh] (lut_ur) {LUT $u_r$};
+  \node[block={1.5cm}{0.8cm}, below=0.2 of lut_uh] (lut_d)  {LUT $d$};
+
+  \node[block={1.5cm}{0.8cm}, right=1.1 of lut_ur] (fj_ur) {FJ $u_h$};
+  \node[block={1.5cm}{0.8cm}, right=1.1 of lut_uh] (fj_uh) {FJ $u_r$};
+  \node[block={1.5cm}{0.8cm}, right=1.1 of lut_d]  (fj_d)  {FJ $d$};
+
+  \node[block={1.5cm}{2.9cm}, right=0.2 of fj_uh] (int) {\rotatebox{90}{Interferometers}};
+
+  \node[block={1.5cm}{2.9cm}, right=0.6 of int] (Js) {\rotatebox{90}{\parbox[c]{2.0cm}{\centering Forward Kinematics}}};
+
+  \node[block={1.5cm}{2.9cm}, right=1.1 of Js] (Ja) {\rotatebox{90}{\parbox[c]{2.0cm}{\centering Inverse Kinematics}}};
+
+  % Signals
+  \draw[->] ($(traj.west) + (-0.7, 0)$)node[above right]{$\theta$} -- (traj.west);
+
+  \draw[->] (lut_ur-|traj.east) --node[midway, above]{$r_{u_r}$} (lut_ur.west);
+  \draw[->] (lut_uh-|traj.east) --node[midway, above]{$r_{u_h}$} (lut_uh.west);
+  \draw[->] (lut_d -|traj.east) --node[midway, above]{$r_{d}$}   (lut_d.west);
+
+  \draw[->] (lut_ur.east) --node[midway, above]{$u_{u_r}$} (fj_ur.west);
+  \draw[->] (lut_uh.east) --node[midway, above]{$u_{u_h}$} (fj_uh.west);
+  \draw[->] (lut_d.east)  --node[midway, above]{$u_{d}$}   (fj_d.west);
+
+  \draw[->] (int.east) -- (Js.west);
+
+  \draw[->] (Js.east|-fj_ur) --node[midway, above]{$d_z$} (Ja.west|-fj_ur);
+  \draw[->] (Js.east|-fj_uh) --node[midway, above]{$r_y$} (Ja.west|-fj_uh);
+  \draw[->] (Js.east|-fj_d)  --node[midway, above]{$r_x$} (Ja.west|-fj_d) ;
+
+  \draw[->] (Ja.east|-fj_ur) -- ++(1.0, 0)node[above left]{$d_{u_r}$};
+  \draw[->] (Ja.east|-fj_uh) -- ++(1.0, 0)node[above left]{$d_{u_h}$};
+  \draw[->] (Ja.east|-fj_d)  -- ++(1.0, 0)node[above left]{$d_{d}$};
+
+  \begin{scope}[on background layer]
+    \node[fit={(fj_d.south west) (int.north east)}, fill=myblue!20!white, draw, inner sep=6pt] (dcm) {};
+    \node[fit={(Js.south west) (Js.north east)}, fill=myyellow!20!white, draw, inner sep=6pt] (speedgoat) {};
+    \node[fit={(lut_d.south west) (lut_ur.north east)}, fill=myred!20!white, draw, inner sep=6pt] (icepap) {};
+    \node[fit={(traj.south west) (traj.north east)}, fill=mygreen!20!white, draw, inner sep=6pt] (bliss_1) {};
+    \node[fit={(Ja.south west) (Ja.north east)}, fill=mygreen!20!white, draw, inner sep=6pt] (bliss_2) {};
+    \node[above] at (dcm.north) {DCM};
+    \node[above] at (speedgoat.north) {Speedgoat};
+    \node[above] at (icepap.north) {IcePAP};
+    \node[above] at (bliss_1.north) {BLISS};
+    \node[above] at (bliss_2.north) {BLISS};
+  \end{scope}
+\end{tikzpicture}
+#+end_src
+
+#+name: fig:block_diagram_lut_stepper
+#+caption: Block diagram of the experiment to create the Lookup Table
+#+RESULTS:
+[[file:figs/block_diagram_lut_stepper.png]]
+
+** TODO Repeatability of the motion                                :noexport:
+
+** Patterns in the Fast Jack motion errors
+In order to understand what should be the "sampling distance" for the lookup table of the stepper motor, we have to analyze the displacement errors induced by the stepper motor.
+
+Let's load the measurements of one bragg angle scan without any LUT.
+#+begin_src matlab
+%% Load Data of the new LUT method
+ol_bragg = (pi/180)*1e-5*double(h5read('Qutools_test_0001.h5','/33.1/instrument/trajmot/data')); % Bragg angle [rad]
+ol_dzw   = 10.5e-3./(2*cos(ol_bragg)); % Wanted distance between crystals [m]
+
+ol_dz    = 1e-9*double(h5read('Qutools_test_0001.h5','/33.1/instrument/xtal_111_dz_filter/data'));  % Dz distance between crystals [m]
+ol_dry   = 1e-9*double(h5read('Qutools_test_0001.h5','/33.1/instrument/xtal_111_dry_filter/data')); % Ry [rad]
+ol_drx   = 1e-9*double(h5read('Qutools_test_0001.h5','/33.1/instrument/xtal_111_drx_filter/data')); % Rx [rad]
+
+ol_t = 1e-6*double(h5read('Qutools_test_0001.h5','/33.1/instrument/time/data')); % Time [s]
+
+ol_ddz   = ol_fj-ol_dz; % Distance Error between crystals [m]
+#+end_src
+
+#+begin_src matlab :exports none
+%% Orientation and Distance error of the Crystal measured by the interferometers
+figure;
+hold on;
+plot(180/pi*ol_bragg, 1e6*ol_drx, 'DisplayName', '$R_x$')
+plot(180/pi*ol_bragg, 1e6*ol_dry, 'DisplayName', '$R_y$')
+hold off;
+legend('location', 'northwest');
+xlabel('Bragg Angle [deg]'); ylabel('Angle Error [$\mu$rad]');
+yyaxis right
+plot(180/pi*ol_bragg, 1e6*ol_ddz, 'DisplayName', '$\epsilon D_z$')
+ylabel('Distance Error [$\mu$m]');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/exp_without_lut_xtal_pos_errors.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:exp_without_lut_xtal_pos_errors
+#+caption: Orientation and Distance error of the Crystal measured by the interferometers
+#+RESULTS:
+[[file:figs/exp_without_lut_xtal_pos_errors.png]]
+
+Now let's convert the errors from the frame of the crystal to the frame of the fast jacks (inverse kinematics problem) using the Jacobian matrix.
+#+begin_src matlab
+%% Compute Fast Jack position errors
+% Jacobian matrix for Fast Jacks and 111 crystal
+J_a_111 = [1,  0.14, -0.1525
+           1,  0.14,  0.0675
+           1, -0.14,  0.0425];
+
+ol_de_111 = [ol_ddz'; ol_dry'; ol_drx'];
+
+% Fast Jack position errors
+ol_de_fj = J_a_111*ol_de_111;
+
+ol_fj_ur = ol_de_fj(1,:);
+ol_fj_uh = ol_de_fj(2,:);
+ol_fj_d  = ol_de_fj(3,:);
+#+end_src
+
+#+begin_src matlab :exports none
+%% Fast Jack position errors
+figure;
+hold on;
+plot(180/pi*ol_bragg, 1e6*ol_fj_ur, 'DisplayName', '$\epsilon_{u_r}$')
+plot(180/pi*ol_bragg, 1e6*ol_fj_uh, 'DisplayName', '$\epsilon_{u_h}$')
+plot(180/pi*ol_bragg, 1e6*ol_fj_d , 'DisplayName', '$\epsilon_{d}$')
+hold off;
+xlabel('Bragg Angle [deg]'); ylabel('Distance Error [$\mu$m]');
+legend('location', 'northwest');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/exp_without_lut_fj_pos_errors.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:exp_without_lut_fj_pos_errors
+#+caption: Estimated motion errors of the fast jacks during the scan
+#+RESULTS:
+[[file:figs/exp_without_lut_fj_pos_errors.png]]
+
+Let's now identify this pattern as a function of the fast-jack position.
+
+As we want to done frequency Fourier transform, we need to have uniform sampling along the fast jack position.
+To do so, the function =resample= is used.
+#+begin_src matlab
+Xs = 0.1e-6; % Sampling Distance [m]
+
+%% Re-sampled data with uniform spacing [m]
+ol_fj_ur_u = resample(ol_fj_ur, ol_dzw, 1/Xs);
+ol_fj_uh_u = resample(ol_fj_uh, ol_dzw, 1/Xs);
+ol_fj_d_u  = resample(ol_fj_d,  ol_dzw, 1/Xs);
+
+ol_fj_u = Xs*[1:length(ol_fj_ur_u)]; % Sampled Jack Position
+#+end_src
+
+#+begin_src matlab :exports none
+% Only take first 500um
+ol_fj_ur_u = ol_fj_ur_u(ol_fj_u<0.5e-3);
+ol_fj_uh_u = ol_fj_uh_u(ol_fj_u<0.5e-3);
+ol_fj_d_u  = ol_fj_d_u (ol_fj_u<0.5e-3);
+ol_fj_u    = ol_fj_u   (ol_fj_u<0.5e-3);
+#+end_src
+
+The result is shown in Figure [[fig:exp_without_lut_fj_pos_errors_distance]].
+
+#+begin_src matlab :exports none
+%% Fast Jack position errors
+figure;
+hold on;
+plot(1e3*ol_fj_u, 1e6*ol_fj_ur_u, 'DisplayName', '$\epsilon_{u_r}$')
+plot(1e3*ol_fj_u, 1e6*ol_fj_uh_u, 'DisplayName', '$\epsilon_{u_h}$')
+plot(1e3*ol_fj_u, 1e6*ol_fj_d_u , 'DisplayName', '$\epsilon_{d}$')
+hold off;
+xlabel('Fast Jack Position [mm]'); ylabel('Distance Error [$\mu$m]');
+legend('location', 'northwest');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/exp_without_lut_fj_pos_errors_distance.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:exp_without_lut_fj_pos_errors_distance
+#+caption: Position error of fast jacks as a function of the fast jack motion
+#+RESULTS:
+[[file:figs/exp_without_lut_fj_pos_errors_distance.png]]
+
+Let's now perform a Power Spectral Analysis of the measured displacement errors of the Fast Jack.
+#+begin_src matlab
+% Hanning Windows with 250um width
+win = hanning(floor(400e-6/Xs));
+
+% Power Spectral Density [m2/(1/m)]
+[S_fj_ur, f] = pwelch(ol_fj_ur_u-mean(ol_fj_ur_u), win, 0, [], 1/Xs);
+[S_fj_uh, ~] = pwelch(ol_fj_uh_u-mean(ol_fj_uh_u), win, 0, [], 1/Xs);
+[S_fj_d,  ~] = pwelch(ol_fj_d_u -mean(ol_fj_d_u ),  win, 0, [], 1/Xs);
+#+end_src
+
+As shown in Figure [[fig:exp_without_lut_wavenumber_asd]], we can see a fundamental "reciprocal length" of $5 \cdot 10^4\,[1/m]$ and its harmonics.
+This corresponds to a length of $\frac{1}{5\cdot 10^4} = 20\,[\mu m]$.
+
+#+begin_src matlab :exports none
+%%
+figure;
+hold on;
+plot(f, sqrt(S_fj_ur), 'DisplayName', '$u_r$');
+plot(f, sqrt(S_fj_uh), 'DisplayName', '$u_h$');
+plot(f, sqrt(S_fj_d), 'DisplayName', '$d$');
+hold off;
+set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
+xlabel('Reciprocal Length [1/m]');
+ylabel('ASD [$\frac{m}{1/\sqrt{m}}$]')
+legend('location', 'northeast');
+xlim([1e4, 2e6]); ylim([1e-13, 2e-9]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/exp_without_lut_wavenumber_asd.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:exp_without_lut_wavenumber_asd
+#+caption: Spectral content of the error as a function of the reciprocal length
+#+RESULTS:
+[[file:figs/exp_without_lut_wavenumber_asd.png]]
+
+
+
+Instead of looking at that as a function of the reciprocal length, we can look at it as a function of the spectral distance (Figure [[fig:exp_without_lut_spectral_content_fj_error]]).
+
+We see that the errors have a pattern with "spectral distances" equal to $5\,[\mu m]$, $10\,[\mu m]$, $20\,[\mu m]$ and smaller harmonics.
+#+begin_src matlab :exports none
+figure;
+hold on;
+plot(1e6./f, sqrt(S_fj_ur), 'DisplayName', '$u_r$');
+plot(1e6./f, sqrt(S_fj_uh), 'DisplayName', '$u_h$');
+plot(1e6./f, sqrt(S_fj_d), 'DisplayName', '$d$');
+hold off;
+set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
+xlabel('Spectral Distance [$\mu$m]');
+ylabel('Spectral Content [$\frac{m}{1/\sqrt{m}}$]')
+legend('location', 'northwest');
+xlim([0.5, 200]); ylim([1e-12, 1e-8]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/exp_without_lut_spectral_content_fj_error.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:exp_without_lut_spectral_content_fj_error
+#+caption: Spectral content of the error as a function of the spectral distance
+#+RESULTS:
+[[file:figs/exp_without_lut_spectral_content_fj_error.png]]
+
+Let's try to understand these results.
+One turn of the stepper motor corresponds to a vertical motion of 1mm.
+The stepper motor has 50 pairs of poles, therefore one pair of pole corresponds to a motion of $20\,[\mu m]$ which is the fundamental "spectral distance" we observe.
+
+#+begin_src matlab
+CPS_ur = flip(-cumtrapz(flip(f), flip(S_fj_ur)));
+CPS_uh = flip(-cumtrapz(flip(f), flip(S_fj_uh)));
+CPS_d  = flip(-cumtrapz(flip(f), flip(S_fj_d)));
+#+end_src
+
+From Figure [[fig:exp_without_lut_cas_pos_error]], we can see that if the motion errors with a period of $5\,[\mu m]$ and $10\,[\mu m]$ can be dealt with the lookup table, this will reduce a lot the positioning errors of the fast jack.
+
+#+begin_src matlab :results none
+%% Cumulative Spectrum
+figure;
+hold on;
+plot(1e6./f, sqrt(CPS_ur), 'DisplayName', '$u_r$');
+plot(1e6./f, sqrt(CPS_uh), 'DisplayName', '$u_j$');
+plot(1e6./f, sqrt(CPS_d),  'DisplayName', '$d$');
+hold off;
+set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
+xlabel('Spectral Distance [$\mu m$]'); ylabel('Cumulative Spectrum [$m$]')
+xlim([1, 500]); ylim([1e-9, 1e-5]);
+legend('location', 'northwest');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/exp_without_lut_cas_pos_error.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:exp_without_lut_cas_pos_error
+#+caption: Cumulative spectrum from small spectral distances to large spectral distances
+#+RESULTS:
+[[file:figs/exp_without_lut_cas_pos_error.png]]
+
+** Experimental Data - Current Method
+The current used method is an iterative one.
+
+#+begin_src matlab
+%% Load Experimental Data
+ol_bragg    = double(h5read('first_beam_0001.h5','/31.1/instrument/trajmot/data'));
+ol_drx      = h5read('first_beam_0001.h5','/31.1/instrument/xtal_111_drx_filter/data');
+
+lut_1_bragg = double(h5read('first_beam_0001.h5','/32.1/instrument/trajmot/data'));
+lut_1_drx   = h5read('first_beam_0001.h5','/32.1/instrument/xtal_111_drx_filter/data');
+
+lut_2_bragg = double(h5read('first_beam_0001.h5','/33.1/instrument/trajmot/data'));
+lut_2_drx   = h5read('first_beam_0001.h5','/33.1/instrument/xtal_111_drx_filter/data');
+
+lut_3_bragg = double(h5read('first_beam_0001.h5','/34.1/instrument/trajmot/data'));
+lut_3_drx   = h5read('first_beam_0001.h5','/34.1/instrument/xtal_111_drx_filter/data');
+
+lut_4_bragg = double(h5read('first_beam_0001.h5','/36.1/instrument/trajmot/data'));
+lut_4_drx   = h5read('first_beam_0001.h5','/36.1/instrument/xtal_111_drx_filter/data');
+#+end_src
+
+The relative orientation of the two =111= mirrors in the $x$ directions are compared in Figure [[fig:lut_old_method_exp_data]] for several iterations.
+We can see that after the first iteration, the orientation error has an opposite sign as for the case without LUT.
+
+#+begin_src matlab :exports none
+%% Plot Drx for all the LUT iterations
+figure;
+hold on;
+plot(1e-5*ol_bragg,    1e-3*ol_drx   , ...
+     'DisplayName', sprintf('$i=0$, $\\delta_{R_x} = %.0f$ [nrad rms]', rms(ol_drx-mean(ol_drx))))
+plot(1e-5*lut_1_bragg, 1e-3*lut_1_drx, ...
+     'DisplayName', sprintf('$i=1$, $\\delta_{R_x} = %.0f$ [nrad rms]', rms(lut_1_drx-mean(lut_1_drx))))
+plot(1e-5*lut_2_bragg, 1e-3*lut_2_drx, ...
+     'DisplayName', sprintf('$i=2$, $\\delta_{R_x} = %.0f$ [nrad rms]', rms(lut_2_drx-mean(lut_2_drx))))
+plot(1e-5*lut_4_bragg, 1e-3*lut_4_drx, ...
+     'DisplayName', sprintf('$i=4$, $\\delta_{R_x} = %.0f$ [nrad rms]', rms(lut_4_drx-mean(lut_4_drx))))
+hold off;
+xlabel('Bragg Angle [deg]'); ylabel('$R_x$ error [$\mu$rad]');
+legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/lut_old_method_exp_data.pdf', 'width', 'wide', 'height', 'tall');
+#+end_src
+
+#+name: fig:lut_old_method_exp_data
+#+caption: $R_x$ error with the current LUT method
+#+RESULTS:
+[[file:figs/lut_old_method_exp_data.png]]
+
 ** Simulation
 In this section, we suppose that we are in the frame of one fast jack (all transformations are already done), and we wish to create a LUT for one fast jack.
 
@@ -303,6 +667,72 @@ exportFig('figs/compare_old_new_lut_motion.pdf', 'width', 'wide', 'height', 'nor
 #+RESULTS:
 [[file:figs/compare_old_new_lut_motion.png]]
 
+** Experimental Data - Proposed method
+The new proposed method has been implemented and tested.
+
+The result is shown in Figure [[fig:lut_comp_old_new_experiment]].
+After only one iteration, the result is close to the previous method.
+
+#+begin_src matlab
+%% Load Data of the new LUT method
+ol_new_bragg = double(h5read('Qutools_test_0001.h5','/33.1/instrument/trajmot/data'));
+ol_new_drx   = h5read('Qutools_test_0001.h5','/33.1/instrument/xtal_111_drx_filter/data');
+
+lut_new_bragg = double(h5read('Qutools_test_0001.h5','/34.1/instrument/trajmot/data'));
+lut_new_drx   = h5read('Qutools_test_0001.h5','/34.1/instrument/xtal_111_drx_filter/data');
+#+end_src
+
+#+begin_src matlab :exports none
+%% Plot Drx for new and old method
+figure;
+hold on;
+plot(1e-5*ol_new_bragg,    1e-3*ol_new_drx   , ...
+     'DisplayName', sprintf('$i=0$, $\\delta_{R_x} = %.0f$ [nrad rms]', rms(ol_new_drx-mean(ol_new_drx))))
+plot(1e-5*lut_new_bragg, 1e-3*lut_new_drx, ...
+     'DisplayName', sprintf('New LUT $i=1$, $\\delta_{R_x} = %.0f$ [nrad rms]', rms(lut_new_drx-mean(lut_new_drx))))
+plot(1e-5*lut_4_bragg, 1e-3*lut_4_drx, ...
+     'DisplayName', sprintf('Old LUT $i=4$, $\\delta_{R_x} = %.0f$ [nrad rms]', rms(lut_4_drx-mean(lut_4_drx))))
+hold off;
+xlabel('Bragg Angle [deg]'); ylabel('$R_x$ error [$\mu$rad]');
+legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/lut_comp_old_new_experiment.pdf', 'width', 'wide', 'height', 'tall');
+#+end_src
+
+#+name: fig:lut_comp_old_new_experiment
+#+caption: Comparison of the $R_x$ error for the current LUT method and the proposed one
+#+RESULTS:
+[[file:figs/lut_comp_old_new_experiment.png]]
+
+If we zoom on the 20deg to 25deg bragg angles, we can see that the new method has much less "periodic errors" as compared to the previous one which shows some patterns.
+
+#+begin_src matlab :exports none
+%% Plot Drx for new and old method
+figure;
+hold on;
+set(gca,'ColorOrderIndex',2)
+plot(1e-5*lut_new_bragg, lut_new_drx, ...
+     'DisplayName', 'New LUT')
+plot(1e-5*lut_4_bragg, lut_4_drx, ...
+     'DisplayName', 'Old LUT')
+hold off;
+xlabel('Bragg Angle [deg]'); ylabel('$R_x$ error [nrad]');
+legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
+xlim([20, 25]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/lut_comp_old_new_experiment_zoom.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:lut_comp_old_new_experiment_zoom
+#+caption:
+#+RESULTS:
+[[file:figs/lut_comp_old_new_experiment_zoom.png]]
+
+
 * Attocube Calibration
 :PROPERTIES:
 :header-args:matlab+: :tangle matlab/dcm_attocube_lut.m
@@ -310,6 +740,64 @@ exportFig('figs/compare_old_new_lut_motion.pdf', 'width', 'wide', 'height', 'nor
 <<sec:dcm_attocube_lut>>
 ** Introduction                                                      :ignore:
 
+The idea is to calibrate the periodic non-linearity of the interferometers, a known displacement must be imposed and the interferometer output compared to this displacement.
+This should be performed over several periods in order to characterize the error.
+
+We here suppose that we are already in the frame of the Attocube (the fast-jack displacements are converted to Attocube displacement using the transformation matrices).
+We also suppose that we are at a certain Bragg angle, and that the stepper motors are not moving: only the piezoelectric actuators are used.
+
+The setup is schematically with the block diagram in Figure [[fig:block_diagram_lut_attocube]].
+The signals are:
+- $u$: Actuator Signal (position where we wish to go)
+- $d$: Disturbances affecting the signal
+- $y$: Displacement of the crystal
+- $y_g$: Measurement of the crystal motion by the strain gauge with some noise $n_g$
+- $y_a$: Measurement of the crystal motion by the interferometer with some noise $n_a$
+
+#+begin_src latex :file block_diagram_lut_attocube.pdf
+\definecolor{myblue}{rgb}{0, 0.447, 0.741}
+\definecolor{myred}{rgb}{0.8500, 0.325, 0.098}
+
+\begin{tikzpicture}
+  \node[block] (G) at (0,0){$G(s)$};
+  \node[addb, right=1 of G] (addd) {};
+  \node[block, align=center, right=1 of addd] (non_linearity) {Periodic\\Non-linearity};
+  \node[addb, right=1 of non_linearity] (addna) {};
+  \node[addb, below=1.8 of addna] (addnsg) {};
+
+  \draw[->] ($(G.west) + (-1.0, 0)$) node[above right]{$u$} -- (G.west);
+  \draw[->] (G.east) -- (addd.west);
+  \draw[->] (addd.east) -- (non_linearity.west);
+  \draw[->] ($(addd.north) + (0, 1.0)$) node[below right]{$d$} -- (addd.north);
+  \draw[->] (non_linearity.east) -- (addna.west);
+  \draw[->] (addna.east) -- ++(1.2, 0) node[above left]{$y_a$};
+  \draw[->] ($(addna.north) + (0, 1.0)$) node[below right](na){$n_a$} -- (addna.north);
+  \draw[->] ($(addd.east) + (0.4, 0)$)node[branch]{} node[above]{$y$} |- (addnsg.west);
+  \draw[->] (addnsg.east) -- ++(1.2, 0) node[above left]{$y_g$};
+  \draw[->] ($(addnsg.north) + (0, 1.0)$) node[below right](nsg){$n_{g}$} -- (addnsg.north);
+
+  \begin{scope}[on background layer]
+    \node[fit={(non_linearity.south west) (na.north east)}, fill=myblue!20!white, draw, inner sep=6pt] (attocube) {};
+    \node[fit={(non_linearity.west|-addnsg.south) (nsg.north east)}, fill=myred!20!white, draw, inner sep=6pt] (straingauge) {};
+    \node[below right] at (attocube.north west) {Attocube};
+    \node[below right] at (straingauge.north west) {Strain Gauge};
+  \end{scope}
+\end{tikzpicture}
+#+end_src
+
+#+name: fig:block_diagram_lut_attocube
+#+caption: Block Diagram schematic of the setup used to measure the periodic non-linearity of the Attocube
+#+RESULTS:
+[[file:figs/block_diagram_lut_attocube.png]]
+
+The problem is to estimate the periodic non-linearity of the Attocube from the imperfect measurments $y_a$ and $y_g$.
+
+Then a Lookup Table (LUT) is build.
+
+The wavelength of the Attocube is 1530nm, therefore the non-linearity has a period of 765nm.
+The amplitude of the non-linearity can vary from one unit to the other (and maybe from one experimental condition to the other).
+It is typically between 5nm peak to peak and 20nm peak to peak.
+
 ** 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>>
@@ -333,6 +821,21 @@ exportFig('figs/compare_old_new_lut_motion.pdf', 'width', 'wide', 'height', 'nor
 
 
 
+** Simulations
+We have some constrains on the way the motion is imposed and measured:
+- We want the frequency content of the imposed motion to be at low frequency in order not to induce vibrations of the structure.
+  We have to make sure the forces applied by the piezoelectric actuator only moves the crystal and not the fast jack below.
+  Therefore, we have to move much slower than the first resonance frequency in the system.
+- As both $y_a$ and $y_g$ should have rather small noise, we have to filter them with low pass filters.
+  The cut-off frequency of the low pass filter should be high as compared to the motion (to not induce any distortion) but still reducing sufficiently the noise.
+  Let's say we want the noise to be less than 1nm ($6 \sigma$).
+
+Suppose we have the power spectral density (PSD) of both $n_a$ and $n_g$.
+
+- [ ] Take the PSD of the Attocube
+- [ ] Take the PSD of the strain gauge
+- [ ] Using 2nd order low pass filter, estimate the required low pass filter cut-off frequency to have sufficiently low noise
+
 * Helping Functions                                                 :noexport:
 ** Initialize Path
 #+NAME: m-init-path

figs/block_diagram_lut_attocube.svg

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